Experimental comparisons of auctions under single- and multi-unit demand.

Alsemgeest, Paul ; Noussair, Charles ; Olson, Mark 等

I. INTRODUCTION

The study of multiple-unit auctions is important because of their widespread use and the high value of goods sold. Examples include auctions of U.S. treasury debt, cut flowers, wine, foreign exchange, and other financial and monetary instruments. Moreover, since auctions are often proposed by economists as a method of sale for other goods, as in McMillan [1994], the use of auctions can be expected to increase in the future. Many different auction rules are used to sell these diverse commodities and auction rule changes are often suggested, usually with the intention of increasing revenue or improving the allocations, and thus there is a considerable and increasing number of known auction formats.

Evaluating new or comparing different auction mechanisms using field tests can be very costly and risky. However, experimental methods can and have been used by economists to study the behavioral characteristics of different auction procedures; Kagel [1995] provides a recent comprehensive survey. Most experimental studies consider the behavior of auctions when there is a single unit to be sold. A few studies, such as Cox et al. [1984; 1985] and McCabe et al. [1990; 1991] study multiple-unit auctions in environments in which each bidder demands at most one object from a set of identical objects being sold. The studies of Smith [1967], Belovicz [1979], Miller and Plott [1985] and Burns [1985] consider multi-unit demand environments. These are environments in which bidders have demand for multiple (identical) units and are permitted to purchase multiple units of the commodity sold. The extension to multi-unit demand is important since the presence of multi-unit demand is a characteristic of many auctions in the field, including all of the examples above.

In this study we consider the behavior of two types of multi-unit auction under both single-unit and multi-unit demand. We construct a laboratory environment which we use to study an English clock (EC), which is a special type of English auction, and a sealed-bid auction with lowest-accepted-bid pricing (SB). In the SB each bidder pays a per-unit price equal to the lowest accepted bid. The EC was chosen because it is known to provide highly efficient allocations in auctions of single-unit goods and therefore is likely to be suggested for many field applications. The SB was chosen because of its widespread use which makes its outcomes a useful basis of comparison for the outcomes of the English clock auction.(1) However, we do not attempt to represent a particular field situation here. Instead, since our interest is in understanding the basic properties of the auctions, we chose an experimental design that enables us to make simple comparisons across treatments and with previous (and possibly future) studies. The design enables us to perform simple testing of game-theoretic models in some of the treatments.

In our view the difference between single- and multi-unit demand is critical. As we discuss in section II, many of the results of auction theory that are valid in single-unit demand environments no longer apply in multi-unit demand environments, suggesting that the observed behavior of auctions might also differ between the two environments. Therefore, we isolate the effects of multi-unit demand by conducting single-unit demand experiments under identical market demand and supply conditions as the multi-unit demand experiments. These and other aspects of the experimental design are described in detail in section III. The results and a discussion are given in sections IV and V respectively.

II. INSTITUTIONS, THEORY AND PREVIOUS STUDIES

The English Clock Auction

This variant of the English auction was modeled by Milgrom and Weber [1982] and studied by McCabe et al. [1990; 1991]. In our version, the auction begins with all bidders publicly and simultaneously announcing their initial quantity demanded at price zero. If there is excess demand, the price is increased by a small increment, and the bidders then announce new quantities demanded. These new quantities must be less than or equal to the previous quantity announced, and therefore exit is irrevocable. The procedure continues until quantity demanded and quantity supplied are equal. The market-clearing price becomes the uniform per-unit price charged to demanders. Each bidder's profit on each item equals the difference between his valuation for the item and the price he pays to obtain it. Under multi-unit demand, if exactly j units are obtained by a bidder, he is awarded the difference between his j highest valuations and j times the purchase price.(2)

In an environment in which demanders with independent private values wish to buy at most one unit from a group of identical units being sold, each bidder has a dominant strategy to reduce his quantity demanded from one to zero, once the price reaches his valuation. The bidding stops at a level approximately equal (within one price increment) to the (K +1)st highest valuation. The resulting allocation is Pareto optimal. The demanders with the K highest valuations receive units and pay a per-unit price equal to the (K + 1)st highest valuation.

In multi-unit demand environments, however, Pareto-optimality cannot always be guaranteed by assuming profit-maximizing behavior on the part of bidders.(3) Demanders may have an incentive to behave strategically by underrevealing their demands. In the case of two-unit demand it is a dominant strategy for a bidder to reduce quantity demanded from one unit to zero units when the price reaches her higher valuation. However, it is a best response for the bidder to reduce quantity demanded from two units to one unit prior to the price reaching the bidder's second highest valuation.(4)

Sealed-Bid Auction

In SB, each subject is permitted to submit one or two sealed bids, depending on the environment. The bids are collected by the auctioneer, who accepts only the K highest bids. The per-unit price equals the lowest accepted bid. The items are awarded to the subjects who placed the accepted bids.(5)

Each subject's profit equals the difference between her valuations and the price she pays for the units she receives. In an independent private values environment with K units to be sold, if each demander is risk-neutral and draws one valuation from a common uniform distribution in which the lowest possible valuation is zero (the conditions present in our experiment), a symmetric Bayes-Nash equilibrium for the auction is given by

[Mathematical Expression Omitted], [for every] i,

where [Mathematical Expression Omitted] is the common equilibrium strategy of all bidders, i indexes the bidders, [v.sub.i] is bidder i's valuation, N is the number of bidders and K is the number of units sold. See Alsemgeest et al. [1995] for a derivation. Since [Mathematical Expression Omitted] is monotone increasing in [v.sub.i], the outcome is Pareto optimal. In the special case where K = 1, we obtain the equilibrium bidding function for the first price auction (in which the highest bidder gets the one item sold and pays the amount of her bid) first derived by Vickrey [1961]. This auction has not been modeled to date in an independent private values environment in which demanders may wish to buy more than one unit. However, under two-unit demand, the incentive exists to submit bids less than valuations for both units.

Revenue

The Revenue Equivalence Theorem as proven by Myerson [1981], Engelbrecht-Wiggans [1988] and Bulow and Klemperer [1994] states that under risk neutrality, independent private values and single-unit demand, any auction mechanism with a Nash equilibrium where the bidders with the highest valuations always receive the items, and a bidder with the lowest possible valuation has an expected payoff of zero, has the same expected revenue to the seller in the Nash equilibrium. However, Tenorio [1994] shows that when there is multi-unit demand, this result no longer applies to many simple auctions because inefficient allocations can occur in equilibrium.

In the laboratory, the revenue-generating properties of different multi-unit auctions have been studied in the single-unit demand case by Cox et al. [1984; 1985] and McCabe et al. [1990, 1991].(6) McCabe et al. [1991] report that the English clock generates prices very close to the competitive equilibrium level in a single-unit demand, independent private-values environment. Cox et al. [1985] compare the revenue generated by the uniform-price sealed-bid auction with highest-rejected-bid pricing (a sealed-bid auction in which the highest K bids are accepted and the winners pay a per-unit price equal to the (K + 1)st highest bid) and the discriminative auction (in which each of the K winners pays the amount she bid) in a single-unit demand, independent private-value environment. Although they find that the revenue generated by both of the auctions is below the Nash equilibrium level, they do not reject the hypothesis that revenue is equal between the two auctions. They find that previous experience with one of the auctions can affect subsequent bidding behavior under the other auction. For example, bids are lower in the uniform-price auction, in which there exists a dominant strategy of bidding one's valuation, when subjects have previously bid in a discriminative auction, in which it is optimal to bid below one's valuation, than when they have not bid previously in a discriminative auction. The uniform-price auction generates more revenue relative to the discriminative auction when the uniform-price auction is the first treatment in the sequence of two treatments in a session, than when it is the second treatment.

Multi-unit demand, private-value environments have been studied by Miller and Plott [1985]. They compare SB and the discriminative auction and find that when demand is inelastic at the competitive equilibrium, the discriminative auction generates higher revenue than SB; this result is reversed when demand is elastic. SB generates the competitive equilibrium revenue and subjects tend to bid their valuations. However, their environment differs from ours in that market demand is stationary from period to period, and in that for their parameters, the strategy profile consisting of all demanders bidding truthfully always constitutes a Nash equilibrium, resulting in the competitive equilibrium revenue. Our interest here is in environments, such as the independent private-values environment, in which market demand is stochastic and unilateral gains from underrevelation generally exist.(7)

Efficiency

The welfare measure used in this study to compare the allocations in our various treatments is the percentage of the maximum possible gains from trade which is realized by the allocation process. This fraction is called the efficiency of the allocation. It is computed as the sum of the producer and consumer surplus (producer surplus is the revenue to the auctioneer) resulting from the allocation process divided by the sum of the producer and consumer surplus which would be realized if the demanders with the K highest valuations receive the K units.

McCabe et al. [1991] report a very high mean efficiency of 99.98% for the English clock in single-unit demand, multiple-unit auctions. Simultaneous sealed-bid auctions have not generated high efficiency as consistently under single-unit demand as the English clock. Cox et al. [1985] report an average efficiency of 97.2% for the uniform-price sealed-bid auction with highest-rejected-bid pricing and 97.7% for the discriminative auction. Under multi-unit demand, Miller and Plott [1985] find that bidding in the SB converges to truthful demand revelation, and therefore efficient allocations, with replication of the market period.

Ill. EXPERIMENTAL DESIGN

The Design

We conducted eight experimental sessions, each consisting of 20 periods. Four sessions (two with each auction) were run "by hand" at a blackboard and four sessions (two with each auction) were run on a computer network. In each session, exactly one of the auctions was used. We used a crossover design, in which one group is measured under treatment A and then measured under treatment B, and a separate group is measured under B and then A. Four sessions started with ten periods in which six bidders each had single-unit demand, followed by ten periods in which three bidders each had two-unit demand, thus keeping total demand constant at six units. In the other four sessions, the order was reversed. Thus, subjects in the single-unit demand auctions in periods 11-20 had already participated in 10 two-unit demand auctions in periods 1-10 under the same auction rule (and vice-versa.) A benefit of a crossover design is that the treatment comparison is made within subjects. A disadvantage of a crossover design is that earlier treatments may have an effect on later treatments.(8)

Supply and Demand

In the experiment there is a completely inelastic supply of four homogeneous items to be allocated in each of the 20 periods of each experimental session. Bidders' valuations in terms of the experimental currency are independently drawn integers from the discrete uniform distribution with support on the [1,1000] interval. In the two-unit case, the valuations consist of two integers drawn independently and ordered from higher to lower so that demand is downward sloping. The same set of 120 valuations, six valuations for each of the 20 periods, is used for each of the eight experimental sessions. This yields identical market demand (the same six valuations) in the period possessing the same period number in any session, allowing us to use data with a common period number as paired observations from different treatments. For example, period 17 in all eight sessions had the same market demand so any differences across treatments in period 17 could not be attributed to a difference in market demand.

Information Structure

In order to prevent explicit collusion, which is assumed not to exist in any of the theoretical models we have reviewed, communication between subjects is not allowed during the experiment. It is common knowledge that each subject knows his own valuations and the distribution from which each player's valuations are drawn, but not the actual valuations of his competitors. That is, an independent private-values information structure is present. Furthermore, each subject knows how many items there are for sale in any period, how many competitors he has (two or five) and how many valuations each of them has on his valuation sheet.

During the course of the session, additional information becomes available. In the English clock sessions run by hand, at the beginning of each period each subject's initial demand is posted on the blackboard. During the period, whenever a bidder reduces her stated quantity demanded, the price at which demand was reduced and the identification number of the demander is posted. Therefore, throughout the period, each subject knows the quantity demanded at the current price by each demander. The above information remains posted for the duration of the session.

In the sealed-bid sessions run by hand, at the end of the market period the market price and the identification numbers of the bidders who are awarded units are posted. The information remains posted for the entire session. Subjects are not informed of the bids of other participants, but are aware that the final price is the fourth highest bid. In the computerized sessions of both auctions the above information [TABULAR DATA FOR TABLE I OMITTED] is accessible to subjects on their computer screens.(9)

Experimental Sessions

All sessions were conducted at the CREED laboratory at the University of Amsterdam and all subjects were undergraduate economics and business majors at the university. Subjects volunteered and were not allowed to participate in more than one session in this series. Each subject received a 10 guilder (1 Dutch guilder = 55 U.S. cents) fee for arriving at the session. Subjects also received profits during each period equal to the sum of their valuations for the units they purchased minus the total amount they spent to acquire the units.(10) Sessions lasted about 75 minutes including instruction, two periods of practice, and 20 periods that counted toward subjects' earnings.

IV. RESULTS

Treatment Effects

Table I presents means and between-session standard deviations of efficiency and revenue by treatment. The initial observation evident from Table I, which we state as our first result, is that efficiencies are very high in all sessions and treatments. The basis for characterizing the efficiency as high is a comparison with the efficiency of the allocations resulting from the simulated behavior of "zero-intelligence" traders as described by Gode and Sunder [1993].(11)

TABLE II

Statistics on Deviations from Bayes-Nash Equilibrium Revenue

Session Periods Average Std. Dev. t-statistic

1 1-10 9.20 68.64 0.42
2 11-20 68.73 62.34 3.49
3 1-10 90.00 65.05 4.38
4 11-20 -0.07 101.50 -.002

RESULT 1. For all sessions and treatments the efficiency of allocation is very high.

Support. In all sessions at least 75% of the periods and in all treatments at least 80% of the periods attain 100% efficiency. For all sessions average efficiency of allocation is close to 0.99. The allocations resulting from the zero-intelligence simulated traders mentioned above have an average efficiency of 0.923.

To check whether the differences in revenue across treatments evident in Table I are statistically significant, we construct a statistical model and perform appropriate tests. The analysis leads to the following result.

RESULT 2. SB generates more revenue than EC under both single-unit and two-unit demand. The EC generates more revenue under single-unit than under two-unit demand. The revenue generated by SB is not significantly different between single-unit and two-unit demand. Prior experience in a different treatment can negate the differences between treatments.

Support. A repeated-measures ANOVA was conducted on the difference between the observed price and the competitive equilibrium price, which equals the fifth highest valuation.(12) In periods 1-10, the type of auction and the demand environment have significant effects on price at the 1% and 10% levels respectively. However, the difference in revenue between single-and two-unit demand is specific to EC and is not significant for SB. In periods 11-20, there is no effect of mechanism or environment on price. See Alsemgeest et al. [1995] for details.

Evaluating Equilibrium Predictions

In previous studies, in single-unit demand environments, the English clock auction tends to generate the competitive equilibrium outcome, which occurs if all demanders use their dominant strategy. In our experiments (with the exception of one of the sessions to be discussed below), we replicate this result.

RESULT 3. The English clock auction produces per-unit prices no different from the 5th highest valuation, the competitive equilibrium price, under single-unit demand.

Support. In 31 of the 40 market periods, the final price is less than one price increment (10 units of the experimental currency) away from the competitive equilibrium level.

There is no dominant strategy in SB, but in our single-unit demand treatment, if all bidders are risk neutral, it is a symmetric Bayes-Nash equilibrium for each demander to submit a bid equal to two-thirds of his valuation. We find some evidence of bidding higher than the equilibrium level in our single-unit demand treatment.

RESULT 4. The SB produces prices greater than the risk-neutral Bayes-Nash equilibrium prediction under single-unit demand in two out of four sessions. In the other two sessions prices do not differ from the prediction.

Support. Table II contains the average deviations from the Bayes-Nash equilibrium for each of the sessions in which SB was used.

[TABULAR DATA FOR TABLE III OMITTED]

In sessions 2 and 3 the mean is significantly higher than the equilibrium prediction, whereas it is not significantly different from the equilibrium prediction in sessions 1 and 4.

Individual Bidding Strategies

Sealed Bid Auction. Many possible strategy profiles can lead to similar market prices. We therefore take a closer look at the strategies used by bidders. For each bidder in the SB sessions we estimate simple bidding equations. For the single-unit demand data, we estimate one equation which requires the bidding strategy to be linear in the valuation, but allows the strategy to have a non-zero intercept at valuation 0. For the two-unit demand data, we estimate two linear equations for each bidder, one with the higher bid made by the bidder each period as the dependent variable and an analogous equation for the lower bid. Each of these two components of the bidding strategy is allowed to depend on both valuations and to have a non-zero intercept.

For the single-unit demand environment the linear specification yields a simple test of the symmetric Bayes-Nash equilibrium, which makes the prediction that the intercept equals zero, and the coefficient of valuation equals 2/3 for each subject. The average estimated strategy is [B.sub.i]([v.sub.i]) = 25.73 + .77[v.sub.i]. Only four of the 24 bidders have estimated intercepts that are significantly different from zero at the 5% level. For ten of the 24 bidders in the single-unit demand treatment, the slope coefficient is not significantly different from 2/3 at the 5% level. For 11 of the 24 bidders, the coefficient of the valuation is greater and for the remaining three bidders is less than 2/3 at the 5% level of significance.

In the two-unit demand treatment the estimated average bidding strategies are [Mathematical Expression Omitted] and [Mathematical Expression Omitted], where [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are demander i's higher and lower bids respectively, and [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are his higher and lower valuations respectively. Only one of 24 estimated intercept terms is significantly different from zero. The higher valuation does not have a significant effect on lower bids for any bidder, and the lower valuation has a significant effect on the higher bid for only two out of 12 bidders. The average coefficients of [Mathematical Expression Omitted] in the [Mathematical Expression Omitted] equation and [Mathematical Expression Omitted] in the [Mathematical Expression Omitted] equation in the two-unit case and [v.sub.i] in the single-unit case are very close together. Our two-unit demanders behaved like two separate single-unit demanders.

English Clock. In the English clock auction, subjects could attempt to influence prices by exiting the bidding process at prices below their valuations, an action we refer to as underrevelation. Table III gives the size of the average underrevelation (Avg. Und.) when it occurred, the number of occurrences (Ocs.) of underrevelation, and the total number of observations (Obs., total number of exits from the auction) for each demand environment and treatment order.(13)

Underrevelation rarely occurs in the single-unit demand environment except for during one session in which there were 11 observed underrevelations by an average amount of - 124. There are at least three reasonable interpretations of the data from this session. Because all but one of the 11 underrevelations were for units with valuations of less than 400, they may have been "throw away" bids, which are demand reductions by bidders who view their probability of obtaining the unit to be very low. A second possible explanation is that, because the extensive underrevelation occurred in periods 11-20, it was due to a hysteresis effect from the first 10 periods, in which the same subjects were in the [EC.sub.2] treatment, in which there was a unilateral incentive to underreveal. A third possible explanation is that it may have been an effort to cooperate to lower price.(14)

As anticipated by theory, underrevelation occurs much more frequently for the lower-valued units in the two-unit demand treatment than in the single-unit demand treatment. However, subjects are much less likely to underreveal demand in the two-unit demand environment if they have participated in 10 periods of [EC.sub.1] already than if they have not. In periods 1-10, underrevelation occurred in 30 out of 36 observations on demanders' first exits in a period of [EC.sub.2], but in periods 11-20, in which [EC.sub.2] was always preceded by 10 periods of [EC.sub.1], underrevelation occurred in nine of the 36 observations. Hysteresis effects are also evident in the data in Table I. In the first 10 periods, every period of [EC.sub.1] resulted in an efficient allocation and prices were very close to the equilibrium level, whereas [EC.sub.2] resulted in an inefficient allocation 25% of the time and lower prices than [EC.sub.1]. In contrast, in periods 11-20, the revenue and the number of efficient periods from [EC.sub.1] and [EC.sub.2] are very close to each other.

V. DISCUSSION

From the standpoint of efficiency, both auctions perform very well under both types of demand. Both auctions generate high efficiency (86% of the auctions led to the optimal allocation) and the welfare loss from strategic behavior in the two-unit demand environment seems to be quite small. Almost all possible gains from trade are realized. We are in agreement with McCabe et al. [1990; 1991] who find that the English clock auction generates highly efficient outcomes in their single-unit demand environment.

The SB leads to higher revenues than the English clock under both single-unit and two-unit demand. The higher revenue in the sealed-bid auctions under single-unit demand is consistent with the presence of risk aversion on the part of subjects, which would lead demanders to make higher bids to try to increase their probability of winning at the cost of lowering profit in the event of winning. Our single-unit demand data provides another example of the violation of the assumptions of the revenue equivalence theorem in addition to those documented by previous studies.

In the English clock auction, single-unit demand leads to higher revenue than two-unit demand. In the English clock single-unit demand data, we generally observe the outcome resulting from the dominant-strategy equilibrium-strategy profile. The dominant strategy in this game seems to be transparent to subjects. In the two-unit demand English clock, subjects who have not participated in single-unit demand auctions seem to have little difficulty realizing their strategic incentive to reduce their quantities demanded from two to one at prices lower than their lower valuations, in order to lower the prices they pay for the one unit they may yet receive.

We find evidence of widespread bidding higher than the risk-neutral Bayes-Nash prediction in the sealed-bid auction. The subjects' bidding strategies in the two-unit demand treatment are separable in the sense that demanders' higher (lower) bids do not seem to depend on their lower (higher) valuations. This separability is an intuitive result. Underbidding is beneficial to a demander only in the event that his bid is the fourth highest bid overall. Therefore, the incentive to lower one's bid on the more highly valued unit should not depend on the lower value, since the lower bid is not accepted when the higher bid is the fourth highest. Likewise, if the lower bid is the fourth highest overall, it means that the higher bid must be accepted, and thus there is no reason that the lower bid should depend on the amount of the higher valuation (but only on the fact that a higher bid exists). We also find that the bidding behavior on the higher-valued unit was the same in both the single-and the multi-unit demand treatments. The existence of a second valuation (both for the demanders themselves and on the part of other demanders) did not affect demanders' strategies. Revenue did not differ significantly between single-and two-unit demand for SB.

Subjects' prior experience can affect their behavior in subsequent auctions. We see effects similar to those reported by Cox et al. [1985]. They find that the uniform price auction with highest-rejected-bid pricing generates more revenue than the discriminative auction when it is first in the sequence of treatments. Here we find that the revenue of the English clock under single-unit demand exceeds that under two-unit demand for the first treatment in the sequence but not for the second.

In the English clock, subjects had a strong tendency to reveal demand in the single-unit demand environment. Moreover, when the multi-unit demand treatment followed ten periods of single-unit demand, subjects in the multi-unit demand treatment were much more likely to reveal demand than when they did not have previous experience in the single-unit demand treatment. The hysteresis effect is strong enough to offset the difference in revenue between single-and multi-unit demand observed in the first ten periods of the sessions, in which there are no hysteresis effects. It may be the case that subjects can be conditioned to think in terms of a particular game-theoretic solution concept (e.g. dominant strategies) and that it may take some time to (perhaps subconsciously) begin thinking in terms of another type of solution concept (e.g. Nash equilibrium, rationalizable strategies).

Any inferences concerning the use of one of the two auctions analyzed here for particular problems in the field must take into account the limitations of our study. Claims about the robustness of our results in other multi-unit demand environments, and indeed for other parameters within the same environment, must await further research. The extension of game-theoretic modeling of auctions to multi-unit demand environments, with their applicability to large classes of parameters, can help to make the argument that experimental results such as ours have wide applicability.

ABBREVIATIONS

EC: English clock SB: Sealed-bid pricing

Some of the material in this article is drawn from Paul Alsemgeest's master's thesis (Doctoraalscriptie) from Erasmus University, Rotterdam, the Netherlands, 1993. We are grateful to the CREED laboratory for financial support and thank David Fahrner, John Kagel, and three anonymous referees for comments.

1. Examples of the use of SB include markets for foreign exchange in many countries. See Tenorio [1993] for an interesting analysis of foreign exchange markets in Zambia. SB is also widely used for selling government debt. Umlauf [1993] analyzes auctions for Mexican treasury securities, in which SB is used. The U.S. Treasury uses uniform price auctions with highest-rejected bid pricing, but each bid is for such a large quantity that it is generally the case that the highest rejected bid is the same as the lowest accepted bid. Bergsten et al. [1987] report that in Australia import quotas are auctioned using SB.

2. If the quantity demanded becomes strictly less than the quantity supplied, a tie-breaking rule is applied. Of the four sessions in which the English clock was run, two were conducted by hand and two were conducted by computer. In the two computerized sessions, the following tie-breaking rule was used. Let K equal the number of units for sale and let t indicate a round of the English clock auction. If at price [p.sub.t+1], the quantity demanded, [[q.sup.d].sub.t+1], is strictly less than the quantity supplied, K, the price is "backed up" to the previous price of [p.sub.t]. The [[q.sup.d].sub.t+1] items are then allocated to those who demanded items at [P.sub.t+1] and the remaining K - [[q.sup.d].sub.t+1] items are allocated randomly at price [p.sub.t] to those who demanded more items at [p.sub.t] than at [p.sub.t+1].

In the sessions run by hand, the tie-breaking rule used is the following: If there is excess supply at price [p.sub.t+1], the price is reduced in small increments and subjects can voluntary increase their quantity demanded up to their [p.sub.t] level. If total market quantity demanded does not reach K before the price reaches [p.sub.t], subjects who previously demanded more units at [p.sub.t] than at the current iteration are randomly reallocated the units they dropped earlier until market quantity demanded equals K units.

3. It is well known that theoretical results and intuition derived from single-unit demand environments do not necessarily carry over to multi-unit demand environments. Vickrey [1961], Wilson [1979], Hansen [1988], Maskin and Riley [1992], Back and Zender [1993] and Ausubel and Cramton [1996] have modeled auctions in environments in which demanders have continuous preferences for a perfectly divisible good; others, such as Bikchandani [1988], Tenorio [1994], and Noussair [1995] have analyzed environments in which demanders have positive valuations for multiple indivisible units of a good. None of these papers study the properties of the auctions we use here in an environment with the same structure as our laboratory environment.

4. See Ausubel and Cramton [1996] for an analysis of strategic demand reduction in uniform-price auctions in the case of continuous demand for a divisible good. As an illustration of demand reduction, consider the very simple situation in which there are two bidders and two units to be sold. Suppose that Bidder I has a valuation of 10 per-unit for obtaining up to two units and 0 for obtaining additional units and Bidder 2 draws a valuation v for obtaining up to one unit where P(v = 1) = P(v = 9) = 1/2. Bidder 1 knows the distribution of v and also knows that Bidder 2 has a positive valuation for exactly one unit. Suppose that ties are broken in Bidder 1's favor. Bidder 2 has a dominant strategy of dropping out of the bidding when the price reaches her valuation. Bidder 1's strategy is a number s such that he continues to demand two units until the price reaches s: he demands only one unit for all prices between s and 10, and demands zero units at prices greater than 10. It is clear upon reflection that there are only two candidates for Bidder 1's optimal strategy, s = 1 and s = 9. Consider the strategy s = 9. If v = 9, he purchases two units at price 9; if v = 1, he purchases two units at price 1. His expected profit is .5(2 + 18) = 10. Suppose he uses strategy s = 1. If v = 9, he purchases one unit at price 1; if v = 1, he purchases two units at price 1. His expected profit is .5(9 + 18) = 13.5. If Bidder 2's valuation is 9, the final allocation, in which each bidder obtains one unit, is suboptimal. In the optimal allocation Bidder 1 would obtain both units.

5. Ties for the Kth highest bid are broken by randomly allocating the unit(s) to the tied bidders with equal probability.

6. When there is a single unit to be sold, experimental evidence documents several violations of the revenue equivalence theorem. See, for example, Coppinger et al. [1980] or Kagel and Levin [1993].

7. Smith [1967] and Belovicz [1979] also compare the SB and the discriminative auction. Both study environments where the auction is for a security which has a common value to all bidders and therefore the independent private-values structure is not present. Smith finds that the difference in revenue of the two auctions depends on the amount of excess demand. When there is a large amount of excess demand, SB generates more revenue, and when there is a small amount of excess demand the discriminative auction generates more revenue. Belovicz finds that greater excess demand leads to higher prices in SB under single-unit demand but not under multi-unit demand. Burns [1985] compares revenue in multi-unit English auctions, in which the items are auctioned sequentially, one by one, under single-and multi-unit demand and finds that revenue is higher under multi-unit demand than under single-unit demand. In her design, there is an incentive to wait until later auctions when there may be an opportunity to satisfy demand at a lower price. The waiting carries a higher opportunity cost for multi-unit demanders, which may make them less likely to underbid.

8. In the sessions in which the single-unit demand treatment with six subjects was run first, three subjects were chosen randomly after the tenth period to participate in the two-unit demand treatment and the rest were paid and allowed to leave. In the sessions in which the two-unit demand treatment was first and the experiment was run by hand, three subjects were chosen randomly to participate in the first ten periods. The other three observed the auctions and were paid the average of the three bidders' earnings. In the sessions in which the two-unit demand treatment was first and the sessions were run by computer, the subjects were divided into two groups of three and two separate auctions were run. However, because of a computer programming error, the data were only kept for one of the two groups. The error caused the data from one of the groups to overwrite the data generated by the other group for the entire 10 periods. Since the two groups of subjects were drawn from the same population, there is no reason to suppose that any biasing of the data occurred because of the error.

9. In the sessions that were run by computer, in both auctions, bidders did not know the identification numbers of the bidders who performed any of the actions they observed during the course of the session. Otherwise the information available to subjects was the same in the computerized sessions as in the sessions run by hand. In the computerized English clock, they knew the market quantity demanded at each price and could access the data from previous periods. In the computerized SB sessions, they could observe the history of market prices and how many units they won in previous periods.

10. In the single-unit demand treatments 250 units of the experimental currency corresponded to 1 U.S. dollar. In the two-unit demand treatment, the conversion rate was 600 to 1. The different conversion rates for the single-unit and two-unit demand treatments were intended to equalize the expected payoff to subjects across treatments. Since subjects can purchase more units and there were theoretical reasons to expect prices to be lower in the two-unit demand environment, the different conversion rates were expected to offset the greater profit in terms of the experimental currency in the two-unit demand environment.

11. The zero-intelligence efficiency estimates for each period were generated by drawing six random bids, one bid assigned to each of the six valuations drawn that period. Each random bid was drawn from a uniform distribution with a lower bound of zero and an upper bound of the assigned valuation. The efficiencies were computed by taking the sum of the valuations assigned to the four highest bids and dividing by the four highest valuations. This algorithm was repeated 4,000 times for each period to obtain a distribution of efficiencies for the period. The average, standard deviation, and probability of 100% efficiency for a period were calculated from these distributions. The mean efficiency (average over all 20 periods) for an auction composed of ZI traders is .923 with a standard deviation of .02. The highest mean efficiency for any period was for period 11, in which there are valuations of 19 and 22. The lowest mean efficiency was .888 for period 20. The overall probability that ZI traders attain 100% efficiency is .411. The probability is lowest in period nine (.171) and highest in period 11 (.840).

12. This transformation is made to normalize for variations in the competitive equilibrium price between periods resulting from random variations in the valuations drawn by demanders.

13. For the multi-unit demand treatment, the observation is attributed to the Lower Valuation (LV) if it is the first exit by the bidder in the period, and the Higher Valuation (HV) if it is the second exit by the bidder.

14. Cox et al. [1984] identify a region of the (N,K) parameter space in which bidding behavior in single-unit demand discriminative auctions is consistent with violations of the hypothesis of noncooperative behavior. The region includes the parameters in our study. The 11 underrevelations did occur in a session run by hand in which participants were visible to each other, the conditions under which it is probably easiest for subjects to behave cooperatively, since subjects can observe the identity of those performing actions.

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Alsemgeest: Erasmus University, Stadthoudersweg, Rotterdam, The Netherlands, Phone 31-10-466-1124 E-mail palsemgee@gend.nl

Noussair: Assistant Professor, Department of Economics Krannert School of Management, Purdue University West Lafayette, Ind., Phone 1-765-494-4416 Fax 1-765-494-9658 E-mail noussair@mgmt.purdue.edu

Olson: Assistant Professor, Department of Economics Krannert, School of Management, Purdue University West Lafayette, Ind., Phone 1-765-494-4416 Fax 1-765-494-9658 E-mail molson@mgmt.purdue.edu