《博弈论讲义》英文版 Problem Set 3: Due Thursday, October 28

Eco514-Game Theory Problem Set 3: Due Thursday, October 28 1. Asymmetric auctions Consider an interdependent-values auction with two bidders, each of whom observes an i.i.d uniform signal s; E [ 0, 1]. Bidder i' s valuation for the object is equal to v (si, s-i)=Ai si+s-i where Ai >1 but in general A1+ A2 (a) Construct a Bayesian Nash equilibrium of the second-price auction in this environ- ent. is the outcome efficient? b)Now construct a Bayesian Nash equilibrium of the first-price auction in this environ- ment. Is the outcome efficient? Does revenue equivalence hold? 2. Auctions with a reserve price In the setting of the previous exercise, suppose that the seller decides to set a positive reserve price r>0: that is, all bids below the reserve price are discarded(so, in particular, it may be the case that both bids are discarded). Assume that the value of r is publicly announced before the auction begins, so it is commonly known Construct an equilibrium of the second-price auction in this environment, and express the expected revenues to the seller as a function of r. Does setting r=0 maximize the seller's expected revenues? 3. From or:99.1.101.3.108.1

Eco514—Game Theory Problem Set 3: Due Thursday, October 28 1. Asymmetric Auctions Consider an interdependent-values auction with two bidders, each of whom observes an i.i.d. uniform signal si ∈ [0, 1]. Bidder i’s valuation for the object is equal to vi(si , s−i) = λisi+s−i , where λi > 1 but in general λ1 6= λ2. (a) Construct a Bayesian Nash equilibrium of the second-price auction in this environ￾ment. Is the outcome efficient? (b) Now construct a Bayesian Nash equilibrium of the first-price auction in this environ￾ment. Is the outcome efficient? Does revenue equivalence hold? 2. Auctions with a reserve price In the setting of the previous exercise, suppose that the seller decides to set a positive reserve price r > 0: that is, all bids below the reserve price are discarded (so, in particular, it may be the case that both bids are discarded). Assume that the value of r is publicly announced before the auction begins, so it is commonly known. Construct an equilibrium of the second-price auction in this environment, and express the expected revenues to the seller as a function of r. Does setting r = 0 maximize the seller’s expected revenues? 3. From OR: 99.1, 101.3, 108.1 1