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1. General A is defending a territory which is accessible by two mountain passes against general B. A has three divisions at her disposal and B has two divisions. Each general allocates her divisions between the two passes. A wins the battle at a pass if and only if she assigns at least as many divisions to the pass as does B. The payo of a general is the number of battles she wins minus the number of battles that she loses. a) Formulate this situation as a strategic game, by setting out the strategy sets and payo matrices. b) Dene a weakly dominated strategy. What are the weakly dominated strategies for each general? c) Show that this game does not have a pure strategy Nash equilibrium. By iteratively eliminating weakly dominated strategies or otherwise, solve for a mixed strategy Nash equilibrium.

2. Consider the following payo table, where the payo to the two players, ROW and COLUMN, only depends upon the actions taken by the two players. L R T 2,2 0,0 B 3,0 1,1 a) Consider the game where the players choose actions simultaneously. Solve for a Nash equilibrium in this game. b) Suppose now that ROW chooses his action rst; COLUMN observes ROWs chosen action, and then chooses his own action. Payos depend upon the action prole that results, as set out in the above table. Set out the extensive form of this game. Solve for a subgame perfect equilibrium. c) Suppose now that ROW chooses his action rst. With probability 3 4 , COLUMN observes ROWs chosen action. With probablility 1 4 , COLUMN does not observe ROWs chosen action. In either case, whether he observes or not, COLUMN then chooses his own action. Set out the extensive form of this game, setting out clearly COLUMNs information sets. Solve for a subgame perfect equilibrium where ROW chooses T: d) For the game in part (c), is there a subgame perfect equilibrium where ROW chooses B?

3. Consider a public goods provision game, with n individuals. Each indi- vidual must choose whether or not to contribute to the public good, and the public good is provided if and only if at least one individual contributes. The value of the good is the same for every individual and equals 1: If individual i provides the good, he incurs a cost ci; where ci is independently and identically distributed across individuals, and is uniformly distributed on [0; 1]: The total payo to an individual is the value of the good (if provided) minus the cost of provision (which is ci if the individual provides the good, and zero otherwise). a) Solve for a symmetric Bayesian Nash equilibrium of this game where each individual provides the good if and only if ci is below a critical threshold c: (You need not solve for c explicitly, but you need to set out the equation that it must satisfy). How does the probability that an individual provides the good vary with n? b) Set out an expression for the probability that the good is provided at all, as a function of c and n: c) Briey set out one economic or social situation that this model illustrates. 4. Consider the following strategic situation, involving player 1 and player

2: i) Nature chooses player 1 to be either weak (type w); or strong (type s); where a weak type has probability 0.4. Player 1 observes natures choice, while player 2 does not (it is common knowledge that the probability of w is 0.4). ii) Player 1 chooses breakfast, from the set fbeer, quicheg: iii) Player 2 observes player 1s choice of breakfast and chooses from the set fDUEL, NO}. The payos to the two players are as given in the gure on the next page, where the rst entry denotes the payo of player 1. a)Solve for all the pooling equilibria of this game. Which equilibrium is more plausible? b) Show that this game does not have a separating equilibrium.