1 Game Theory. By the end of this section, you should be able to…. ► In a simultaneous game played only once, find and define: the Nash equilibrium.

11

Game TheoryGame Theory

By the end of this section, By the end of this section, you should be able to….you should be able to….

► In a simultaneous game played only In a simultaneous game played only once, find and define:once, find and define: the Nash equilibriumthe Nash equilibrium dominant and dominated strategiesdominant and dominated strategies the Pareto Optimumthe Pareto Optimum

►Discuss strategies in infinitely Discuss strategies in infinitely repeated games.repeated games.

2

3

What is What is Game TheoryGame Theory??

►DFN:DFN: A way of describing various A way of describing various possible outcomes in any situation possible outcomes in any situation involving two or more interacting involving two or more interacting individuals.individuals.

►A game is described by:A game is described by: 1. Players1. Players 2. Strategies of those players2. Strategies of those players 3. Payoffs: the utility/profit for each of the 3. Payoffs: the utility/profit for each of the

strategy combinations.strategy combinations.

4

Assumptions in Game TheoryAssumptions in Game Theory

►1. 1. Perfect InformationPerfect Information – Players – Players observe all of their rivals’ previous observe all of their rivals’ previous moves.moves.

►2. 2. Common KnowledgeCommon Knowledge – All players – All players know the structure of the game, know know the structure of the game, know that their rivals know it and their rivals that their rivals know it and their rivals know that they know it.know that they know it.

5

Prisoner’s Dilemma GamePrisoner’s Dilemma Game► There at 2 players: Player 1, and Player 2.There at 2 players: Player 1, and Player 2.► Each has 2 possible strategies: Confess (C) or Do Not Confess Each has 2 possible strategies: Confess (C) or Do Not Confess

(DNC).(DNC).► Players only play the games once.Players only play the games once.► Payoffs are years in jail, so they are expressed as negative Payoffs are years in jail, so they are expressed as negative

numbers. Both players want the least amount of years in jail numbers. Both players want the least amount of years in jail they can have.they can have.

Player 2Player 2

Player 1Player 1CC DNCDNC

CC -6 , -6-6 , -6 0 , -90 , -9

DNCDNC -9 , 0-9 , 0 -1 , -1-1 , -1

C

C

DNC

DNC

6

Prisoner’s Dilemma GamePrisoner’s Dilemma Game

►How do we “solve” this game (predict How do we “solve” this game (predict which set of strategies will be played)?which set of strategies will be played)?

►1. Look for Strictly Dominant Strategies 1. Look for Strictly Dominant Strategies and Strictly Dominated Strategies and Strictly Dominated Strategies Strictly Dominant StrategiesStrictly Dominant Strategies - the best - the best

strategy regardless of what other players strategy regardless of what other players do.do.

Strictly Dominated StrategiesStrictly Dominated Strategies – a strategy – a strategy in which another strategy yields the player in which another strategy yields the player a higher payoff regardless of what other a higher payoff regardless of what other players do.players do.

7

Prisoner’s Dilemma GamePrisoner’s Dilemma Game► 2. Eliminate Strictly Dominated Strategies 2. Eliminate Strictly Dominated Strategies

(Player 1’s Strategy if Player 2 Confesses)(Player 1’s Strategy if Player 2 Confesses) If Player 2 C and Player 1 C, Player 1 gets 6 years.If Player 2 C and Player 1 C, Player 1 gets 6 years. If Player 2 C and Player 1 DNC, Player 1 gets 9 If Player 2 C and Player 1 DNC, Player 1 gets 9

years in jail.years in jail. If Player 2 is going to Confess, # of years in jail if If Player 2 is going to Confess, # of years in jail if

Player 1 C < # of years in jail if Player 1 DNCPlayer 1 C < # of years in jail if Player 1 DNC Thus if Player 1 thinks Player 2 is going to Thus if Player 1 thinks Player 2 is going to

confess, Player 1 is better off confessing too.confess, Player 1 is better off confessing too.

Player 2Player 2


CC -6 , -6-6 , -6 0 , -90 , -9

DNCDNC -9 , 0-9 , 0 -1 , -1-1 , -1

8


(Player 1’s Strategy if Player 2 Does Not Confess)(Player 1’s Strategy if Player 2 Does Not Confess) If Player 2 DNC and Player 1 C, Player 1 gets 0 If Player 2 DNC and Player 1 C, Player 1 gets 0

years.years. If Player 2 DNC and Player 1 DNC, Player 1 gets 1 If Player 2 DNC and Player 1 DNC, Player 1 gets 1

year in jail.year in jail. If Player 2 is going to Not Confess, # of years in If Player 2 is going to Not Confess, # of years in

jail if Player 1 C < # of years in jail if Player 1 DNCjail if Player 1 C < # of years in jail if Player 1 DNC Thus if Player 1 thinks Player 2 does not confess, Thus if Player 1 thinks Player 2 does not confess,

Player 1 is better off confessing.Player 1 is better off confessing. Player 2Player 2


CC -6 , -6-6 , -6 0 , -90 , -9

DNCDNC -9 , 0-9 , 0 -1 , -1-1 , -1

9


(Player 1’s Dominant Strategy)(Player 1’s Dominant Strategy) If Player 2 C, Player 1 is better off confessing.If Player 2 C, Player 1 is better off confessing. If Player 2 DNC, Player 1 is better off confessing.If Player 2 DNC, Player 1 is better off confessing. Regardless of what Strategy Player 2 uses, Player Regardless of what Strategy Player 2 uses, Player

1 is better off confessing.1 is better off confessing. Thus, Confessing is a dominant strategy for Player Thus, Confessing is a dominant strategy for Player

1 and Do Not Confess is a dominated strategy.1 and Do Not Confess is a dominated strategy.

Player 2Player 2


CC -6 , -6-6 , -6 0 , -90 , -9

DNCDNC -9 , 0-9 , 0 -1 , -1-1 , -1

10


(Player 2’s Dominant Strategy)(Player 2’s Dominant Strategy) Since we know it is strategic for Player 1 to play Since we know it is strategic for Player 1 to play

Confess, to determine Player 2’s dominant Confess, to determine Player 2’s dominant strategy we compare Player 2’s years in jail.strategy we compare Player 2’s years in jail.

Since Player 1 C, Player 2 is better off confessing.Since Player 1 C, Player 2 is better off confessing. Thus, Confessing is a dominant strategy for Player Thus, Confessing is a dominant strategy for Player

2 and Do Not Confess is a dominated strategy.2 and Do Not Confess is a dominated strategy.

Player 2Player 2


CC -6 , -6-6 , -6 0 , -90 , -9

DNCDNC -9 , 0-9 , 0 -1 , -1-1 , -1

11

Prisoner’s Dilemma GamePrisoner’s Dilemma Game► 3. Solution to the Game 3. Solution to the Game

Both Players playing the strategy which is best Both Players playing the strategy which is best for them given what the other person does yields for them given what the other person does yields a solution at Confess, Confess.a solution at Confess, Confess.

After all dominated strategies are eliminated, After all dominated strategies are eliminated, what’s left is a what’s left is a Nash EquilibriumNash Equilibrium..

You can eliminate Strictly Dominated Strategies You can eliminate Strictly Dominated Strategies in any order and will get the same result.in any order and will get the same result.

Player 2Player 2


CC -6 , -6-6 , -6 0 , -90 , -9

DNCDNC -9 , 0-9 , 0 -1 , -1-1 , -1

12

Nash EquilibriumNash Equilibrium

►DFN:DFN: The result of all players playing The result of all players playing their best strategy given what their their best strategy given what their competitors are doing.competitors are doing. Player 1 knew it is a strictly dominant Player 1 knew it is a strictly dominant

strategy for Player 2 to Confess. Thus strategy for Player 2 to Confess. Thus Player 1 will confess because they do best Player 1 will confess because they do best under that strategy knowing what Player 2 under that strategy knowing what Player 2 will do and vice versa.will do and vice versa.

13

Another Way to Solve a Another Way to Solve a GameGame

► Star the highest payoff for one of the Players Star the highest payoff for one of the Players given the other Player is locked into each given the other Player is locked into each strategy and vice versa.strategy and vice versa.

► Suppose Player 1 is locked into Confessing, Suppose Player 1 is locked into Confessing, Player 2 is better off Confessing. Player 2 is better off Confessing.

► So we put a star above Player 2’s Payoff.So we put a star above Player 2’s Payoff.

Player 2Player 2


CC -6 , -6-6 , -6 0 , -90 , -9

DNCDNC -9 , 0-9 , 0 -1 , -1-1 , -1

14


►Suppose Player 1 is locked into Suppose Player 1 is locked into Not Confessing, Player 2 is better Not Confessing, Player 2 is better off Confessing. off Confessing.

►So we put a star above Player 2’s So we put a star above Player 2’s Payoff.Payoff.

Player 2Player 2


CC -6 , -6-6 , -6 0 , -90 , -9

DNCDNC -9 , 0-9 , 0 -1 , -1-1 , -1

15


►Suppose Player 2 is locked into Suppose Player 2 is locked into Confessing, Player 1 is better off Confessing, Player 1 is better off Confessing. Confessing.

►So we put a star above Player 1’s So we put a star above Player 1’s Payoff.Payoff.

Player 2Player 2


CC -6 , -6-6 , -6 0 , -90 , -9

DNCDNC -9 , 0-9 , 0 -1 , -1-1 , -1

16


►Suppose Player 2 is locked into Not Suppose Player 2 is locked into Not Confessing, Player 1 is better off Confessing, Player 1 is better off Confessing. Confessing.

►So we put a star above Player 1’s Payoff.So we put a star above Player 1’s Payoff.►(Confess, Confess) is a (Confess, Confess) is a Nash EquilibriumNash Equilibrium

because it has two starsbecause it has two stars

Player 2Player 2


CC -6 , -6-6 , -6 0 , -90 , -9

DNCDNC -9 , 0-9 , 0 -1 , -1-1 , -1

17

Nash Equilibrium vs. Pareto Nash Equilibrium vs. Pareto OutcomeOutcome

►Nash EquilibriumNash Equilibrium is the result when both is the result when both players act strategically given what the players act strategically given what the other is going to do (Confess, Confess).other is going to do (Confess, Confess).

►Pareto OptimumPareto Optimum is the result that is the result that benefits both players the most (DNC, benefits both players the most (DNC, DNC).DNC).

Player 2Player 2


CC -6 , -6-6 , -6 0 , -90 , -9

DNCDNC -9 , 0-9 , 0 -1 , -1-1 , -1

18

Another GameAnother Game

► Suppose now there are two Players, Row Suppose now there are two Players, Row and Column, with two Strategies each.and Column, with two Strategies each. Row can go Up or DownRow can go Up or Down Column can go Left or RightColumn can go Left or Right

Column PlayerColumn Player

Row Row PlayerPlayer

LeftLeft RightRight

UpUp 5 , 115 , 11 1 , 101 , 10

DownDown 10 , 710 , 7 2 , 22 , 2

19

Another Game – Eliminating Another Game – Eliminating Strictly Dominated StrategiesStrictly Dominated Strategies

►Down is a Dominant Strategy and Up is a Down is a Dominant Strategy and Up is a Dominated Strategy. (10>5 and 2>1)Dominated Strategy. (10>5 and 2>1)

► Left is a Dominant Strategy and Right is a Left is a Dominant Strategy and Right is a Dominated Strategy. (7>2)Dominated Strategy. (7>2)

► Thus, (Down, Left) is a Thus, (Down, Left) is a Nash EquilibriumNash Equilibrium..



LeftLeft RightRight

UpUp 5 , 115 , 11 1 , 101 , 10

DownDown 10 , 710 , 7 2 , 22 , 2

20

Another Game – StarsAnother Game – Stars► If Column chooses Left, Row is better choosing Down (10>5)If Column chooses Left, Row is better choosing Down (10>5)

Star Row’s payoff for (Down, Left)Star Row’s payoff for (Down, Left)► If Column chooses Right, Row is better choosing Down (2>1)If Column chooses Right, Row is better choosing Down (2>1)

Star Row’s payoff for (Down, Right)Star Row’s payoff for (Down, Right)► If Row chooses Up, Column is better choosing Left (11>10)If Row chooses Up, Column is better choosing Left (11>10)

Star Column’s payoff for (Up, Left)Star Column’s payoff for (Up, Left)► If Row chooses Down, Column is better choosing Left (7>2)If Row chooses Down, Column is better choosing Left (7>2)

Star Row’s payoff for (Down, Left)Star Row’s payoff for (Down, Left)► Thus (Down, Left) is the Thus (Down, Left) is the Nash EquilibriumNash Equilibrium (2 stars) (2 stars)



LeftLeft RightRight

UpUp 5 , 115 , 11 1 , 101 , 10

DownDown 10 , 710 , 7 2 , 22 , 2

21

Another Game – Nash vs. ParetoAnother Game – Nash vs. Pareto

►Notice the Nash Equilibrium has the highest Notice the Nash Equilibrium has the highest total society payoff (Pareto Outcome).total society payoff (Pareto Outcome).



LeftLeft RightRight

UpUp 5 , 115 , 11 1 , 101 , 10

DownDown 10 , 710 , 7 2 , 22 , 2

22

Another Type of GameAnother Type of Game

► Coordination Coordination ► There are 2 Nash EquilibriumsThere are 2 Nash Equilibriums

Friend 2Friend 2

Statue of Statue of LibertyLiberty

Empire Empire State BldgState Bldg

Friend 1Friend 1 Statue of Statue of LibertyLiberty

8 , 88 , 8 0 , 00 , 0

Empire Empire State BldgState Bldg

0 , 00 , 0 3 , 33 , 3

23

Another Type of Game IIAnother Type of Game II

► Battle of the SexesBattle of the Sexes► There are 2 Nash Equilibriums.There are 2 Nash Equilibriums.

MaleMale

BalletBallet GameGame

FemaleFemale BalletBallet 8 , 38 , 3 0 , 00 , 0

GameGame 0 , 00 , 0 3 , 83 , 8

24

Infinitely Repeated GamesInfinitely Repeated Games

► Strategies Players can play:Strategies Players can play: 1. Always play Pareto (Co-operate)1. Always play Pareto (Co-operate) 2. Always play Nash (Strategic)2. Always play Nash (Strategic) 3. Grimm Strategy (Punish) – play Pareto until 3. Grimm Strategy (Punish) – play Pareto until

the other player diverges from Pareto, then play the other player diverges from Pareto, then play Nash.Nash.

4. Tit-for-Tat (Reciprocate) – play what the other 4. Tit-for-Tat (Reciprocate) – play what the other player played last round.player played last round.

►One of two things will happen:One of two things will happen: 1. Players Converge on Nash Equilibrium by 1. Players Converge on Nash Equilibrium by

strategically playing Dominant Strategies.strategically playing Dominant Strategies. 2. Players could end up “co-operating” for the 2. Players could end up “co-operating” for the

greater good of all play Pareto. greater good of all play Pareto.

25

A Final Note on Nash A Final Note on Nash EquilibriumEquilibrium

►Nash Equilibrium predictions are only Nash Equilibrium predictions are only accurate if each player correctly predicts accurate if each player correctly predicts what the other player is going to do.what the other player is going to do.

► For a player to accurately predict what the For a player to accurately predict what the other player is going to do and act on it, other player is going to do and act on it, both players must act strategically and NOT both players must act strategically and NOT select Strictly Dominated Strategies.select Strictly Dominated Strategies.

► But, with some other knowledge about the But, with some other knowledge about the other player (relationship, partner before, other player (relationship, partner before, etc.), it could be strategic to play other etc.), it could be strategic to play other strategies.strategies.

1 Game Theory. By the end of this section, you should be able to…. ► In a simultaneous game played only once, find and define: the Nash equilibrium.

Documents

strategies player

strategy player

c c dnc slide

dominated strategies

dominant strategies

dominated strategy

cdnc c

game theory slide