Notes on interdependent values values-regime: the wallet game. In particular, assume there are only two bidders.pendent I will limit my remarks to the simplest possible example of bidding in an interde Let us consider the second-price auction first. A crucial difference between private- and interdependent-values settings is that players no longer have weakly dominant strategies With private values, a player who bids less than her valuation risks losing the object and gains nothing conditional upon winning because her payment is not determined by her bid, but by her opponent With interdependent values, her valuation conditional upon winning the object depends on her opponents bid function, because the latter determines the set of types against which she wins; thus, indirectly, her own bid determines her valuation conditional upon winning and the entire logic of the Vickrey-Groves-Clark mechanism breaks down Indeed, "truthful bidding"is not even defined, if not perhaps with reference to a fixed quilibrium belief! Thus, we can at best hope to obtain efficiency in equilibrium, not as a consequence of the sole assumption that players do not choose weakly dominated actions Indeed, we shall see that efficiency may fail to obtain in equilibrium Let us solve the wa S1+S2. I claim that game. We need only check that 2s is a best reply for type s against the putative equilibrium belief. Her expected payoff when she bids a is given by U(a, s)= (+s2-2s2)ds2=1。1 which has a maximum at a=2s, as one can easily check; the second-order conditions are also easily verified. Also note that, since values are purely common, any allocation mechanism is trivially efficient. However, there are examples of three-bidder games with interdependent values in which efficiency fails Note that revenues equal twice the expected value of the minimum of two i i.d. uniform random variables,or(from our previous analysis)3 Let us now consider a first-price auction in this setting. I claim than now(s1, s2)is an quilibrium profile. To see this, note that as-a+ and again the claim is proved. From our previous analysis, expected revenues equal the expected value of the maximum of two i.i. d. uniform variates, or(again)3.This, ho is not generally the case for interdependent-values settingsNotes on interdependent values I will limit my remarks to the simplest possible example of bidding in an interdependent values-regime: the wallet game. In particular, assume there are only two bidders. Let us consider the second-price auction first. A crucial difference between private- and interdependent-values settings is that players no longer have weakly dominant strategies. With private values, a player who bids less than her valuation risks losing the object, and gains nothing conditional upon winning because her payment is not determined by her bid, but by her opponent’s. With interdependent values, her valuation conditional upon winning the object depends on her opponent’s bid function, because the latter determines the set of types against which she wins; thus, indirectly, her own bid determines her valuation conditional upon winning, and the entire logic of the Vickrey-Groves-Clark mechanism breaks down. Indeed, “truthful bidding” is not even defined, if not perhaps with reference to a fixed equilibrium belief! Thus, we can at best hope to obtain efficiency in equilibrium, not as a consequence of the sole assumption that players do not choose weakly dominated actions. Indeed, we shall see that efficiency may fail to obtain in equilibrium. Let us solve the wallet game under the second-price payment rule: recall that vi(s) = s1 + s2. I claim that the profile (2s1, 2s2) is a Bayesian Nash equilibrium of the associated game. We need only check that 2s is a best reply for type s against the putative equilibrium belief. Her expected payoff when she bids a is given by U(a, s) = Z 1 2 a 0 (s + s2 − 2s2)ds2 = 1 2 as − 1 2 a 2 which has a maximum at a = 2s, as one can easily check; the second-order conditions are also easily verified. Also note that, since values are purely common, any allocation mechanism is trivially efficient. However, there are examples of three-bidder games with interdependent values in which efficiency fails. Note that revenues equal twice the expected value of the minimum of two i.i.d. uniform random variables, or (from our previous analysis) 2 3 . Let us now consider a first-price auction in this setting. I claim than now (s1, s2) is an equilibrium profile. To see this, note that U(a, s) = Z a 0 (s + s2 − a)ds2 = a(s − a) + 1 2 a 2 and again the claim is proved. From our previous analysis, expected revenues equal the expected value of the maximum of two i.i.d. uniform variates, or (again) 2 3 . This, however, is not generally the case for interdependent-values settings. 8