Game Theory

212.430 Game Theory and Applications 2001

In Ho Lee

Mid-Term Examination

October 24, 2001

1. Is the following game dominance solvable? Identify the Nash equilibrium in pure strategies. U D L 4, 3 5, 5 M 2, 7 5, -1 R 0, 4 -4, -2

2. There are k witnesses to a crime. All of the witnesses would like to see the perpetrator apprehended. This will happen only if the crime is reported to the police. Since all of the witnesses are busy people, they would rather have someone else report the crime than reporting it themselves. To be specific, assume that the utility that the players get when the criminal is caught is 4 (and the criminal is caught if and only if a report is made) and zero if not caught. The disutility from reporting is 1. All the witnesses decide simultaneously on whether to report. (a) Model this story as a strategic form game for k = 2. (b) Find the pure strategy Nash equilibria of the game. (c) In the case of k = 2, calculate the mixed strategy Nash equilibrium. (d) Calculate the symmetric mixed strategy equilibrium for general k and calculate also the probability of getting caught as a function of k. (e) Is the mixed strategy equilibrium efficient? Explain why or why not.

3. Discuss briefly. In Centipede game of Rosenthal (1981), the unique subgame perfect equilibrium is that each player plays D whenever a move is given. Rosenthal criticized the equilibrium concept in that following a deviation from the equilibrium strategy by either player, the rationality of either player is not sustainable. Explain the meaning of rationality in the criticism. Why does the observation cause difficulty for the subgame perfect equilibrium concept? Can you suggest a reconciliation of the problem?

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