# 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.

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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  dominant and dominated strategies  the Pareto Optimum ► Discuss strategies in infinitely repeated games. 2

3 What is Game Theory? ► DFN: A way of describing various possible outcomes in any situation involving two or more interacting individuals. ► A game is described by:  1. Players  2. Strategies of those players  3. Payoffs: the utility/profit for each of the strategy combinations.

4 Assumptions in Game Theory ► 1. Perfect Information – Players observe all of their rivals’ previous moves. ► 2. Common Knowledge – All players know the structure of the game, know that their rivals know it and their rivals know that they know it.

5 Prisoner’s Dilemma Game ► There at 2 players: Player 1, and Player 2. ► Each has 2 possible strategies: Confess (C) or Do Not Confess (DNC). ► Players only play the games once. ► Payoffs are years in jail, so they are expressed as negative numbers. Both players want the least amount of years in jail they can have. Player 2 Player 2 Player 1 CDNC C -6, -6 0, -9 DNC -9, 0 -1, -1 C C DNC

6 Prisoner’s Dilemma Game ► How do we “solve” this game (predict which set of strategies will be played)? ► 1. Look for Strictly Dominant Strategies and Strictly Dominated Strategies  Strictly Dominant Strategies - the best strategy regardless of what other players do.  Strictly Dominated Strategies – a strategy in which another strategy yields the player a higher payoff regardless of what other players do.

7 Prisoner’s Dilemma Game ► 2. Eliminate Strictly Dominated Strategies (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 DNC, Player 1 gets 9 years in jail.  If Player 2 is going to Confess, # of years in jail if Player 1 C < # of years in jail if Player 1 DNC  Thus if Player 1 thinks Player 2 is going to confess, Player 1 is better off confessing too. Player 2 Player 2 Player 1 CDNC C -6, -6 0, -9 DNC -9, 0 -1, -1

8 Prisoner’s Dilemma Game ► 2. Eliminate Strictly Dominated Strategies (Player 1’s Strategy if Player 2 Does Not Confess)  If Player 2 DNC and Player 1 C, Player 1 gets 0 years.  If Player 2 DNC and Player 1 DNC, Player 1 gets 1 year in jail.  If Player 2 is going to Not Confess, # of years in jail if Player 1 C < # of years in jail if Player 1 DNC  Thus if Player 1 thinks Player 2 does not confess, Player 1 is better off confessing. Player 2 Player 2 Player 1 CDNC C -6, -6 0, -9 DNC -9, 0 -1, -1

9 Prisoner’s Dilemma Game ► 2. Eliminate Strictly Dominated Strategies (Player 1’s Dominant Strategy)  If Player 2 C, Player 1 is better off confessing.  If Player 2 DNC, Player 1 is better off confessing.  Regardless of what Strategy Player 2 uses, Player 1 is better off confessing.  Thus, Confessing is a dominant strategy for Player 1 and Do Not Confess is a dominated strategy. Player 2 Player 2 Player 1 CDNC C -6, -6 0, -9 DNC -9, 0 -1, -1

10 Prisoner’s Dilemma Game ► 2. Eliminate Strictly Dominated Strategies (Player 2’s Dominant Strategy)  Since we know it is strategic for Player 1 to play Confess, to determine Player 2’s dominant strategy we compare Player 2’s years in jail.  Since Player 1 C, Player 2 is better off confessing.  Thus, Confessing is a dominant strategy for Player 2 and Do Not Confess is a dominated strategy. Player 2 Player 2 Player 1 CDNC C -6, -6 0, -9 DNC -9, 0 -1, -1

11 Prisoner’s Dilemma Game ► 3. Solution to the Game  Both Players playing the strategy which is best for them given what the other person does yields a solution at Confess, Confess.  After all dominated strategies are eliminated, what’s left is a Nash Equilibrium.  You can eliminate Strictly Dominated Strategies in any order and will get the same result. Player 2 Player 2 Player 1 CDNC C -6, -6 0, -9 DNC -9, 0 -1, -1

12 Nash Equilibrium ► DFN: The result of all players playing their best strategy given what their competitors are doing.  Player 1 knew it is a strictly dominant strategy for Player 2 to Confess. Thus Player 1 will confess because they do best under that strategy knowing what Player 2 will do and vice versa.

13 Another Way to Solve a Game ► Star the highest payoff for one of the Players given the other Player is locked into each strategy and vice versa. ► Suppose Player 1 is locked into Confessing, Player 2 is better off Confessing. ► So we put a star above Player 2’s Payoff. Player 2 Player 2 Player 1 CDNC C -6, -6 0, -9 DNC -9, 0 -1, -1

14 Another Way to Solve a Game ► Suppose Player 1 is locked into Not Confessing, Player 2 is better off Confessing. ► So we put a star above Player 2’s Payoff. Player 2 Player 2 Player 1 CDNC C -6, -6 0, -9 DNC -9, 0 -1, -1

15 Another Way to Solve a Game ► Suppose Player 2 is locked into Confessing, Player 1 is better off Confessing. ► So we put a star above Player 1’s Payoff. Player 2 Player 2 Player 1 CDNC C -6, -6 0, -9 DNC -9, 0 -1, -1

16 Another Way to Solve a Game ► Suppose Player 2 is locked into Not Confessing, Player 1 is better off Confessing. ► So we put a star above Player 1’s Payoff. ► (Confess, Confess) is a Nash Equilibrium because it has two stars Player 2 Player 2 Player 1 CDNC C -6, -6 0, -9 DNC -9, 0 -1, -1

17 Nash Equilibrium vs. Pareto Outcome ► Nash Equilibrium is the result when both players act strategically given what the other is going to do (Confess, Confess). ► Pareto Optimum is the result that benefits both players the most (DNC, DNC). Player 2 Player 2 Player 1 CDNC C -6, -6 0, -9 DNC -9, 0 -1, -1

18 Another Game ► Suppose now there are two Players, Row and Column, with two Strategies each.  Row can go Up or Down  Column can go Left or Right Column Player Column Player Row Player LeftRight Up 5, 11 1, 10 Down 10, 7 2, 2

19 Another Game – Eliminating Strictly Dominated Strategies ► Down is a Dominant Strategy and Up is a Dominated Strategy. (10>5 and 2>1) ► Left is a Dominant Strategy and Right is a Dominated Strategy. (7>2) ► Thus, (Down, Left) is a Nash Equilibrium. Column Player Column Player Row Player LeftRight Up 5, 11 1, 10 Down 10, 7 2, 2

20 Another Game – Stars ► If Column chooses Left, Row is better choosing Down (10>5)  Star Row’s payoff for (Down, Left) ► If Column chooses Right, Row is better choosing Down (2>1)  Star Row’s payoff for (Down, Right) ► If Row chooses Up, Column is better choosing Left (11>10)  Star Column’s payoff for (Up, Left) ► If Row chooses Down, Column is better choosing Left (7>2)  Star Row’s payoff for (Down, Left) ► Thus (Down, Left) is the Nash Equilibrium (2 stars) Column Player Column Player Row Player LeftRight Up 5, 11 1, 10 Down 10, 7 2, 2

21 Another Game – Nash vs. Pareto ► Notice the Nash Equilibrium has the highest total society payoff (Pareto Outcome). Column Player Column Player Row Player LeftRight Up 5, 11 1, 10 Down 10, 7 2, 2

22 Another Type of Game ► Coordination ► There are 2 Nash Equilibriums Friend 2 Friend 2 Statue of Liberty Empire State Bldg Friend 1 Statue of Liberty 8, 8 0, 0 Empire State Bldg 0, 0 3, 3

23 Another Type of Game II ► Battle of the Sexes ► There are 2 Nash Equilibriums. Male Male BalletGame FemaleBallet 8, 3 0, 0 Game 3, 8

24 Infinitely Repeated Games ► Strategies Players can play:  1. Always play Pareto (Co-operate)  2. Always play Nash (Strategic)  3. Grimm Strategy (Punish) – play Pareto until the other player diverges from Pareto, then play Nash.  4. Tit-for-Tat (Reciprocate) – play what the other player played last round. ► One of two things will happen:  1. Players Converge on Nash Equilibrium by strategically playing Dominant Strategies.  2. Players could end up “co-operating” for the greater good of all play Pareto.

25 A Final Note on Nash Equilibrium ► Nash Equilibrium predictions are only accurate if each player correctly predicts what the other player is going to do. ► For a player to accurately predict what the other player is going to do and act on it, both players must act strategically and NOT select Strictly Dominated Strategies. ► But, with some other knowledge about the other player (relationship, partner before, etc.), it could be strategic to play other strategies.

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