Determine the bayesian nash equilibrium of a game, Game Theory

Stanley is auctioning an item that he values at zero. Betty and Billy, the two potential buyers, each have independent private values which are drawn from a uniform distribution, Pr (v < x) = x. The auction is a first price auction; the high bidder gets the good and pays his or her bid to Stanley.

(a) Suppose Billy bids B (v) = v. What is Betty's expected payoff from bidding b when her value is v?

(b) What is Betty's best response to Billy's strategy B (v) = v?

(c) What is the BNE of this game? Recall that the seller's profit in this setting is 1 3 Now suppose that Stanley has a crush on Betty, and so he gives her an advantage in the auction. First Stanley asks Billy for a bid. Then he shows Billy's bid to Betty, and lets her either match Billy's bid, or not. If she decides to match, then Betty pays Billy's bid and receives the object. If she decides not to match, then Billy pays his bid to receive the item. Of course, Billy is aware that this is the procedure when he bids.

(d) Suppose Betty's value is vT and Billy bids b. What must be true for Betty to choose to match Billy's bid? What must be true for Billy to purchase the item?

(e) If Billy bids b, what is the probability that he will be able to purchase the item?

(f) If his value is vB what will Billy choose to bid?

(g) What is Stanley's profit under this new procedure? (Hint: Stanley will never collect anything more than Billy's bid).

(h) How much is Stanley hurt because he decided to favor Betty?

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