Week 8: Basic concepts in game theory Part 1: Examples of games We introduce here the basic objects involved in game theory. To specify a game ones gives The players. The set of all possible strategies for each player. The payos: if each player picks a certain strategy then each player receive a payorepresented by a number. The payofor every player depends, in general, from the strategies of all other players. Payos have many dierent meanings, e.g., an amount of money, a number of years of happiness, the tness in biology, etc… Our convention is that the highest payois deemed the most desirable one. In the rest of this section we introduce some of the standard games in game theory, together with the “story behind them” The prisoner’s dilemma:. This is one of the most famous game in game theory. One possible story associated to it goes as follows. A prosecutor seeks two arrest two bank robbers. He lacks proofs of their involvement in the heist but has managed to have them arrested for a minor fraud charge. He oers to each them the following choices. If you confess but your accomplice does not then you will go free and your accomplice will be punished 8 years in jail. If you both confess then you will each get 6 years in jail. Finally, if none confess then they both will convicted for the minor fraud charge, say for 1 year in prison. We will represent payos and strategies using a table. We will call the player Robert (R is for the “row” player) and Collin (C is for “column” player). For both Robert and Collin the strategies are confess or not confess. Robert Collin confess not confess confess -6 -6 -8 0 not confess -6 -8 -1 -1 1
The rows represents the strategies of Robert, the columns the strategies of Collin. The numerical entries are understood as follows: in each box the lower left entry is the payofor Robert while the the upper right entry is the payofor Collin. Solution of the prisoner’s dilemma: To see what is the best option for the prisoner’s dilemma one…