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Game Theory

By Ben Cutting & Rohit Venkat. Game Theory. Game Theory: General Definition. Mathematical decision making tool Used to analyze a competitive situation in order to determine the optimal course of action Involves at least two players who usually must choose an action from at least two options

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Game Theory

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  1. By Ben Cutting & RohitVenkat Game Theory

  2. Game Theory: General Definition • Mathematical decision making tool • Used to analyze a competitive situation in order to determine the optimal course of action • Involves at least two players who usually must choose an action from at least two options • A player’s payoff (what they gain/lose from the game) is determined by both their own choice and the choices of other players • Players act “rationally” in their decision making, try to maximize their payoff

  3. History • John von Neumann published a series of papers in 1928 pertaining to game theory • Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern (1944) • Initially developed to analyze competitions in which one individual does better at another’s expense (zero sum games) • Developed extensively in the 1950s by many scholars to treat a wide class of interactions

  4. Key Terms • Nash Equilibrium – state in which each player has a given strategy that provides them with their maximum payoff. Therefore no player has an incentive to change their strategy unilaterally • Strategy – a player’s plan of action that accounts for all possible game scenarios. Completely describes a player’s behavior

  5. Representations • Two classical representations: matrix form and tree form • Matrix form is traditionally associated with simultaneous move games Player B 1 2 1 Player A 2

  6. Representations (cont.) • Tree form • Outcomes often change if the type of game is changed Player A 1 2 1 2 1 2 Player B 1 2 3 4 4 2 -1 -1

  7. Types of Games • Symmetric games • Zero Sum games • Cooperative games • Imperfect Information games • Continuous games

  8. Symmetric Games • Strategies of both players are the same • Common in many classical 2x2 games such as the Prisoners Dilemma • Nash equilibrium is where both confess and betray the other • Both have the same strategy: Always choose to confess Prisoner B Not Confess Confess Not Confess Prisoner A Confess

  9. Zero Sum Games • Game in which all payoffs add to zero • Example: Matching pennies game • Each player chooses either odd or even before flipping their pennies simultaneously • If both pennies come up either heads or tails, Even wins. Otherwise Odd wins *Notice the total sum of the payoffs = 0 Odd Heads Tails Heads Even Tails

  10. Cooperative Games • A game is cooperative if the players are able to form binding commitments • Communication among players is allowed in cooperative games • Players coordinate their strategies to attain the maximum combined payoff Player B Cooperate Defect Cooperate Player A Defect

  11. Imperfect information • Using earlier example, except now Player B does not know Player A’s choice of action • In this case Player B will be tempted to choose option 2 to get a payoff of 4 (assuming Player A chooses option 1), not knowing A’s strategy Player A 1 2 1 2 1 2 Player B 1 2 3 4 4 0 -1 -1

  12. Continuous Games • Games in which there is not a discrete number of players, moves, and/or outcomes • The strategy set for each player is also continuous • Example: Cops and Robbers (pursuit & evasion game) • A group of players trying to capture another group (the number of players varies) • Game does not have a finite length or outcome (some robbers may never get caught)

  13. Applications • Economics • Bargaining, duopolies, fair division, etc. • Political Science • Political economy, public choice, social choice theory, etc. • Biology • Animal behavior • Computer Science & Logic • Interactive computations, multi-agent systems • Philosophy • Social norms

  14. Limitations • Assumptions made by game theorists are sometimes violated • Human behavior often deviates from game theory models due to irrationality and different motives (altruism)

  15. What is the equilibrium outcome of this game? • Chip (C) and Dale (D) are negotiating over how to divide a pile of 100 acorns. The order of events is: • First Round: C makes D an initial offer. D accepts or rejects. If D accepts, the game ends and C and D get their acorns. If D rejects, 10 acorns rot because of the delay and the game continues with 90 acorns to be divided. • Second Round: D makes an offer. C accepts or rejects. If C accepts, the game ends and C and D get their acorns. If C rejects, 10 acorns rot because of the delay and the game continues with 80 acorns to be divided. • Third Round: C makes a final offer. D accepts or rejects. If D accepts, then C and D get their acorns. If D rejects, the game ends and neither C nor D get any acorns.

  16. Works Cited • http://www.answers.com/topic/game-theory • http://en.wikipedia.org/wiki/Game_theory • http://plato.stanford.edu/entries/game-theory • http://william-king.www.drexel.edu/top/eco/game/zerosum.html

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