Multi-Unit Auctions: a Comparison of Static and Dynamic Mechanisms

Transcription

1 Multi-Unit Auctions: a Comparison of Static and Dynamic Mechanisms Alejandro M. Manelli Department of Economics, Arizona State University, Tempe, Az Martin Sefton Department of Economics, University of Newcastle, Newcastle-upon-Tyne, NE1 7RU, United Kingdom and Benjamin S. Wilner LECG/Navigant Consulting 1603 Orrington, Suite 1500, Evanston, Il60201 Version: October 1999 Abstract We compare, experimentally, the Vickrey auction and an ascending-price auction recently introduced by Ausubel (1997). We evaluate the relative performance of both auctions in terms of efficiency and revenue in multi-unit environments where valuations either have a common-value component or are private information. We find that the Ausubel auction is less prone to overbidding and may yield higher revenue than the Vickrey auction. The gain in revenue seems to be coupled with a loss of efficiency. We are grateful to Jorge Aseff for excellent research assistance, to Lawrence Ausubel, Joyce Berg, Robert Forsythe, Thomas Rietz, and Rafael Tenorio for useful conversations and suggestions, to Douglas DeJong for support and assistance with programming issues, to Gary Fethke for support and encouragement, and to Sue Bremner for exact editorial advice. We are also grateful to the participants in the 1998 Maryland Auction Conference where a preliminary version of this paper was presented. Financial support from the National Science Foundation under Grant SBR is gratefully acknowledged. 1

2 1 Introduction Because of their numerous actual and potential applications, ranging from the sale of U.S. treasury bills and spectrum licenses to initial public offerings and the allocation of office space, multi-object auctions have received considerable attention from practitioners and economists. Despite this attention, there is still a definite need to identify adequate multi-unit trading institutions that implement efficient outcomes and generate high seller s revenue in various environments. Recently, Ausubel (1997) introduced a dynamic, multi-unit auction that is sufficiently simple and appealing to be of practical use, and that has many desirable features such as efficient outcomes, and higher revenues than many commonly used auctions. In this paper, we compare experimentally the relative performance, in terms of revenue and efficiency, of the Vickrey auction and the Ausubel auction in multi-unit environments. We find that the Ausubel auction exhibits promising experimental characteristics that make it a strong contender in applications. Our experiments are best evaluated in the context of some well-known properties of auctions. Classical English auctions of a single object have several advantages over second-price, sealed-bid auctions. When bidders have private valuations i.e., a bidder s valuation for the object is not affected by the information that other bidders may possess both types of auctions implement sincere bidding in dominant strategies, generate efficient outcomes, and with adequate reserve prices, maximize the seller s expected revenue. When bidders valuations have a common-value component i.e, a bidder s valuation for the object is affected by the private information of other bidders theory predicts that English auctions generate higher expected revenue than second-price auctions. 1 Experimental evidence indicates substantial misbehavior in second-price auctions even with private values: bids systematically higher than predicted, divergences between predicted and actual bids, and considerable variability in the degree of this divergence across individuals. Repetition, the use of experienced subjects, and various other design features have little or no effect on these departures from predicted behavior. In contrast, English auctions work well under a wide variety of conditions; observed behavior quickly converges to predicted behavior, and there is evidence that bidders make inferences from opponents actions in pure common-value settings (Kagel, Harstad, and Levin (1987), Levin, Kagel, and Richard (1996)). 2 1 This is due to the linkage effect (Milgrom and Weber (1985)). See, for instance, McAfee and McMillan (1987) or Klemperer (1999) and their references for precise statements of results and assumptions. 2 See also Kagel s comments on Tests of the Strategic Equivalence of Second-Price and English Auctions (1995). Levin, Kagel, and Richard (1996) find in addition that that an alternative hypothesis fits their data 2

3 The extension of the second-price auction to multi-unit environments has been known for sometime (Vickrey (1961)). Recently Ausubel (1997) provided a multi-unit extension of the English auction that inherits many of its desirable theoretical properties. We find that the Ausubel mechanism also inherits some of the desirable experimental properties of the English auction. Overbidding, conservatively defined as bidding above the maximum possible value of an object, although present in both mechanisms is much more prevalent and pervasive in the Vickrey auction. As in the single-unit English auction, participants in the Ausubel mechanism seem to understand the consequences of overbidding better than participants in the Vickrey mechanism. Nonetheless, bidders in the Ausubel auction engage in a particular type of overbidding, strategic in nature, whose sole purpose is to influence their opponent s behavior either through beliefs or assignments. We discuss this observation at length in Section 5. We also find evidence of a potential trade-off between eficiency and revenue. The Vickrey auction achieves higher efficiency but yields lower revenue than the Ausubel auction. Our experimental setting consists of a seller with three identical objects, and three buyers, each of them with demand for two units. In Vickrey auctions, participants simultaneously submit bids for different units. Bids are ordered and objects are assigned to participants with the highest bids. If a bidder receives n units she pays the sum of the highest n bids (other than her own) that her bids displace. common second-price auction.) (In a single-unit setting, this describes the In Ausubel auctions, the auctioneer begins by quoting a low price. Bidders announce the quantities they wish to purchase at the announced price. Assignments at the quoted price are determined by a simple procedure. For instance to compute bidder A s assignment, the difference between the available supply and the aggregate demand of A s opponents is calculated. This difference is used to fulfill A s demand at the quoted price. (If the difference is negative, A gets zero units.) If some units remain unsold, the auctioneer quotes a slightly higher price and the process continues. While participants in Vickrey auctions bid values for the various units; participants in Ausubel auctions bid quantities at various posted prices. We compare both auctions when valuations have a common-value component (CVC), and when valuations are private information (PV). Since most of the Ausubel auction s desirable properties only hold when buyers have non-increasing marginal valuations for the different units, we circumscribe our experiments to this case and ignore important alternatives such as superadditive valuations. 3 better than Nash equilibrium theory. 3 A variety of alternative trading institutions have been investigated in environments with supperadditive valuations, see for instance, Brenner and Morgan (1997), Isaac and James (1997), Ledyard, Porter and Rangel 3

4 With private values, sincere bidding is an equilibrium in both mechanisms. Hence both institutions yield the same revenue and allocation. The theoretical advantages of the Ausubel mechanism are realized when valuations have a common-value component. In this case, Vickrey auctions have no efficient equilibria in a general class of models (Ausubel (1997)). The winner s curse roughly states that a bidder s expected valuation conditional on winning is less than her unconditional expected valuation. In the Vickrey auction, a bidder s conditional expected valuation decreases with the number of units she wins. Hence, bidders have an incentive to engage in demand reduction, i.e, in equilibrium, bidders shade their bids for a second object more than they shade their bids for the first object. Demand reduction may lead to inefficient outcomes: a bidder may not obtain a second object even though her marginal valuation for the second unit is higher than that of another winning bidder. In contrast, the Ausubel auction has an efficient Nash equilibrium. Because participants in the Ausubel auction observe their rivals actions at previously quoted prices, they possess more information when bidding. This information may be used to make inferences about rivals private information. This mitigates the winner s curse effect, leading to efficient outcomes and higher revenue. 4 Although the experimental literature on auctions is vast, most studies have considered single-object models (See Kagel (1995) and its references). Two notable studies of auctions with multi-unit demands are Alsemgeest, Noussair, and Olson (1996), and Kagel and Levin (1997). 5 Kagel and Levin study three trading institutions in private-value environments, the sealed-bid uniform-price, the uniform-price clock, and the Ausubel auction. They are mainly concerned with the extent of demand reduction in all three mechanisms. In their experiments a subject with a two-unit demand competes against computer simulated rivals with singleunit demands. They find evidence of considerable demand reduction in both uniform-price mechanisms but not in the Ausubel mechanism. They also find that the sealed-bid uniform auction generates more revenue but less efficient outcomes than the Ausubel auction. Alsemgeest et al. test for demand reduction in an English clock auction with uniform pricing, and in a sealed-bid auction with lowest-accepted-bid pricing. They compare the performance of these auctions in two private-value settings. In one of them bidders demand (1997), Plott (1997), Rothkopf, Pekec, and Harstad (1997). 4 Engelbrecht-Wiggans and Kahn (1995), Katzman (1995), Noussair (1995), and Tenorio (1996) find equilibria of uniform-price auctions that exhibit demand reduction; Bolle (1995) and Ausubel and Cramton (1996) show with additional generality that the equilibria of uniform-price and discriminatory auctions are inefficient in a variety of environments. 5 A further study of note is a field experiment by List and Lucking-Reiley (1999) in which significant demand reduction is observed in a uniformed-price auction. 4

5 at most one unit; in the other bidders demand at most two units. The authors report that both auctions produce reasonably efficient allocations (in 86% of all auctions the allocation is efficient) and that the welfare loss from demand reduction in the two-unit environment is relatively small. They also find that the sealed-bid auction generates higher revenue than the English clock auction with both one-unit and two-unit demands. Early studies of auctions with multi-unit demands (e.g., Smith (1967), Belovicz (1979), Miller and Plott (1985)), focussed mainly on the revenue properties of discriminatory and uniform-price auctions. Other studies (e.g., Cox, Smith, and Walker (1984) (1985); McCabe, Rassenti, and Smith (1990) (1991)) consider multi-unit environments but where each bidder demands at most one object. Demand reduction, and its potential efficiency loss, is not possible when bidders have unit demands. The rest of the paper is organized as follows. In the next two Sections we introduce our experimental model, discuss its theoretical implications, and illustrate the functioning of both auctions with an example. In Section 4 we describe the experimental design and in Section 5 the results. Section 6 concludes. 2 The Model Our two main experiments, one conducted at Arizona State University (ASU) and the other at The University of Iowa (UI), compare slightly different versions of the Vickrey and Ausubel mechanisms in various settings. Most equilibrium properties of the Ausubel auction have been obtained in a continuum model, i.e., a model in which private information, bids, and quoted prices may take a continuum of values. In experimental settings the mentioned variables have, at most, finitely many values. The discretization of the continuum model, necessary in the experimental setting, introduces various difficulties. First, since the experimental setting does not conform to the theoretical model, the latter cannot be invoked to establish properties of the former. Furthermore, it is well known that a continuum model and its discrete versions (even if arbitrarily close), may have very different equilibrium properties. Thus, it cannot be sustained without further theoretical arguments that a fine discretization will inherit the desirable properties of the continuum model. We determine some equilibrium properties in the ASU experiment by laboriously computing pure strategy equilibria. Second, the mechanism itself, originally defined in a continuum environment, must be adapted to the discrete setting. We introduce below the experimental environments and then describe both auctions. 5

6 Environment: In each auction, three certificates are auctioned off to three potential buyers, bidders A, B, and C. In the ASU experiment, each bidder i observes her type v i drawn from the set {1, 4, 7, 8, 9, 10}. All types v i are equally likely but no repetition of types is possible: there is at most a single individual with a given type. A bidder s valuation for each unit depends on her own type and those of her opponents. (We justify our modeling choice after completing the description of the environment and the formal presentation of both mechanisms.) The first two units acquired by any bidder, say bidder A, have the same valuation v A, obtained as a weighted average of all participants types: v A = βv A + (1 β)(1/2)(v B + v C ) where β = 0.5 represents environments with a common value component, and β = 1 environments with private values. The third unit acquired by any bidder has value zero. Thus, bidders have demand for two objects. We believe the model described has not been used in laboratory studies. In the UI experiment, the distribution of types is different. Each bidder s type is drawn independently and with equal probability from the set {1, 2, 3, 8, 9, 10}. Thus, repetitions are possible. Only valuations with a common value component (i.e., β = 0.5) are considered in the UI experiment. Ausubel auction: Each bidder i in an Ausubel auction observes her private information v i. The auctioneer, the computer in our case, quotes the price p = 0.5. Then bidders, simultaneously and independently, indicate how many units they are willing to purchase at the quoted price. To determine, for instance, how many objects, bidder A receives at a price p = 0.5, the auctioneer computes the difference between the total available supply (3 in this case) and the aggregate demand of bidders B and C. If the difference is strictly positive, it is used to fulfill bidder A s demand. Otherwise A receives no units at that price. Other bidders assignments are determined in a parallel manner. If some units are unassigned the auctioneer increases the price to p = 1.5, informs all bidders of the bids and assignments made at previous prices, and the process is repeated. (Price increments are always of one unit.) The auction ends when all goods have been assigned, all bidders demand zero at the posted price, or the price reaches the value p =

7 A careful reading of the last paragraph reveals that some units may not be assigned in case of ties such as when two or more bidders drop out of the auction at the same price. Imagine, for example, that the three bidders request two units at price p 1 but request zero units at price p. At price p, the available supply is three and the aggregate demand is zero, at previous prices the aggregate demand exceeds the available supply. The procedure passes from a state of excess demand to a state of excess supply and therefore, there are unassigned, unrequested units at the last quoted price. There are different ways to treat these unrequested units. In the ASU experiment, the unrequested units are not assigned and the auction ends. In the UI experiment the unrequested units are assigned randomly among active bidders in proportion to their stated demands at the previous price. Thus, all units are always assigned in the UI experiment. Although the procedure adopted in the UI experiment best resembles Ausubel s design, it generates, in conjunction with the discrete nature of the price-increase, some undesirable incentives. If all bidders are expected to drop out at price p, then bidders have an incentive to overstate their demands at p 1 in order to obtain a better share of the unassigned goods. This effect disappears when prices are increased continuously, or when unrequested units are unassigned as is the case in the ASU experiment. Vickrey Auction: Each bidder i in a Vickrey auction observes her private information v i, and submits three bids, one for each available unit. Bids are restricted to be integers between 0 and 12, and a bid for the j th unit must be no higher than the bid for the (j 1) th unit (for j = 2, 3). Bids are then ordered from highest to lowest and the three highest bids are assigned one unit each. To see how prices are determined suppose bidder A is assigned a single unit. Then, A pays the highest unaccepted bid of B or C, plus 0.5. If A is assigned two units, A pays the sum of the two highest unaccepted bids of B or C, plus 0.5 per unit, and similarly for three units. The transaction cost of 0.5 per unit is intended to address the discrete nature of the experiment and to make it comparable with the Ausubel auction (see the example in Section 3). The software converts bids into the corresponding bids in an Ausubel auction, and then applies Ausubel s algorithm. In the ASU experiment, some units may be unassigned in case of ties; in the UI experiment, all three units are always assigned. We summarize below some theoretical properties of the Vickrey and Ausubel auctions in the continuum model, and then describe the properties that obtain in our discrete setting. 6 6 The interested reader should consult Ausubel (1997) for precise assumptions, statements, and additional properties. 7

8 1. Sincere bidding, i.e., bidding one s own valuation, is a weakly dominant strategy of the Vickrey auction with private values. 2. Sincere bidding is a Nash equilibrium of the Ausubel auction with private values. 3. With private values and sincere bidding both auctions generate the same revenue and allocation. 4. In a class of environments with a common value component, there is an efficient equilibrium for the Ausubel mechanism but not for the Vickrey mechanism. Furthermore, the efficient equilibrium of the Ausubel mechanism maximizes seller s revenue among all those incentive compatible and individually rational mechanisms that assign all three units. In the design of the ASU experiment we purposely considered alternative parameters i.e., distribution of types, value of β, minimum price-increase in the Ausubel auction and transaction cost in the Vickrey auction, etc. to obtain a discrete experimental model with desirable properties. It is straightforward to verify that properties (1-3) hold in the ASU setting. Property (4), however, only holds partially. The Ausubel mechanism has several efficient equilibria (and several other equilibria with small inefficiencies, such as ties in the second stage of the ascending price auction, after a first object has been assigned), but although most equilibria of the Vickrey auction (ASU, CVC) are inefficient, there are a few equilibria in which demand reduction is not sufficient to cause inefficiencies. The parameters selected for our ASU experiment constitute a compromise among somewhat contradictory goals. First, the difference between a subject s type and valuation must be significant in order to observe whether bidders update their conditional valuations with new information (the linkage effect). Second, players valuations must be sufficiently different to facilitate the test of efficiency. Finally, types and posted prices must be few to keep the length of the experiment within reasonable limits. 3 An Example: To illustrate the functioning of the Vickrey and Ausubel auctions we present a simple example. There are three units available and three bidders, A, B, and C. Bidders observe their own valuations, indicated in the table below. 8

9 A B C First Unit Second Unit Third Unit For instance, the first two units acquired by bidder A bring her a payoff of 10 per unit, the last one a payoff of zero. There is no common-value component in this example. Consider first the Vickrey auction. As in the single-unit environment, it is a dominant strategy to bid sincerely. Suppose accordingly that players indeed bid their own valuations. Ordering the bids of all players from highest to lowest yields the sequence 10, 10, 7, 7, 4, 4, 0, 0. Units are assigned to bidders with the three highest bids: A receives two units (for first and second place), B receives one unit (for third place); C receives none. The two highest bids (of B and C) that A displaces are 7 and 4, so A pays a total of = 12. Participant B displaces a bid of 4 belonging to C, thus B pays 4.5; C pays zero. Consider now the Ausubel auction and suppose that participants bid sincerely, an equilibrium of the game. At each price p quoted by the auctioneer, bidders announce their desired quantities: p A B C At p = 3.5 (or lower), no units are assigned to any bidder: the total demand of A s opponents is = 4, which exceeds the total available supply; a similar argument applies to other bidders. The first assignment occurs at p = 4.5. At this price, there are three units available and the total demand of B and C is only = 2. Hence, A receives one unit at p = 4.5. By similar logic, B receives one unit at p = 4.5; C receives no units. The auctioneer then quotes a higher price. Participants bid the following quantities: p A B C Note that given her marginal valuations, bidder A would be willing to purchase two units at p = 5.5, but since she has already acquired one object, she only has use for a single 9

10 additional unit. There is now a single unit available. Thus, no assignments are made till the price reaches p = 7.5. At this price, Bidder A receives a second unit: B and C s demand totals = 0. Bidders B and C receive no units. Since all units have been assigned the procedure ends. Summarizing, participant A receives two units and pays a total of = 12. Participant B receives a single unit and pays 4.5. The outcome of both auctions is the same. 4 Experimental Design Subjects were randomly selected from a pool of undergraduates taking Business courses at Arizona State University or The University of Iowa. Our main experiments were conducted in a series of sessions and no subject took part in more than one session. Some of these subjects subsequently participated in a related experiment to investigate the effect of experience. Each experimental session consists of 20 rounds of either a Vickrey auction or an Ausubel auction. In each round, three certificates are auctioned off to three bidders, A, B, C. By using only three bidders we increase the informational content of any single bidder s actions, and facilitate the detection of bidders responses to new information. In any given session the same three bidders participate in all twenty auctions. At least three sessions are run simultaneously in the same laboratory room, so that no individual is able to identify his opponents with certainty. Although this precaution does not eliminate potential supergame effects, it seems reasonable to expect that it will make implicit collusion somewhat more difficult. In the Ausubel auction, each bidding screen provides full information about bids and purchases at previous prices but without disclosing the identity of the bidders. All subjects are provided with a set of instructions that include examples illustrating the functioning of the relevant auction. They are also provided with tables indicating a bidder s valuation given her type and all possible combinations of opponents types. The experimenter reads the instructions aloud and then answers questions. Subjects are then given a quiz to test their understanding of the auction mechanism. After discussing the answers, the experimenter entertains a final round of questions. The types of all 3 bidders in the 20 auctions were drawn in advance from the distributions described in Section 2. The same types were then used in all sessions of both auctions. This facilitates comparison of results, and eliminates variability due to differences in draws. Although bidding above one s maximum possible valuation may be unadvisable, we do not prevent subjects from doing so. 10

11 As is customary, to account for potential losses, participants are provided with an endowment of points at the beginning of the session. At the end of the session, accumulated points were exchanged for dollars at a fixed, publicly-known, and predetermined rate or 25 cents per point. A fee of $5 was guaranteed to individuals for attending a session. Earnings varied between $5 and $38. In all sessions the instructional phase lasted approximately 20 minutes. The decision making phase lasted an additional 60 minutes (Vickrey) and (75) minutes (Ausubel). A questionnaire was conducted at the end of the 20 rounds. Subjects appeared well motivated by the cash incentives as evidenced by their willingness to participate in further experiments. 5 Results The Vickrey and Ausubel auctions have multiple Bayesian Nash equilibria in the environments considered. When valuations have a common value component, this multiplicity is somewhat disturbing for there appears to be no focal equilibrium strategy. As noted earlier, with private values sincere bidding is an equilibrium of the game tested in our ASU experiment. Given the lack of suitable theoretical predictions, we choose not to compare observed bidding behavior with equilibrium strategies. Instead we judge bidding behavior in reference to rationality postulates such as the use of dominated strategies. A salient experimental feature of second-price auctions even in the single-unit, privatevalue case is that subjects often bid above their valuations. The excessive bids common in second-price auctions rarely occur in single-unit English auctions. Davis and Holt (1993, p. 279) outline the following explanation for this difference (emphasis in the original): [In the English auction] the bidder with the highest values discovers that it is not necessary to pay more than the price at which the second highest bidder drops out. In contrast no such learning takes place during a second-price auction. As in single-unit settings, our experiments show extensive overbidding but also reveal a more complex bidding pattern. Result 1 (ASU CVC, PV ) Participants in Vickrey auctions overbid at a much higher rate than participants in Ausubel auctions (Table 1, row 1). The difference is statistically significant at the 1% level. 7 (ASU CVC) In environments with a common value component, the proportion of auctions 7 Unless specified otherwise, all tables contain group-level data: average over the last ten rounds of auctions in which each group of three bidders participates. 11

12 in which a single bidder acquires three units is higher with the Ausubel than with the Vickrey mechanism (Table 1, row 2). The difference is statistically significant at the 5% level. (ASU PV) This difference disappears in environments with private values. We use two measures of overbidding to illustrate our findings. The first measure is the number of bids (for the first two objects) that exceed the bidder s maximum possible usevalue given the bidder s private information. In any given auction, the maximum number of such bids is six, two per bidder. The second measure is the proportion of auctions in which a bidder acquires all three units. Neither measure involves comparisons with equilibrium behavior. Acquiring a third unit, in itself, cannot benefit a bidder; the third unit has value zero for all bidders. Nonetheless, we observe that participants in Ausubel auction obtain a third unit more frequently than participants in Vickrey auctions. Exit questionnaires provide an explanation. Several participants in Ausubel auctions (both PV and CVC) describe their strategy along the following lines: demand as many units as possible at any reasonable price until the desired two units are secured and then drop out immediately; they argue that following this strategy impedes any assignment to rivals and indicates their determination to prevail. This strategy is not without risk: a bidder may end up with three units when her opponents exit the auction simultaneously. Data from individual bidding behavior support the explanation based on exit surveys: more than 60% of bidders that acquire 3 units in Ausubel auctions do so because their opponents simultaneously dropped out. Note that bidders following the described strategy may not incur net losses, even when acquiring three units, because profits from the first two units may compensate losses from the third. In turn, this may prevent the type of learning adduced by Davis and Holt in the single-unit English auction. The data shows no appreciable difference between behavior in the first and last ten-rounds of auctions in either mechanism. 8 As one might expect, overbidding tends to raise prices, and therefore revenue tends to be greater in the mechanism that induces more overbidding: Result 2 (ASU CVC, PV) The Vickrey mechanism yields higher seller s revenue than the Ausubel mechanism. The difference, statistically significant at the 2.5% level, seems due to important overbidding in the Vickrey mechanism. A simple computation of seller s revenue (ASU PV) illustrates the pervasiveness of overbidding in the Vickrey mechanism and the superior performance of the Ausubel mechanism 8 A similar observation has been made before regarding second-price auctions: overbids may be unsuccessful and hence costless ex post, or they may not entail a loss even if successful. This may explain the slow rate at which subjects learn to avoid overbidding. 12

13 Table 1: ASU Experiment group level data CVC PV Vickrey Ausubel t/ p-value Vickrey Ausubel t/ p-value n=12 n=14 n= 12 n=12 Overbid (.689) (.407) (1.085) (.404) Three units (.083) (.039) (.167) (.138) Revenue (.081) (.126) (.098) (.133) Efficiency (.065) (.082) (.077) (.070) Overbid: number of bids on the first two units that exceed the maximum possible value of the object given the bidder s private information. Since there are three bidders, the maximum number of oberbids is six. There is a difficulty in comparing overbidding across mechanisms: In an Ausubel auction there may be bidders that intend to overbid but do not do so because the auction ends before the price is sufficiently high to engage in overbidding. In a Vickrey auction, since all bids are submitted simultaneously, all overbids can be identified. To make results from both auctions comparable, we convert Vickrey bids into the implied drop prices in an Ausubel mechanism. Overbids in the Vickrey auction that would not have been observed in a corresponding Ausubel auction are censored. Three units: proportion of auctions in which one bidder obtains all three units. Revenue: proportion of potential gains from trade captured by the seller. Efficiency: proportion of potential gains from trade realized. Group-level data: each fundamental observation is the average over the last ten auctions played by a group. This results in a sample of 12 observations in the case of Vickrey mechanisms and 14 in the case of Ausubel mechanisms. For each sample the table reports averages, standard deviations in parentheses, and the t-statistic and p-value for equality of means. Neither the results nor their significance are altered when the Wilcoxon rank-sum test for equality of medians is performed. in this regard. Sincere bidding is, with private values, an equilibrium strategy in both mechanisms, and a dominant strategy in the Vickrey auction. With sincere bidding, seller s revenue is 0.49% of potential gains from trade in either mechanism. As indicated in Table 1, row 3, seller s revenue constitutes 64% of potential gains from trade in the Vickrey, and 52% in the Ausubel mechanism. Table 2, displaying the average bid per type, provides evidence of demand reduction and considerable variability in bids per type in the Vickrey auction. This should result, as predicted, in an efficiency loss from missasigned units in the Vickrey auction. The efficiency implications of overbidding are less clear; overbidding does not cause inefficiencies if bids are monotonic on type. The higher rate at which single bidders acquire three units in Ausubel 13

14 Table 2: Average bid Vickrey Auction (ASU) ASU CVC bid for unit ASU PV bid for unit type type n=60 (2.303) (2.018) (1.854) 60 (3.276) (2.490) (1.658) n=60 (1.928) (1.730) (1.978) 60 (2.146) (1.680) (1.927) n=48 (1.259) (1.458) (2.431) 48 (1.967) (1.833) (2.652) n=48 (1.650) (1.750) (2.684) 48 (1.748) (1.907) (2.520) n=48 (1.331) (1.829) (2.401) 48 (1.655) (2.436) (3.038) n=96 (1.757) (1.931) (2.467) 96 (1.597) (2.572) (3.229) Standard deviations in parentheses. auctions also suggests an efficiency loss. A simple search of the data reveals that, indeed, auctions in which a single bidder acquires three units are the auctions in which the Ausubel mechanism experiences its highest efficiency loss. Result 3 (ASU CVC, PV) Efficiency is similar in both auctions, approximately 85% of potential gains from trade. Given that the relative performance of the mechanisms is influenced by considerable overbidding, we conducted some additional sessions to investigate the effects of experience and bidder sophistication. We invited subjects who had previously taken part in privatevalue sessions to take part in a related experiment. Volunteers then participated in another session using the same mechanism they had already experienced, although this time in an environment with a common value component. We found that experience, both as defined above, or as measured by comparing results of the first-ten rounds with those of the last-ten rounds, does not make any difference. We also recruited subjects from the economics Ph.D. program. These subjects were not familiar with the theory of multi-unit auctions, but would have had a basic training in microe- 14

15 Table 3: UI Experiment session level data Vickrey Ausubel t/ p-value n=12 n=12 Overbidding (.665) (.281) Three units (.100) (.156) Revenue (.172) (.191) Efficiency (.050) (.044) See Table 1 for the definition of variables. conomic theory. Although the sample was relatively small (3 groups of Vickrey and 4 groups of Ausubel mechanisms) we found no appreciable difference with the undergraduate population. One notable feature of the graduate sessions was that some subjects tried to implement repeated-game strategies that were inappropriate for the environment. Casual observation suggests that the graduates are less motivated by earnings than the undergraduate subjects. We now turn to the results of the UI experiment in which Vickrey and Ausubel auctions were conducted only in environments with a common value component. The model features two main departures from the one used in ASU, the way ties are handled and the distribution of types. As explained in Section 2, while in the ASU experiment ties may imply that some units are not assigned, in the UI experiment ties are resolved by allocating units randomly. With regard to the distribution of types, in the UI experiment, each bidder s type is drawn independently and with equal probability from the set {1, 2, 3, 8, 9, 10}. Result 4 (UI CVC) As in the ASU-experiments, participants in Vickrey auctions overbid at a much higher rate than participants in Ausubel auctions (Table 3, row 1). The difference is statistically significant at the 10% level. In the Ausubel mechanism, there is a higher proportion of auctions where a single individual acquires three units (Table 3, row 2). The difference is statistically significant at the 2.5% level. As compared with the ASU experiment, both mechanisms exhibit considerably fewer instances of overbidding. We speculate that the simpler distribution of types a bidder s type, 15

16 Table 4: Average Bid: UI CVC Vickrey Auction bid for unit type/n n = 72 (3.184) (2.157) (0.904) n= 48 (2.856) (1.919) (1.571) n= 48 (2.302) (1.555) (1.350) n= 60 (2.261) (2.521) (1.648) n= 60 (2.064) (2.506) (1.651) n= 72 (2.061) (2.663) (1.700) Average bid: average bid over all bidders with a given type. Standard deviations in parenthseses. roughly, is either low (i.e. 1, 2, 3) or high (i.e., 8, 9, 10) is responsible for this difference through two effects. First, simpler distributions are easier to comprehend and use. Indeed, the average bids per type (Table 4, column 1) suggest that bidders with either the three lowest types or the three highest types behaved similarly. Second, given a bidder s private information, her maximum use-value is higher under the UI distribution; this makes overbidding more difficult. Once overbidding in both mechanisms subsides, the Ausubel auction outperforms the Vickrey auction in terms of revenue but is outperformed in terms of efficiency. As in the ASU experiments, after isolating the individual auctions in which the Ausubel mechanism yields a considerable efficiency loss, we find that they are consistently auctions in which a bidder acquires three units. Result 5 (UI CVC) The data suggests an apparent trade-off between revenue and efficiency: the Ausubel auction generates higher seller s revenue than the Vickrey auction; the difference is approximately 8% of the potential gains from trade (table 3, row 3). The difference is statistically significant at the 15% level. 16

17 Result 6 (UI CVC) Both mechanisms achieve high level of efficiency, above 90% of the potential gains from trade (table 3, row 4). The Ausubel mechanism achieves about 4% less of the potential gains from trade than the Vickrey mechanism. The difference is statistically significant at the 5% level. 6 Conclusions We find that, as in single-unit environments, overbidding is pervasive in the Vickrey mechanism. The magnitude of overbidding, specially in the case with private valuations, may reflect a lack of understanding of the underlying game or an inability of the subjects to identify optimal strategies. It would be of some interest to investigate whether the use of dominated strategies concomitant with overbidding is reduced by more detailed explanations of the auction and its strategic implications. (In actual applications such alternative explanations are readily available.) In the Ausubel mechanism overbidding is manifested, mainly, in the number of bidders that acquire a worthless third unit. The acquisition of a third unit appears to be the unfortunate by-product of a strategy designed to influence rivals behavior. This type of overbidding is strategic in nature and cannot be dismissed without adherence to an equilibrium theory. Given the strategic nature of overbidding in the Ausubel auction, one might expect different results in an experimental setting in which a bidder is matched with computer-generated rivals such as the one used by Kagel and Levine (1997): a bidder playing a computer program might conclude that it may be very difficult to influence her rival s behavior. Indeed it would be interesting to repeat our experiments in such environment. Some characteristics of the tested model three bidders and three available objects may have contributed to the strategic overbidding observed in the Ausubel auction: by demanding three units, a bidder prevents its rivals from securing any objects and only risks the acquisition of a single undesirable unit. If, for instance, ten objects were available, it would be more costly to achieve the same objective. The dynamic structure of the Ausubel mechanism, responsible for its theoretical advantages, also adds an element of strategic risk not present in Vickrey auctions. Casual observation of bidding strategies in the Ausubel mechanism (UI and ASU CVC) suggests that, on occasions, excessive bids by low-type participants mislead other players into exuberant bidding. Thus, the bidder incurring the loss may not be the bidder making the error which may retard the learning process. When making an inference about valuations from an opponent s actions, a bidder must take into account the possibility of an opponent s mistake. 17

18 A comparison of both mechanisms in terms of revenue and efficiency suggests the existence of a trade-off. Overbidding, which is much more prevalent in Vickrey auctions, tends to increase revenue. Notwithstanding this tendency, in the UI experiment the Ausubel mechanism yields higher revenue than Vickrey mechanism (approximately 12% of potential gains from trade). This revenue gain comes at the cost of efficiency: the Vickrey mechanism yields a small efficiency gain over the Ausubel mechanism (approximately 4% of potential gains from trade). Both auctions, however, achieve high efficiency, perhaps because of a subjects tendency to use monotonic strategies. We conclude that the Ausubel mechanism has good potential in applications. 18

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