Eco514-Game Theory Lecture 8.5: More on Auctions; PS#1 Marciano siniscalchi October 14. 1999 Introduction These notes essentially tie up a few loose ends in Lecture 8; in particular, I exhibit examples of inefficiencies in first- and second-price auctions. I would also like to briefly comment on Questions 1 and 2 in Problem Set 2 The first-price auction may be inefficient even with private values Both examples I am going to show are due to Eric Maskin(to the best of my knowledge) The first point I wish to make is that, even in a private-values setting, asymmetries may render a first-price auction inefficient Consider a two-bidder situation in which the value of the first bidder is uniformly dis- tributed on 0, 1, and the value of the second bidder is uniformly distributed on 0, 2. We look for a(necessarily asymmetric)equilibrium in increasing, continuous bid fund (using a convenient shorthand notation) maps a: [0, i-R+ First, note that ai(0)=0 in any equilibrium. For suppose w l.o.g. that a1(0)=a>0 Then it must be the case that in equilibrium a wins with probability zero, so that also a2(0)=a. But since we are assuming that bid functions are increasing and continuous types e>0 with e a expect to win with positive probability and pay at least a< aile contradiction Now, I claim that there is no such equilibrium in which a2(2)> 1. Note that the previous argument, together with the assumptions that bid functions are increasing and continuous implies that positive bids win with positive probability. Hence bidder 1 will never bid above 1, her maximum value. But then, from the point of view of bidder 2, any bid a> 1 may be improved upon by bidding Hence, bidder 2 never bids more than 1. But this means that for some realization of the signals close to(1, 2), by continuity we will have a2(u2)< a101 even if u2>U1; the first-price auction assigns the object to bidder 1Eco514—Game Theory Lecture 8.5: More on Auctions; PS#1 Marciano Siniscalchi October 14, 1999 Introduction These notes essentially tie up a few loose ends in Lecture 8; in particular, I exhibit examples of inefficiencies in first- and second-price auctions. I would also like to briefly comment on Questions 1 and 2 in Problem Set 2. The first-price auction may be inefficient even with private values Both examples I am going to show are due to Eric Maskin (to the best of my knowledge). The first point I wish to make is that, even in a private-values setting, asymmetries may render a first-price auction inefficient. Consider a two-bidder situation in which the value of the first bidder is uniformly dis￾tributed on [0, 1], and the value of the second bidder is uniformly distributed on [0, 2]. We look for a (necessarily asymmetric) equilibrium in increasing, continuous bid functions, i.e. (using a convenient shorthand notation) maps ai : [0, i] → R+. First, note that ai(0) = 0 in any equilibrium. For suppose w.l.o.g. that a1(0) = a > 0. Then it must be the case that in equilibrium a wins with probability zero, so that also a2(0) = a. But since we are assuming that bid functions are increasing and continuous, types  > 0 with  < a expect to win with positive probability and pay at least a < ai(): contradiction. Now, I claim that there is no such equilibrium in which a2(2) > 1. Note that the previous argument, together with the assumptions that bid functions are increasing and continuous, implies that positive bids win with positive probability. Hence bidder 1 will never bid above 1, her maximum value. But then, from the point of view of bidder 2, any bid a > 1 may be improved upon by bidding a+1 2 . Hence, bidder 2 never bids more than 1. But this means that, for some realization of the signals close to (1,2), by continuity we will have a2(v2) < a1(v1) even if v2 > v1; the first-price auction assigns the object to bidder 1. 1