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Chapter 14 – Game Theory
14.1 Nash Equilibrium14.2 Repeated Prisoners’ Dilemma
14.3 Sequential-Move Games and
Strategic Moves
2
Game Theory and LifeYou are on a first date with the love of
your dreams. You can propose 2 activities:
1)Safe activity (Coffee)2)Exciting Activity (Waterpark)
Your date could either want a safe activity or an exciting activity. There are different results if your ideas match up or clash:
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Mr/MissRightMr/MissRight
YouYou
Chapter Fourteen
First Date GameWhat is the outcome of this game?Payoff format is (Left, Top)
4Chapter Fourteen
Game Theory ComponentsPlayers: agents participating in the game (You and Your Date
Strategies: Actions that each player may take under any possible circumstance (Coffee, Waterpark)
Outcomes: The various possible results of the game (four, each represented by one cell of the payoff matrix)
Payoffs: The benefit that each player gets from each possible outcome of the game (the profits entered in each cell of the payoff matrix)
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Best ResponsesIn all game theory games, players choose
strategies without knowing with certainty what the opposing player will do.
Players construct BEST RESPONSES
-optimal actions given all possible actions of other players
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Mr/MissRightMr/MissRight
YouYou
Chapter Fourteen
First Date Game Best ResponsesIf you know your date will pick coffee, you should pick coffee, since 10 > -5If you know your date will pick waterpark, you should pick waterpark, since 20 > 0
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Mr/MissRightMr/MissRight
YouYou
Chapter Fourteen
First Date Game Best ResponsesIf your date knows you will pick coffee, they should pick coffee, since 10 > -5If your date knows you will pick waterpark, they should pick waterpark, since 20 > 0
Note that this game is SYMMETRICAL
8Chapter Fourteen
Nash Equilibrium
Definition: A Nash Equilibrium occurs when each player chooses a strategy that gives him/her the highest payoff, given the strategy chosen by the other player(s) in the game. ("rational self-interest")
Nash Equilibria occur when best responses line up
The Date Game:
Nash equilibria: Each proposes coffee or each proposes waterpark.
Definition: A Nash Equilibrium occurs when each player chooses a strategy that gives him/her the highest payoff, given the strategy chosen by the other player(s) in the game. ("rational self-interest")
Nash Equilibria occur when best responses line up
The Date Game:
Nash equilibria: Each proposes coffee or each proposes waterpark.
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Game Theory
•A special kind of Best Response:
•Strategy that is best no matter what the other player does.
DOMINANT STRATEGY
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Advertising
A’s profit= $50 000
A’s loss =
$25 000
A’s profit= $75 000
A’s profit = $10 000
B’s profit = $50 000
B’s profit = $75 000
B’s loss = $25 000
B’s profit = $10 000
Don’t advertise AdvertiseB’s
STRATEGY
A’s STRATEGY
Don’t advertise
Advertise
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Dominant Strategy
A’s profit= $50 000
A’s loss =
$25 000
A’s profit= $75 000
A’s profit = $10 000
B’s profit = $50 000
B’s profit = $75 000
B’s loss = $25 000
B’s profit = $10 000
Don’t advertise AdvertiseB’s
dominant strategy is advertise
A’s dominant strategy is advertise
Don’t advertise
Advertise
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Prisoner’s Dilemma
• This is an example of a prisoner’s dilemma type of game.– There is dominant strategy. – The dominant strategy does not result in the best
outcome for either player.– It is hard to cooperate even when it would be
beneficial for both players to do so– Cooperation between players is difficult to maintain
because cooperation is individually irrational.
• eg., The dominant strategy: advertise
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Classic Prisoners’ DilemmaRocky’s strategies
ConfessDeny
Ginger’sstrategies
Confess
5 yearsPrison
5 yearsPrison
7 yearsPrison
Go free
1 yearPrison
1 yearPrison
7 yearsPrison
Go freeDenyDominant strategy: confess, even though they would both be better off if they both kept their mouths shut.
Dominant Strategy Equilibrium
Definition: A Dominant Strategy Equilibrium occurs when each player uses a dominant strategy.
HondaHonda
ToyotaToyota
15Chapter Fourteen
Dominated Strategy
Definition: A player has a dominated strategy when the player has another strategy that gives it a higher payoff no matter what the other player does.
HondaHonda
ToyotaToyota
16Chapter Fourteen
Dominant or Dominated Strategy
Why look for dominant or dominated strategies?
A dominant strategy equilibrium is particularly compelling as a "likely" outcome
Similarly, because dominated strategies are unlikely to be played, these strategies can be eliminated from consideration in more complex games. This can make solving the game easier.
Why look for dominant or dominated strategies?
A dominant strategy equilibrium is particularly compelling as a "likely" outcome
Similarly, because dominated strategies are unlikely to be played, these strategies can be eliminated from consideration in more complex games. This can make solving the game easier.
17
HondaHonda
Dominated Strategy
ToyotaToyota
"Build Large" is dominated for each playerBy eliminating the dominated strategies, we can reduce the game matrix.
18Chapter Fourteen
Finding Nash Equilibrium Cases
1) Nash Equilibrium where Dominant Strategies overlap
2) Nash Equilibrium with one Dominant Strategy
3) Nash Equilibrium by eliminating Dominated Strategy
4) Nash Equilibrium through Best Responses
1) Nash Equilibrium where Dominant Strategies overlap
2) Nash Equilibrium with one Dominant Strategy
3) Nash Equilibrium by eliminating Dominated Strategy
4) Nash Equilibrium through Best Responses
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StudentStudent
Nash Equilibrium – Dominant Overlap
ProfessorProfessor
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StudentStudent
Nash Equilibrium – One Dominant
ProfessorProfessor
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StudentStudent
Nash Equilibrium – Eliminate Dominated
ProfessorProfessor
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StudentStudent
Nash Equilibrium – Best Responses
ProfessorProfessor
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Nash Equilibrium
• However it is found, a Nash Equilibrium ALWAYS occurs where Best Responses line up
• If Multiple Nash Equilibria exist, we can’t conclude WHICH outcome will occur, only the possible outcomes that can occur
• Also, it is often APPEARS that no Nash Equilibria exist:
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BarneyBarney
No Nash Equilibrium
FredFred
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Mixed StrategiesPure Strategy – A specific choice of a strategy from the player’s possible strategies in a game. (ie: Rock)
Mixed Strategy – A choice among two or more pure strategies according to pre-specified probabilities. (ie: Rock, Paper or Scissors each 1/3rd of the time)
If Pure Strategies can’t produce a Nash Equilibrium, Mixed Strategies can:
If both players randomize each choice 1/3rd of the time, nether have an incentive to deviate.