Private Value Auctions: A First Look

We begin the formal analysis by considering equilibrium bidding behavior in the four common auction forms in an environment with independently and identically distributed private values. In the previous chapter we argued that the open descending price (or Dutch) auction is strategically equivalent to the first-price sealed-bid auction. When values are private, the open ascending price (or English) auction is also equivalent to the second-price sealed-bid auction, albeit in a weaker sense. Thus, for our purposes, it is sufficient to consider the two sealed-bid auctions.

This chapter introduces the basic methodology of auction theory. We postulate an informational environment consisting of (1) a valuation structure for the bidders—in this case, that of private values—and (2) a distribution of information available to the bidders—in this case, it is independently and identically distributed. We consider different auction formats—in this case, first- and second-price sealed-bid auctions. Each auction format now determines a game of incomplete information among the bidders and, keeping the informational environment fixed, we determine a Bayesian-Nash equilibrium for each resulting game. When there are many equilibria, we usually select one on some basis—dominance, perfection, or symmetry—but make sure that the criterion is applied uniformly to all formats. The relative performance of the auction formats on grounds of revenue or efficiency is then evaluated by comparing the equilibrium outcomes in one format versus another.

2.1 THE SYMMETRIC MODEL

There is a single object for sale, and N potential buyers are bidding for the object. Bidder i assigns a value of Xi to the object—the maximum amount a bidder is willing to pay for the object. Each Xi is independently and identically distributed on some interval [0,ω] according to the increasing distribution function F. It is assumed that F admits a continuous density f ≡ F′ and has full support. We allow for the possibility that the support of F is the nonnegative real line [0,∞) and if that is so, with a slight abuse of notation, write ω = ∞. In any case, it is assumed that E [Xi] < ∞.

Bidder i knows the realization xi of Xi and only that other bidders' values are independently distributed according to F. Bidders are risk neutral; they seek to maximize their expected profits. All components of the model other than the realized values are assumed to be commonly known to all bidders. In particular, the distribution F is common knowledge, as is the number of bidders.

Finally, it is also assumed that bidders are not subject to any liquidity or budget constraints. Each bidder i has sufficient resources so if necessary, he or she can pay the seller up to his or her value xi. Thus, each bidder is both willing and able to pay up to his or her value.

We emphasize that the distribution of values is the same for all bidders, and we will refer to this situation as one involving symmetric bidders.

In this framework, we examine two major auction formats:

I. A first-price sealed-bid auction, where the highest bidder gets the object and pays the amount he bid

II. A second-price sealed-bid auction, where the highest bidder gets the object and pays the second highest bid

Each of these auction formats determines a game among the bidders. A strategy for a bidder is a function i:0,ω→R+, which determines his or her bid for any value. We will typically be interested in comparing the outcomes of a symmetric equilibrium—an equilibrium in which all bidders follow the same strategy—of one auction with a symmetric equilibrium of the other. Given that bidders are symmetric, it is natural to focus attention on symmetric equilibria. We ask the following questions:

What are symmetric equilibrium strategies in a first-price auction (I) and a second-price auction (II)?

From the point of view of the seller, which of the two auction formats yields a higher expected selling price in equilibrium?

2.2 SECOND-PRICE AUCTIONS

Although the first-price auction format is more familiar and even natural, we begin our analysis by considering second-price auctions. The strategic problem confronting bidders in second-price auctions is much simpler than that in first-price auctions, so they constitute a natural starting point. Also recall that in the private values framework, second-price auctions are equivalent to open ascending price (or English) auctions.

In a second-price auction, each bidder submits a sealed bid of bi, and given these bids, the payoffs are:

i=xi−maxj≠ibjifbi>maxj≠ibj0ifbi<maxj≠ibj

We also assume that if there is a tie, so bi = maxj≠i bj, the object goes to each winning bidder with equal probability. Bidding behavior in a second-price auction is straightforward.

Proposition 2.1. In a second-price sealed-bid auction, it is a weakly dominant strategy to bid according to βII (x) = x.

Proof. Consider bidder 1, say, and suppose that p1 = maxj≠1 bj is the highest competing bid. By bidding x1, bidder 1 will win if x1 > p1 and not if x1 < p1 (if x1 = p1, bidder 1 is indifferent between winning and losing). Suppose, however, that he bids an amount z1 < x1. If x1 > z1 ≥ p1, then he still wins, and his profit is still x1 − p1. If p1 > x1 > z1, he still loses. However, if x1 > p1 > z1, then he loses, whereas if he had bid x1, he would have made a positive profit. Thus, bidding less than x1 can never increase his profit but in some circumstances may actually decrease it. A similar argument shows that it is not profitable to bid more than x1.

It should be noted that the argument in Proposition 2.1 relied neither on the assumption that bidders' values were independently distributed nor the assumption that they were identically so. Only the assumption of private values is important, and Proposition 2.1 holds as long as this is the case.

With Proposition 2.1 in hand, let us ask how much each bidder expects to pay in equilibrium. Fix a bidder—say, 1—and let the random variable Y1 ≡ Y1(N−1) denote the highest value among the N − 1 remaining bidders. In other words, Y1 is the highest-order statistic of X2,X3,…,XN (see Appendix C). Let G denote the distribution function of Y1. Clearly, for all y, G(y) = F(y)N−1. In a second-price auction, the expected payment by a bidder with value x can be written as

IIx=ProbWin×E2nd highest bid|xis the highest bid=ProbWin×E2nd highest value|xis the highest value=Gx×EY1|Y1<x

2.3 FIRST-PRICE AUCTIONS

In a first-price auction, each bidder submits a sealed bid of bi, and given these bids, the payoffs are

i=xi−biifbi>maxj≠ibj0ifbi<maxj≠ibj

As before, if there is more than one bidder with the highest bid, the object goes to each such bidder with equal probability.

In a first-price auction, equilibrium behavior is more complicated than in a second-price auction. Clearly, no bidder would bid an amount equal to his or her value, since this would only guarantee a payoff of 0. Fixing the bidding behavior of others, at any bid that will neither win for sure nor lose for sure, the bidder faces a simple trade-off. An increase in the bid will increase the probability of winning while, at the same time reducing the gains from winning. To get some idea about how these effects balance off, we begin with a heuristic derivation of symmetric equilibrium strategies.

Suppose that bidders j ≠ 1 follow the symmetric, increasing, and differentiate equilibrium strategy βI ≡ β. Suppose bidder 1 receives a signal, X1 = x, and bids b. We wish to determine the optimal b.

First, notice that it can never be optimal to choose a bid b > β(ω), since in that case, bidder 1 would win for sure and could do better by reducing his bid slightly, so he still wins for sure but pays less. So we need only consider bids b ≤ β(ω). Second, a bidder with value 0 would never submit a positive bid, since he would make a...