17. (A very brief) Introduction to game theory

1. 17. (A very brief) Introduction to game theory Varian, Chapters 28, 29

2. Interacting decisions • In most of our discussion of behavior, we’ve looked at either • Individual actions – e.g., consumer choice • Market interaction – e.g., Edgeworth box, with the help of the Walrasian auctioneer • In some cases, we have allowed an agent’s decisions to depend on her prediction of another agent’s behavior • E.g. monopolist (recall B’s price offer curve)

3. Strategic interaction • In many cases, every agent must predict the behavior of the others …..as well as anticipating their responses to her own behavior • E.g.s, • Mozilla vs Internet Explorer • Sotheby’s vs Christie’s • US vs Iran?

4. A simple game: pet the dog C’s strategies Player C J’s strategies Player J J’s payoffs in blue C’s payoffs in red

5. What will J do? Player C • If C plays stay • Pet(2) is better than don’t(1) • If C plays flee • Pet(1) is better than don’t(0) Player J

6. Jhas a dominant strategy • No matter what C does, J should play Pet Player C Player J

7. What will C do? Player C Player J • If J plays Pet • Stay(2) is better than flee (1) • If B plays Bottom • Stay(1) is better than flee(0)

8. Calso has a dominant strategy Player C Player J • No matter what J does, C should play Left

9. So, outcome of the game is: • The dominant strategy equilibrium is (Pet, Stay) Player C Player J

10. A game without dominant strategies Girlfriend Boyfriend • If Girlfriend plays E • E(2) is better than D (0) • If Girlfriend plays D • D (1) is better than E(0) Boyfriend is not sure what to do

11. Girlfriend’s choice Girlfriend Boyfriend • If boyfriendplays E • E (1) is better than D (0) • If boyfriend plays D • D (2) is better than E(0) Girlfriendis not sure what to do

12. Nash equilibrium • A pair of strategies (xA, xB) constitutes a Nash Equilibrium if and only if A’s best choice is xA, given that B is choosing xB; and B’s best choice is xB, given that A is choosing xA. • No player has an incentive to unilaterally deviate from the equilibrium strategy

13. Nash equilibrium of the game • (Top,Left) is a Nash equilibrium of this game Player B Player A

14. The prisoners’ dilemma • A’s dominant strategy is to confess • B’s dominant strategy is to confess also Player B Outcome is not Pareto efficient! Player A Cooperation is much better for both

15. Repeated games • Suppose the prisoners played their game twice • Each could adopt a strategy for the repeated game of the form: • Deny in the first stage; • In the second stage, do what the other guy did in the previous stage • This is called the “tit-for-tat” strategy i.e., cooperate i.e., continue to cooperate, or punish

16. Does this work? No! • At stage two: • Each player thinks of himself as playing the original game, so they play (Confess, Confess) • At stage one: • Each player realizes that at stage two, good behavior will be punished anyway, so he might as well not cooperate

17. Indefinitely repeated games can solve the prisoners’ dilemma • If there is always the prospect of a “next” round, then cooperation can be sustained • Strategy is: • co-operate as long as other guy co-operates; • punish forever after he cheats • Each player weighs short-term gain from cheating against long-term loss from non-cooperation • So players need to value future payoffs high enough for tit-for-tat to work

18. Cartel enforcement – OPEC Payoffs are profits in squillions of dollars Rest of OPEC Saudi Arabia Nash equilibrium Cooperative outcome

19. Game with no Nash equilibrium* • Best responses are underlined • *Nash equilibrium is to randomly choose pitch/swing Batter Pitcher