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Game Theory
By
Ali Farahani Rad Benjamin Ghassemi
2
What is Game Theory Game theory is a branch of applied
mathematics that is often used in the context of economics It studies strategic interactions between agents
In economics an agent is an actor in a model that (generally) solves an optimization problem In this sense it is equivalent to the term player which is also used in economics but is more common in game theory
3
What is Game Theory In strategic games agents choose
strategies that will maximize their return given the strategies the other agents choose
The essential feature is that it provides a formal modeling approach to social situations in which decision makers interact with other agents Game theory extends the simpler optimization approach developed in neoclassical economics
4
What is Game Theory Neoclassical economics refers to a general
approach in economics focusing on the determination of prices outputs and income distributions in markets through supply and demand These are mediated through a hypothesized maximization of income-constrained utility by individuals and of cost-constrained profits of firms employing available information and factors of production
Antonietta Campus (1987) marginal economics The New Palgrave A Dictionary of Economics v 3 p 323
5
Applications of Game Theory
Mathematics Computer Science Biology Economics Political Science International
Relations
Psychology Law Military Strategy Management Sports Game Playing Philosophy
6
Representation of games The games studied by game theory are well-defined mathematical objects A game consists of a set of players a set of moves (or strategies) available to those players and a specification of payoffs for each combination of strategies
7
Extensive form Games here are often presented as trees Here each vertex (or node) represents a point of choice for a player The player is specified by a number listed by the vertex The lines out of the vertex represent a possible action for that player The payoffs are specified at the bottom of the tree
8
Extensive form
9
Normal form The normal (or strategic form) game is usually
represented by a matrix which shows the players strategies and payoffs More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions
Player 2chooses Left
Player 2chooses Right
Player 1chooses Up
4 3 ndash1 ndash1
Player 1chooses Down
0 0 3 4
Normal form or payoff matrix of a 2-player 2-strategy game
10
Normal form When a game is presented in normal form
it is presumed that each player acts simultaneously or at least without knowing the actions of the other If players have some information about the choices of other players the game is usually presented in extensive form
11
Types of games Cooperative or non-cooperative
Symmetric and asymmetric
Zero sum and non-zero sum
Simultaneous and sequential
Perfect information and imperfect information
Infinitely long games
Discrete and continuous games
Meta games
12
Cooperative or non-cooperative A game is cooperative if the players are able to form binding
commitments For instance the legal system requires them to adhere to their promises In non-cooperative games this is not possible
Often it is assumed that communication among players is allowed in cooperative games but not in non-cooperative ones This classification on two binary criteria has been rejected (Harsanyi 1974)
Of the two types of games non-cooperative games are able to model situations to the finest details producing accurate results Cooperative games focus on the game at large Considerable efforts have been made to link the two approaches The so-called Nash-program has already established many of the cooperative solutions as non-cooperative equilibrium
13
Symmetric and asymmetric A symmetric game is a game where the
payoffs for playing a particular strategy depend only on the other strategies employed not on who is playing them If the identities of the players can be changed without changing the payoff to the strategies then a game is symmetric
14
Zero sum and non-zero sum Zero sum games are a special case of
constant sum games in which choices by players can neither increase nor decrease the available resources In zero-sum games the total benefit to all players in the game for every combination of strategies always adds to zero (more informally a player benefits only at the expense of others)
15
Simultaneous and sequential Simultaneous games are games where both players move
simultaneously or if they do not move simultaneously the later players are unaware of the earlier players actions (making them effectively simultaneous) Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions This need not be perfect information about every action of earlier players it might be very little knowledge For instance a player may know that an earlier player did not perform one particular action while he does not know which of the other available actions the first player actually performed
The difference between simultaneous and sequential games is captured in the different representations discussed above Normal form is used to represent simultaneous games and extensive form is used to represent sequential ones
16
Perfect information and imperfect information An important subset of sequential games consists
of games of perfect information A game is one of perfect information if all players know the moves previously made by all other players Thus only sequential games can be games of perfect information since in simultaneous games not every player knows the actions of the others
Perfect information is often confused with complete information which is a similar concept Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions
17
Infinitely long games Games as studied by economists and real-world
game players are generally finished in a finite number of moves Pure mathematicians are not so constrained and set theorists in particular study games that last for infinitely many moves with the winner (or other payoff) not known until after all those moves are completed
18
Discrete and continuous games Most of the objects treated in most branches of
game theory are discrete with a finite number of players moves events outcomes etc However the concepts can be extended into the realm of real numbers This branch has sometimes been called differential games because they map to a real line usually time although the behaviors may be mathematically discontinuous Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics
19
Meta games These are games the play of which is the
development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
2
What is Game Theory Game theory is a branch of applied
mathematics that is often used in the context of economics It studies strategic interactions between agents
In economics an agent is an actor in a model that (generally) solves an optimization problem In this sense it is equivalent to the term player which is also used in economics but is more common in game theory
3
What is Game Theory In strategic games agents choose
strategies that will maximize their return given the strategies the other agents choose
The essential feature is that it provides a formal modeling approach to social situations in which decision makers interact with other agents Game theory extends the simpler optimization approach developed in neoclassical economics
4
What is Game Theory Neoclassical economics refers to a general
approach in economics focusing on the determination of prices outputs and income distributions in markets through supply and demand These are mediated through a hypothesized maximization of income-constrained utility by individuals and of cost-constrained profits of firms employing available information and factors of production
Antonietta Campus (1987) marginal economics The New Palgrave A Dictionary of Economics v 3 p 323
5
Applications of Game Theory
Mathematics Computer Science Biology Economics Political Science International
Relations
Psychology Law Military Strategy Management Sports Game Playing Philosophy
6
Representation of games The games studied by game theory are well-defined mathematical objects A game consists of a set of players a set of moves (or strategies) available to those players and a specification of payoffs for each combination of strategies
7
Extensive form Games here are often presented as trees Here each vertex (or node) represents a point of choice for a player The player is specified by a number listed by the vertex The lines out of the vertex represent a possible action for that player The payoffs are specified at the bottom of the tree
8
Extensive form
9
Normal form The normal (or strategic form) game is usually
represented by a matrix which shows the players strategies and payoffs More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions
Player 2chooses Left
Player 2chooses Right
Player 1chooses Up
4 3 ndash1 ndash1
Player 1chooses Down
0 0 3 4
Normal form or payoff matrix of a 2-player 2-strategy game
10
Normal form When a game is presented in normal form
it is presumed that each player acts simultaneously or at least without knowing the actions of the other If players have some information about the choices of other players the game is usually presented in extensive form
11
Types of games Cooperative or non-cooperative
Symmetric and asymmetric
Zero sum and non-zero sum
Simultaneous and sequential
Perfect information and imperfect information
Infinitely long games
Discrete and continuous games
Meta games
12
Cooperative or non-cooperative A game is cooperative if the players are able to form binding
commitments For instance the legal system requires them to adhere to their promises In non-cooperative games this is not possible
Often it is assumed that communication among players is allowed in cooperative games but not in non-cooperative ones This classification on two binary criteria has been rejected (Harsanyi 1974)
Of the two types of games non-cooperative games are able to model situations to the finest details producing accurate results Cooperative games focus on the game at large Considerable efforts have been made to link the two approaches The so-called Nash-program has already established many of the cooperative solutions as non-cooperative equilibrium
13
Symmetric and asymmetric A symmetric game is a game where the
payoffs for playing a particular strategy depend only on the other strategies employed not on who is playing them If the identities of the players can be changed without changing the payoff to the strategies then a game is symmetric
14
Zero sum and non-zero sum Zero sum games are a special case of
constant sum games in which choices by players can neither increase nor decrease the available resources In zero-sum games the total benefit to all players in the game for every combination of strategies always adds to zero (more informally a player benefits only at the expense of others)
15
Simultaneous and sequential Simultaneous games are games where both players move
simultaneously or if they do not move simultaneously the later players are unaware of the earlier players actions (making them effectively simultaneous) Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions This need not be perfect information about every action of earlier players it might be very little knowledge For instance a player may know that an earlier player did not perform one particular action while he does not know which of the other available actions the first player actually performed
The difference between simultaneous and sequential games is captured in the different representations discussed above Normal form is used to represent simultaneous games and extensive form is used to represent sequential ones
16
Perfect information and imperfect information An important subset of sequential games consists
of games of perfect information A game is one of perfect information if all players know the moves previously made by all other players Thus only sequential games can be games of perfect information since in simultaneous games not every player knows the actions of the others
Perfect information is often confused with complete information which is a similar concept Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions
17
Infinitely long games Games as studied by economists and real-world
game players are generally finished in a finite number of moves Pure mathematicians are not so constrained and set theorists in particular study games that last for infinitely many moves with the winner (or other payoff) not known until after all those moves are completed
18
Discrete and continuous games Most of the objects treated in most branches of
game theory are discrete with a finite number of players moves events outcomes etc However the concepts can be extended into the realm of real numbers This branch has sometimes been called differential games because they map to a real line usually time although the behaviors may be mathematically discontinuous Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics
19
Meta games These are games the play of which is the
development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
3
What is Game Theory In strategic games agents choose
strategies that will maximize their return given the strategies the other agents choose
The essential feature is that it provides a formal modeling approach to social situations in which decision makers interact with other agents Game theory extends the simpler optimization approach developed in neoclassical economics
4
What is Game Theory Neoclassical economics refers to a general
approach in economics focusing on the determination of prices outputs and income distributions in markets through supply and demand These are mediated through a hypothesized maximization of income-constrained utility by individuals and of cost-constrained profits of firms employing available information and factors of production
Antonietta Campus (1987) marginal economics The New Palgrave A Dictionary of Economics v 3 p 323
5
Applications of Game Theory
Mathematics Computer Science Biology Economics Political Science International
Relations
Psychology Law Military Strategy Management Sports Game Playing Philosophy
6
Representation of games The games studied by game theory are well-defined mathematical objects A game consists of a set of players a set of moves (or strategies) available to those players and a specification of payoffs for each combination of strategies
7
Extensive form Games here are often presented as trees Here each vertex (or node) represents a point of choice for a player The player is specified by a number listed by the vertex The lines out of the vertex represent a possible action for that player The payoffs are specified at the bottom of the tree
8
Extensive form
9
Normal form The normal (or strategic form) game is usually
represented by a matrix which shows the players strategies and payoffs More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions
Player 2chooses Left
Player 2chooses Right
Player 1chooses Up
4 3 ndash1 ndash1
Player 1chooses Down
0 0 3 4
Normal form or payoff matrix of a 2-player 2-strategy game
10
Normal form When a game is presented in normal form
it is presumed that each player acts simultaneously or at least without knowing the actions of the other If players have some information about the choices of other players the game is usually presented in extensive form
11
Types of games Cooperative or non-cooperative
Symmetric and asymmetric
Zero sum and non-zero sum
Simultaneous and sequential
Perfect information and imperfect information
Infinitely long games
Discrete and continuous games
Meta games
12
Cooperative or non-cooperative A game is cooperative if the players are able to form binding
commitments For instance the legal system requires them to adhere to their promises In non-cooperative games this is not possible
Often it is assumed that communication among players is allowed in cooperative games but not in non-cooperative ones This classification on two binary criteria has been rejected (Harsanyi 1974)
Of the two types of games non-cooperative games are able to model situations to the finest details producing accurate results Cooperative games focus on the game at large Considerable efforts have been made to link the two approaches The so-called Nash-program has already established many of the cooperative solutions as non-cooperative equilibrium
13
Symmetric and asymmetric A symmetric game is a game where the
payoffs for playing a particular strategy depend only on the other strategies employed not on who is playing them If the identities of the players can be changed without changing the payoff to the strategies then a game is symmetric
14
Zero sum and non-zero sum Zero sum games are a special case of
constant sum games in which choices by players can neither increase nor decrease the available resources In zero-sum games the total benefit to all players in the game for every combination of strategies always adds to zero (more informally a player benefits only at the expense of others)
15
Simultaneous and sequential Simultaneous games are games where both players move
simultaneously or if they do not move simultaneously the later players are unaware of the earlier players actions (making them effectively simultaneous) Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions This need not be perfect information about every action of earlier players it might be very little knowledge For instance a player may know that an earlier player did not perform one particular action while he does not know which of the other available actions the first player actually performed
The difference between simultaneous and sequential games is captured in the different representations discussed above Normal form is used to represent simultaneous games and extensive form is used to represent sequential ones
16
Perfect information and imperfect information An important subset of sequential games consists
of games of perfect information A game is one of perfect information if all players know the moves previously made by all other players Thus only sequential games can be games of perfect information since in simultaneous games not every player knows the actions of the others
Perfect information is often confused with complete information which is a similar concept Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions
17
Infinitely long games Games as studied by economists and real-world
game players are generally finished in a finite number of moves Pure mathematicians are not so constrained and set theorists in particular study games that last for infinitely many moves with the winner (or other payoff) not known until after all those moves are completed
18
Discrete and continuous games Most of the objects treated in most branches of
game theory are discrete with a finite number of players moves events outcomes etc However the concepts can be extended into the realm of real numbers This branch has sometimes been called differential games because they map to a real line usually time although the behaviors may be mathematically discontinuous Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics
19
Meta games These are games the play of which is the
development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
4
What is Game Theory Neoclassical economics refers to a general
approach in economics focusing on the determination of prices outputs and income distributions in markets through supply and demand These are mediated through a hypothesized maximization of income-constrained utility by individuals and of cost-constrained profits of firms employing available information and factors of production
Antonietta Campus (1987) marginal economics The New Palgrave A Dictionary of Economics v 3 p 323
5
Applications of Game Theory
Mathematics Computer Science Biology Economics Political Science International
Relations
Psychology Law Military Strategy Management Sports Game Playing Philosophy
6
Representation of games The games studied by game theory are well-defined mathematical objects A game consists of a set of players a set of moves (or strategies) available to those players and a specification of payoffs for each combination of strategies
7
Extensive form Games here are often presented as trees Here each vertex (or node) represents a point of choice for a player The player is specified by a number listed by the vertex The lines out of the vertex represent a possible action for that player The payoffs are specified at the bottom of the tree
8
Extensive form
9
Normal form The normal (or strategic form) game is usually
represented by a matrix which shows the players strategies and payoffs More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions
Player 2chooses Left
Player 2chooses Right
Player 1chooses Up
4 3 ndash1 ndash1
Player 1chooses Down
0 0 3 4
Normal form or payoff matrix of a 2-player 2-strategy game
10
Normal form When a game is presented in normal form
it is presumed that each player acts simultaneously or at least without knowing the actions of the other If players have some information about the choices of other players the game is usually presented in extensive form
11
Types of games Cooperative or non-cooperative
Symmetric and asymmetric
Zero sum and non-zero sum
Simultaneous and sequential
Perfect information and imperfect information
Infinitely long games
Discrete and continuous games
Meta games
12
Cooperative or non-cooperative A game is cooperative if the players are able to form binding
commitments For instance the legal system requires them to adhere to their promises In non-cooperative games this is not possible
Often it is assumed that communication among players is allowed in cooperative games but not in non-cooperative ones This classification on two binary criteria has been rejected (Harsanyi 1974)
Of the two types of games non-cooperative games are able to model situations to the finest details producing accurate results Cooperative games focus on the game at large Considerable efforts have been made to link the two approaches The so-called Nash-program has already established many of the cooperative solutions as non-cooperative equilibrium
13
Symmetric and asymmetric A symmetric game is a game where the
payoffs for playing a particular strategy depend only on the other strategies employed not on who is playing them If the identities of the players can be changed without changing the payoff to the strategies then a game is symmetric
14
Zero sum and non-zero sum Zero sum games are a special case of
constant sum games in which choices by players can neither increase nor decrease the available resources In zero-sum games the total benefit to all players in the game for every combination of strategies always adds to zero (more informally a player benefits only at the expense of others)
15
Simultaneous and sequential Simultaneous games are games where both players move
simultaneously or if they do not move simultaneously the later players are unaware of the earlier players actions (making them effectively simultaneous) Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions This need not be perfect information about every action of earlier players it might be very little knowledge For instance a player may know that an earlier player did not perform one particular action while he does not know which of the other available actions the first player actually performed
The difference between simultaneous and sequential games is captured in the different representations discussed above Normal form is used to represent simultaneous games and extensive form is used to represent sequential ones
16
Perfect information and imperfect information An important subset of sequential games consists
of games of perfect information A game is one of perfect information if all players know the moves previously made by all other players Thus only sequential games can be games of perfect information since in simultaneous games not every player knows the actions of the others
Perfect information is often confused with complete information which is a similar concept Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions
17
Infinitely long games Games as studied by economists and real-world
game players are generally finished in a finite number of moves Pure mathematicians are not so constrained and set theorists in particular study games that last for infinitely many moves with the winner (or other payoff) not known until after all those moves are completed
18
Discrete and continuous games Most of the objects treated in most branches of
game theory are discrete with a finite number of players moves events outcomes etc However the concepts can be extended into the realm of real numbers This branch has sometimes been called differential games because they map to a real line usually time although the behaviors may be mathematically discontinuous Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics
19
Meta games These are games the play of which is the
development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
5
Applications of Game Theory
Mathematics Computer Science Biology Economics Political Science International
Relations
Psychology Law Military Strategy Management Sports Game Playing Philosophy
6
Representation of games The games studied by game theory are well-defined mathematical objects A game consists of a set of players a set of moves (or strategies) available to those players and a specification of payoffs for each combination of strategies
7
Extensive form Games here are often presented as trees Here each vertex (or node) represents a point of choice for a player The player is specified by a number listed by the vertex The lines out of the vertex represent a possible action for that player The payoffs are specified at the bottom of the tree
8
Extensive form
9
Normal form The normal (or strategic form) game is usually
represented by a matrix which shows the players strategies and payoffs More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions
Player 2chooses Left
Player 2chooses Right
Player 1chooses Up
4 3 ndash1 ndash1
Player 1chooses Down
0 0 3 4
Normal form or payoff matrix of a 2-player 2-strategy game
10
Normal form When a game is presented in normal form
it is presumed that each player acts simultaneously or at least without knowing the actions of the other If players have some information about the choices of other players the game is usually presented in extensive form
11
Types of games Cooperative or non-cooperative
Symmetric and asymmetric
Zero sum and non-zero sum
Simultaneous and sequential
Perfect information and imperfect information
Infinitely long games
Discrete and continuous games
Meta games
12
Cooperative or non-cooperative A game is cooperative if the players are able to form binding
commitments For instance the legal system requires them to adhere to their promises In non-cooperative games this is not possible
Often it is assumed that communication among players is allowed in cooperative games but not in non-cooperative ones This classification on two binary criteria has been rejected (Harsanyi 1974)
Of the two types of games non-cooperative games are able to model situations to the finest details producing accurate results Cooperative games focus on the game at large Considerable efforts have been made to link the two approaches The so-called Nash-program has already established many of the cooperative solutions as non-cooperative equilibrium
13
Symmetric and asymmetric A symmetric game is a game where the
payoffs for playing a particular strategy depend only on the other strategies employed not on who is playing them If the identities of the players can be changed without changing the payoff to the strategies then a game is symmetric
14
Zero sum and non-zero sum Zero sum games are a special case of
constant sum games in which choices by players can neither increase nor decrease the available resources In zero-sum games the total benefit to all players in the game for every combination of strategies always adds to zero (more informally a player benefits only at the expense of others)
15
Simultaneous and sequential Simultaneous games are games where both players move
simultaneously or if they do not move simultaneously the later players are unaware of the earlier players actions (making them effectively simultaneous) Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions This need not be perfect information about every action of earlier players it might be very little knowledge For instance a player may know that an earlier player did not perform one particular action while he does not know which of the other available actions the first player actually performed
The difference between simultaneous and sequential games is captured in the different representations discussed above Normal form is used to represent simultaneous games and extensive form is used to represent sequential ones
16
Perfect information and imperfect information An important subset of sequential games consists
of games of perfect information A game is one of perfect information if all players know the moves previously made by all other players Thus only sequential games can be games of perfect information since in simultaneous games not every player knows the actions of the others
Perfect information is often confused with complete information which is a similar concept Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions
17
Infinitely long games Games as studied by economists and real-world
game players are generally finished in a finite number of moves Pure mathematicians are not so constrained and set theorists in particular study games that last for infinitely many moves with the winner (or other payoff) not known until after all those moves are completed
18
Discrete and continuous games Most of the objects treated in most branches of
game theory are discrete with a finite number of players moves events outcomes etc However the concepts can be extended into the realm of real numbers This branch has sometimes been called differential games because they map to a real line usually time although the behaviors may be mathematically discontinuous Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics
19
Meta games These are games the play of which is the
development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
6
Representation of games The games studied by game theory are well-defined mathematical objects A game consists of a set of players a set of moves (or strategies) available to those players and a specification of payoffs for each combination of strategies
7
Extensive form Games here are often presented as trees Here each vertex (or node) represents a point of choice for a player The player is specified by a number listed by the vertex The lines out of the vertex represent a possible action for that player The payoffs are specified at the bottom of the tree
8
Extensive form
9
Normal form The normal (or strategic form) game is usually
represented by a matrix which shows the players strategies and payoffs More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions
Player 2chooses Left
Player 2chooses Right
Player 1chooses Up
4 3 ndash1 ndash1
Player 1chooses Down
0 0 3 4
Normal form or payoff matrix of a 2-player 2-strategy game
10
Normal form When a game is presented in normal form
it is presumed that each player acts simultaneously or at least without knowing the actions of the other If players have some information about the choices of other players the game is usually presented in extensive form
11
Types of games Cooperative or non-cooperative
Symmetric and asymmetric
Zero sum and non-zero sum
Simultaneous and sequential
Perfect information and imperfect information
Infinitely long games
Discrete and continuous games
Meta games
12
Cooperative or non-cooperative A game is cooperative if the players are able to form binding
commitments For instance the legal system requires them to adhere to their promises In non-cooperative games this is not possible
Often it is assumed that communication among players is allowed in cooperative games but not in non-cooperative ones This classification on two binary criteria has been rejected (Harsanyi 1974)
Of the two types of games non-cooperative games are able to model situations to the finest details producing accurate results Cooperative games focus on the game at large Considerable efforts have been made to link the two approaches The so-called Nash-program has already established many of the cooperative solutions as non-cooperative equilibrium
13
Symmetric and asymmetric A symmetric game is a game where the
payoffs for playing a particular strategy depend only on the other strategies employed not on who is playing them If the identities of the players can be changed without changing the payoff to the strategies then a game is symmetric
14
Zero sum and non-zero sum Zero sum games are a special case of
constant sum games in which choices by players can neither increase nor decrease the available resources In zero-sum games the total benefit to all players in the game for every combination of strategies always adds to zero (more informally a player benefits only at the expense of others)
15
Simultaneous and sequential Simultaneous games are games where both players move
simultaneously or if they do not move simultaneously the later players are unaware of the earlier players actions (making them effectively simultaneous) Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions This need not be perfect information about every action of earlier players it might be very little knowledge For instance a player may know that an earlier player did not perform one particular action while he does not know which of the other available actions the first player actually performed
The difference between simultaneous and sequential games is captured in the different representations discussed above Normal form is used to represent simultaneous games and extensive form is used to represent sequential ones
16
Perfect information and imperfect information An important subset of sequential games consists
of games of perfect information A game is one of perfect information if all players know the moves previously made by all other players Thus only sequential games can be games of perfect information since in simultaneous games not every player knows the actions of the others
Perfect information is often confused with complete information which is a similar concept Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions
17
Infinitely long games Games as studied by economists and real-world
game players are generally finished in a finite number of moves Pure mathematicians are not so constrained and set theorists in particular study games that last for infinitely many moves with the winner (or other payoff) not known until after all those moves are completed
18
Discrete and continuous games Most of the objects treated in most branches of
game theory are discrete with a finite number of players moves events outcomes etc However the concepts can be extended into the realm of real numbers This branch has sometimes been called differential games because they map to a real line usually time although the behaviors may be mathematically discontinuous Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics
19
Meta games These are games the play of which is the
development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
7
Extensive form Games here are often presented as trees Here each vertex (or node) represents a point of choice for a player The player is specified by a number listed by the vertex The lines out of the vertex represent a possible action for that player The payoffs are specified at the bottom of the tree
8
Extensive form
9
Normal form The normal (or strategic form) game is usually
represented by a matrix which shows the players strategies and payoffs More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions
Player 2chooses Left
Player 2chooses Right
Player 1chooses Up
4 3 ndash1 ndash1
Player 1chooses Down
0 0 3 4
Normal form or payoff matrix of a 2-player 2-strategy game
10
Normal form When a game is presented in normal form
it is presumed that each player acts simultaneously or at least without knowing the actions of the other If players have some information about the choices of other players the game is usually presented in extensive form
11
Types of games Cooperative or non-cooperative
Symmetric and asymmetric
Zero sum and non-zero sum
Simultaneous and sequential
Perfect information and imperfect information
Infinitely long games
Discrete and continuous games
Meta games
12
Cooperative or non-cooperative A game is cooperative if the players are able to form binding
commitments For instance the legal system requires them to adhere to their promises In non-cooperative games this is not possible
Often it is assumed that communication among players is allowed in cooperative games but not in non-cooperative ones This classification on two binary criteria has been rejected (Harsanyi 1974)
Of the two types of games non-cooperative games are able to model situations to the finest details producing accurate results Cooperative games focus on the game at large Considerable efforts have been made to link the two approaches The so-called Nash-program has already established many of the cooperative solutions as non-cooperative equilibrium
13
Symmetric and asymmetric A symmetric game is a game where the
payoffs for playing a particular strategy depend only on the other strategies employed not on who is playing them If the identities of the players can be changed without changing the payoff to the strategies then a game is symmetric
14
Zero sum and non-zero sum Zero sum games are a special case of
constant sum games in which choices by players can neither increase nor decrease the available resources In zero-sum games the total benefit to all players in the game for every combination of strategies always adds to zero (more informally a player benefits only at the expense of others)
15
Simultaneous and sequential Simultaneous games are games where both players move
simultaneously or if they do not move simultaneously the later players are unaware of the earlier players actions (making them effectively simultaneous) Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions This need not be perfect information about every action of earlier players it might be very little knowledge For instance a player may know that an earlier player did not perform one particular action while he does not know which of the other available actions the first player actually performed
The difference between simultaneous and sequential games is captured in the different representations discussed above Normal form is used to represent simultaneous games and extensive form is used to represent sequential ones
16
Perfect information and imperfect information An important subset of sequential games consists
of games of perfect information A game is one of perfect information if all players know the moves previously made by all other players Thus only sequential games can be games of perfect information since in simultaneous games not every player knows the actions of the others
Perfect information is often confused with complete information which is a similar concept Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions
17
Infinitely long games Games as studied by economists and real-world
game players are generally finished in a finite number of moves Pure mathematicians are not so constrained and set theorists in particular study games that last for infinitely many moves with the winner (or other payoff) not known until after all those moves are completed
18
Discrete and continuous games Most of the objects treated in most branches of
game theory are discrete with a finite number of players moves events outcomes etc However the concepts can be extended into the realm of real numbers This branch has sometimes been called differential games because they map to a real line usually time although the behaviors may be mathematically discontinuous Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics
19
Meta games These are games the play of which is the
development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
8
Extensive form
9
Normal form The normal (or strategic form) game is usually
represented by a matrix which shows the players strategies and payoffs More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions
Player 2chooses Left
Player 2chooses Right
Player 1chooses Up
4 3 ndash1 ndash1
Player 1chooses Down
0 0 3 4
Normal form or payoff matrix of a 2-player 2-strategy game
10
Normal form When a game is presented in normal form
it is presumed that each player acts simultaneously or at least without knowing the actions of the other If players have some information about the choices of other players the game is usually presented in extensive form
11
Types of games Cooperative or non-cooperative
Symmetric and asymmetric
Zero sum and non-zero sum
Simultaneous and sequential
Perfect information and imperfect information
Infinitely long games
Discrete and continuous games
Meta games
12
Cooperative or non-cooperative A game is cooperative if the players are able to form binding
commitments For instance the legal system requires them to adhere to their promises In non-cooperative games this is not possible
Often it is assumed that communication among players is allowed in cooperative games but not in non-cooperative ones This classification on two binary criteria has been rejected (Harsanyi 1974)
Of the two types of games non-cooperative games are able to model situations to the finest details producing accurate results Cooperative games focus on the game at large Considerable efforts have been made to link the two approaches The so-called Nash-program has already established many of the cooperative solutions as non-cooperative equilibrium
13
Symmetric and asymmetric A symmetric game is a game where the
payoffs for playing a particular strategy depend only on the other strategies employed not on who is playing them If the identities of the players can be changed without changing the payoff to the strategies then a game is symmetric
14
Zero sum and non-zero sum Zero sum games are a special case of
constant sum games in which choices by players can neither increase nor decrease the available resources In zero-sum games the total benefit to all players in the game for every combination of strategies always adds to zero (more informally a player benefits only at the expense of others)
15
Simultaneous and sequential Simultaneous games are games where both players move
simultaneously or if they do not move simultaneously the later players are unaware of the earlier players actions (making them effectively simultaneous) Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions This need not be perfect information about every action of earlier players it might be very little knowledge For instance a player may know that an earlier player did not perform one particular action while he does not know which of the other available actions the first player actually performed
The difference between simultaneous and sequential games is captured in the different representations discussed above Normal form is used to represent simultaneous games and extensive form is used to represent sequential ones
16
Perfect information and imperfect information An important subset of sequential games consists
of games of perfect information A game is one of perfect information if all players know the moves previously made by all other players Thus only sequential games can be games of perfect information since in simultaneous games not every player knows the actions of the others
Perfect information is often confused with complete information which is a similar concept Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions
17
Infinitely long games Games as studied by economists and real-world
game players are generally finished in a finite number of moves Pure mathematicians are not so constrained and set theorists in particular study games that last for infinitely many moves with the winner (or other payoff) not known until after all those moves are completed
18
Discrete and continuous games Most of the objects treated in most branches of
game theory are discrete with a finite number of players moves events outcomes etc However the concepts can be extended into the realm of real numbers This branch has sometimes been called differential games because they map to a real line usually time although the behaviors may be mathematically discontinuous Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics
19
Meta games These are games the play of which is the
development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
9
Normal form The normal (or strategic form) game is usually
represented by a matrix which shows the players strategies and payoffs More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions
Player 2chooses Left
Player 2chooses Right
Player 1chooses Up
4 3 ndash1 ndash1
Player 1chooses Down
0 0 3 4
Normal form or payoff matrix of a 2-player 2-strategy game
10
Normal form When a game is presented in normal form
it is presumed that each player acts simultaneously or at least without knowing the actions of the other If players have some information about the choices of other players the game is usually presented in extensive form
11
Types of games Cooperative or non-cooperative
Symmetric and asymmetric
Zero sum and non-zero sum
Simultaneous and sequential
Perfect information and imperfect information
Infinitely long games
Discrete and continuous games
Meta games
12
Cooperative or non-cooperative A game is cooperative if the players are able to form binding
commitments For instance the legal system requires them to adhere to their promises In non-cooperative games this is not possible
Often it is assumed that communication among players is allowed in cooperative games but not in non-cooperative ones This classification on two binary criteria has been rejected (Harsanyi 1974)
Of the two types of games non-cooperative games are able to model situations to the finest details producing accurate results Cooperative games focus on the game at large Considerable efforts have been made to link the two approaches The so-called Nash-program has already established many of the cooperative solutions as non-cooperative equilibrium
13
Symmetric and asymmetric A symmetric game is a game where the
payoffs for playing a particular strategy depend only on the other strategies employed not on who is playing them If the identities of the players can be changed without changing the payoff to the strategies then a game is symmetric
14
Zero sum and non-zero sum Zero sum games are a special case of
constant sum games in which choices by players can neither increase nor decrease the available resources In zero-sum games the total benefit to all players in the game for every combination of strategies always adds to zero (more informally a player benefits only at the expense of others)
15
Simultaneous and sequential Simultaneous games are games where both players move
simultaneously or if they do not move simultaneously the later players are unaware of the earlier players actions (making them effectively simultaneous) Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions This need not be perfect information about every action of earlier players it might be very little knowledge For instance a player may know that an earlier player did not perform one particular action while he does not know which of the other available actions the first player actually performed
The difference between simultaneous and sequential games is captured in the different representations discussed above Normal form is used to represent simultaneous games and extensive form is used to represent sequential ones
16
Perfect information and imperfect information An important subset of sequential games consists
of games of perfect information A game is one of perfect information if all players know the moves previously made by all other players Thus only sequential games can be games of perfect information since in simultaneous games not every player knows the actions of the others
Perfect information is often confused with complete information which is a similar concept Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions
17
Infinitely long games Games as studied by economists and real-world
game players are generally finished in a finite number of moves Pure mathematicians are not so constrained and set theorists in particular study games that last for infinitely many moves with the winner (or other payoff) not known until after all those moves are completed
18
Discrete and continuous games Most of the objects treated in most branches of
game theory are discrete with a finite number of players moves events outcomes etc However the concepts can be extended into the realm of real numbers This branch has sometimes been called differential games because they map to a real line usually time although the behaviors may be mathematically discontinuous Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics
19
Meta games These are games the play of which is the
development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
10
Normal form When a game is presented in normal form
it is presumed that each player acts simultaneously or at least without knowing the actions of the other If players have some information about the choices of other players the game is usually presented in extensive form
11
Types of games Cooperative or non-cooperative
Symmetric and asymmetric
Zero sum and non-zero sum
Simultaneous and sequential
Perfect information and imperfect information
Infinitely long games
Discrete and continuous games
Meta games
12
Cooperative or non-cooperative A game is cooperative if the players are able to form binding
commitments For instance the legal system requires them to adhere to their promises In non-cooperative games this is not possible
Often it is assumed that communication among players is allowed in cooperative games but not in non-cooperative ones This classification on two binary criteria has been rejected (Harsanyi 1974)
Of the two types of games non-cooperative games are able to model situations to the finest details producing accurate results Cooperative games focus on the game at large Considerable efforts have been made to link the two approaches The so-called Nash-program has already established many of the cooperative solutions as non-cooperative equilibrium
13
Symmetric and asymmetric A symmetric game is a game where the
payoffs for playing a particular strategy depend only on the other strategies employed not on who is playing them If the identities of the players can be changed without changing the payoff to the strategies then a game is symmetric
14
Zero sum and non-zero sum Zero sum games are a special case of
constant sum games in which choices by players can neither increase nor decrease the available resources In zero-sum games the total benefit to all players in the game for every combination of strategies always adds to zero (more informally a player benefits only at the expense of others)
15
Simultaneous and sequential Simultaneous games are games where both players move
simultaneously or if they do not move simultaneously the later players are unaware of the earlier players actions (making them effectively simultaneous) Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions This need not be perfect information about every action of earlier players it might be very little knowledge For instance a player may know that an earlier player did not perform one particular action while he does not know which of the other available actions the first player actually performed
The difference between simultaneous and sequential games is captured in the different representations discussed above Normal form is used to represent simultaneous games and extensive form is used to represent sequential ones
16
Perfect information and imperfect information An important subset of sequential games consists
of games of perfect information A game is one of perfect information if all players know the moves previously made by all other players Thus only sequential games can be games of perfect information since in simultaneous games not every player knows the actions of the others
Perfect information is often confused with complete information which is a similar concept Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions
17
Infinitely long games Games as studied by economists and real-world
game players are generally finished in a finite number of moves Pure mathematicians are not so constrained and set theorists in particular study games that last for infinitely many moves with the winner (or other payoff) not known until after all those moves are completed
18
Discrete and continuous games Most of the objects treated in most branches of
game theory are discrete with a finite number of players moves events outcomes etc However the concepts can be extended into the realm of real numbers This branch has sometimes been called differential games because they map to a real line usually time although the behaviors may be mathematically discontinuous Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics
19
Meta games These are games the play of which is the
development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
11
Types of games Cooperative or non-cooperative
Symmetric and asymmetric
Zero sum and non-zero sum
Simultaneous and sequential
Perfect information and imperfect information
Infinitely long games
Discrete and continuous games
Meta games
12
Cooperative or non-cooperative A game is cooperative if the players are able to form binding
commitments For instance the legal system requires them to adhere to their promises In non-cooperative games this is not possible
Often it is assumed that communication among players is allowed in cooperative games but not in non-cooperative ones This classification on two binary criteria has been rejected (Harsanyi 1974)
Of the two types of games non-cooperative games are able to model situations to the finest details producing accurate results Cooperative games focus on the game at large Considerable efforts have been made to link the two approaches The so-called Nash-program has already established many of the cooperative solutions as non-cooperative equilibrium
13
Symmetric and asymmetric A symmetric game is a game where the
payoffs for playing a particular strategy depend only on the other strategies employed not on who is playing them If the identities of the players can be changed without changing the payoff to the strategies then a game is symmetric
14
Zero sum and non-zero sum Zero sum games are a special case of
constant sum games in which choices by players can neither increase nor decrease the available resources In zero-sum games the total benefit to all players in the game for every combination of strategies always adds to zero (more informally a player benefits only at the expense of others)
15
Simultaneous and sequential Simultaneous games are games where both players move
simultaneously or if they do not move simultaneously the later players are unaware of the earlier players actions (making them effectively simultaneous) Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions This need not be perfect information about every action of earlier players it might be very little knowledge For instance a player may know that an earlier player did not perform one particular action while he does not know which of the other available actions the first player actually performed
The difference between simultaneous and sequential games is captured in the different representations discussed above Normal form is used to represent simultaneous games and extensive form is used to represent sequential ones
16
Perfect information and imperfect information An important subset of sequential games consists
of games of perfect information A game is one of perfect information if all players know the moves previously made by all other players Thus only sequential games can be games of perfect information since in simultaneous games not every player knows the actions of the others
Perfect information is often confused with complete information which is a similar concept Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions
17
Infinitely long games Games as studied by economists and real-world
game players are generally finished in a finite number of moves Pure mathematicians are not so constrained and set theorists in particular study games that last for infinitely many moves with the winner (or other payoff) not known until after all those moves are completed
18
Discrete and continuous games Most of the objects treated in most branches of
game theory are discrete with a finite number of players moves events outcomes etc However the concepts can be extended into the realm of real numbers This branch has sometimes been called differential games because they map to a real line usually time although the behaviors may be mathematically discontinuous Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics
19
Meta games These are games the play of which is the
development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
12
Cooperative or non-cooperative A game is cooperative if the players are able to form binding
commitments For instance the legal system requires them to adhere to their promises In non-cooperative games this is not possible
Often it is assumed that communication among players is allowed in cooperative games but not in non-cooperative ones This classification on two binary criteria has been rejected (Harsanyi 1974)
Of the two types of games non-cooperative games are able to model situations to the finest details producing accurate results Cooperative games focus on the game at large Considerable efforts have been made to link the two approaches The so-called Nash-program has already established many of the cooperative solutions as non-cooperative equilibrium
13
Symmetric and asymmetric A symmetric game is a game where the
payoffs for playing a particular strategy depend only on the other strategies employed not on who is playing them If the identities of the players can be changed without changing the payoff to the strategies then a game is symmetric
14
Zero sum and non-zero sum Zero sum games are a special case of
constant sum games in which choices by players can neither increase nor decrease the available resources In zero-sum games the total benefit to all players in the game for every combination of strategies always adds to zero (more informally a player benefits only at the expense of others)
15
Simultaneous and sequential Simultaneous games are games where both players move
simultaneously or if they do not move simultaneously the later players are unaware of the earlier players actions (making them effectively simultaneous) Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions This need not be perfect information about every action of earlier players it might be very little knowledge For instance a player may know that an earlier player did not perform one particular action while he does not know which of the other available actions the first player actually performed
The difference between simultaneous and sequential games is captured in the different representations discussed above Normal form is used to represent simultaneous games and extensive form is used to represent sequential ones
16
Perfect information and imperfect information An important subset of sequential games consists
of games of perfect information A game is one of perfect information if all players know the moves previously made by all other players Thus only sequential games can be games of perfect information since in simultaneous games not every player knows the actions of the others
Perfect information is often confused with complete information which is a similar concept Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions
17
Infinitely long games Games as studied by economists and real-world
game players are generally finished in a finite number of moves Pure mathematicians are not so constrained and set theorists in particular study games that last for infinitely many moves with the winner (or other payoff) not known until after all those moves are completed
18
Discrete and continuous games Most of the objects treated in most branches of
game theory are discrete with a finite number of players moves events outcomes etc However the concepts can be extended into the realm of real numbers This branch has sometimes been called differential games because they map to a real line usually time although the behaviors may be mathematically discontinuous Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics
19
Meta games These are games the play of which is the
development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
13
Symmetric and asymmetric A symmetric game is a game where the
payoffs for playing a particular strategy depend only on the other strategies employed not on who is playing them If the identities of the players can be changed without changing the payoff to the strategies then a game is symmetric
14
Zero sum and non-zero sum Zero sum games are a special case of
constant sum games in which choices by players can neither increase nor decrease the available resources In zero-sum games the total benefit to all players in the game for every combination of strategies always adds to zero (more informally a player benefits only at the expense of others)
15
Simultaneous and sequential Simultaneous games are games where both players move
simultaneously or if they do not move simultaneously the later players are unaware of the earlier players actions (making them effectively simultaneous) Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions This need not be perfect information about every action of earlier players it might be very little knowledge For instance a player may know that an earlier player did not perform one particular action while he does not know which of the other available actions the first player actually performed
The difference between simultaneous and sequential games is captured in the different representations discussed above Normal form is used to represent simultaneous games and extensive form is used to represent sequential ones
16
Perfect information and imperfect information An important subset of sequential games consists
of games of perfect information A game is one of perfect information if all players know the moves previously made by all other players Thus only sequential games can be games of perfect information since in simultaneous games not every player knows the actions of the others
Perfect information is often confused with complete information which is a similar concept Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions
17
Infinitely long games Games as studied by economists and real-world
game players are generally finished in a finite number of moves Pure mathematicians are not so constrained and set theorists in particular study games that last for infinitely many moves with the winner (or other payoff) not known until after all those moves are completed
18
Discrete and continuous games Most of the objects treated in most branches of
game theory are discrete with a finite number of players moves events outcomes etc However the concepts can be extended into the realm of real numbers This branch has sometimes been called differential games because they map to a real line usually time although the behaviors may be mathematically discontinuous Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics
19
Meta games These are games the play of which is the
development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
14
Zero sum and non-zero sum Zero sum games are a special case of
constant sum games in which choices by players can neither increase nor decrease the available resources In zero-sum games the total benefit to all players in the game for every combination of strategies always adds to zero (more informally a player benefits only at the expense of others)
15
Simultaneous and sequential Simultaneous games are games where both players move
simultaneously or if they do not move simultaneously the later players are unaware of the earlier players actions (making them effectively simultaneous) Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions This need not be perfect information about every action of earlier players it might be very little knowledge For instance a player may know that an earlier player did not perform one particular action while he does not know which of the other available actions the first player actually performed
The difference between simultaneous and sequential games is captured in the different representations discussed above Normal form is used to represent simultaneous games and extensive form is used to represent sequential ones
16
Perfect information and imperfect information An important subset of sequential games consists
of games of perfect information A game is one of perfect information if all players know the moves previously made by all other players Thus only sequential games can be games of perfect information since in simultaneous games not every player knows the actions of the others
Perfect information is often confused with complete information which is a similar concept Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions
17
Infinitely long games Games as studied by economists and real-world
game players are generally finished in a finite number of moves Pure mathematicians are not so constrained and set theorists in particular study games that last for infinitely many moves with the winner (or other payoff) not known until after all those moves are completed
18
Discrete and continuous games Most of the objects treated in most branches of
game theory are discrete with a finite number of players moves events outcomes etc However the concepts can be extended into the realm of real numbers This branch has sometimes been called differential games because they map to a real line usually time although the behaviors may be mathematically discontinuous Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics
19
Meta games These are games the play of which is the
development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
15
Simultaneous and sequential Simultaneous games are games where both players move
simultaneously or if they do not move simultaneously the later players are unaware of the earlier players actions (making them effectively simultaneous) Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions This need not be perfect information about every action of earlier players it might be very little knowledge For instance a player may know that an earlier player did not perform one particular action while he does not know which of the other available actions the first player actually performed
The difference between simultaneous and sequential games is captured in the different representations discussed above Normal form is used to represent simultaneous games and extensive form is used to represent sequential ones
16
Perfect information and imperfect information An important subset of sequential games consists
of games of perfect information A game is one of perfect information if all players know the moves previously made by all other players Thus only sequential games can be games of perfect information since in simultaneous games not every player knows the actions of the others
Perfect information is often confused with complete information which is a similar concept Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions
17
Infinitely long games Games as studied by economists and real-world
game players are generally finished in a finite number of moves Pure mathematicians are not so constrained and set theorists in particular study games that last for infinitely many moves with the winner (or other payoff) not known until after all those moves are completed
18
Discrete and continuous games Most of the objects treated in most branches of
game theory are discrete with a finite number of players moves events outcomes etc However the concepts can be extended into the realm of real numbers This branch has sometimes been called differential games because they map to a real line usually time although the behaviors may be mathematically discontinuous Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics
19
Meta games These are games the play of which is the
development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
16
Perfect information and imperfect information An important subset of sequential games consists
of games of perfect information A game is one of perfect information if all players know the moves previously made by all other players Thus only sequential games can be games of perfect information since in simultaneous games not every player knows the actions of the others
Perfect information is often confused with complete information which is a similar concept Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions
17
Infinitely long games Games as studied by economists and real-world
game players are generally finished in a finite number of moves Pure mathematicians are not so constrained and set theorists in particular study games that last for infinitely many moves with the winner (or other payoff) not known until after all those moves are completed
18
Discrete and continuous games Most of the objects treated in most branches of
game theory are discrete with a finite number of players moves events outcomes etc However the concepts can be extended into the realm of real numbers This branch has sometimes been called differential games because they map to a real line usually time although the behaviors may be mathematically discontinuous Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics
19
Meta games These are games the play of which is the
development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
17
Infinitely long games Games as studied by economists and real-world
game players are generally finished in a finite number of moves Pure mathematicians are not so constrained and set theorists in particular study games that last for infinitely many moves with the winner (or other payoff) not known until after all those moves are completed
18
Discrete and continuous games Most of the objects treated in most branches of
game theory are discrete with a finite number of players moves events outcomes etc However the concepts can be extended into the realm of real numbers This branch has sometimes been called differential games because they map to a real line usually time although the behaviors may be mathematically discontinuous Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics
19
Meta games These are games the play of which is the
development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
18
Discrete and continuous games Most of the objects treated in most branches of
game theory are discrete with a finite number of players moves events outcomes etc However the concepts can be extended into the realm of real numbers This branch has sometimes been called differential games because they map to a real line usually time although the behaviors may be mathematically discontinuous Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics
19
Meta games These are games the play of which is the
development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
19
Meta games These are games the play of which is the
development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
20
Key Elements of a Game
Players Who is interacting
Strategies What are their options
Payoffs What are their incentives
Information What do they know
Rationality How do they think
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
21
Cigarette Advertising on TV
All US tobacco companies advertised heavily on
television
Surgeon General issues official warning
bullCigarette smoking may be hazardous
Cigarette companiesrsquo reaction
bullFear of potential liability lawsuits
Companies strike agreement
bullCarry the warning label and cease TV advertising in exchange for
immunity from federal lawsuits
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
22
Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits
Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
23
Payoff Table
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
24
Best responses
Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise
Advertise is dominant strategy This is another Prisonersrsquo Dilemma
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
25
What Happened
After the 1970 agreement cigarette advertising decreased by $63 million
Profits rose by $91 million
Whyhow were the firms able to escape from the Prisonerrsquos Dilemma
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
26
Changing the Game through Government-Enforced Collusion
The agreement with the government forced the firms not to advertise
The preferred outcome (No Ad No Ad) then was all that remained feasible
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
27
Rationality
Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by
1forming a belief about the world
2choosing an action that maximizes your welfare given that belief
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
28
And Common Knowledge of Rationality Most game-theoretic analysis makes the
further assumption that playersrsquo rationality is common knowledge
bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each
player is rationalbullEach player knows that each player knows that each
player knows thateach player is rationalbullEach player knows that each player knows that each
player knows that each player knows that each player is rational
bullEtc etc etc
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
29
And Correct Beliefs
Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow
Implicitly this is saying that in novel strategic situations each player knows what the other believes
Requires all players to thoroughly understand each other
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
30
Nash Equilibrium Nash Equilibrium
bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies
Best Response
bullThe strategy that maximizes my payoff given othersrsquo strategies
Everybody is playing a best response
bullNo incentive to unilaterally change my strategy
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
31
Dominant StrategiesRecall Cigarette Ad Game
Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
32
Dominant Strategies and Rationality
If you are rational you should play your dominant strategy Period
No need to think about whether others are rational etc
Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
33
Dominant Strategies and Rationality
Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may
fail to be ldquorationalrdquo
Yet dominant strategies give a clear prescription of what to do regardless
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
34
Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co
slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo
--Wall Street Journal January 31 2003
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
35
SUV Price Wars The Game
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
36
SUV Price Wars Outcome
Each firm has a unilateral incentive to discount but neither achieves a pricing advantage
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
37
Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game
1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)
2 But both firms are worse off when they both Discount than if they both Donrsquot
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
38
Prisonersrsquo Dilemma Game
Key features
bullBoth players have a dominant strategy to Confess
bullBUT both players better off if they both donrsquot
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
39
Prisonersrsquo Dilemma Game
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
40
Reaction Curves in Prisonersrsquo Dilemma
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
41
Evolution in Prisonersrsquo Dilemma (One Population)
Row and Col players are drawn from the same population
Those who Confess get higher payoff so Confess dominates the population
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
42
Loyal Servant Game
Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same
thing as Master
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
43
Loyal Servant Game
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
44
Reaction Curves in Loyal Servant Game
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
45
Evolution in Loyal Servant Game (Two Populations)
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
46
Bluffing in Poker Set-Up
Player A will be drawing on an inside straight flush
Player A will have the best hand if bullflush (another club 9 cards total) or bullstraight (any 2 or 7 additional 6 cards)
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
47
Winning Cards
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
48
Bluffing Game Rules Each player has put $100 into the pot
After receiving the fifth card player A will either Raise $100or Not
If Raise Player B then either Calls (adds $100 more) or Folds (automatically losing $100 already in pot)
Player A wins the pot if either A gets winning card or B folds
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
49
Bluffing Game Rules
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
50
Analysis of Bluffing Game You get Good Card 1548 about 13 What do you do with Bad Card
If you never raise player B will always Fold when you have a Good Card
bullget +100 when Good -100 when Bad
bullaverage payoff about ndash33 If you always raise player B will always Call
you on it (even worse)
bullget + 200 when Good -200 when Bad
bullaverage payoff about -67
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
51
How Often to Raise in Equilibrium Need to Raise enough for Player B to be
indifferent between Fold and Call B gets ndash100 if Folds
B gets either ndash200 or +200 if CallsbullBy Call B ldquorisks 100 to gain 300rdquo relative to Fold
bullSo we need Prob(Bluff| Raise) = 25
15 Good Cards so we Bluff on 5 Bad CardsbullSo Raise with 533 Bad Cards
bullWhen 13 chance of Good Card Bluff with prob 16
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
52
How Often to Fold in Equilibrium Need to Fold enough for Player A to be
indifferent between Raise and Not with Bad Card
A gets ndash100 if Not Raise A gets either ndash200 or +100 if Raise
bullBy raising A ldquorisks 100 to gain 200rdquo So we Fold 33
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
53
Payoffs in Equilibrium Player B Folds 33 of time
Good Card 33(+100)+67(+200) so get 167 when Good Card
hellipamp Player A indifferent to Raise or Not given a Bad Card ndash100 when Bad Card
Overall payoff is about ndash11for A much better than alwaysnever bluffing
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
54
Best responses in bluffing
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
55
Next Step hellip
Strategies and game theory
from to
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
56
References 1 David McAdamsGame Theory for
Managers MIT openCourseWare 2 Adam M Brandenburger and Barry J
Nalebuff The Right Game Use Game Theory to Shape Strategy Harvard Buseness Review July - August 1995
3 The Internet wwwwikipediaorg
