Consider a first-price sealed-bid auction for a single

Question

Question: Consider a first-price sealed-bid auction for a single indivisible good with two bidders, i=1,2. Bidders i's valuation, v_i, is given by

v_i=t_i +0,5

where t_i is a random variable drawn independently from a commonly known uniform distribution over [0,1]. The realization of t_i is privately observed by bidder i, the object is awarded to the highest bidder who pays her own bid (i.e. first price). The losing bidder pays nothing. In the case of equal bids, each bidder wins the object with equal probability and pays her own bid.

a. Derive the symmetric Bayesian Nash equilibrium of this game where each bidder uses a linear function of her private information, i.e of the form b_i=sigmat_i+betta.

b. What is the conditional expected payoff of bidder i with type t_i in this equilibrium?