[Incidentally, this argument does not rule out the possibility that equilibria in non- increasing or discontinuous bid functions might exist. Such equilibria can be ruled out under certain conditions: see Lizzeri and Persico. Games and Economic behavior. 1999 for details The second-price auction can be inefficient with interdependent values and more than two bidders This example is really clever. Consider a three-bidder, interdependent-values environment with signals distributed according to some law on a support which includes the point (1, 1, 1) The actual distribution of signals does not matter(you can rig things so you get the same conclusion even if signals are correlated), but to keep things simple, assume that signals are independen Suppose valuations are as follows 1 S1+-S2+ t2(s1,s2,S3)=82+71+s3 Consider(s1, S2, s3)in a(small)neighborhood of(1, 1, 1). For ex-post efficiency to obtain the object should be assigned to either bidder l or 2, depending on whether s3>l or ss <1 That is: efficient assignment depends on s3 But in a second-price auction, a bid depends only on a buyers own signal; thus, whether bidder 1 or bidder 2 receives the object depends on the relative magnitude of s1 and $2 Hence, for some realization of the signals in a neighborhood of (1, 1, 1), the outcome of the auction will be inefficient In case you were wondering, Dasgupta and Maskin offer a simple solution to this problem allow players to submit bids contingent on the signals of all the other players. That is, an action for bidder 1 with signal s1 is a function f(s1)(s2, s3)to be interpreted as follows: "If I, bidder 1, knew that my opponents' signal were S2 and S3 respectively, then I would bid f(s1(s2, S3). Every bidder is thus asked to report, not just a single bid, but a full bid function The seller then computes a fixpoint of these bid functions, and takes the fixpoint to be the actual vector of signals; the allocation is determined with respect to this vector. It turns out that it is an equilibrium to report truthfully: that is, to report f(s1)($2, S3)=U1(s1, $2, $3). On Problem set 2[Incidentally, this argument does not rule out the possibility that equilibria in non￾increasing or discontinuous bid functions might exist. Such equilibria can be ruled out under certain conditions: see Lizzeri and Persico, Games and Economic Behavior, 1999 for details.] The second-price auction can be inefficient with interdependent values and more than two bidders This example is really clever. Consider a three-bidder, interdependent-values environment, with signals distributed according to some law on a support which includes the point (1,1,1). The actual distribution of signals does not matter (you can rig things so you get the same conclusion even if signals are correlated), but to keep things simple, assume that signals are independent. Suppose valuations are as follows: v1(s1, s2, s3) = s1 + 1 2 s2 + 1 4 s3 v2(s1, s2, s3) = s2 + 1 4 s1 + 1 2 s3 v3(s1, s2, s3) = s3 Consider (s1, s2, s3) in a (small) neighborhood of (1, 1, 1). For ex-post efficiency to obtain, the object should be assigned to either bidder 1 or 2, depending on whether s3 > 1 or s3 < 1. That is: efficient assignment depends on s3. But in a second-price auction, a bid depends only on a buyer’s own signal; thus, whether bidder 1 or bidder 2 receives the object depends on the relative magnitude of s1 and s2. Hence, for some realization of the signals in a neighborhood of (1,1,1), the outcome of the auction will be inefficient. [In case you were wondering, Dasgupta and Maskin offer a simple solution to this problem: allow players to submit bids contingent on the signals of all the other players. That is, an action for bidder 1 with signal s1 is a function f(s1)(s2, s3) to be interpreted as follows: “If I, bidder 1, knew that my opponents’ signal were s2 and s3 respectively, then I would bid f(s1)(s2, s3).” Every bidder is thus asked to report, not just a single bid, but a full bid function. The seller then computes a fixpoint of these bid functions, and takes the fixpoint to be the actual vector of signals; the allocation is determined with respect to this vector. It turns out that it is an equilibrium to report truthfully: that is, to report f(s1)(s2, s3) = v1(s1, s2, s3).] On Problem Set 2 2