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Characterization of equilibrium in pay-as-bid auctions for multiple units.

Article Excerpt
Abstract Equilibrium bidding strategies under most multi-unit auction rules cannot be obtained as closed form expressions. Research in multi-unit auctions has, therefore, depended on implicit characterization of equilibrium strategies using the first-order conditions of the bidders' expected payoff maximization problem. In this paper we consider the pay-as-bid auction with diminishing marginal values for two units and show that any symmetric equilibrium in continuous strategies has the necessary properties to allow such a characterization. Moreover, any increasing solution to the system of differential equations that is used to characterize the equilibrium strategies describes an equilibrium strategy.

Keywords Multi-unit auctions * Pay-as-bid rule

JEL Classification Numbers D44

1 Introduction

A large number of auctions involve multiple units of the same item. Some of them like the Treasury bill auctions and the procurement auctions together amount to trillions of dollars annually making studies into the working of such auctions a matters of practical interest. At the same time, it has been shown that results developed for single-unit auctions do not generally extend to the multi-unit set-up (Engelbrecht-Wiggans and Kahn, 1998a,b; Ausubel and Cramton, 1998). This has resulted in a recent interest in developing a separate theory for multi-unit auctions.

The exploration of multi-unit auctions has been impeded by the mathematical complexity involved in analyzing even the basic models. Most of the standard auctions do not have a closed form expression for the equilibrium bidding strategies making the analysis of the bidding behavior difficult. Many authors have, therefore, bypassed explicit calculations of the equilibrium bidding strategies, and resorted to less explicit characterizations of the equilibrium bidding behavior. For instance, Noussair (1995), Katzman (1995, 1999), Engelbrecht-Wiggans and Kahn (1998a,b) have considered auctions where two indivisible units of an object are on sale, and each bidder has diminishing marginal values for the successive units. Equilibrium bidding strategies in such auctions do not have closed form expressions under most auction rules. They have therefore analyzed the auctions with the equilibrium strategies implicitly characterized by (a system of) differential equations based on the first-order conditions of bidders' expected payoff maximization problem. Engelbrecht-Wiggans and Kahn (1998a) even reduced the equations for a pay-as-bid auction1 to a characterization on just the boundary of the bidders' (two-dimensional) set of (marginal) values to make it numerically solvable. Under appropriate conditions such a reduced characterization is sufficient for extending the solution of the system to the interior of the set of values, thereby making the relatively simpler characterization an important tool for analyzing the auction. Chakraborty (2004) showed that this approach can be useful even for characterizing auctions where bidders have increasing marginal values.

Such an approach to analyzing auctions through the reduced characterization, however, leaves a crucial gap. First, the assumptions under which the reduced characterization is valid, restricts the set of equilibria that can be considered making it less likely that such an equilibrium will in fact exist. Thus, the robustness of an analysis based on such a characterization will increase significantly if the characterization is shown to be valid in the absence of some of these assumptions. Second, a numerically computed solution (like Engelbrecht-Wiggans and Kahn 1998a) and analysis may not mean much if the first-order condition based characterization yields multiple solutions some of which do not describe the equilibrium. Imagine the worst scenario where the equilibrium of interest may not even exist. If a solution to the equations exists in that situation it describes anything but an equilibrium. An analysis based on a numerical description or an implicit characterization of this solution will be misleading. In short, the literature begs for results that examine the extent of the necessity and sufficiency of such characterizations as descriptions of some broad range of equilibria. The objective of this paper is to do that in the context of the pay-as-bid auction rule.

The model of multi-unit auction studied in this paper is that of Engelbrecht-Wiggans and Kahn (1998a) (see also Katzman 1995) where two identical and indivisible units of an object are sold to a finite number of symmetric bidders, each with diminishing marginal values for the successive units. This approach to multi-unit auction differs from that taken by Ausubel and Cramton (1998) and Back and Zender (1993), among others. They consider auctions for a perfectly divisible object where bidders submit continuous demand/bid schedules for different shares of the object. While their work provide valuable insights into the bidding behavior and revenue ranking of alternative rules for multi-unit auctions, their approach avoids the multi-dimensionality problem of multi-unit auctions by considering Characterization of equilibrium in pay-as-bid auctions for multiple units 199 only single-dimensional bidder types. The resulting simplification of the strategic environment makes their approach complementary rather than a substitute.

This paper is organized as follows: Section 2 provides the details of the model of multi-unit auction that we consider, and a representative bidder's problem. In Section 3 we prove the properties of equilibrium strategies that are vital for a reduced characterization of the equilibrium strategies of a pay-as-bid auction on the boundary of the set of bidder values. In Section 4 we show that any increasing solution of the system of differential equations is also an equilibrium of the multi-unit auction. Finally, we conclude in Section 5. The proofs of all results are gathered in Section 6.

2 The model

Two identical units of an object are to be auctioned to n (symmetric) bidders through a sealed bid auction without a reserve. We follow Engelbrecht-Wiggans and Kahn (1998a) and assume that bidder i has privately known diminishing (marginal) values [V.sub.1i] and [V.sub.2i] for the successive units. From the perspective of the other bidders, [V.sub.1i] and [V.sub.2i] are distributed according to a continuous distribution F([v.sub.1], [v.sub.2]), where [v.sub.1] and [v.sub.2] are realizations of the random...

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