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1 Game Theory

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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

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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.

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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.

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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

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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.

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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

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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

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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

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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

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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

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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.

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.

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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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