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

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)

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?

A simple game: pet the dog StayFlee Don’t1,21,20,10,1 Pet2,12,11,01,0 Player J Player C C’s strategies J’s strategies J’s payoffs in blue C’s payoffs in red

What will J do? If C plays stay –Pet (2) is better than don’t (1) If C plays flee –Pet (1) is better than don’t (0) StayFlee Don’t1,21,20,10,1 Pet2,12,11,01,0 Player J Player C

J has a dominant strategy No matter what C does, J should play Pet StayFlee Don’t1,21,20,10,1 Pet2,12,11,01,0 Player J Player C

What will C do? If J plays Pet –Stay (2) is better than flee (1) If B plays Bottom –Stay (1) is better than flee (0) StayFlee Don’t1,21,20,10,1 Pet2,12,11,01,0 Player J Player C

C also has a dominant strategy No matter what J does, C should play Left StayFlee Don’t1,21,20,10,1 Pet2,12,11,01,0 Player J Player C

So, outcome of the game is: The dominant strategy equilibrium is (Pet, Stay) StayFlee Don’t1,21,20,10,1 Pet2,12,11,01,0 Player J Player C

Expend- ables Die Hard Expend- ables 2,12,10,00,0 Die Hard 0,00,01,21,2 A game without dominant strategies If Girlfriend plays E –E (2) is better than D (0) If Girlfriend plays D –D (1) is better than E (0) Expend- ables Note- book Expend- ables 2,12,10,00,0 Note- book 0,00,01,21,2 Boyfriend Girlfriend Boyfriend is not sure what to do

Girlfriend’s choice If boyfriend plays E –E (1) is better than D (0) If boyfriend plays D –D (2) is better than E (0) Expend- ables Die Hard Expend- ables 2,12,10,00,0 Die Hard 0,00,01,21,2 Boyfriend Girlfriend Girlfriend is not sure what to do

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

Nash equilibrium of the game (Top,Left) is a Nash equilibrium of this game LeftRight Top2,12,10,00,0 Bottom0,00,01,21,2 Player A Player B

The prisoners’ dilemma A’s dominant strategy is to confess B’s dominant strategy is to confess also ConfessDeny Confess-3,-30,-6 Deny-6,0-1,-1 Player A Player B Outcome is not Pareto efficient! Cooperation is much better for both

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

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

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

Cartel enforcement – OPEC Low priceHigh price Low price3,33,37,17,1 High price1,71,75,55,5 Saudi Arabia Rest of OPEC Payoffs are profits in squillions of dollars Nash equilibriumCooperative outcome

SwingDon’t Fastball0,10,11,01,0 Curve- ball 1,01,00,10,1 Game with no Nash equilibrium* Best responses are underlined *Nash equilibrium is to randomly choose pitch/swing Pitcher Batter SwingDon’t Fastball0,10,11,01,0 Curve- ball 1,01,00,10,1