Game Theory Game theory models strategic behavior by agents who understand that their actions affect the actions of other agents. Been used to study –Market competition – Cournot Oligopoly. –Military strategies. –Bargaining/Negotiations –Biology A game consists of players, strategies, and payoffs.

Battle of Bismarck Sea Imamura Kenny 1 North South North South Imamura wants to transport troops. Kenny wants to bomb Japanese troops.. North route is two days, Southern route is three days. It takes one day for Kenny to switch routes.

Imamura wants to run convoy from Rabaul to Lae

The Prisoners’ Dilemma Clyde Bonnie C S C S Bonnie and Clyde are caught. They can confess or be silent.

War Mars Venus Shoot Not Shoot Shoot Not Shoot Are we doomed to the bad outcome? Not in trench warfare of WWI. This happens since the game is repeated. Tit-for-tat can work. Not so easy if uncertainty of action.

Coordination Problem Jim Sean 0.5 Beta VHS Beta VHS Jim and Sean want to have the same VCR. Beta is a better technology than VHS.

Nash equilibrium A Nash equilibrium is a set of strategies – Where each player has no incentive to deviate. –Given other equilibrium strategies a player would choose his equilibrium strategy. –A best response to a best response. A pure strategy equilibrium is where each player only chooses a particular strategy with certainty. What are the pure-strategy equilibria in Prisoners’ dilemma and the coordination game. Is there always a pure strategy equilibrium?

Penalty Kick Goalie Kicker 1 Kick R Dive L Dive R Kick L A Kicker can kick a ball left or right. A Goalie can dive left or right.

Mixed Strategy equilibrium Happens in the Penalty kick game. Notice that if the Kicker kicks (.5L+.5R), the Goalie is indifferent to diving left or right. If the Goalie dives (.5L+.5R), the Kicker is indifferent to kicking left or right. Thus, (.5L+.5R,.5L+.5R) is a mixed-strategy N.E. Nash showed that there always exists a Nash equilibrium.

Do you believe it? Do they really choose only L or R? Yes. Kickers 93.8% and Goalie 98.9%. Kickers are either left or right footed. Assume R means kick in “easier” direction. Below is percentage of scoring. Nash prediction for (Kicker, Goalie)=(41.99L+58.01R, 38.54L+61.46R) Actual Data =(42.31L+57.69R, 39.98L+60.02R) Kick R Dive L Dive R Kick L