1. GAME THEORY By Ben Cutting & Rohit Venkat

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 1, 1 3, 4 4, 2 -1, -1 Player A 12 1 2 Player B

6. Representations (cont.)  Tree form  Outcomes often change if the type of game is changed Player A Player B 1 1 1 2 2 2 1212 3434 4242

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 1, 1 10, 0 0, 10 5, 5 Prisoner B Prisoner A Confess Not 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 1, -1 -1, 1 1, -1 Heads TailsHeads Tails Even Odd

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 3, 3 0, 5 5, 0 1, 1 Player B Player A Defect Cooperate

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 1 1 1 2 2 2 1212 3434 4040 Player A Player B

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