درآمد و کارایی مزایده هایی با قیمت یکنواخت چندواحدی

This paper analyzes efficiency in a uniform-price multi-unit auction with a positive reservation price. I demonstrate that the reservation price is an important policy tool that may increase efficiency (or welfare) in multi-unit uniform-price auctions. I show that the higher the reservation price is, the higher is the sellerʼs revenue and the higher is the efficiency of a final allocation of units that could be attained in a multi-unit uniform-price auction. The reservation price increases the bidderʼs equilibrium strategy in a specific way that is inherent to the uniform-price auction. Thus the reservation price effect on efficiency is in contrast to other auction formats; e.g., the reservation price decreases efficiency in the Vickrey auction and single-unit auctions with symmetric bidders. Therefore the main result can be added to the list of results from mechanism design and auction theory that fail to extend the single-unit/single-dimensional context to the multi-unit/multi-dimensional one.

An auction is an exchange mechanism with asymmetric information. It can be treated as a game in which the seller offers one or more units (of the same type) to the participants. The seller does not know the bidderʼs value of any particular unit, but he can set up an explicit set of institutional rules determining resource allocation and prices on the basis of bids from the auction participants. In Vickrey auctions (Vickrey, 1961) bidders reveal the true valuation of each unit and the final allocation is efficient. Krishna (2002) formulates conditions when the equilibrium in a multi-unit auction is efficient. However, the equilibrium strategies in a multi-unit uniform-price auction do not satisfy these conditions (see Morgan, 2001). The effect of reservation prices on a multi-unit auction is difficult to assess in general (see Zhan, 2008). I show that in a private value multi-unit uniform-price auction, a positive reservation price increases both efficiency and revenue. Therefore it can be added to the list of results from mechanism design and auction theory that fail to extend the single-unit/single-dimensional context to the multi-unit/multi-dimensional one, e.g., Armstrong (1996), Perry and Reny (1999), and Levin (2004). For a benchmark of an auction game I follow the model with symmetric risk-neutral bidders who have independent-private values and where the payment is a function of bids alone as suggested by McAfee and McMillan (1987). The only additional assumptions are that the seller offers more than one unit for sale and the bidders demand two units that I call “initial” and “subsequent.” This model has been analyzed in Engelbrecht-Wiggans and Kahn (1998). I focus on the difference between the bidderʼs true value and the submitted bid which is called “bid shading” or “demand reduction” in the literature. In a uniform-price auction with no reservation price, a different shading in strategies on initial and subsequent units is present and prevents the seller from reaching a Pareto-efficient distribution of units. I show that when the seller sets some specific (optimal) reservation price, the difference in shading on the initial and subsequent unit decreases, which can prevent some inefficient allocations of units and, moreover, the seller gains higher revenue. For illustration, imagine two bidders in an auction with 2 units for sale with zero reservation price. Each bidder has two values and submits two bids. I denote the values of the first bidder v1v1, v2v2, the values of the second bidder as View the MathML sourcev1′, View the MathML sourcev2′ and the bids of the first bidder as b1b1, b2b2 and of the second bidder as View the MathML sourceb1′, View the MathML sourceb2′. If the values are such that View the MathML sourcev1′>v2′>v1>v2, then it is efficient if the first bidder wins both units. But in many cases equilibrium strategic behavior forces the bidders to submit bids with the ordering View the MathML sourceb1′>b1>b2′>b2, and the seller does not allocate the 2 units efficiently. When the seller increases the reservation price above v1v1, the second bidder does not submit a bid above the reservation price, and the first bidder wins both units, which is an efficient outcome.2 At the same time revenue typically increases. This reasoning is valid for the multi-unit uniform-price and to some extent for other multi-unit auctions if View the MathML sourceb1>b2′ and View the MathML sourcev1<v2′. On the other hand, setting the reservation price too high introduces inefficiency when the supply is greater than the number of submitted bids (e.g., View the MathML sourcev1′>R>v2′>v1>v2). In summary, the seller faces a trade off between these two sources of inefficiency and the total effect is ambiguous. In this paper I show that the expected efficiency typically increases when the seller increases the reservation price above 0 in the uniform-price auction. In addition, the results of this paper also contribute to the literature on efficient multi-unit auction design. Krishna (2002, Proposition 13.3) argues that equilibrium strategies cannot be efficient if a shading difference across units is present.3 The seller who uses submitted bid ordering to allocate units cannot attain efficient allocations when bidders use different shading across initial and subsequent units. This paper, additionally, supports Krishnaʼs proposition that any means that decrease shading differences can improve efficiency of the final allocation. The reservation price is an example of such a mean in a multi-unit uniform-price auction when the number of bidders is small. The paper is organized as follows. At first I discuss the relationship of this paper to other studies. Then, in the next section, I develop a model of a uniform-price auction for n bidders, k units of supply, and a reservation price R. I also derive expressions for the expected efficiency measures. In the rest of the paper, I disentangle two sources of inefficiency, show the general conditions when the optimal reservation price that maximizes efficiency is positive, and demonstrate this main contribution of the paper with a simple example. Finally, I briefly discuss generalizations and conclude the paper.

This paper shows the significance of reservation prices in a multi-unit auction with independent-private-value bidders, not only for revenue-gaining, but also for efficiency or social welfare reasons. In a typical case, the outcome of the multi-unit auction is not efficient. The reason for this is that bidders, although symmetric, use different strategies on the initial and subsequent unit they demand. The difference in strategies distorts efficiency in many cases (see Krishna, 2002). A similar distortion is well-known in an English single-unit auction with asymmetric bidders or in the case when the seller favors one group of bidders over the rest of the bidders (see McAfee and McMillan, 1987). But in a multi-unit auction the difference in strategies occurs even though the bidders are symmetric, risk-neutral, independent private-valued, and payment is a function of the bids alone (cf. with assumptions A1–A4 in McAfee and McMillan, 1987). One of the principal goals of the literature survey by Zhan (2008) is a trade off between efficiency and revenue in auctions. In this paper I use a differential equation approach to derive a comparative statics result on the effect of the reservation price on efficiency and revenue in the uniform-price multi-unit auction. I disentangle two sources of efficiency loss because the reservation price has two effects: (1) it excludes bidders with values below the reservation price from the auction; and (2) it motivates the bidders to bid closer to their true values above the reservation price. The former effect decreases revenue and introduces efficiency loss if not enough bidder values, and hence bids, are above the reservation price. The latter effect improves both the efficiency and the revenue if enough values, and hence the bids, are submitted above the reservation price. For low reservation prices, the latter case occurs more often and, therefore, an increase of the reservation price improves the efficiency. This is in contrast to the single-unit first- and second-price auction models in McAfee and McMillan (1987) with symmetric bidders. Palfrey (1983) shows that the seller who sells units in bundles may increase revenue at the expense of efficiency. In contrast, I show that a positive reservation price not far from zero increases both revenue and efficiency in the uniform-price auction. The influence of the reservation price depends on the level of bid shading or demand reduction on each unit. The bid shading varies for each auction format. For the uniform-price auction, the effect of the reservation price on efficiency is opposite to that for the Vickrey auction, when the reservation price is not far from 0. A question for future research is what effect prevails for other auction formats (e.g., pay-your-bid) that have been only partially studied in the multi-unit auction literature (Swinkels, 1999, and Lebrun and Tremblay, 2003). It seems that proper reservation price setting is an important mechanism design tool to set up an optimal multi-unit auction when the number of bidders is not large. The efficient optimal reservation price depends on the number of bidders. The marginal increase in efficiency is not too great if there are many bidders in the auction, but it determines the kind of lower bound of the optimal reservation price which maximizes both revenue and efficiency. Therefore, the government as a seller who cares about both revenue and efficiency should never set the reservation price below this lower bound. Although I used an independent-private-value assumption, this model is applicable to other real uniform-price auctions because they are typically regarded as a mixture of independent private-value and common-value paradigms. In this case there is no general auction mechanism to achieve ex post efficiency. One application is to the auctions of licenses for the radio-frequency spectrum (PCS) by the Federal Communication Commission (certainly there are a lot of institutional details that make the analysis more complicated, see Krishna and Rosenthal, 1996). Another important application is the uniform-price auction of T-bill securities (T-bills) if the common-value assumption seems not to be appropriate. Hortacsu and Kastl (2012) could not reject the hypotheses of the private value component in 3-months T-bills. Moreover, T-bill auctions can be treated as a partially independent value auction if the secondary market is far enough from perfect liquidity, which is typical for emerging markets (e.g., the T-bill market in the Czech Republic and other CEEC). It seems, at least intuitively, that the reservation price increases efficiency even if we enrich the model by a more complicated demand curve that has uncertainty in every demanded unit or assuming interdependencies among the bidder values (or signals). This intuition supports the result of Engelbrecht-Wiggans and Kahn (1998), who formed the transformation of a class of 3 unit demand models into 2 unit demand models. Finally, although comparisons between the uniform-price and pay-your-bid auctions are difficult (cf. Katzman, 1999) it seems that the uniform-price auction requires more information gathering and strategic considerations from the seller to design the auction because the auction outcome is more sensitive to an optimal reservation price than the pay-your-bid auction.