# Nash Equilibrium

1) Consider the following game between players One and Two. First, One decides whether to wear a swimsuit or regular clothes, and Two observes this. Next, the players simultaneously decide whether to go the Beach or to the Opera. If One is wearing regular clothes, the payoffs are as in the standard Battle of the Sexes game where One has a preference for the beach and Two has a preference for the opera (i.e., One’s payoff is 2 if they both go to the beach, 1 if they both go to the Opera, and 0 if they go to different places, etc.). If One is wearing a swimsuit, the payoffs are the same as above except that One’s payoff is -1from going to the Opera. (a) Draw the extensive form of this game. (b) Find the pure strategy Subgame Perfect Nash Equilibria. Describe explicitly the equilibrium strategies. (c) Briefly explain why there is no Subgame Perfect Nash Equilibrium outcome where both players go to the opera. (d) Find a Nash Equilibrium in which both players go to the opera.

2) Consider a game in which players A and B both have two actions, Left and Right. Suppose the game is played twice. For each case below, (i) draw the corresponding game tree with the appropriate information sets; (ii) give the number of strategies each player has; (iii) give the number of subgames of the game. (a) A moves first, B observes A’s action, then B moves. A then observes B’s action, and the game is repeated in a similar fashion.In what follows, assume that players simultaneously decide between Left and Right, and this simultaneous game is played twice. (b) Once the game has been played for the first time, players’ actions are publicly revealed. Players then play again. (c) Once the game has been played for the first time, player B learns the action A has taken. Player A learns nothing. Players then play again. (d) Once the game has been played for the first time, player A learns the action B has taken. Not only does player B not learn anything, but he forgets what action he chose. Players then play again.

Two firms play the following game. Firm 1 chooses a quantity q1, and this is observed by Firm 2, who in turn chooses his quantity q2. The inverse demand function is P(q1 + q2) = 5 – (q1 + q2) and both firms’ marginal cost is 1. Firms’ profit at the end of the game is their revenue minus their cost. (a) Find the quantities produced in a Subgame Perfect Nash Equilibrium and each firm’s profit. (b) How much would Firm 2 be willing to pay to change the order of play and become the first one to move?

1.  Consider the following extensive form game (attached): (a) Give the normal form representation of this game. (b) Find all pure strategy Nash equilibria. (c) Using Backward Induction, find the Subgame Perfect Nash Equilibrium of this game

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