Properties of scoring auctions.

This article studies scoring auctions, a procedure commonly used to buy differentiated products: suppliers submit offers on all dimensions of the good (price, level of nonmonetary attributes), and these are evaluated using a scoring rule. We provide a systematic analysis of equilibrium behavior in scoring auctions when suppliers' private information is multidimensional (characterization of equilibrium behavior and expected utility equivalence). In addition, we show that scoring auctions dominate several other commonly used procedures for buying differentiated products, including menu auctions, beauty contests, and price-only auctions with minimum quality thresholds.

1. Introduction

* In many procurement situations, the buyer cares about attributes other than price when evaluating the offers submitted by suppliers. Examples of nonmonetary attributes that buyers care about include lead time, time to completion, and quality. Buyers have adopted several practices for dealing with these situations. Some use detailed request-for- quotes that specify minimum standards that the offers need to satisfy, and then evaluate the submitted bids based on price only. Others select a small set of potential suppliers and negotiate on all dimensions of the contract with each of them.

A third option is to combine the competition induced by a request-for-quote with the flexibility in terms of contract specification offered by negotiation. Several procedures belong to this category. In a "menu auction," the buyer lets suppliers submit menus of price and nonmonetary attributes, and choose the combination that best suits his needs. In a "beauty contest," the buyer tells suppliers he cares about other attributes than price but requests a single offer from them. Again, he chooses the offer he prefers from the received offers. In a scoring auction, the buyer announces the way he will rank the different offers, that is, the scoring rule; suppliers submit an offer on all dimensions of the product, and the contract is awarded to the supplier who submitted the offer with the highest score according to the scoring rule.

In this article, we study the properties of scoring auctions in which price enters linearly into the scoring rule. Examples of such scoring auctions include "A + B bidding" for highway construction work in the United States, where the highway procurement authorities evaluate offers on the basis of their costs as well as time to completion, weighted by a road user cost, (1) and auctions for electricity reserve supply (Bushnell and Oren, 1994; Wilson, 2002). The European Union has recently adopted a new public procurement directive. The new law allows for two different award criteria: lowest cost and best economic value. The new provisions require that the procurement authority publishes ex ante the relative weighting of each criterion used when best economic value is the basis for the award. (2) In effect, the new law mandates the use of scoring auctions. This is significant, as public procurement in the European Union is estimated at about 16% of GDP. (3) The use of scoring auctions is also gaining favor in the private sector, with several procurement software developers incorporating scoring capability in their auction designs.

A distinguishing feature of our model is that suppliers' private information about their cost is multidimensional. This means that the low-cost supplier for the base option is not necessarily the low-cost supplier when it comes to increasing quality on some other dimension. It allows us to consider the likely situation where firms differ in their fixed and variable costs of production. Our motivation for allowing multidimensional private information is to build a model of scoring auctions that can generate equilibrium predictions that mimic what is observed in the data. When private information is one-dimensional, equilibrium offers can be parameterized by a single parameter and describe a curve in the price-attributes space (Che, 1993). Our model does not suffer from this severe limitation.

We derive two sets of results. First, we characterize equilibrium behavior in scoring auctions when private information is multidimensional and the scoring rule is linear in price. We prove that the multidimensionality of suppliers' private information can be reduced to a single dimension (their "pseudotype") that is sufficient to characterize equilibrium outcomes in these auctions (Theorem 1). This allows us to establish a correspondence between the set of scoring auctions and the set of standard single-object one-dimensional independent private value (IPV) auction environments (Corollary 1). The equilibrium in the scoring auction inherits the properties of the corresponding standard IPV auction (existence and uniqueness of equilibrium, efficiency, etc.). We also prove a new expected utility theorem for the buyer when private information is multidimensional and independently distributed, and the scoring rule is linear in price (Theorem 2).

Our second set of results compares scoring auctions to other common procedures used to buy differentiated products. We show that, from the buyer's perspective, scoring auctions strictly dominate price-only auctions with minimum quality standards. They weakly dominate a menu auction and a beauty contest when an open ascending format is used (the open ascending format is often used for online procurement). When a sealed-bid "second-price" format is used, they weakly dominate a menu auction and strictly dominate a beauty contest. Finally, the ranking between the first-price scoring auction and the first-price menu auction is ambiguous: we find that some buyers prefer the menu auction whereas others prefer the scoring auction. Moreover, we establish that first-price menu auctions are always inefficient. Note that our purpose in this article is not to determine how optimally to buy a differentiated product but, instead, to study the properties of a commonly used and simple procedure for doing so, the scoring auction. Thus, our second set of results provides a motivation for focusing on the scoring auction given its attractive properties.

[] Related literature. There are several papers studying scoring auctions. Most papers note, as we do, that, once the scoring rule is given, the maximum level of social welfare a supplier can produce (in our paper, the pseudotype) can be used to construct an equilibrium in these auctions. This involves a benign change of variables when private information is one- dimensional as in Che (1993) and Branco (1997), but the operation is not so anodyne when private information is multidimensional. Specifically, we show that such a reduction in dimensionality requires that (i) the scoring rule be linear in price, and (ii) that private information be independently distributed across suppliers, unless the auction format admits a dominant strategy equilibrium. The papers we are aware of that allow for multidimensional private information, Bushnell and Oren (1994; 1995), happen to satisfy these conditions (these papers derive the scoring rule that induces productive efficiency in an environment with multidimensional private information). There is also a series of papers on scoring auctions published in the computer science and operations research literature. The focus there is on implementability through practical online/iterative processes (see, e.g., Bichler and Kalagnanam, 2003; Parkes and Kalagnanam, 2005). (4)

Several recent papers study other auction environments with multidimensional private information. In some environments, bidder preferences, the structure of information, or the specific allocation mechanism suggest the locus of types likely to use the same bidding strategies at equilibrium. These pseudotypes are used to construct an equilibrium (see, e.g., Che and Gale, 1998; Fang and Parreiras, 2002; de Frutos and Pechlivanos, 2006). Our approach is identical, except for the fact that, in addition, we prove that no other relevant equilibrium exists. This allows us to derive a utility equivalence theorem and to leverage the analogy between our environment and the standard IPV environment. (5)

Che (1993) and Asker and Cantillon (2006) derive the optimal buying mechanism when quality matters. A scoring auction in which price enters linearly into the scoring rule implements the optimal scheme when private information is one-dimensional. Under some conditions on the crosspartial derivative of costs, the optimal scoring rule underweighs quality relative to the true preference of the buyer. When private information is multidimensional, Asker and Cantillon show that the buyer is still interested in distorting qualities away from their efficient levels. However, the optimal scheme can no longer be implemented by a scoring auction with a scoring rule that is linear in price. Nevertheless, they provide numerical examples suggesting that such scoring auctions perform almost as well as the optimal scheme.