Properties of scoring auctions.

Finally, we carry out one last simplification of the problem. Let [s.sub.i] = [x.sub.i][t.sup.W.sub.i] + (1 - [x.sub.i])[t.sup.L.sub.i] in (2). Given suppliers' risk neutrality and the linearity of the scoring rule, there is no loss in defining the outcome of a scoring auction as the pair ([x.sub.i], [s.sub.i]), rather than ([x.sub.i], [t.sup.W.sub.i], [t.sup.L.sub.i]). Suppliers' expected payoff is thus given by

[x.sub.i]k ([[theta].sub.i) - [s.sub.i], (5)

Notation. For the remainder, we adopt the following notation and conventions. The outcome function of a scoring auction is a vector of probabilities of winning ([x.sub.1], ..., [x.sub.N]) and scores to fulfill by each supplier, ([s.sub.i], ..., [S.sub.N]). (If the outcome in a given scoring auction is stochastic, these are distributions over vectors of probabilities of winning and scores.) The arguments in these functions are the bids submitted by all suppliers, {([pi.sub.i], [Q.sub.i]).sup.N.sub.i=1]. (9) Later in the article, we will switch to a direct revelation mechanism approach where the outcome will be a function of suppliers' pseudotypes, ([k.sub.1], ..., [k.sub.N]) [member of] [R.sub.N]. To avoid introducing too much new notation, we shall make these the arguments of the x and s functions. We shall also write [x.sub.i]([k.sub.i]) to denote the expectation of [x.sub.i] over the types of the other suppliers, [E.sub.k-i] [x.sub.i] ([k.sub.i] - [K.sub.-i]). The arguments will be spelled out whenever confusion is possible.

3. A sufficient statistics result

Suppliers' pseudotypes are sufficient statistics in this environment if knowing the distribution of suppliers' pseudotypes is all one needs in order to describe the set of equilibrium outcomes of the auction and evaluate the buyer's expected payoff.

In this section, we prove that pseudotypes are sufficient statistics. Proving this result requires two steps. First, we show that all equilibria of the scoring auction are outcome equivalent to an equilibrium where suppliers are forced to submit bids only as a function of their pseudotypes. We define two equilibria as outcome equivalent if they both lead to the same distribution of outcomes ([x.sub.1], ..., [x.sub.N]) and ([s.sub.1], ..., [S.sub.N]) in the aggregate. Because outcome equivalence is not enough to guarantee that the buyer is indifferent among these equilibria, we next prove the stronger result that the equilibria in the scoring auction and in the constrained scoring auction have the same distribution of outcomes, conditional on types.

Lemma 1. All equilibria of a quasilinear scoring auction are outcome equivalent to an equilibrium where bidders with the same pseudotypes adopt the same strategies.

Proof. Consider any equilibrium ([[epsilon].sub.i], ..., [[epsilon].sub.i]), where [[epsilon].sub.i] is a mapping from [[theta].sub.i] to a distribution over (p, Q) [member of] [R.sup.M+1]. Then, for all [[theta].sub.i], for all [[theta].sub.i] and all ([p.sup.*.sub.i], [Q.sup.*.sub.i]) in the support of supplier i's equilibrium strategy,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where the expression for supplier i's expected profit derives from (5). In (6), suppliers' private information enters their objective function only through their pseudotypes. Thus, supplier i is indifferent among the strategies played by all the realizations of supplier i's type with the same pseudotype.

We can construct a new equilibrium, ([[??].sub.1], ..., [[??].sub.N]), such that:

1. [[??].sub.i]([[theta].sub.i) = k([[??].sub.i]) whenever k([[theta].sub.i]) = k([[??].sub.i]).

2. Define [[THETA].sub.i](k) = {[theta].sub.i] [member of] | k([[theta].sub.i]) = k}. For each k in the support of bidder i's pseudotypes, the distribution over (p, Q) generated under [[??].sub.i] for a given [[theta].sub.i] [member of] [member of] [[THETA].sub.i](k) replicates the aggregate distribution over (p, Q) over all [[theta].sub.i] [member of] [[THETA].sub.i] (k) under [[epsilon].sub.i].

By construction, the distribution of bidder i's opponents' strategies is the same as before from bidder i's perspective. Moreover, [[??].sub.i] is a best response for bidder i. Hence it is an equilibrium, and bidders' strategies are only a function of their pseudotypes. By construction, ([[??].sub.1], ... [[??].sub.N]) and ([[epsilon].sub.1], ..., [[epsilon].sub.N]) lead to the same aggregate distribution of (p, Q) and therefore scores and probabilities of winning. Q.E.D.

An aspect of Lemma 1 worth stressing is the role played by the assumption that types are independent across bidders. From the expression of suppliers' expected profit, [x.sub.i]k([[theta].sub.i]) - [s.sub.i], we already know that their payoffs are only a function of their pseudotypes. Independence ensures that their beliefs are also independent of their types beyond their pseudotypes (actually, independence is stronger: it makes suppliers' beliefs independent of their types and pseudotypes). Without independence, bidders' private information would enter their expected payoff in (6), both through their pseudotypes and through their expectations over their opponents' types.

Lemma 1 implies that the set of possible outcomes ([x.sub.1], ..., [x.sub.N]) and ([s.sub.1], ..., [s.sub.N]) Can be generated by equilibria where suppliers bid exclusively on the basis of their pseudotypes. However, it does not imply that nothing is lost by restricting attention to these equilibria. Outcome equivalence does not imply utility equivalence for the buyer. To see this, consider the following example.

Consider two equally likely types, [[theta].sub.i] and [[??].sub.1] (this assumption is inessential for the argument) such that k([[theta].sub.i]) = k([[??].sub.i]) and suppose that, in equilibrium, they get a different outcome: ([x.sub.i], [s.sub.i]) and ([[??].sub.i], [[??].sub.i]). By definition, these two types generate expected utility f([[theta].sub.i])[s.sub.i] + [f.sub.i]([[??].sub.i])[[??].sub.i] for the buyer, according to the scoring rule. However, this differs from true expected utility if [phi](.) [not equal to] v(., t). To know how much expected utility the suppliers generate for the buyer, we need to know how they will satisfy their obligations. Let Q and Q be the choice of [[theta].sub.i] and [[??].sub.i], respectively (these are independent of [s.sub.i] and [[??].sub.i]). The total monetary transfer from the buyer to the suppliers is then given by [x.sub.i] [phi](Q) - [s.sub.i] [phi]([??]) - [[??].sub.i], and the buyer's true expected utility is given by

[f.sub.i]([[theta].sub.i])[[x.sub.i](v(Q, t) - [theta](Q)) + [s.sub.i] + [[??].sub.i](v([??], t) - [phi](Q)) + [[??].sub.i].

This equilibrium is outcome equivalent to an equilibrium where type [[theta].sub.i] adopts [[??].sub.i]'s strategy and vice versa. In that equilibrium, the buyer's true expected utility is given by

[f.sub.i]([[theta].sub.i])[[??].sub.i](v(Q, t) - [phi](Q)) + [[??].sub.i] + [x.sub.i](v([??], t) - [phi]([??])) + [s.sub.i]].

Clearly, the buyer is not indifferent between these two equilibria unless [x.sub.i] = [[??].sub.i] or v(Q, t) = [phi](Q). The next result ensures that suppliers with the same pseudotypes receive the same equilibrium outcome function ([x.sub.i], [s.sub.i]). This rules out the situation described in the previous example.

Consider any equilibrium ([[epsilon].sub.i], ..., [[epsilon].sub.N]). Define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Similarly, define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], ([p.sup.*.sub.-i], [Q.sup.*.sub.-i]))] . Let [[s.bar].sub.i](k), [[bar.s].sub.i](k) be the resulting scores to fulfill. In words, [[bar.x].sub.i](k) is the lowest expected probability of winning the contract among all the bids in the support of bidder i's strategy when he has pseudotype k. Similarly, [[bar.x].sub.i](k) is his highest expected probability of winning.

Lemma 2. In any equilibrium, [[x.bar].sub.i](k) = [[bar.x].sub.i](k) and (k) = [[s.bar].sub.i] (k) for all k except possibly on a set of measure zero.

Proof. Define [U.sub.i](k) as supplier i's equilibrium expected payoff when he has pseudotype k. Incentive compatibility (IC) implies that (10,11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence [[x.bar].sub.i] (k) is monotonically increasing in k. The same argument applies to [[bar.x].sub.i] (k). Hence [[x.bar].sub.i](k) and [[bar.x].sub.i](k) are almost everywhere continuous. A similar argument based on the IC constraint establishes that [[x.bar].sub.i](k) [greater than or equal to] ([??]) for all [??] < k. Together with the a.e. continuity of these functions, this implies that [[x.bar].sub.i](k) = [[bar.x].sub.i] (k) (and [[s.bar].sub.i] (k) = [[s.bar].sub.i] (k)) almost everywhere. Q.E.D.

Define two equilibria as typewise outcome equivalent if they generate the same distribution of outcomes ([x.sub.1], ..., [x.sub.N]) and ([s.sub.1], ..., [s.sub.N]), conditional on types in [[THETA].sub.1] x ... x [[THETA].sub.N] . We are now able to prove the main result of this section.

Theorem 1. Every equilibrium in the scoring auction is typewise outcome equivalent to an equilibrium in the scoring auction where suppliers are constrained to bid only on the basis of their pseudotypes, and vice versa.

Proof. All equilibria in the constrained auction are also equilibria in the scoring auction because bidders' preferences and beliefs are entirely determined by their pseudotypes. Lemma 2 implies that all types with the same pseudotype get the same x and s a.e. in all equilibria. Q.E.D.