ECO290E: Game Theory

1. ECO290E: Game Theory Lecture 2 Static Games and Nash Equilibrium Lecture 2

2. 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. Lecture 2

3. What is Game? Lecture 2

4. 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. Lecture 2

5. Normal-form Games The normal-form (strategic-form) representation of a game specifies: • The players in the game. • The strategies available to each player. • The payoff received by each player (for each combination of strategies that could be chosen by the players). Lecture 2

6. 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. Lecture 2

7. 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. Lecture 2

8. Payoff Bi-Matrix Lecture 2

9. 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? Lecture 2

10. 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. Lecture 2

11. 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). Lecture 2

12. 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. Lecture 2

13. Applications of PD Lecture 2

14. Example: Coordination Game Lecture 2

15. 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.” Lecture 2

16. Other Examples • Battle of the sexes • Corruption Game • Stag Hunt Game • Migration Game • Hawk-Dove (Chicken) Game • Land Tenure Game Lecture 2