Publication: RAND Journal of Economics
Publication Date: 22-MAR-07
Delivery: Immediate Online Access
Author: Fevrier, Philippe ; Linnemer, Laurent ; Visser, Michael

Article Excerpt
We report results from an experiment on two-unit sequential auctions with and without a buyer's option (which allows the winner of the first auction to buy the second unit). The four main auction institutions are studied. Observed bidding behavior is close to Nash equilibrium bidding in the auctions for the second unit, but not in the auctions for the first unit. Despite these deviations, the buyer's option is correctly used in most cases. The revenue ranking of the four auctions is the same as in single-unit experiments. Successive prices are declining when the buyer's option is available.

1. Introduction

* In sales of multiple units of a particular good, auction houses often choose to sell the items sequentially, i.e., the items are auctioned separately, one after the other. The advantage of a sequential auction is that it may fit the needs of both small and large buyers, whereas the alternative auction procedure that consists in selling all available units simultaneously, in one shot, may exclude buyers who set low values on the items (which reduces competition at the auction). The main disadvantage of a sequential method is that it can be time consuming, especially when the total number of units for sale is large. For this reason, auctioneers sometimes provide a so-called buyer's option, which gives the winner of the first auction the right to buy any number of units (1,2, ..., or all units available). For each unit, she must pay the winning price established at the first auction. If the winning bidder decides to purchase only part of the total quantity, the remaining items are reauctioned, in the same manner, through a second auction; and this scheme is repeated until all units are eventually sold.

The buyer's option thus clearly offers the best of both worlds: it allows the auctioneer to speed up sales, while keeping the auction mechanism sufficiently flexible to be of interest for different types of buyers. Not surprisingly, therefore, the buyer's option is used in many auctions throughout the world. Cassady (1967) describes how the buyer's option is practiced in fur auctions in Leningrad and London and in fish auctions in English port markets. At the auction market in Aalsmeer, The Netherlands, huge quantities of flowers are sold through sequential descending auctions with a buyer's option (see Van den Berg, Van Ours, and Pradhan, 2001). Well-known auction houses, such as Christie's and Sotheby's (see Ashenfelter, 1989; Ginsburgh, 1998) and Drouot (see Fevrier, Roos, and Visser, 2005), systematically use the buyer's option in their sequential ascending auctions of fine wines.

Despite the practical importance of the buyer's option, little attention has been paid to the subject in the literature. The only theoretical article we are aware of is Black and De Meza (1992). They consider the independent private value (IPV) paradigm and derive equilibrium bidding strategies in two-unit sequential second-price auctions with and without the buyer's option. All buyers in their model have either decreasing demand for the two units (the additional value of the second unit is less than the value of the first unit) or flat demand (both units are valued the same). Empirical studies are also rare. Ashenfelter (1989) and Ginsburgh (1998) report that the option is exercised by many buyers in ascending wine auctions at Christie's and Sotheby's. Van den Berg, Van Ours, and Pradhan (2001) use data on sequential descending auctions of roses to study the declining price phenomenon. Finally, Fevrier, Roos, and Visser (2005), using data on ascending auctions of wine held at Drouot, structurally estimate their bidding model, and find that the seller's revenue in a system where items are auctioned sequentially is the same as in a system based on the buyer's option.

This article studies two-unit sequential auctions and looks in particular at the role of the buyer's option. We adopt the IPV paradigm and assume that the two units are sold to two risk-neutral buyers. Buyers desire both units, and their demand for the items is either decreasing, flat, or increasing (implying that the value of the second unit exceeds the value of the first unit). The four main auction institutions are considered: first-price, descending (Dutch), second-price (Vickrey), and ascending (English) auctions. Although there are few field examples of first-price and second-price sequential auctions with or without a buyer's option, (1) it is nonetheless of interest to study sealed-bid auctions. As in standard one-unit auction theory, we show that first-price (respectively, second-price) and Dutch (respectively, English) sequential auctions with or without a buyer's option are theoretically isomorphic. Furthermore, the four auction formats generally generate the same expected revenue. By analogy with experimental studies on single-unit auctions (see Kagel, 1995, for a survey), our experimental design thus allows us to test whether bidding behavior is identical and whether there is revenue equivalence.

Other theoretical predictions are confronted with the experimental data as well. We test whether observed bidding behavior corresponds to risk-neutral Nash equilibrium bidding and whether the buyer's option has the predicted effect on first-auction bidding behavior. We also analyze to what extent the experimental subjects exercise their option (do first-auction winners directly buy the second unit, or instead wait and attempt to obtain the additional unit in the second auction?) and test if observed frequencies of using the option correspond to predicted frequencies. Predictions on the degree of efficiency of auction outcomes are also tested, and we compare observed price patterns with their predicted counterparts.

Our main empirical findings are the following. Observed bidding behavior matches the predictions of the theory well in the auctions for the second unit. In the auctions for the first unit, there are, however, important deviations between the Nash equilibrium strategies and the observed bidding strategies. Although the experimental subjects tend to adjust their bids in the direction that theory predicts, the adjustments are generally too modest. Despite the deviations in the auctions for the first unit, the buyer's option is correctly used in most cases. Our results also indicate that the revenue ranking of the four canonical auction institutions is the same as that found in single-unit experiments. The ordering of the auctions' mechanisms in terms of expected revenue is thus robust to the sequential two-unit extension considered here. Finally, we find that successive prices in sequential auctions are declining when the buyer's option is available.

Experimental work on multiunit sequential auctions is rare. (2) Burns (1985) considers sequential English auctions. The experiment is designed to mimic the Australian wool market, and the article's main objective is to study the effect of market size on auction prices. The observed behavior is not confronted with equilibrium predictions. Keser and Olson (1996) consider sequential first-price auctions and suppose that buyers have single-unit demand functions. Their main objective is to compare observed price sequences with the predicted patterns derived in Weber (1983), under different design parameters. As in Burns (1985), the article focuses on one particular auction mechanism, and no attempt is made to examine outcomes under alternative institutions. Robert and Montmarquette (1999) consider several auction institutions and also provide theoretical foundations for each of them. In their models, the number of items desired by each buyer is a random variable and demand functions are decreasing. They consider sequential Dutch, English, and mixed auctions, and compare observed behavior with predicted behavior. None of these experimental articles on sequential auctions analyzes the buyer's option. The only experimental article that looks at the buyer's option is by Katok and Roth (2004). They experimentally study (among other things) two-unit sequential Dutch auctions with a buyer's option. Their model has two small bidders (single-unit demand) and one larger bidder (two-unit increasing demand). This asymmetry between bidders is essential in Katok and Roth's work, as they focus their analysis on the "free-rider problem."

The article proceeds as follows. In the next section, the theoretical background is presented. In deriving the risk-neutral Nash equilibrium bidding functions for the different auction institutions, we partly draw on Black and De Meza (1992), Donald, Paarsch, and Robert (1997), and Fevrier (2000). Most results in Section 2 are new. In Section 3 we describe the experimental design and in Section 4 the experimental results. Section 5 concludes.

2. Theoretical background

* Suppose that two units of a good are auctioned to two potential buyers. Each buyer is assumed to be risk neutral and desires to purchase both units. Adopting the IPV paradigm, let [v.sub.i] denote the value that buyer i places on the first unit. The value [v.sub.i] and the value of i's opponent are independently drawn from a uniform distribution on the interval [0, [bar.v]]. It is assumed that the value that i places on the second unit is k[v.sub.i]. The parameter k can take three values: k [member of] {1/2, 1, 2}. The value of k is common knowledge. Note that k = 1/2 implies that the second unit is valued less than the first unit (decreasing demand), k = 1 that both units are valued the same (flat demand) and k = 2 that the second unit is valued more than the first (increasing demand).

The two units are sold sequentially. The first unit of the good is sold in the first auction. The manner in which it is auctioned depends on the auction institution. Let a indicate the auction institution, a [member of] {D, E, F, S}, where D stands for Dutch auction, E for English auction, F for first-price auction and S for second-price auction, and let [p.sub.1] denote the price the winner of the first auction has to pay for the first unit. When a [member of] {D, E}, the unit is auctioned using a clock, When a = D, the clock starts very high and descends until one of the players stops the clock. This player wins the unit and [p.sub.1] equals the price at which the clock was stopped. When a = E, the clock starts at 0, and increases until one of the players stops the clock. Here the winner of the auction is the player who did not stop the clock. The price [p.sub.1] she has to pay for the first unit is again the amount at which the clock stopped. When a [member of] {F, S}, the unit is sold via sealed-bid auctions. Both players submit their sealed bid to the auctioneer, who awards the unit to the highest bidder. When a = F, the winner pays his own bid, i.e., [p.sub.1] equals the highest submitted bid. When a = S, the winner pays the bid of his opponent, i.e., here [p.sub.1] equals the second-highest submitted bid. For all institutions a, the price [p.sub.1] is revealed to both players once the first auction has ended.

The way in which the second unit is sold depends on whether the buyer's option is available or not. Let o be the indicator for the availability of the buyer's option, o = N if it cannot be used, and o = Y otherwise. For any auction institution a, if o = N, the second unit is auctioned under the prevailing rules of institution a. Let [p.sub.2] be the price paid for the second unit. If instead o = Y, the winner of the first auction has the option to buy one or two units, at the price of pl per unit. When he decides to purchase only one unit, a second auction is held under the conditions of institution a. When he exercises the buyer's option, no second auction is held. Note that, in this case, we automatically have [p.sub.2] = [p.sub.1].

The theoretical model presented here is essentially based on the framework of Black and De Meza (1992). These authors, however, consider only the second-price auction (a = S), and they do not analyze the case of increasing marginal valuation (k = 2).

For any given value of a, o, and k, let G(a, o, k) denote the two-stage game described above. We are looking for perfect Bayesian equilibria of the game G(a, o, k) in pure and symmetric strategies in the first auction. Let [b.sub.1](v) denote the equilibrium strategy of the bidders in the first auction. If o = Y, let bo([p.sub.1]) [member of] {0, 1} indicate whether the winner exercises the buyer's option or not given the auction price [p.sub.1], with bo([p.sub.1]) = 1 meaning that he uses his option and bo([p.sub.1]) = that he does not. Finally, let [b.sup.w.sub.2](v, [p.sub.1]) denote the second auction strategy of the winner of the first auction and [b.sup.l.sub.2](v, [p.sub.1]) the second auction strategy of the loser of the first auction. For practical reasons, these strategies are confronted with the data only when the buyer's option is not available. In the following proposition, the strategies are therefore given only for o = N. But in the proof of the proposition (in the Appendix), explicit use is made of the strategies for o = Y.

Proposition 1. A perfect Bayesian equilibrium (in pure and symmetric strategies in the first auction) of the game G(a, o, k) is

(i) If a [member of] {E, S} o = N, and k [member of] {1/2, 1, 2}, then [b.sub.1](v) = kv, [b.sup.l.sub.2](v, [p.sub.1]) = v, [b.sup.w.sub.2], [p.sub.1]) = kv.

(ii) If a [member of] {D, F}, o = N and k [member of] {1/2, 1}, then no such equilibrium exists.

(iii) If a [member of] {D, F}, o = N and k = 2, then [b.sub.1](v) = (1/2)v, [b.sup.l.sub.2](v, [p.sub.1]) = [b.sup.w.sub.2](v, [p.sub.1]) = v.

(iv) If a [member of] {E, S}, o = Y and k = 1/2, then [b.sub.1](v) is a solution of [b.sub.1](v) - (v/2) = 2[lambda](v - [b.sub.1](v))[b'.sub.1](v), with [lambda] = if [b.sub.1](v) [greater than or equal to] (1/2)[bar.v] and [lambda] = 1...



NOTE: All illustrations and photos have been removed from this article.



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