Note that this approach does not establish existence directly: first, no profile of bid functions satisfying the derived conditions may exist(so the second part of the argument may be empty ) second, even if they do exist, they may fail to be increasing and / or differentiable so the first part of the argument may be unwarranted ). To establish existence, one must invoke results from the theory of differential equations With this in mind, note that Bidder i's payoff function when her opponents follow the equilibrium bid functions can be written as follows Iai, si= M/+1P{s-:H(a,(a(s)≠)=JU{}s-a小 That is: if she wins the object by bidding ai, Bidder i's utility is si-ai; for any realization of the signals, she may be the only high bidder, or there may be another high bidder,or there may be two, etc, in which case she wins with correspondingly smaller probability The assumptions we have made simplify the analysis considerably. First, since the model perfectly symmetric, we can drop all indices from actions and signals Second, since we are assuming that the common equilibrium bid function is increasing and that G has a density, ties occur with probability zero( i.e. there is G-almost always only one winner). Thus, only one term(corresponding to =N\i) is nonzero in the above summation. Third, since the equilibrium bid function is increasing and signals are independent, it can be inverted, and the probability that the bid a is the winning bid is given by G(a-(a))N-l; this is the probability that all opponents observe signals which induce them (in the equilibrium under consideration) to bid at most a. Thus, we get U(a,s)=[G(a-1(a)-s-a Now, if a is a best reply to the equilibrium belief, then it must satisfy the first-order condition (N-1)G(a()2ga( s-a-Ga (a)) 0. If the profile under consideration is an equilibrium then the above restriction must hold for a= a(s); this allows one to simplify the expression somewhat. Moreover, since G strictly positive by assumption, we can divide throughout by G(s)-2 to get (N-1)9(ss-a(s)=G(s)a(s); the interpretation in terms of cost-benefit tradeoff should be clear. Another necessary con- dition for a to be a best-reply is a(0)=0. Thus, we have a first-order linear differential equation and a boundary condition; given our regularity conditions, a solution exists and is determined by the above necessary conditionsNote that this approach does not establish existence directly: first, no profile of bid functions satisfying the derived conditions may exist (so the second part of the argument may be empty); second, even if they do exist, they may fail to be increasing and/or differentiable (so the first part of the argument may be unwarranted). To establish existence, one must invoke results from the theory of differential equations. With this in mind, note that Bidder i’s payoff function when her opponents follow the equilibrium bid functions can be written as follows: Ui(ai , si) = X J⊂N\{i} 1 |J| + 1 Pr {s−i : H(ai ,(a(sj ))j6=i) = J ∪ {i}} [si − ai ]. That is: if she wins the object by bidding ai , Bidder i’s utility is si − ai ; for any realization of the signals, she may be the only high bidder, or there may be another high bidder, or there may be two, etc., in which case she wins with correspondingly smaller probability. The assumptions we have made simplify the analysis considerably. First, since the model is perfectly symmetric, we can drop all indices from actions and signals. Second, since we are assuming that the common equilibrium bid function is increasing, and that G has a density, ties occur with probability zero (i.e. there is G-almost always only one winner). Thus, only one term (corresponding to J = N \ {i}) is nonzero in the above summation. Third, since the equilibrium bid function is increasing and signals are independent, it can be inverted, and the probability that the bid a is the winning bid is given by G(a −1 (a))N−1 ; this is the probability that all opponents observe signals which induce them (in the equilibrium under consideration) to bid at most a. Thus, we get U(a, s) = [G(a −1 (a))]N−1 [s − a] Now, if a is a best reply to the equilibrium belief, then it must satisfy the first-order condition (N − 1)G(a −1 (a))N−2 g(a −1 (a)) 1 a 0 (a −1 (a))[s − a] − [G(a −1 (a))]N−1 = 0. If the profile under consideration is an equilibrium then the above restriction must hold for a = a(s); this allows one to simplify the expression somewhat. Moreover, since G is strictly positive by assumption, we can divide throughout by G(s) N−2 to get (N − 1)g(s)[s − a(s)] = G(s)a 0 (s); (1) the interpretation in terms of cost-benefit tradeoff should be clear. Another necessary con￾dition for a to be a best-reply is a(0) = 0. Thus, we have a first-order linear differential equation and a boundary condition; given our regularity conditions, a solution exists and is determined by the above necessary conditions. 6