Buy or wait, that is the option

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1 Buy or wait, that is the option The buyer s option in sequential laboratory auctions Philippe Février, Laurent Linnemer and Michael Visser Abstract This paper reports the results from an experiment on two-unit sequential auctions with and without a buyer s option. The demand for the two items is either decreasing, flat, or increasing. The 4 main auction institutions (firstprice, Dutch, second-price, English) are studied. Observed bidding behavior is studied both in the strategy space and the payoff space. In the strategy space, the experimental bids are close to Nash equilibrium bidding in the auctions for the second unit, but there are substantial deviations in the auctions for the first unit. The costs of these deviations are measured in the payoff space via three metrics. We find that experimental subjects could have taken advantage of the misbehavior of their opponents. The revenue-ranking of the 4 auction institutions is the same as in single-unit experiments. The buyer s option decreases (resp. increases) revenue in first-price (resp. second-price) auctions, but there is no significant effect in the oral auctions. The buyer s option causes declining price patterns in our experimental auctions. Keywords: Experimental economics, sequential auctions, buyer s option. JEL Classification: C91; D44. We thank two referees for very helpful suggestions. We also wish to thank our colleagues at CREST-LEI for their comments on the experimental design during a pilot study, Marie-Laure Allain, Romain Lesur, and Philippe Donnay for their help in conducting the experiment, students at ENSAE for their cooperation, and the ENSAE administrative staff for their material support. The paper has benefited from remarks by participants at the 5th Experimental Economics Conference in Strasbourg, the seminars of the University of Paris I and GATE, Lyon, the Workshop on auctions at the Free University of Brussels, and the ESEM We are grateful to Philippe Donnay who has very carefully and diligently written the computer programs. Financial support from CNRS, GENES, and INRA is gratefully acknowledged. CREST-LEI-GRECSTA and INSEE. philippe.fevrier@insee.fr. CREST-LEI-GRECSTA and University of Lille 2. linnemer@ensae.fr. Mail address: LEI-ENPC, 28, rue des Saints Pères, Paris, France. Tel: Fax: CREST-LEI-GRECSTA. visser@ensae.fr. 105

2 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 well fits the needs of both small and large buyers, whereas the alternative auction procedure that consists in selling all available units simultaneously, in one shot, typically excludes 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 very time-consuming, especially when the total number of units on 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 he/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 re-auctioned, 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 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)). Wellknown auction houses such as Christie s and Sotheby s (see Ashenfelter (1989) and Ginsburgh (1998)) and Drouot (see Février, Roos, and Visser (2001)) 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 optimal bidding strategies in two-unit sequential secondprice auctions with and without the buyer s option. All buyers in their model have 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 (1999) study price patterns at sequential descending auctions of roses and argue that the presence of the option is the main determinant of the observed price decline. Finally, Février, Roos, and Visser (2001), using data on ascending auctions of wine held at Drouot, structurally estimate their optimal bidding model, and find that the seller s revenue in a system 106

3 where items are auctioned sequentially is the same as in a system based on the buyer s option. This paper 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 2 units are sold to 2 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 4 main auction institutions are considered: first-price, descending (Dutch), second-price (Vickrey), and ascending (English) auctions. Although we are not aware of 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 these sealed-bid auctions. Like in standard one-unit auction theory, it is shown in this paper that first-price (resp. second-price) and Dutch (resp. English) sequential auctions with or without a buyer s option are theoretically isomorphic. Furthermore the 4 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 an equivalence in revenue. 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 buyer s 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 quite 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 (relatively to their values) in the direction that theory predicts, the adjustments are generally too modest. Furthermore, observed first-auction bidding is almost never affected by the presence of the buyer s option, whereas theory generally predicts that there should be an impact. By re-analyzing the data in the payoff space (instead of the strategy space), we evaluate what are the costs of this misbehavior. Considering the same metric as Harrison (1989), we find that while the costs of not playing the equilibrium strategy are negligible under decreasing or flat demand, they are substantial when the demand function is increasing. 1 An exception is Cassady (1967, p. 197) who describes the electronic auction market in Osaka, Japan, where lots of fruit and vegetables are sold via sequential first-price auctions. 107

4 Introducing two complementary metrics, we also find that participants in our experiment could have earned much more money by anticipating the observed behavior of their competitors and by maximizing against this observed pattern. Our results also indicate that the revenue-ranking of the 4 canonical auction institutions is the same as the one found in the single-unit experimental literature. The ordering of the auctions mechanisms in terms of expected revenue is thus robust to the sequentialtwo-unit extension considered in this paper. Finally we find that while successive prices in sequential auctions are generally constant in the absence of the buyer s option, they are (significantly) declining once the option is introduced. Therefore, the buyer s option appears to be responsible for the declining price anomaly in our experiment. Experimental work on multi-unit sequential auctions is still very rare. 2 Burns (1985) considers sequential English auctions. The experiment is designed to mimic the Australian wool market, and the paper s main objective is to study the effect of market size on auction prices. The paper is essentially theory-free in that observed behavior is not confronted with any equilibrium bidding behavior. Keser and Olson (1996) consider sequential first-price auctions and suppose that buyers have singleunit demand functions. Their main objective is to compare observed price-sequences with the predicted patterns derived in Weber (1983), under different design parameters. Similarly as in Burns, the paper focuses on one particular auction mechanism, and no attempt is made to examine outcomes under alternative institutions. Robert and Montmarquette (1999) do 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 3 experimental papers on sequential auctions analyzes the buyer s option. The only experimental paper that does look at the buyer s option is a recent paper by Katok and Roth (2002). 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 paper as they focus their analysis on the free rider problem. The paper proceeds as follows. In the next section the theoretical background is presented. In deriving the risk-neutral Nash equilibrium bidding functions and the expected revenues in the different auction institutions, we partly draw on Black and De Meza (1992), Donald, Paarsch, and Robert (1997) and a recent paper by Février (2000). But most results in this section are actually new. Section 3 describes the 2 Spurred by the recent FCC auctions, experimental papers on all sorts of simultaneous multidemand auctions are, however, flourishing (see for example Kagel and Levin (2001) and the references therein, and the special issue of the Journal of Economics & Management Strategy (1997, Number 3)). 108

5 experimental design, section 4 the experimental results, and section 5 concludes. 2 Theoretical background Suppose that 2 units of a good are auctioned to 2 potential buyers. Each buyer is assumed to be risk-neutral and desires to purchase both units. Adopting the IPV paradigm, let v i denote the value that buyer i places on the first unit. The value v i and the value of i s opponent are independently drawn from a uniform distribution on the interval [0, v]. It is assumed that the value that i places on the second unit is kv i. The parameter k can take three values: k { 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 2 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 {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 1 denote the price the winner of the first auction has to pay for the first unit. When a {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 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 1 he/she has to pay for the first unit is again the amount at which the clock stopped. When a {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/her own bid, i.e. p 1 equals the highest submitted bid. When a = S the winner pays the bid of his opponent, i.e. here p 1 equals the second highest submitted bid. For all institutions a, the price p 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 2 be the price paid for the second unit. If instead o = Y the winner of the first auction has the option to buy 1 or 2 units, at the price of p 1 per unit. When he decides to purchase only 1 unit, a second auction is held under the conditions of institution a. When he/she exercises the buyer s option, no second auction is held. Note that in this case we automatically have p 2 = p 1. The theoretical model presented here is essentially based on the framework built 109

6 by Black and De Meza (1992). These authors, however, only considered 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 bayesian two-stage game described above. We are looking for perfect bayesian equilibria of the game G (a, o, k) with pure and symmetric strategies in the first auction. Let b 1 (v) denote the equilibrium strategy of the bidders in the first auction. If o = Y, let bo (p 1 ) {0, 1} indicate whether the winner exercises the buyer s option or not given the auction price p 1, with bo (p 1 ) = 1 meaning that he/she uses his/her option, and bo (p 1 ) = 0 that he/she does not. Finally, let b w 2 (v, p 1) denote the second auction strategy of the winner of the first auction, and b l 2 (v, p 1) the second auction strategy of the loser of the first auction. For practical reasons, these strategies are only confronted with the data when the buyer s option is not available. In the following proposition, the strategies are therefore only given for o = N. But in the proof of the proposition (appendix A), explicit use is made of the strategies for o = Y. Proposition 1. A symmetric-first-auction perfect bayesian equilibrium of the game G (a, o, k) is: 1. If a {E, S}, o = N, and k { 1 2, 1, 2}, then b 1 (v) = kv, b l 2 (v, p 1) = v, b w 2 (v, p 1) = kv. 2. If a {D, F }, o = N, and k { 1 2, 1}, then no such equilibrium exists. 3. If a {D, F }, o = N, and k = 2, then b 1 (v) = 1 2 v, bl 2 (v, p 1) = b w 2 (v, p 1) = v. 4. If a {E, S}, o = Y, and k = 1 2, then b 1 (v) is solution of b 1 (v) v 2 = 2λ (v b 1 (v)) b 1 (v), with λ = 0 if b 1 (v) 1 2 v and λ = 1 otherwise; bo (p 1) = 1 if p v and bo (p 1) = 0 if p 1 > 1 2 v. 5. If a {E, S}, o = Y, and k = 1, then b 1 (v) = v, bo (p 1 ) [0, 1]. 6. If a {E, S}, o = Y, and k = 2, then b 1 (v) = 2v, bo (p 1 ) = If a {D, F }, o = Y, and k { 1 2, 1, 2}, then b 1 (v) = 1+k 4 v, bo (p 1) = 1. Let us first comment on the predictions for the English and second-price auctions. As mentioned in the introduction, the behavioral predictions are always the same for these 2 mechanisms. When o = N, theory requires bidders to bid kv in the first auction, that is they have to bid the value for the second unit. While this result is intuitive for flat demand, it is less so when demand is decreasing or increasing. With decreasing demand, bid shading is required because losing the first auction is not necessarily bad news, as it implies a weaker rival in the second auction. With increasing demand, over-bidding is required as the winner of the first auction is also 110

7 going to be the winner of the second auction. In the second auction (still when o = N), it is a dominant strategy for each player to bid the value of the unit for which he/she is bidding. That is, the loser of the first auction should bid v, and the winner of the first auction kv. When o = Y and k {1, 2}, optimal first-auction bidding is the same as in the absence of the buyer s option. Put in other words, the buyer s option has no effect on first-auction bidding behavior. However, when k = 1 2, first-auction bidding should be more aggressive than in the absence of the option. The optimal use of the buyer s option is fairly simple when k = 1 2 or k = 1. In the former case it should be used if the first-auction price is lower than the second unit value, and in the latter case the first-auction winner is indifferent between exercising the option or not, which is the meaning of bo (p 1 ) [0, 1]. When k = 2 it is not optimal to use the option because the loser of the first auction is expected to bid less aggressively in the second auction, so the first-auction winner has a higher expected gain by waiting for the second auction. Let us next comment on the predictions for the Dutch and first-price auctions. Again theory predicts that behavior is strictly identical under the 2 institutions. When o = N, there does not exist a symmetric pure strategy equilibrium for k { 1 2, 1}. An explanation for this result is the following. If such an equilibrium were to exist, the loser of the first auction would learn the valuation of the winner (since p 1 is revealed at the end of the first auction). The first-auction winner would then clearly be in an uncomfortable situation in the second auction. The equilibrium in the second auction would take the following form: the winner of the first auction would play a mixed strategy and the loser a pure strategy. However, this secondauction equilibrium is not compatible with a first-auction pure strategy, since we can show that there always exists a profitable deviation. This means that both players should hide their valuation by playing a mixed strategy in the first auction. When o = N and k = 2, a symmetric pure strategy equilibrium does exist for the Dutch and the first-price auctions. This equilibrium is not simple to compute and is not very intuitive as it implies a relatively low first-auction bid. At first sight one might indeed think that it should be rewarding for player 1 to deviate from equilibrium by bidding x 2 (with x > v 1) in the first auction in order to increase the probability to win the first unit, and thereby to enter the second auction with a stronger valuation 2v 1. The proof in appendix A shows however that this deviation is not profitable. Indeed, this deviation decreases the expected gain in the first auction (since bidding half of one s valuation is optimal in a single-unit auction), and it does not affect the expected gain in the second auction. Note that the equilibrium given in the proposition is such that the winner of the first auction, say bidder 1, automatically wins the second auction: his/her valuation for the second unit is 2v 1 while his/her opponent s valuation for the first unit is v 2 v 1 (since the first- 111

8 auction strategy is symmetric), so by bidding v 1 he/she wins the second auction with probability one. Therefore, in equilibrium it is as if both bidders only compete for the first unit. When o = Y, a symmetric pure strategy equilibrium does exist for all values of k. Note that in equilibrium, bidders behave exactly as in standard single-unit Dutch or first-price auctions. Indeed, in equilibrium each player bids 1+k 4 v in the first auction and the winner always exercises his/her option. It is thus as if players submit a single bid equal to 1+k 2 v, for a single good with a value (1 + k)v.3 Note finally that for k = 2, first-auction bidding should be more aggressive when the option is available than when it is not available. 3 Experimental design The experiment was conducted on 28 and 29 March 2001 at the Ecole Nationale de Statistique et de l Administration Economique (ENSAE). 4 Students were recruited through personal s, and fliers that we dispatched in their mailboxes. Seventy four students (out of roughly 360 students that studied at the time at ENSAE) actually participated in the experiment. We organized a total of 10 experimental sessions in the computer rooms at ENSAE, and each student took part in only one session. Only one type of auction mechanism was used per session. Table 1 lists for each session the type of auction mechanism that was studied and the number of participants. From Table 1 it can be seen that 22 students participated in the Dutch auctions, 20 in the English auctions, 16 in the first-price auctions, and 16 in the second-price auctions. The sessions were made up of two parts. In each session, the first part was devoted to sequential auctions without a buyer s option, and the second part to sequential auctions with a buyer s option. This ordering seemed natural as the part with buyer s option is an adjustment of a more basic design. We did not have sessions in which this ordering was reversed. Therefore we can not test for the effect of ordering on subjects behavior. Although such a test would have been interesting, it would have required yet another treatment variable in our design. Given the number of observations that we anticipated to collect in the experiment, we thought that for efficiency reasons this would not have been reasonable. We start by describing the first part of a session. We began by reading aloud the instructions about the auction rules without a buyer s option. Written versions of 3 Recall that, given our model assumptions, the optimal single-unit bid (in first-price and Dutch auctions) for a good valued at v is 1 v. 4 2 The ENSAE is one of the leading French institutions of higher learning in the fields of statistics, economics, finance, and actuarial sciences. After completing the three-year curriculum of this school, graduates have a training comparable to the level attained by first-year Ph.D. students at a good North American university. 112

9 Table 1: Sessions Session Type of auction Number of subjects 1 First-price 8 2 Second-price 8 3 Dutch 6 4 English 6 5 Dutch 10 6 English 8 7 First-price 8 8 Second-price 8 9 Dutch 6 10 English 6 the instructions were distributed to the participants and could be consulted at any time during the experimental session. These instructions can be obtained from the authors upon request. The first part had 12 periods. Since we focus in this paper on auctions with 2 buyers, participants were told that they were in competition with a single person. At the beginning of each period the computer randomly matched each student to another student present in the room (all sessions had an even number of participants), so participants were aware of the fact that their opponent differed from period to period. Participants were also told that in each period 2 units of a fictitious good were sold at auction to each couple. At the start of each period, valuations were independently drawn from a uniform distribution on [0;v]=[0;FFr50.00]. On the computer screen of participant i appeared his/her valuation for the first unit of the good v i, the prevailing value of k, and his/her valuation for the second unit kv i. The value of k changed every 4 periods (k = 1 2 in periods 1-4, k = 1 in periods 5-8, and k = 2 in periods 9-12).5 Participants could observe this information for 30 seconds, after which the first auction started (but the information remained on the screen even during the auction). The manner in which participants could bid depended on the type of auction mechanism that was used during the session. The auction-specific bidding devices will be described later on. Once the first auction was over, some information concerning the first auction was added to the screen of each subject i. It indicated whether i was the winner or not, his/her own bid (if any), the winning price p 1, i.e. the price he/she or his opponent had to pay for the first unit, and his/her gain associated with the auction (v i p 1 if i was the winner, 0 otherwise). Since the identity of the winner of the first 5 For the same reason as above, we did not have sessions in which we changed this ordering. 113

10 auction is crucial knowledge in our experiment, we emphasized this by coloring the box marked Winning bid blue if i had won the first auction, and red otherwise. Note that the exact nature of information released between the two auctions differed slightly with the type of auction mechanism. For instance, for the winner of an English auction the box marked Your bid remained empty, while for the winner of a Dutch auction this box indicated the price at which he had stopped the clock. Before the start of the second auction, participants again had a thirty-seconds reflection period during which they could, if they wished, consult all information on their screen (again, all information remained visualized during the second auction). The second auction functioned in the same way as the first auction. We stressed the fact that the gain associated with the second auction depended on the outcome of the first auction. Thus, winner i of the second auction had a gain of kv i p 2 if he had also won the first auction, and a gain equal to v i p 2 if he had lost the first auction. Once the second auction was terminated for all couples in the room, we proceeded with the next period. The 12 periods of the first part of each experimental session were preceded by 6 dry periods (2 for each value of k). This gave participants the opportunity to familiarize themselves with the bidding method, determine their strategy for the different values of k, and ask questions to the experimenter. Next we describe the second part of the session, the one that was designed to study the buyer s option. We began by reading aloud the instructions about this part of the experiment. Like the first part it consisted of 12 periods. Each period started exactly like in the first part of the experiment: the valuations and the value of k (the values of k alternated as in the first part) showed up on the screen, the first auction started after 30 seconds, and once the first auction was over for player i and his/her rival, their screens updated them on the relevant auction results. Unlike the first part of the session, subjects were told that the winner of the first auction could, if he/she desired, use the buyer s option. If winner i chose to execute his/her option, the period was over for him and his/her opponent, and his/her total gain in the period was (v i p 1 ) + (kv i p 1 ) = (1 + k)v i 2p 1. If he/she chose not to do so, his/her gain associated with the first auction was v i p 1, and a second auction was held after the thirty-seconds pause. The second auction was in all respects identical to the second auction conducted in the first part of the experiment. The 12 wet periods of the second part of each experimental session were again preceded by dry periods, but now just 3 of them (1 for each value of k) since, at least from a practical point of view, the second part differed little from the first. As mentioned above, the way in which participants had to submit their bids depended on the auction format. In the first-price and second-price auctions participants could submit their bid by entering a number in a box marked Submit your bid here. The number could be any positive real integer, i.e. we did not forbid 114

11 subjects to bid in excess of their valuations. In the Dutch and English auctions bidding took place via numerical clocks. After the 30-seconds reflection period, the clock appeared on the screens of the participants. In the English auctions the clock started at 0.00FFr, augmented continuously at a rate of 50.00FFr per minute, and stopped automatically at FFr The clock started and operated simultaneously on the screens of participant i and his/her rival. They could stop the clock at any time by pressing the Enter key or Space bar, or click on a window marked Stop the clock. If neither i nor his rival had stopped the clock before it reached FFr120.00, the computer randomly selected i or his/her rival as the winner (actually this never happened during our experiments). In the Dutch auctions the clock started at FFr60.00 (if k = 1 2 or k = 1) or FFr (if k = 2), descended continuously at the speed of 50.00FFr per minute, and stopped automatically at FFr0.00. The Dutch clock started and operated simultaneously for subject i and his/her opponent and they could stop it, at any time, as the English clock. If neither i nor his/her rival had stopped the clock before it reached FFr0.00, there was no auction winner (again, this never occurred during our experiments). Note that as in the sealed-bid auctions, subjects could bid above their valuations (up to a reasonable limit) in the clock auctions. At the start of an experimental session, i.e., at the beginning of the first period, all participants were given a capital balance of FFr At the end of each period, the gains made during the period were added to the balance, and losses were subtracted from it. We informed the experimental subjects that if the end-of-period balance of a participant was negative (as a result of his/her bidding behavior in the period), the balance would immediately be readjusted to 0. We stressed that balances would only be readjusted at the end of a period, in view of the end-of-period balance, and not at some point during a period. The reason for censoring the start-of-period balances at 0 is to incite subjects to play well all along the experiment. 6 As it turned out, for none of the experimental subjects the capital balance went negative, so it was never necessary to implement the readjustment procedure. At the end of the session participants were paid in cash their final capital balance divided by two. From a theoretical point of view this 50% cut does not affect bidding behavior. On average we paid FFr229 to the students, the minimum payment was FFr60, and the maximum payment FFr360. Experimental sessions lasted between 1.5 and 2 hours (including about 20 minutes for the instructions). Before turning to the experimental results, we want to comment on the number of treatment levels in our experiment. In each session subjects went through 2 3 different treatments. This relatively high number of treatment levels is a drawback of our design as it might have created hysteresis effects. However, we do not think 6 Had we not done this, a subject with a very negative balance at the beginning of period 24, would clearly not have been incited to behave optimally in this last period. 115

12 that this occurred. Each change in the value of k was clearly indicated both on the screen and orally by the experimenter. Moreover, the introduction of the buyer s option was made very clear since we began the second part of the experiment by oral instructions about the rules of this mechanism. We have no evidence that subjects have been confused by the shifts in the value of k nor by the introduction of the buyer s option. On the other hand, one advantage of having several treatments within a session is that the estimation of treatment effects is facilitated since it is not necessary to control for inter-individual differences. 4 Experimental results This section presents our experimental results. First we confront the observed bidding behavior with the Nash equilibrium bidding strategies given in proposition 1. We then study the impact of the buyer s option on the behavior of the experimental subjects, and analyze how well the option is used in the experiment. In order to get a better understanding of the observed deviations from equilibrium, we proceed by re-analyzing the data in the payoff space instead of the strategy space. Finally, we study the revenue, the price patterns, and the efficiency of the 4 auction mechanisms. 4.1 Bidding behavior Figures 5-28 (in appendix H) show all first-auction bids for the different values of k for the 4 auction formats without buyer s option (Figures 5-16) and with buyer s option (13-24). 7 They depict the losing bids for the English auctions, the winning bids for the Dutch auctions, and both winning and losing bids for the sealed-bid auctions. Whenever there is a theoretical prediction (see Proposition 1), the optimal equilibrium bid function b 1 (.) is drawn in dashes. For instance, in Figure 6 (secondprice auction, k = 1 2 ) the dashed line is the function b 1(v) = 1 2v, but in Figure 5 (first-price auction, k = 1 2 ) no dashed line is drawn since no prediction is available. For second-price and English auctions with decreasing demand the solution (approximated using simulations) of the differential equation given in Proposition 1 is b 1 (v) = 0.99v 0.009v 2. This explains why the dashed line in Figures 18 and 20 is curved. In each figure we also draw a fitted line in solid. The solid line in Figures 18 and 20 is the fitted line ˆβ 1 v + ˆβ 2 v 2 where ˆβ 1 and ˆβ 2 are the OLS estimates of the coefficients in the regression b 1it = β 1 v it +β 2 vit 2 +ε it, where b 1it and v it are i s bid and valuation in period t, and ε it an error term that is assumed independent over i and t. In all other figures the solid line represents the fitted line ˆβv where ˆβ is the OLS estimate of the coefficient in the regression b 1it = βv it + ε it. A comparison of the dashed and solid lines is therefore a quick eyeball test of the theoretical predictions. 7 To economize on space, the figures corresponding to the second-unit bids are not given here but they can be obtained from the authors upon request. 116

13 Tables 3 and 4 (in appendix B) report the OLS results of all the first-auction bid regressions, that is the solid lines in Figures Tables 5 and 6 (in appendix C) report the OLS results of the second-auction bid regressions when the buyer s option is not available: the results in Table 5 are based on the second-auction bids submitted by the first-auction losers, and those in Table 6 are based on the second-auction bids of the first-auction winners. Since winners and losers of the first auction should generally behave differently in equilibrium (see Proposition 1), the results for the 2 groups are presented in these 2 separate tables. The OLS results based on the the second-auction bids with buyer s option are not reported. The reason is that the buyer s option has been frequently used by our experimental subjects, so relatively few second auctions were actually held during periods of the experiment, leaving us, in most cases, with too few data to reliably estimate the second-auction bidding strategies when the option is available. Each of the 4 tables also reports the predicted slopes of the optimal bidding strategies, the number of observations used for the regressions, the R 2 of the regressions, and test results of the hypothesis that observed behavior is in line with predicted behavior. The null hypothesis is tested simply by testing whether the coefficient β equals some specific value using the T-test. For example, in Table 3, the OLS estimate ˆβ equals 0.83 for the second-price auctions with decreasing demand, the estimated standard error is 0.04, the number of observations in the regression is 64 (16 subjects 4 periods), the R 2 is 0.87 (defined for a regression model without a constant), the predicted slope is 1 2, and the null hypothesis that β = 1 2 is rejected at the 5% level. 8 Note that for each value of k and auction mechanism, the sum of the number of observations in Table 5 plus those in Table 6 is equal to the number of observations reported in Table 3. This follows automatically because for each given round in a session, the number of first-auction losers plus the number of first-auction winners necessarily equals the number of subjects in the session. Note also that the number of observations reported in the 2 tables are equal for the sealed-bid auctions but differ for the oral auctions. The equality for the sealed-bid auctions follows from the fact that both losing and winning bids are observed for these auction institutions. The difference for the oral auctions follows because in a Dutch (resp. English) auction only the losing (resp. winning) bid is observed. Fact 1. In sequential auctions, subjects only partially understand the strategic implications of the Nash equilibrium in the auction for the first unit. Their observed bidding behavior is however close to optimal bidding behavior in the auction for the second unit. 8 In the case of Figures 18 and 20, the joint hypothesis β 1 = 0.99 and β 2 = is tested using a standard Fisher-test. As Table 4 indicates, the theory is not accepted for a = S, but is accepted for a = E (not at the 5% level, but at the 1% level, which is indicated by Yes in the table. 117

14 Evidence for Fact 1 comes from the analysis of the bid-regressions. We first study how the experimental subjects behaved in the auctions for the second unit, and then we consider their bidding behavior in the auctions for the first unit. As Tables 5 and 6 show, second-auction bidding in the English and second-price auctions is in line with theory in the majority of cases: 8 out of 10 times we can accept the hypothesis that subjects have played, on average, as the equilibrium strategy predicts. Recall that here the optimal strategy calls for submitting one s valuation. The 2 cases where the theory is rejected both concern the first-auction losers of the English and second-price auction under increasing demand: in these 2 cases observed bidding is significantly above the valuation. We discuss this deviation more in detail in fact 3. The tables also indicate that observed bidding in the Dutch and first-price auctions with increasing demand is also quite well in accordance with predicted behavior: subjects have played, on average, according to theory 3 times out of 4. In all English and second-price auctions for the second unit, it is optimal for players to bid their own valuation (which is in fact a dominant strategy). Thus, like in standard single-unit English and second-price auctions, it is optimal for bidders to reveal their valuation. It is therefore of interest to compare our results with those obtained in the experimental literature on single-unit English and single-unit secondprice auctions. The results of Coppinger and ann A. Titus (1980), Kagel, Harstad, and Levin (1987), and Kagel and Levin (1993), indicate that subjects bid according to equilibrium behavior in single-unit English auctions. Most experimental studies show, however, that in single-unit second-price auctions subjects tend to bid above their value (see for instance Kagel, Harstad, and Levin (1987), and Kagel and Levin (1993)). 9 Our results are thus compatible with the previous literature on single-unit English auctions. Compared to the single-unit second-price literature, our findings appear somewhat more in accordance with theory. It should be stressed however that bidding in our experiment and bidding in the single-unit experiments took place in different contexts. 10 These contextual differences can explain the small differences between our results and those obtained in the earlier literature. Next we study the results of the auctions for the first unit. Looking at the tests of the theory for all 4 auction mechanisms, Tables 3 and 4 show that the null hypothesis is accepted just 4 out of 20 times. The theory is mainly accepted in cases where the optimal strategies are relatively transparent. Indeed, the null is accepted 9 Cox, Roberson, and Smith (1982) find that average bidding is below value (but not always significantly). Kagel, Harstad, and Levin (1987) and Kagel and Levin (1993) point out that a likely explanation for these conflicting findings is that, unlike their experiments (and our s!), the design of Cox, Roberson, and Smith (1982) did not allow subjects to bid in excess of their valuation. 10 First because our experimental subjects were more informed about their opponents (when subjects in our experiment submitted their second-auction bid, they knew the first-auction price p 1 ) and second because, under decreasing and increasing demand, bidders in the second auction are no longer symmetric as in the single-unit experiments. 118

15 3 times for the English and second-price auction mechanism with flat demand. The optimal strategies are relatively transparent in these cases because the outcome of the first auction has no impact whatsoever on the second auction. Our experimental subjects have understood that in this relatively simple setup it is optimal to bid their valuation in the first auction. The only exception is the second-price auction with buyer s option. Observed bidding is significantly above the valuation in this case, a deviation from theory that is consistent with the literature discussed above. The equilibrium predictions are massively rejected when the demand function is either decreasing or increasing, that is precisely in the situations where our sequential auctions are inherently more complex. They are more complex to solve in these situations because, in determining the optimal bidding strategies, agents should anticipate and take into account the fact that the valuation of the first-auction winner is modified at the start of the second auction. Although observed first-auction bidding deviates from equilibrium behavior, our results show that subjects tend to adjust their bids relative to their values in a fashion that theory predicts, thereby showing that they acknowledge, at least partially, the strategic implications of non-flat demand functions. Thus in all second-price and English auctions with decreasing (resp. increasing) demand, subjects have indeed understood that optimal behavior calls for bid shading (resp. bidding above value), but the extent to which they do this is too modest to fully fit the theory. Similarly, although observed bidding is significantly above equilibrium bidding in all Dutch and first-price auctions with buyer s option, our experimental subjects have well understood the impact of k on the bidding strategies. Indeed, as k increases, they bid more aggressively as predicted by the theory. In summary, observed bidding behavior in the second auction is mostly in line with equilibrium predictions, but bidding in the first auction is generally off-equilibrium. The subjects thus exhibit bounded rationality in the sense that they solve correctly the final stage of the auction games but are apparently unable to apply correctly the backward-induction-arguments necessary to solve the first stage of the game. Fact 2. Bidding behavior in Dutch (resp. English) auctions is statistically identical to bidding behavior in the first-price (resp. second-price) auction. Evidence for Fact 2 follows from pairwise comparisons of mean bids between Dutch (resp. English) and first-price (resp. second-price) auctions. A total of 24 comparisons are made (there are 12 Dutch/first-price couples and 12 English/secondprice couples; see Tables 3-6): we thus test for bidding equivalence not only in cases where theory explicitly predicts 2 auction mechanisms to be isomorphic, but also in cases where there are no theoretic predictions. Two-sample T-tests on the equality of means show that the null is accepted 23 times (there is significantly lower bidding in English versus second-price auctions when k = 1 2, o = N). Unlike the single-unit 119

16 experimental literature, where bidders are generally found to be more aggressive in first-price and second-price auctions compared to respectively Dutch and English auctions (see Kagel (1995)), our findings suggest that in sequential auctions these mechanisms are behaviorally equivalent. Fact 3. In English and second-price auctions for the second unit under increasing demand and without buyer s option, the losers of the first auction adopt a punitive behavior by bidding significantly above their valuation. Subjects use weakly dominated strategies when a dominant strategy is also available. Table 5 shows that first-auction losers bid more than 20% above their valuation (and dominant strategy) v. The most plausible explanation for this substantial degree of over-bidding is that first-auction losers wish to punish their competitors. Indeed, in sequential English and second-price auctions under increasing demand, first-auction losers know that they have no chance of winning the second auction. They can also anticipate that the winner has no reason to bid less than in the first auction. This allows first-auction losers, without taking much personal risk, to reduce their competitor s gain by bidding above their own valuation. First-auction losers might want to hurt their opponents out of feelings of envy or because of fairness considerations. Such punitive behavior is also observed in other types of experiments. For instance, in sequential bargaining experiments subjects tend to punish competitors past behavior that they see as unfair (see Roth (1995)). Also, Zizzo and Oswald (2001) report the results of an experiment in which subjects first earn money (via gifts from the experimenter, or by playing lotteries) and then have the opportunity to burn the income of other players. It turns out that participants are willing to reduce the income of their opponents even if this reduces their own personal gain in the experiment. It can be formally shown that there exists a Nash equilibrium in weakly dominated strategies where a punitive behavior is adopted in the second auction. In this equilibrium, players bid somewhere between v and 2v (depending on the degree of punitive behavior among subjects) in the first auction to compensate for the smaller gain in the second one. As Figures 14 and 16 show, this line of reasoning is well supported by the data, since practically all first-auction bids are indeed between v and 2v (with some exceptions for the second-price auction). We thus find that our experimental subjects use dominated strategies in a game where a dominant strategy is also available. This finding is contrary to what is sometimes predicted in the mechanism design literature. Kreps (1990) (p. 698) for instance argues that mechanism designers may safely expect players to settle on the dominant strategy even when other equilibria (in weakly dominated strategies) exist. But our finding is in line with the results obtained by Kagel, Kinross, and Levin (2001). From their experiments on several multi-object auction institutions, they 120

17 report closer conformity to equilibrium bidding strategies in the Ausubel auction (in which the equilibrium solution is based on iterated deletion of weakly-dominated bidding strategies), versus the sealed-bid Vickrey auction (which generates sincere bidding as a dominant strategy). Note that the same deviations are observed in the first-unit auctions with buyer s auction (a {E, S}, k = 2 and o = Y ). Indeed, Figures 26 and 28 show that the majority of English and second-price bids are again between v and 2v. These deviations from the equilibrium strategy of Proposition 1 can be explained similarly as above: players anticipate future punitive behavior by submitting a bid somewhere between v and 2v. Furthermore, under punitive behavior, it can be shown that it is optimal for first-auction winners to always exercise the buyer s option. This justifies why the option has been massively used in the English and second-price auctions under increasing demand, a point to which we return in the next section. 4.2 The buyer s option In this subsection we analyze to what extent the buyer s option has been exercised by the subjects in our experiment, and we study the effect of the buyer s option on bidding behavior in the first auction. The results regarding the use of the buyer s option can be found in Table 2. The second column reports the relative number of times the buyer s option has been used for each auction institution, and the third column gives the optimal frequencies. For example, in the English auctions under decreasing demand, the option was used in 96% of the cases when p 1 1 2v, and it was never used otherwise; here theory stipulates that the option should be used when the winning price is lower than 1 2 v, and that it should not be used otherwise. The optimal use of the buyer s option relies on the comparison of the gain that can be made when the option is exercised, (kv p 1 ), and the expected gain when it is not exercised. It turns out that in most cases this comparison is relatively simple because the loser of the first unit is likely to be more aggressive in the second auction than in the first one. Consequently, it is in the interest of the winner of the first auction to exercise the option whenever kv p 1 is positive. This reasoning holds regardless of whether the first-auction winner has deviated in the first auction or not. For k = 2 and a = E, S, however, the loser of the first auction is likely to bid less aggressively in the auction for the second unit, and consequently the winner must not exercise the option. Again, this argument is valid whatever the behavior in the first auction. The predictions given in Table 2 can therefore be viewed as the predictions of the theory conditionally on the observed behavior in the first auction. Fact 4. The buyer s option is correctly used in most of the cases. As explained above, when k = 1 2 or k = 1, it is optimal to use the buyer s 121