Proposition 0.1 In any second-price auctions with private values, bidding one's valuation is the unique weakly dominant action. Therefore, the profile of "truthful" bid functions defined by ai(si)= si for every i E N, is a Bayesian Nash equilibrium of the associated bidding game Thus, truthful bidding is a Bayesian Nash equilibrium of the second-price auction regardless of the assumptions one makes about the underlying signal structure. This is viewed as a particularly comforting result: truthful bidding is certainly"focal"in this setting Proposition 0. 1 shows that it follows from the assumption that players do not choose weakly dominated actions. You already know that the(complete-information version of the) game has other Bayesian Nash equilibria If participants bid truthfully, then no bidder has a higher valuation for the object than the winner. Thus, under truthful bidding, the second-price auction places the object in the right "hands; that is, it is an efficient mechanism. 4 Incidentally, you may already know that second-price auctions are special cases of Vickrey- Groves-Clark direct mechanisms. With private values, truthful reporting is weakly dominant in such mechanisms, and always leads to efficient outcomes. Thus, for example, the k-good extension of the second-price auction(known as the Vickrey auction) also achieves efficiency First-Price Auctions with Independent Private values The analysis of first-price auctions requires more work, and more restrictive assumptions. In particular, we need to assume a lot of symmetry: in particular, assume that ui (si, s-i)=v(si) for all s=(Si, S_i)E 0, 1(so the same function v: 0, 1-R+ defines valuations for all players) and Fi= F for all i E I(so signals are i.i. d ) Also assume that v(1)=i<oo and v(0)=0. Finally, we assume that v and F are continuously differentiable, and that the density f= F is bounded away from zero Now observe that since the valuation function is bounded, we can renormalize it without loss of generality so that v (1)=1. Having done this, since both v and F are continuous and v is increasing, we can further simplify both the notation and the algebra, again without loss of generality, by considering an equivalent setting in which v(s)=s and F is replaced with a new cdf G such that the density of any valuation E [0, 1]under the original model f(o(a)), equals the density of x under G, i.e. g(a) We look for a symmetric equilibrium in increasing, differentiable bid function strategy we shall follow is to assume that such an equilibrium profile(alien exists, and derive some conditions which the profile must satisfy in order to be an equilibrium in differentiable bid functions. Then, we show that any profile of differentiable bid functions satisfying the above conditions is indeed an equilibrium aNote well: in this literature, an auction mechanism is deemed efficient if there is at least one equilibrium of the associated game which induces an efficient allocation. Other equilibria may well be inefficientProposition 0.1 In any second-price auctions with private values, bidding one’s valuation is the unique weakly dominant action. Therefore, the profile of “truthful” bid functions defined by ai(si) = si for every i ∈ N, is a Bayesian Nash equilibrium of the associated bidding game. Thus, truthful bidding is a Bayesian Nash equilibrium of the second-price auction— regardless of the assumptions one makes about the underlying signal structure. This is viewed as a particularly comforting result: truthful bidding is certainly “focal” in this setting; Proposition 0.1 shows that it follows from the assumption that players do not choose weakly dominated actions. You already know that the (complete-information version of the) game has other Bayesian Nash equilibria. If participants bid truthfully, then no bidder has a higher valuation for the object than the winner. Thus, under truthful bidding, the second-price auction places the object in the “right” hands; that is, it is an efficient mechanism.4 Incidentally, you may already know that second-price auctions are special cases of Vickrey￾Groves-Clark direct mechanisms. With private values, truthful reporting is weakly dominant in such mechanisms, and always leads to efficient outcomes. Thus, for example, the k-good extension of the second-price auction (known as the Vickrey auction) also achieves efficiency. First-Price Auctions with Independent Private Values The analysis of first-price auctions requires more work, and more restrictive assumptions. In particular, we need to assume a lot of symmetry: in particular, assume that vi(si , s−i) = v(si) for all s = (si , s−i) ∈ [0, 1]N (so the same function v : [0, 1] → R+ defines valuations for all players) and Fi = F¯ for all i ∈ I (so signals are i.i.d.). Also assume that v(1) = ¯v < ∞ and v(0) = 0. Finally, we assume that v and F¯ are continuously differentiable, and that the density ¯f = F¯0 is bounded away from zero. Now observe that, since the valuation function is bounded, we can renormalize it without loss of generality so that v(1) = 1. Having done this, since both v and F¯ are continuous and v is increasing, we can further simplify both the notation and the algebra, again without loss of generality, by considering an equivalent setting in which v(s) = s and F¯ is replaced with a new cdf G such that the density of any valuation x ∈ [0, 1] under the original model, ¯f(v −1 (x)), equals the density of x under G, i.e. g(x). We look for a symmetric equilibrium in increasing, differentiable bid functions. The strategy we shall follow is to assume that such an equilibrium profile (ai)i∈N = (a, . . . , a) exists, and derive some conditions which the profile must satisfy in order to be an equilibrium in differentiable bid functions. Then, we show that any profile of differentiable bid functions satisfying the above conditions is indeed an equilibrium. 4Note well: in this literature, an auction mechanism is deemed efficient if there is at least one equilibrium of the associated game which induces an efficient allocation. Other equilibria may well be inefficient. 5