1. Consider a variation of the Rubinstein bargaining game in which, instead of the players switching roles when an offer is rejected, the identity of the next person to make the offer is determined by flipping a fair coin. So, before any offer is made, a coin is flipped and there is an equal probability that either player will make the next offer. Both players are risk-neutral. Find the subgame perfect equilibrium and the expected payoffs for each player if: (a) both players have the same discount factor δ; (b) the two players have different discount factors δA and δB.
2. In a Cournot duopoly both firms have constant marginal costs that are either $6 or $18 per unit. The cost levels are equally likely for each firm, and each firm observes its own costs but not the costs of the other firm. The demand is given by P = 24 – Q and fixed costs are zero. Firms choose quantities simultaneously. In each of the following 4 cases find the Cournot (Bayesian-Nash) equilibrium outputs and the expected profits, both as a function of type and the unconditional expected profits and outputs (a) Costs are independent (b) Costs are perfectly positively correlated (c) Costs are perfectly negatively correlated (d) Conditional on one firm’s costs being low (or high) the probability that the other’s cost is the same is 2/3.
3. In the independent private values model there are n ≥ 1 risk-neutral bidders and the bidders' private values are iid with distribution F where F has support [0,1] and cumulative distribution function F(x) = x where  > 0 . (a) Find a Bayesian Nash equilibrium in a first-price sealed bid auction. (b) Find a Bayesian Nash equilibrium in a second-price sealed bid auction (c) What is the expected revenue of the seller in each case? Explain
不好意思都是英文