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Game Theory

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. What is Game Theory ?.

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Game Theory

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  1. Game Theory

  2. 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.

  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. C DNC 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.

  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.

  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.

  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.

  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.

  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.

  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.

  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.

  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

  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).

  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

  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.

  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)

  21. Another Game – Nash vs. Pareto • Notice the Nash Equilibrium has the highest total society payoff (Pareto Outcome).

  22. Another Type of Game • Coordination • There are 2 Nash Equilibriums

  23. Another Type of Game II • Battle of the Sexes • There are 2 Nash Equilibriums.

  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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