There are 2 bidders, labeled i = 1, 2. Bidder i has a valuation vi for the good?this means that, if bidder i gets the good and pays the price p, then i's payoff is vi ? p. The two bidder's valuations are independently and uniformly distributed on [0, 1]. Bids are constrained to be nonnegative. The bidders simultaneously submit their bids. The higher bidder wins the good and pays the price she bid; the other bidder gets and pays nothing. In case of a tie, the winner is determined by a flip of a fair coin. The bidders want to maximize expected payoffs.

(a) Suppose that bidder 2 bids a fraction c2 > 0 of their value, that is, b2(v2) = c2v2. Find the best response of a bidder 1 who has valuation v1.

(b) Find a Bayesian Nash equilibrium.

3 First-price sealed bid auction There are 2 bidders, labeled :2 = 1,2. Bidder i has a valuation e,- for the goodthis means that, if bidder 2' gets the good and pays the prioe p, then i's payoE is o,- p. The two bidder's valuations are independently and uniformiy distributed on [0. 1]. Bids are constrained to be nonnegative. The bidders simultaneously submit their bids. The higher bidder wins the good and pays the price she hid; the other bidder gets and pays nothing. In case of a tie, the winner is determined by a. ip of a fair coin. The bidders went to minimize expected payoffs. (a) Suppose that bidder 2 bids a fraction :22 > i] of their value, that is, 520:2) = c2112. Find the best response of a bidder 1 who has valuation :11. (b) Find a Bayesian Nash equilibrium