Question

Game Theory Eco 405 Homework 2 Due February 20, 2020 1. Find all the Nash equilibria you can of the following game. LCDR T 0,1 4,2 1,1 3,1 M 3,3 0,6 1,2 -1,1 B 2.5 1.7 3.8 0.0 2. This question refers to a second-price, simultaneous bid auction with n > 1 bidders. Assume that the bidders' valuations are 1, ,... where > > ... > >0. Bidders simultaneously submit bids, and the winner is the one who has the highest bid. The winner gets the object and pays the second-highest bid. If there is a tie for highest bid, the object is assigned with equal probability to each one of the highest bidders. Bidder i's payoff ist, -p if she wins and pays the price p, and zero if she does not win. Bidder i's payoff is assumed to be (1 ) if k > 2 bidders all bid p and if that is the highest bid. a. Show that the following strategy profiles are Nash equilibria of the second- price, simultaneous-bid auction: i) (0,0,0,...,0) (All bidders bid zero except bidder 1, who bids vu). ii) (us, n. 0,...,0) (Bidders 3 through n bid zero, bidder 1 bids and bidder 2 bids 01). b. Find a Nash equilibrium in which bidder 5 gets the object (assuming there are at least 5 bidders). Show it is a Nash equilibrium. c. Why might the Nash equilibria in which a bidder other than bidder 1 wins seem less reasonable? 3. This question refers to a first-price, simultaneous bid auction, in which the highest bidder gets the object (wins) and pays their own bid. If there is a tie, the lowest-indexed bidder gets the object. The payoff to player iis -p if she wins and pays p, and is zero if she does not win, where is player i's valuation. Again assume that wy > > ... > >0. a. Show that both (0,0,0,...,0) and (,,0....0) (bidders 3 through n bid zero) are Nash equilibria. b. Show that in any Nash equilibrium, the two highest bids are the same, and one of these bids is submitted by player 1. c. Show that in any Nash equilibrium, the highest bid is at least ty and at most d. Show that any strategy profile in which the two highest bids are the same, one of the highest bids is submitted by player 1, and the highest bid is at least uy and at most ry, is a Nash equilibrium.

Answer 1

то, І t Em 3,3 2,5 0,6 1,7 Best response are underlines Two sure strategy Nash qe (T,C) and (B,D) are two Nash egen