Game Theory: Introduction. Game Theory. Game Theory: Applications. Game Theory: Overview

Transcription

1 Game Theory: Introduction Game Theory Game theory A means of modeling strategic behavior Agents act to maximize own welfare Agents understand their actions affect actions of other agents ECON 370: Microeconomic Theory Summer 2004 ice niversity Stanley Gilbert Econ Game Theory 2 Game Theory: Applications Game theory has been applied to analyze Oligopolies Cartels: OPEC Tax competition across jurisdictions, countries Military strategies Externalities: using common resources like fishery Game Theory: Overview A game consists of a set of players (often two) a set of strategies for each player the payoffs to each player for all combinations of possible strategy choices by the players Simplest possible game Two-player game Each chooses between only two strategies Econ Game Theory 3 Econ Game Theory 4

2 Two-Player Game: Example Players are A and A has two strategies: p and own has two strategies: eft and ight Two-Player Game: Matrix Payoff matrix for game Payoff matrix - table showing payoffs to A and for each of four possible strategy combinations If A plays, what will do? Econ Game Theory 5 Econ Game Theory 6 Nash Equilibrium: Introduction Nash equilibrium a play where each strategy is a best response to the other Can be multiple Nash equilibria Example has two Nash equilibria (,) and (,) Nash Equilibrium: Matrix (,) and (,) are both Nash equilibria ut (,) is preferred to (,) by both A & (,) is Pareto preferred to (,) Is (,) the only (likely) equilibrium? NO Econ Game Theory 7 Econ Game Theory 8 2

3 Prisoner s ilemma Game Here: S=Silence, C=Confess, Payoffs are years in jail onnie S C Clyde S C Sequential Games Thus far, simultaneous play games oth players choose strategies simultaneously Sequential play games One player plays before the other player eader - player who plays first Follower - player who plays second May be possible to choose among alternative Nash equilibria Only Nash equilibrium for this game is (C,C), even though (S,S) gives both players better payoffs So, the only Nash equilibrium is inefficient Econ Game Theory 9 Econ Game Theory 0 Sequential Game: Matrix Sequential Game: Extensive Form A plays first plays second A Simultaneous game: (,) and (,) are both Nash equilibria No obvious way to choose between them Econ Game Theory (3,9) (,8) (0,0) (2,) Solution proceeds from the end to the beginning What will do if A chooses /? Then What will A do knowing how will react? Econ Game Theory 2 3

4 Sequential Game: Extensive Form If A plays then plays ; A gets 3 If A plays then plays ; A gets 2 A anticipates, so (,) is likely Nash equilibrium A Econ Game Theory 3 (3,9) (,8) (0,0) (2,) Mixed Strategies Thus far, all pure strategies Players choose a single strategy E.g., player A plays only or only Alternative: Mixed strategies Choose combination of stratgies Example: A chooses with probability 0.25 and with probability 0.75 Econ Game Theory 4 Pure Strategies: New Matrix Mixed Strategies: Suppose chooses mixed strategy with probability π plays p, and with probability π plays own I.e., mixing pure strategies Mixed strategy has probability distribution (π, π ) o pure strategy Nash equilibria exist? No Econ Game Theory 5 Econ Game Theory 6 4

5 Mixed Strategies: Similarly, has mixed strategy with probability distribution (π, π ) with probability π plays eft and with probability π plays ight Nash Equilibrium in Mixed Strategies Each player chooses optimal probabilities, given opponent s probabilities Each set of expectations satisfied in eq m. Econ Game Theory 7 Mixed Strategies Assuming players are risk-neutral They will pick the alternative with the highest expected value If they are randomizing Then they do not clearly prefer one option to another That is, Expected Values of both alternatives must be equal Assume both players randomize plays alternative with probability µ plays alternative with probability λ Econ Game Theory 8 Calculating Mixed Strategies A For player A to be indifferent between and We must have: µ + 0( µ) = 0µ + 3( µ) or µ = 0.75 Calculating Mixed Strategies For player to be indifferent between and We must have: 2λ + 5( λ) = 4λ + 2( λ) or λ = 0.6 Econ Game Theory 9 Econ Game Theory 20 5

6 Payoff A = = Expected Payoffs Existence of Nash Equilibrium Consider game with finite number of players each with a finite number of pure strategies Such a game has at least one (pure or mixed strategy) Nash equilibrium If no pure strategy Nash equilibrium, then must have at least one mixed strategy Nash equilibrium Payoff = = Econ Game Theory 2 Econ Game Theory 22 6