ncidentally, this is where symmetry plays a role. Without it, Equation(1) would be replaced by a system of differential equations, and a more sophisticated machinery (or some clever trick) is required to establish the existence of a solution. We shall compute it explicitly under the additional assumption that G(s=s, i.e. signals are uniformly distributed. Then the above differential equation reduces to (N-1)s-a(s))= sa(s) whence we can verify immediately that a(s)=Nw s is the only solution satisfying a(0)=0 Given this parametrization, it is easy to verify that the profile(x-s,., N s)is indeed Bayesian Nash equilibrium of the first-price auction. Indeed, it is the only symmetric equi- librium in differentiable and increasing strategies. Note that this verification is necessary, because the differential equation derived above was only a necessary condition for the func- tion a( to be a best reply More generally, it can be shown that Equation(1), along with the boundary condition a(0)=0, indeed defines a Bayesian Nash equilibrium of the game In case you are wondering whether invoking, say, concavity of the objective function to conclude that Equation(1)is also a sufficient condition for a to define an equilibrium, think about the following: the concavity of the objective function U(a, s)depends on exogenously given entities such as v and G, but also on the function a itself Two observations are in order. First, note that bidders shade their bids-the latter is always below their valuation, although the difference shrinks to zero in the limit. However, under the same assumptions about signal distributions and valuations, first- and second-price auctions yield the same expected revenues to the seller To see this in our simple example, note that at any state s, the price paid to the seller equals the second-highest signal; hence, expected revenues are equal to the ex pectation of the second order statistic drawn from an i.i. d. uniform sample of size N. This is (N-2)n(s) ()x-21-G(s)y()ds=N(N-1) ds N+1 On the other hand, expected revenues in a first-price auction equaly times the expectation of the first order statistic sNG(s)-lg(sds=N/sds=N N+1 Again, this is actually an instance of a more general fact(see Myerson's paper for details) assuming we are willing to compare equilibria in different games, first- and second-price auctions are revenue-equivalent with private valuesIncidentally, this is where symmetry plays a role. Without it, Equation (1) would be replaced by a system of differential equations, and a more sophisticated machinery (or some clever trick) is required to establish the existence of a solution. We shall compute it explicitly under the additional assumption that G(s) = s, i.e. signals are uniformly distributed. Then the above differential equation reduces to (N − 1)[s − a(s)] = sa 0 (s) whence we can verify immediately that a(s) = N−1 N s is the only solution satisfying a(0) = 0. Given this parametrization, it is easy to verify that the profile ( N−1 N s, . . . , N−1 N s) is indeed Bayesian Nash equilibrium of the first-price auction. Indeed, it is the only symmetric equi￾librium in differentiable and increasing strategies. Note that this verification is necessary, because the differential equation derived above was only a necessary condition for the func￾tion a(·) to be a best reply. More generally, it can be shown that Equation (1), along with the boundary condition a(0) = 0, indeed defines a Bayesian Nash equilibrium of the game. In case you are wondering whether invoking, say, concavity of the objective function to conclude that Equation (1) is also a sufficient condition for a to define an equilibrium, think about the following: the concavity of the objective function U(a, s) depends on exogenously given entities such as v and G, but also on the function a itself. Two observations are in order. First, note that bidders shade their bids—the latter is always below their valuation, although the difference shrinks to zero in the limit. However, under the same assumptions about signal distributions and valuations, first- and second-price auctions yield the same expected revenues to the seller. To see this in our simple example, note that at any state s, the price paid to the seller equals the second-highest signal; hence, expected revenues are equal to the expectation of the second order statistic drawn from an i.i.d. uniform sample of size N. This is Z s N (N − 2)!1!G(s) N−2 [1 − G(s)]g(s)ds = N(N − 1) Z s N−1 (1 − s)ds = N − 1 N + 1 . On the other hand, expected revenues in a first-price auction equal N−1 N times the expectation of the first order statistic, Z sNG(s) N−1 g(s)ds = N Z s N ds = N N + 1 . Again, this is actually an instance of a more general fact (see Myerson’s paper for details): assuming we are willing to compare equilibria in different games, first- and second-price auctions are revenue-equivalent with private values. 7