2008/02/06Lecture 21 ECO290E: Game Theory Lecture 2 Static Games and Nash Equilibrium

2008/02/06Lecture 22 Review of Lecture 1 Game Theory studies strategically inter-dependent situations. provides us tools for analyzing most of problems in social science. employs Nash equilibrium as a solution concept. creates a revolution in Economics.

2008/02/06Lecture 23 What is Game? Complete Information Incomplete Information StaticNash Equilibrium (Lec. 2-5) Bayesian Nash Equilibrium (Lec ) DynamicSubgame Perfect Equilibrium (Lec. 6, 8-9) Perfect Bayesian Equilibrium (Lec )

2008/02/06Lecture 24 Games in Two Forms Static games  The normal/strategic-form representation Dynamic games  The extensive-form representation In principle, static (/ dynamic) games can also be analyzed in an extensive- form (/a normal-form) representation.

2008/02/06Lecture 25 Normal-form Games The normal-form (strategic-form) representation of a game specifies: 1.The players in the game. 2.The strategies available to each player. 3.The payoff received by each player (for each combination of strategies that could be chosen by the players).

2008/02/06Lecture 26 Static Games In a normal-form representation, each player simultaneously chooses a strategy, and the combination of strategies chosen by the players determines a payoff for each player. The players do not necessarily act simultaneously: it suffices that each chooses her own action without knowing others’ choices.  We will also study dynamic games in an extensive-form representation later.

2008/02/06Lecture 27 Example: Prisoners ’ Dilemma Two suspects are charged with a joint clime, and are held separately by the police. Each prisoner is told the following: 1) If one prisoner confesses and the other one does not, the former will be given a reward of 1 and the latter will receive a fine equal to 2. 2) If both confess, each will receive a fine equal to 1. 3) If neither confesses, both will be set free.

2008/02/06Lecture 28 Payoff Bi-Matrix Player 2 Player 1 Silent Confess Silent Confess -2 1

2008/02/06Lecture 29 How to Use Bi-Matrices Any two players game (with finite number of strategies) can be expressed as a bi-matrix. The payoffs to the two players when a particular pair of strategies is chosen are given in the appropriate cell. The payoff to the row player (player 1) is given first, followed by the payoff to the column player (player 2).  How can we solve this game?

2008/02/06Lecture 210 Definition of Nash Equilibrium Nash equilibrium (mathematical definition) A strategy profile s* is called a Nash equilibrium if and only if the following condition is satisfied: Nash equilibrium is defined over strategy profiles, NOT over individual strategies.

2008/02/06Lecture 211 Solving PD Game For each player, u(C,C)>u(S,C) holds.  (Confess, Confess) is a NE. There is no other equilibrium. Playing “Confess” is optimal no matter how the opponent takes “Confess” or “Silent.”  “Confess” is a dominant strategy. The NE is not (Pareto) efficient.  Optimality for individuals does not necessary imply optimality for a group (society).

2008/02/06Lecture 212 Terminology Dominant strategy: A strategy s is called a dominant strategy if playing s is optimal for any combination of other players’ strategies. Pareto efficiency: An outcome of games is Pareto efficient if it is not possible to make one person better off (through moving to another outcome) without making someone else worse off.

2008/02/06Lecture 213 Applications of PD ExamplesPlayersSilentConfess Arms races CountriesDisarmArm International trade policy Countries Lower trade barriers No change Marital cooperation CoupleObedientDemandin g Provision of public goods CitizenContributeFree-ride Deforestatio n Woodmen Restrain cutting Cut down maximum

2008/02/06Lecture 214 Example: Coordination Game Player 2 Player 1 WindowsMac Windows 1 0 Mac 0 2

2008/02/06Lecture 215 Solving Coordination Game There are two equilibria, (W,W) and (M,M).  Games, in general, can have more than one Nash equilibrium. Everybody prefers one equilibrium (M,M) to the other (W,W).  Several equilibria can be Pareto-ranked. However, bad equilibrium can be chosen.  This is called “coordination failure.”

2008/02/06Lecture 216 Other Examples Battle of the sexes  Corruption Game Stag Hunt Game  Migration Game Hawk-Dove (Chicken) Game  Land Tenure Game