1 . Define a strategy s; ES, as weakly dominant for player i in game IN = [I, {Si), fu;(.)}] if for all s;'ES, we have u;(s;, s ) zui(s;', s_ ) for all s. ES.i. Under the previous definition, if a player has two weakly dominant strategies, then for every strategy choice by his opponents, the two strategies yield him equal payoffs. Provide an example of a two-player game in which a player has two weakly dominant strategies but his opponent prefers that he plays one of them rather than the other. Explain. 2. A firm and a worker interact as follows. First, the firm can make 2 contract offers (wage, job type): (w, z =0) and (w, z =1), where z = 0 denotes the "safe" job and z =1 denotes the "risky" job. After observing the firm's contract offer (w, z), the worker accepts or rejects it. If the worker rejects the contract, then he gets a payoff of 100, which corresponds to his outside opportunities. If he accepts the job, then the worker cares about two things: his wage and his status. Then, the worker's payoff is [w + v(x)], where v(x) is the value of status x. The worker's status x depends on how he is rated by his peers, which is influenced by characteristics of his job as well as random events. Specifically, his rating x can be either 1 (poor), 2 (good), or 3 (excellent). If the worker has the safe job, then x = 2 for sure. On the other hand, if the worker has the risky job, then x = 3 with probability q and x = 1 with probability (1 - q). That is, with probability q, the worker's peers think of him as excellent. Assume that v(1) = 0 and v(3) = 100, and let v(2) = y. The worker searches to maximize his expected payoff. The firm obtains a return of (180 - w) when the worker is employed in the safe job. The firm gets a return of (200 - w) when the worker has the risky job. If the worker rejects the firm's offer, then the firm obtains 0. You need to compute the Subgame Perfect Nash Equilibria (SPNE) of this game and analyze which SPNE generate a payoff structure of an optimal contract, by answering the following questions. a. How large must the wage offer be in order for the worker to rationally accept the safe job? What is the firm's maximum payoff in this case? [Hint: The parameter "y" should be included in your answer.] b. How large must the wage offer be in order for the worker to rationally accept the risky job? What is the firm's maximum payoff in this case? [Hint: The parameter q should be included in your answer.] 2 c. What is the firm's optimal contract offer for the case in which q = 1/2? [Hint: Your answer should include an inequality describing conditions under which z = 1 is optimal. ]