Do Qs 4 only. The model is the same as Qs 2 as mentioned in the qs hence why i have included qs 2 as well.

2. (20 points) Consider the two-bidder auction example in class, where 4. (20 points) Consider the auction model with a continuum of possible bidder i's valuation, Vi, is equal to 1 with probability q E (0, 1) and 0 valuations. Bidder i's valuation, Vi, is drawn from the uniform distri- bution on [0, 1], for i = 1, 2, ..., n. In other words, the cdf of Vi, can with probability 1 - q, for i = 1, 2. Assume Vi and V2 are mutually be defined as F(v) = v for v E [0, 1] (and, of course, F(v) = 0 for v 1). Each bidder's valuation is independent of any independent. The object is sold by a first-price auction. The bidders other bidder's valuation. Consider the first-price auction. As I have ar- are allowed to bid any nonnegative price. Whenever there is a tie, each gued in class, the strategy profile in which Bi(v) = B(v) = (n-1).v for all v E [0, 1] and i = 1, 2, ..., n is a Nash equilibrium. For this bidder gets the good with probability 1/2. problem, focus on the case n = 3. (a) Consider bidder 1. Given bidders 2 and 3 bid B(v) = 2v/3 for all (a) Argue that, when Vi = 0, it is weakly dominant for player i to v E [0, 1], show that when Vi = 3/4, the best response for bidder 1 to bid bid Bi(0) = 0 for i = 1, 2. 2 3_ B(1/2) = 3 4 2 (b) Show that there does not exist an equilibrium in which B1 (1) = Hint: Express his payoff as a function of his bid, b, and show that b = 1/3 maximizes his expected payoff. B2(1) = 3/4. (b) Suppose the seller uses a posted price p. What is her expected revenue? Which price maximizes her expected revenue? Hint: (c) Find the mixed strategy Nash equilibrium when q = 1/4. What is the probability of at least one buyer is willing to pay p? (c) Recall that in the first price auction, the seller's expected revenue (d) Argue that the expected revenue of the seller in a first-price auc- s (n-1)/(n+1). Compare the seller's revenue from the first-price tion is lower than that in posted-price selling. auction and that from posted-price selling