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Multiunit Pay-Your-Bid Auction with One-Dimensional ∗ Multiunit Demands

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Multiunit Pay-Your-Bid Auction with One-Dimensional ∗ Multiunit Demands INTERNATIONAL ECONOMIC REVIEW Vol. 44, No. 3, August 2003 MULTIUNIT PAY-YOUR-BID AUCTION WITH ONE-DIMENSIONAL ∗ MULTIUNIT DEMANDS BY BERNARD LEBRUN AND MARIE-CHRISTINE TREMBLAY1 Department of Economics, York University; Finance Canada, Policy Analysis, Ottawa An arbitrary number of units of a good is sold to two bidders through a discriminatory auction. The bidders are homogeneous ex ante and their demand functions are two-step functions that depend on a single parameter. We character- ize the symmetric Bayesian equilibrium and prove its existence and uniqueness. We compare this equilibrium with the equilibrium of the multiunit Vickrey auc- tion and with the equilibria of the single-unit first price and second price auctions. We examine the consequences of bundling all units into one package. We study the impacts that variations of the “relative” supply have on the equilibrium, on the bidders’ average payoffs per unit, and on the efficiency of the equilibrium allocation. 1. INTRODUCTION In the commonly used discriminatory or “pay-your-bid” auction, the bidders submit demand curves for multiple units of a good and the auctioneer acts as a perfectly discriminating monopolist. It has been used to sell, for example, bonds, bills, foreign exchange, import quota licenses, airport landing slots, mineral rights, timber rights (see Tenorio, 1999; Bikhchandani and Huang, 1993; Ausubel and Cramton, 1998), SO2 emission allowances (by the EPA, see Ellerman et al., 2000), and gold (by the IMF, see Feldman and Reinhart, 1995a, 1995b). When the auctioneer sells only one unit, the discriminatory auction is also called the first price auction. Researchers have studied extensively the first price auction and the discriminatory auction with multiple units where every bidder demands only one unity. In this article, we consider the two-bidder case with multiple units where both bidders have multiple-unit demands. In our model, the auctioneer puts n units of a good up for sale and sets a reserve price. A bidder values equally the first m units at vh each and values equally the remaining units at vl = g (vh) ≤ vh each. The function g is strictly increasing and common to both bidders. We refer to a bidder’s m first units as his high valuation 2 units, to his other units as his low valuation units, and to his valuation vh for ∗ Manuscript received November 1998; revised June 2002. 1 Constructive remarks by one referee and the associate editor are gratefully acknowledged. Bernard Lebrun benefited from the financial support of the Social Sciences and Humanities Research Council of Canada. Please address correspondence to: Bernard Lebrun, Department of Economics, York University, 4700 Keele Street, Toronto, ON, Canada M3J 1P3. Phone: 416 736 2100 (ext 33663). Fax: 416 736 5987. E-mail: [email protected]. 2 All n units are identical units of the same homogeneous good. However, because his demand curve is nondecreasing, a bidder values his first units more than his last units. 1135 1136 LEBRUN AND TREMBLAY each of his first m units as his type. Because the bidders’ types are distributed independently according to the same probability distribution F, the bidders are homogeneous ex ante. A bidder knows only his own type. We characterize the symmetric equilibrium where the bidders use the same “pure” strategy satisfying some standard regularity assumptions. We prove ex- istence and uniqueness of this equilibrium. At the equilibrium, bidders submit identical bids for units of identical valuations. When the bidders’ demand curves are flat at least up to the supplied quantity (m ≥ n), they thus submit flat curves and the equilibrium is as if the n units were bundled together. If a bidder’s bids win with strictly positive probability and if his type is differ- ent from the highest possible type, his “high bid”—the common bid on his high valuation units—is strictly larger than his “low bid”—the common bid on his low valuation units. Because the equilibrium of the discriminatory auction “separates” the high valuation units from the low valuation units, it allocates the units more ef- ficiently than the equilibrium of the first price auction where all units are bundled together. However, even without bundling, the allocation at the equilibrium of the dis- criminatory auction can be inefficient. Since bidders shade their bids less on low valuation units than on high valuation units, even if a bidder values his low valua- tion units less than his opponent values his high valuation units, his low bid may be higher than his opponent’s high bid. Indeed, this does occur with strictly positive probability and the equilibrium allocation is inefficient. In the two-unit case, there is a link between the equilibria of the multiple unit discriminatory auction with ex ante homogeneous bidders and the equilibria of the first price auction with ex ante heterogeneous bidders. The equilibrium of the discriminatory auction is as if every bidder was split into two agents, each one bidding only for one unit: a first agent, bidding for the bidder’s first unit; and a second agent, bidding for the bidder’s second unit. Since a bidder wins his first unit if and only if the other bidder does not win his second unit, a bidder’s first agent competes with the other bidder’s second agent in a first price auction. Since the valuations on the first and second units are distributed differently, the agents competing in a first price auction are heteroge- neous ex ante. In the equilibrium of the discriminatory auction, each agent bids according to the equilibrium of the first price auction. Even if a bidder’s agents bid independently, his first agent submits a bid higher than his second agent. The link in the two-unit case between the discriminatory auction with ex ante homogeneous bidders and the first price auction with ex ante heterogeneous bid- ders extends immediately to the case of an even number of units where exactly half are high valuation units. In the general case, there is no such explicit link between equilibria of the discriminatory auction and the first price auction. Nev- ertheless, we successfully apply the methods of proof from the study of the first price auction with ex ante heterogeneous bidders to the study of the discriminatory auction. A general link exists between the equilibria in weakly dominant strategies of the multiunit Vickrey auction, or Vickrey–Groves–Clark mechanism, and the (one unit) second price auction. From this link, we prove that bundling all units into one MULTIUNIT PAY-YOUR-BID AUCTION 1137 package decreases efficiency and increases the seller’s revenue from the Vickrey auction. From the link between the discriminatory auction with two units and the first price auction, we show that no such trade-off between efficiency and revenue exists in the discriminatory auction. With ex ante heterogeneous bidders, there exists no general ranking between the auctioneer’s revenues from the discriminatory and Vickrey auctions. In fact, in the one-unit case, these auctions reduce to the first price and second price auctions and, as it is well known, no general ranking exists between these two auctions when the bidders are heterogeneous ex ante. However, from the revenue equivalence theorem (see Myerson, 1981; Riley and Samuelson, 1981), the first and second price auctions are equivalent when the bidders are homogeneous ex ante. Because of the links mentioned above between the multiunit auctions with ex ante homogeneous bidders and the one-unit auctions with ex ante heterogeneous bidders, this equivalence does not extend to the multiunit case. Moreover, the nonexistence of a general ranking between the revenues from the two one-unit auctions with ex ante heterogeneous bidders implies the nonexistence of such a ranking between the revenues from the multiunit discriminatory and Vickrey auctions, even in the symmetric case of ex ante homogeneous bidders. Since the parameter m determines the position of the demand curves along the quantity axis, the ratio n/m determines the position of the supply relative to the demands.3 The equilibrium bid functions depend only on this “relative supply” n/m or, equivalently, its inverse m/n—the “relative demand” or the proportion of high valuation units in the total supply. An increase of the relative supply always increases the bidders’ per unit averages of their interim and ex ante expected payoffs. When the relative supply is equal to 2, the total supply is, in a sense, the least homogeneous to the bidders since it counts the same number of high valuation units as of low valuation units. The difference in bid shading between high and low valuation units and the probability of an inefficient allocation increase when the relative supply becomes closer to 2. If n/m < 2 or, equivalently, if a majority of units in the total supply are high valuation units to the bidders, an increase of the relative supply decreases the bid function on the high valuation units. It also decreases the “average” of the low- and high-bid probability distributions a bid on a high valuation unit competes against. In general, there are types where the bid function on the low valuation units increases. When n/m ≥ 2 or, equivalently, when a majority of units are low valuation units to the bidders, an increase of the relative supply results in a decrease of the bid function on the low valuation units and of the “average” bid probability distribution a bid on a low valuation unit competes against. The bid function on the high valuation units increases for some types. We study the two-unit case in Section 2. We extend our results to the n unit case in Section 3.
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