In the Hollywood movie “A Beautiful Mind”, Russel Crowe plays John Nash who developed the Nash Equilibrium concept in his PhD thesis at Princeton University. In one of the early scenes of the movie, Nash finds himself in a bar with three of his fellow (male) mathematics PhD students when a group of five women enters the bar. The attention of the PhD students is focused on one of the five women, with each of the four PhD students expressing interest in asking her out. Suppose we simplified the example to one in which it was only Nash and one other student encountering a group of two women. We then have two pure strategies to consider for each PhD student: Pursue woman A or pursue woman B. Suppose that each viewed a date with woman A as yielding a “payoff ” of 10 and a date with woman B as yielding a payoff of 5. Each will in fact get a date with the woman that is approached if they approach different women, but neither will get a date if they approach the same woman in which case they both get a payoff of 0.

1. Write down the payoff matrix of this game. What are the Nash Equilibria of this game?

2. Draw the game tree for this game (assume Nash (player 1) is the first mover). What is the subgame perfect Nash equilibrium of this game?

Assume that there are three herdsmen in a common grazing area. They have to decide whether or not to add an additional cow to the common. The common area is currently feeding 90 animals, this amount of animals do not destroy the land. Note that each herdsman has 30 animals in the commons. If the herdsman decides to add a cow, the percentage loss in value is 8%. The total value of herd is denoted as (N+x)(1-0.04*x) where N represent the number of the herd at the limit of the grazing and x is number of heads of cattle beyond grazing capacity . Obtain the Nash equilibrium of this

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1 The pure strategy equilibria of this game are AB and BA ... View the full answer