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SCIT1003Chapter 3: Prisoner’s Dilemma

Non-Zero Sum Game

Prof. Tsang

Zero-Sum Games

• The sum of the payoffs remains constant during the course of the game.

• Two sides in conflict, e.g. chess, sports• Being well informed always helps a player

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Example of zero-sum game

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Matching Pennies

matcherMis-matcher

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Rock-Paper-Scissors

Military game: attack the easy or hard pass?

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Easy pass Hard pass

Easy pass 0.5, 0.5 0.4, 0.6

Hard pass 0, 1 1, 0

Attacker

Defense side

Payoff is the winning probability.

Games of Conflict

• Two sides competing against each other• Characteristics of zero-sum games: your loss

is my gain• Simultaneous moves: lack of information

about the opponent’s move• Logical circle of reasoning: I think that he

thinks that I think that …

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Zero-sum game matrices are sometimes expressed with only one number in each box, in which case each entry is interpreted as a gain for row-player and a loss for column-player.

Player A

Player B

Non-Zero Sum GamePrisoner’s Dilemma

• A zero-sum game is one in which the players' interests are in direct conflict, e.g. in football, one team wins and the other loses; payoffs sum to zero.

• A game is non-zero-sum, if players interests are not always in direct conflict, so that there are opportunities for both to gain, e.g. games in economics

• For example, when both players choose Don't Confess in the Prisoners' Dilemma

• Most game in reality have aspects of common interests as well as conflict.

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Prisoners’ Dilemma: payoff matrix

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Confess Don’t Confess

Confess -5, -5 0, -10

Don’t Confess

-10, 0 -3, -3

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Imperfect Information

• Partial or no information concerning the opponent is given in advance to the player’s decision, e.g. Prisoner’s Dilemma.

• Imperfect information may be diminished over time if the same game with the same opponent is played repeatedly.

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Games of Co-operationPlayers may improve payoff through• communicating• forming binding coalitions & agreements • do not apply to zero-sum games

Prisoner’s Dilemma with Cooperation

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Strategies• A strategy is a “complete plan of action” that fully

determines the player's behavior, a decision rule or set of instructions about which actions a player should take following all possible histories up to that stage.

• The strategy concept is sometimes (wrongly) confused with that of a move. A move is an action taken by a player at some point during the play of a game (e.g., in chess, moving white's Bishop a2 to b3).

• A strategy on the other hand is a complete algorithm for playing the game, telling a player what to do for every possible situation throughout the game.

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Dominant or dominated strategy • A strategy S for a player A is dominant if it

is always the best strategies for player A no matter what strategies other players will take.

• A strategy S for a player A is dominated if there is at least a strategy better than it no matter what strategies other players will take.

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Rule: If you have a dominant strategy, use it!

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Use strategy 1

Dominance Solvable

• If each player has a dominant strategy, the game is dominance solvable

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COMMANDMENT

If you have a dominant strategy, use it.

Expect your opponent to use his/her dominant strategy if he/she has one.

Only one player has a Dominant Strategy

• For The Economist: – G dominant, S dominated

• Dominated Strategy:• There exists another strategy which always does better regardless

of opponents’ actions

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The EconomistG S

TimeS 100 , 100 0 , 90

G 95 , 100 95 , 90

How to recognize a Dominant Strategy

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To determine if the row player has any dominant strategy1.Underline the maximum payoff in each column2.If the underlined numbers all appear in a row, then it is the dominant strategy for the row player

No dominant strategy for the row player in this example.

How to recognize a Dominant Strategy

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To determine if the column player has any dominant strategy1.Underline the maximum payoff in each row2.If the underlined numbers all appear in a column, then it is the dominant strategy for the column player

There is a dominant strategy for the column player in this example.

If there is no dominant strategy• Does any player have a dominant strategy?• If there is none, ask “Does any player have a

dominated strategy?”• If yes, then

• Eliminate the dominated strategies• Reduce the normal-form game• Iterate the above procedure

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Eliminate any dominated strategy

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Eliminate strategy 2 as it’s dominated by strategy 1

Successive Elimination of Dominated Strategies

• If a strategy is dominated, eliminate it• The size and complexity of the game is

reduced• Eliminate any dominated strategies from the

reduced game• Continue doing so successively

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Example: Two competing Bars • Two bars (bar 1, bar 2) compete each other• Can charge price of $2, $4, or $5 for a drink• 6000 tourists pick a bar randomly• 4000 natives select the lowest price bar

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$2 $4 $5

Bar 1

$2 10 , 10 14 , 12 14 , 15

$4 12 , 14 20 , 20 28 , 15

$5 15 , 14 15 , 28 25 , 25

Bar 2

No dominant strategy for the both players.

Successive Elimination of Dominated Strategies

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$4 $5

Bar 1$4 20 , 20 28 , 15

$5 15 , 28 25 , 25

25 , 25

28 , 15

14 , 15

$5$4

15 , 2815 , 14$5

20 , 2012 , 14$4Bar 114 , 1210 , 10$2

$2

,

,

,

, ,

, , Bar 1, ,

Bar 2

Bar 2

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An example for Successive Elimination of strictly dominated strategies, or the process of iterated dominance

Equilibrium• The interaction of all players' strategies results in an

outcome that we call "equilibrium." • Traditional applications of game theory attempt to

find equilibria in games.• In an equilibrium, each player is playing the strategy

that is a "best response" to the strategies of the other players. No one is likely to change his strategy given the strategic choices of the others.

• Equilibrium is not: • The best possible outcome. Equilibrium in the one-shot

prisoners' dilemma is for both players to confess. • A situation where players always choose the same action.

Sometimes equilibrium will involve changing action choices (known as a mixed strategy equilibrium).

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Definition: Nash Equilibrium“If there is a set of strategies with the property

that no player can benefit by changing his/her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium.”

Source: http://www.lebow.drexel.edu/economics/mccain/game/game.html

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Nash equilibrium• If each player has chosen a strategy and no

player can benefit by changing his/her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium.

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Conditions for Nash equilibrium

• Each player is choosing a best response to what he believes the other players will do.

• Each player’s beliefs are correct. The other players are all doing what everyone else thinks they are doing.

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Assumptions:Rational players“Putting yourself in the other person’s shoes”

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Example: B B Lean vs Rainbow’s Endp.105

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Games with infinitely many strategiesp.124

Rainbow’s End’s price

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Finding Nash equilibria: (a) with strike-outs; (b) with underlinings

(U, L) is not a Nash equilibrium because Player 2 can gain by deviating alone to R; …You can cross out those that are not Nash equilibria.

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L C R

U 0, 2 2, 3 4, 1

M 1, 1 3, 1 0, 2

D 0, 3 1, 0 5, 1

Sometimes, there is no Nash Equilibrium

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No strictly dominant strategies and no strictly dominated strategies.

Sometimes, there are more than one Nash Equilibrium.

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Hunting game: multi-Nash Equilibria

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Stag Bison

Stag 34

00

Bison 00

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Barney’s choice

Fred’schoice

Hunting game with 2-Nash Equilibria

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Ballet

Football

Ballet 34

00

Football 00

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Barney’s choice

Alice’schoice

Battle of sexes

Prisoner’s Dilemma: finding the Nash equilibrium

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Which is a Nash Equilibrium?

Prisoner’s Dilemma : Applications

• Relevant to:– Nuclear arms races. – Dispute Resolution and the decision to hire a

lawyer. – Corruption/political contributions between

contractors and politicians. • How do players escape this dilemma?

– Play repeatedly– Find a way to ‘guarantee’ cooperation– Change payoff structure

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Nuclear arms racesprisoner's dilemma in disguise

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Is there a Nash Equilibrium?

Two countries try to decide whether to build the nuclear bombs.

Cigarette Advertisingprisoner's dilemma in disguise

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Philip Morris

No Ad Ad

Reynolds No Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30

Two companies try to decide whether to run cigarette advertisement.

Price it higher? Lower?prisoner's dilemma in disguise

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B B Lean80 70

Rainbow’s

80 72k , 72k 24k , 110k

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110k , 24k 70k , 70k

p.69

Sustainability of resources sharing

• Community resources sharing is generally viewed as a form of cooperative game similar to Prisoner’s Dilemma by most people.

• However, its consequence is much deeper than the simple (& superficial) payoff matrix would suggest.

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Tragedy of the Commons• When individuals, acting independently & rationally,

will deplete a shared common resource even when doing so is not in their best interest.

• An example to explain overuse of shared resources.• Extend the Prisoner’s Dilemma to more than two

players.• Each member of a group of neighboring farmers

prefers to let his cow/sheep to graze on the commons, rather than keeping it on his own inadequate land, but the commons will be rendered unsuitable for grazing if it is overgrazed.

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In the beginning, there is a nice piece of grassland owned by all villagers. “What a waste!” said a farmer.

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Happy farmers with their well-fed cows.

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“Why not have more cows? Why waste the resource?” said the farmers.

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In the end, sad farmers with their hungry cows.

Tragedy of the Commonsan apparent payoff matrix at the start

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Your neighbor

Add a cow Don’t add

add a cow

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12

Don’t add

2

1

1

1

As long as the common pasture is not overgrazed, adding one more cow is the dominant strategy for everybody.

Tragedy of the Commonsan apparent payoff matrix in between

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Your neighbor

Add a cow Don’t add

add a cow

1.51.5

0.91.8

Don’t add

1.80.9

1 1

When the common pasture is starting to be overgrazed, adding cow is still the dominant strategy for everybody, but the return is getting worse.

Tragedy of the Commonsthe final form of payoff matrix

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Your neighbor

Add a cow Don’t add

add a cow

0.50.5

0.71.1

Don’t add

1.10.7

1 1

Finally when the common pasture is overgrazed, adding cow is no longer the dominant strategy for everybody.

Tragedy of the Commons

• Problem: cost of maintenance is ignored/externalized– Farmers don’t adequately pay for their impact.– Resources are overused due to inaccurate estimates of

cost.

• Relevant to:– Health or other social benefits– Environmental laws, overfishing, whaling, pollution,

etc.– Global warming

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Environmental policyTragedy of the Commons in disguise

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Factory B

pollution No pollution

Factory A

pollution

50 , 50 60 , 20

No pollution

20 , 60 20 , 20

Two factories producing same chemical can choose to pollute (lower production cost) or not to pollute (higher production cost).

Global warmingTragedy of the Commons in disguise

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Country B

Emissions Reduced?

No Yes

Country A No

50 , 50 60 , 20

Yes 20 , 60 20 , 20

Two countries producing CO2 can choose to reduce (higher production cost) or not to reduce (lower production cost).

Cost to press button = 2 units

When button is pressed, food given = 10 units

Another Example:Big & Little Pigs

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Decisions, decisions...

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What’s the best strategy for the little pig? Does he have a dominant strategy?

Does the big pig have a dominant strategy?

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Small Company

research No research

Big Company

research

5 , 1 4 , 4

No research

9 , -1 0 , 0

Research in industriesBig & Little Pigs

in disguise

N-person Investment game

• In a group of N>>2 persons• If you invest $0, you get $0 in return• If you invest $10, you get

– $5 return, if more than 60% of the group invest– -$10 return, if less than 60% of the group invest

• Is there any Nash equilibrium?• Answer: 2, • What are they? Why?

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Example in real life

• Which technology to choose?– Window or Apple– Betamax or VHS (video format in 1980s)

• What else??

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Summary: Ch. 3• A strategy is a “complete plan of action” that fully

determines the player's behavior.• Dominant strategy is the best strategy no matter

what strategies other players will take.• A dominated strategy is one at least there is a

strategy better than it no matter what strategies other players will take.

• If you have a dominant strategy, use it!• Eliminate any dominated strategy

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Assignment 3.1

Assignment 3.1