Fixed revenue auctions: theory and behavior.

Deck, Cary A. ; Wilson, Bart J.

I. INTRODUCTION

There are numerous varieties of auction institutions used in practice and studied in the economics literature. The four standard mechanisms for selling a fixed quantity as in a single unit or lot are the English, Dutch, first-price sealed bid, and second-price sealed bid auctions. The widely known theoretical results are that the English and second-price auctions are equivalent as are the Dutch and first-price auctions. One of the most insightful theoretical results of private value auctions is that under certain assumptions the expected revenue is constant across the four mechanisms (see McAfee and McMillan 1987; Milgrom 1987; Myerson 1981). But behavioral examinations of private value auctions have consistently found that the relationships among the formats are not so straightforward. English clock auctions generate truthful revelation as predicted; however, second-price auctions, which should also generate truthful revelation, do not reliably do so in a laboratory setting (Harstad 2000; Kagel, Harstad, and Levine 1987). Further, revenue equivalence does not hold between first-price and second-price auctions (Coppinger, Smith, and Titus 1980). (1) In part, this is due to the fact that many bidders in first-price auctions act as if they are risk averse. Furthermore, Dutch auctions and first-price sealed bid auctions are not behaviorally isomorphic; observed prices are significantly lower in the Dutch clock auctions than in first-price auctions (Cox, Roberson, and Smith 1982).

Almost exclusively, the focus of previous work has been on auctions determining the selling price for a prespecified lot. (2) However, in some situations, a seller may be more concerned about raising a fixed amount of revenue. For example, a business may sell off just enough inventory to gain the needed liquidity to undertake a particular project. A person may pawn just enough items to secure money with which to pay the monthly bills. Alternatively, in a procurement setting, a buyer may desire to acquire as much as possible for some nonfungible fixed budget. For example, a researcher whose grant is expiring may buy as many supplies as possible with the remaining money or a firm may have a fixed advertising budget with which to buy the most effective campaign. Though not typically thought of in this way, auctions can be used to solve these types of problems as well. (3) We define a fixed revenue auction to be a bidding mechanism in which a prespecified total payment is exchanged for a variable quantity of a good. (4) Wessen and Porter (1997) developed a fixed revenue auction to cover the $326,000 cost of moving antennae for the Cassini mission to Saturn. The auction allowed competing research teams to place bids in terms of the mass they desired on the craft and the price per unit for the mass.

The goal of this paper is to understand the behavioral properties of fixed revenue auctions utilizing the four standard mechanisms. To enable comparisons with the extensive literature on auctions, the environment is designed to allow the maximum similarity between auctions in the two dimensions. This includes placing nontrivial restrictions on buyer values to yield a theoretical equivalence between auction dimensions. Given this, one might be inclined to suppose that the behavioral properties are a forgone conclusion. This need not be the case as evidenced by the aforementioned lack of isomorphism between first-price and Dutch auctions and between English and second-price auctions in the standard setting. Experiments have shown that even simple changes in framing can impact behavior. (5) For example, framing a lottery as a gain or a loss can change the value one places on the lottery and framing the ultimatum game as a buyer-seller interaction can lead to behavior more consistent with material self-interest. A change in auction dimension, however, is more than a simple framing effect; it is a change in the underlying decision problem. A priori, it is not known how a dimensional change will influence behavior. If behavior differs by dimension, the experiments could provide new insights into how people approach these institutions. (6) If behavior is consistent with previous results, then it provides greater confidence in the robustness of previous work.

The next section presents a theoretical treatment of each mechanism in a fixed revenue context. Separate sections discuss the design and results of laboratory experiments investigating behavior in these auctions. A final section contains concluding remarks. As a prelude to our results, we find that under a generalization of the typical assumption regarding values, the theoretical and behavioral properties of the four standard auctions translate to a fixed quantity dimension in a consistent and an intuitive way.

II. THEORETICAL MODEL

We begin by considering the standard, fixed quantity auction format and then identify where and how fixed revenue auctions differ from the familiar model. In the single (fixed)-unit, independent private value auction, there are n bidders who value the lot up for auction. In the English auction, the price starts low and increases until only one bidder remains willing to purchase. The sole remaining bidder buys the item at the final price. (7) The Dutch auction begins with a high price that falls until a bidder agrees to purchase at that price. In contrast, first- and second-price sealed bid auctions are both static in that potential buyers submit sealed bids. For the first-price sealed bid auction, the party submitting the highest bid wins the auction and pays a price equal to his winning bid, whereas in the second-price sealed bid auction, the party submitting the highest bid wins but pays a price equal to the second highest bid.

Assuming a uniform distribution of values over the interval [[v.bar], [bar.v]] and risk-neutral bidders, the following results for the Nash equilibrium bid functions are well known:

(1) b(v) = [v.bar] + [(n- 1)/n](v - [v.bar]) for first-price sealed bid and Dutch clock auctions

and

(2) b(v) = v for second-price sealed bid and English clock auctions.

In the standard auction, the quantity q is set by the seller. To consider fixed revenue auctions, we must generalize the notion of value to be a function of quantity, v(q). Figure 1 shows various possible value functions. Standard, fixed quantity auctions are vertical slice of this figure, as in the dashed line. In a fixed revenue auction, q is the amount of the bid.

Fixed quantity English auctions start with a "low" price that multiple buyers are willing to accept and gradually increase prices thereby becoming less favorable to the bidders. For a fixed revenue auction, a favorable starting position for the buyer would be a large quantity. Making the trade less desirable to the buyer involves reducing the quantity. Thus, in both dimensions, English auctions approach value curves from below. Dutch auctions for a standard, fixed quantity start with a "high" price which gradually decreases, becoming more favorable to the bidders. The parallel for a fixed revenue auction would be to start with an undesirable low quantity which increases to become more favorable to buyers. Thus, in both dimensions, Dutch auctions approach value functions from above.

[FIGURE 1 OMITTED]

The fixed revenue counterparts to the first-and second-price sealed bid auctions would be the last- and penultimate-quantity sealed bid auctions, respectively. In both the first- and the second-price sealed bid auctions, the winner is the agent submitting the bid most favorable to the seller. In the quantity dimension, the most favorable bids are the ones for the smallest quantity. For the last-quantity auction, the winner receives a quantity equal to her bid, while the winner of the penultimate-quantity auction would receive a quantity equal to the second lowest bid.

Before presenting a theoretical model of a fixed revenue auction, we offer the following two comments. First, the fixed revenue auction is a different allocative mechanism in which the bidders face a different decision-problem than in the fixed quantity auction. In a traditional Dutch auction, the price linearly (vertically) approaches an individual's demand curve from above (at a fixed quantity). In a Dutch version of the fixed revenue auction, the price per unit also approaches an individual's demand from above but along a curve as the quantity increases. (8) Second, in what follows, we purposively select a set of assumptions that allows us to compare the theory and behavior of fixed revenue auctions to previous work on standard auctions. We consider this to be a prudent first step in understanding the basic properties of fixed revenue auctions. We are not presuming that these assumptions are appropriate for all or even most applications.

To determine equilibrium behavior in fixed revenue auctions, one cannot simply translate the common assumption of uniform values v(q)~U[[v.bar], [bar.v]] to q~U[[q.bar], [bar.q]] because bidders consider the expected profit from each potential bid. For fixed quantity auctions, a bid reduction of $1 results in an additional profit to the bidder of $1, but in a fixed revenue auction, bidders need to know the value function to determine how an increase of 1 unit impacts the bidder's profits. Thus, as a means for comparing fixed revenue and fixed quantity auctions, we make an additional assumption about the value functions to determine optimal bidding behavior in a fixed revenue auction. A simple functional form which generalizes the uniform distribution assumption is that v = [alpha] + [beta]q, where [alpha]~U[[[alpha].bar], [bar.[alpha]]]. While there are a plethora of assumptions one could make on the form of the value functions, (9) this form simultaneously allows the values associated with a specified quantity to be distributed uniformly and the quantities associated with a specified payment to also be distributed uniformly. Under this assumption, bids are a function of [alpha]. With these value functions, the optimal bids for a standard, fixed quantity auction (Equations 1 and 2) can be rewritten as:

(1') b([alpha]+ [beta]q) = [alpha]+[beta]q + [(n - 1)/n]([alpha] - [[alpha].bar])

and

(2') b([alpha] + [beta]q) = [alpha] + [beta]q.

To determine the optimal bid function in the last-quantity auction with a payment p for a bidder with v = [alpha] + [beta]q, define the breakeven quantity as [??] = (p - [alpha])/[beta]. (10) [??] is thus bounded by [q.bar] = (p - [bar.[alpha]])/[beta] and [bar.q] = (p - [[alpha].bar])/[beta]. Given the one-to-one mapping between [alpha] and [??], let b([??]) be the bid function and [??](b) be the inverse bid function. The probability that the other n - 1 bidders ask for a quantity greater than b is [([bar.q] - [??])/ [[bar.q] - [q.bar]].sup.n-1], and the expected profit from a bid of [bar.b] is ([alpha] + [beta]b - p)[([bar.q] - [??])/[[bar.q] - [q.bar]].sup.n-1]. The first-order condition for profit maximization by a bidder yields Equation (3):

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Through standard manipulation, this yields

b' [beta][([bar.q] - [??]).sup.n-1] - (n - 1) [beta]b[([bar.q] - [??]).sup.n-2] = (n -1)([alpha] - p)[([bar.q] - [??]).sup.n-2[,

which can be simplified to the following bid function:

(4) b = [bar.q] - [(n - a)/n]([bar.q] - [??]).

The similarities between Equations (4) and (1) are clear.

Taking into account that [bar.q] (p - [[alpha].bar])/[beta], Equation (4) can be rewritten as:

b = (p - [[alpha].bar])/[beta] - [(n - 1)/n][([alpha] - [[alpha].bar])/[beta]].

In a first-price, fixed quantity auction, a bidder wants to bid below value to create a profit in the event the bidder wins the auction. The parallel in a fixed revenue auction is to ask for a larger quantity. For a fixed quantity auction, bidders under-reveal by v - b, which can be rewritten as ([alpha] - [[alpha].bar])/n given that v = [alpha] + [beta]q with q fixed. For a bidder with this value function in a fixed revenue auction, the optimal amount of "over-revelation" is

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Equation (5) also yields the optimal stopping rule for the increasing quantity in a Dutch clock fixed revenue auction. The intuition of the isomorphism is the same in the quantity dimension as in the price dimension.

As in the price dimension, truthful revelation is the dominant strategy in the English quantity clock and penultimate-quantity auctions, b = [??] = (p - [alpha])/[beta]. Intuitively, in the penultimate-quantity auction, asking for a larger quantity lowers the likelihood of winning but does not change the amount of the payoff conditional on winning. If asking for a smaller quantity causes a bidder to win that would not have won with truthful revelation, then the bidder would be worse off than having not won the auction. If the bidder would have won anyway, then lowering the bid would not change the payoff. This is also true for the English clock auction.

The familiar expected price in a standard, first-price fixed quantity auction is [v.bar] + [(n - 1)/(n + 1)]([bar.v] - [v.bar]) or replacing v with [alpha] + [beta]q is ([[alpha].bar]+bq)+ [(n - 1)/(n + 1)]([bar.[alpha]]- [[alpha].bar]). The translated calculation for the last-quantity fixed revenue auction gives an expected quantity of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Taking into account that [bar.q] (p - [[alpha].bar])/[beta], the right hand side of Equation (6) can be rewritten as [bar.q] - [(n - 1)/(n + 1)]([bar.q] - [q.bar]). The translation of revenue equivalence holds as well. That is, given the form of the value function and the uniform distribution of the [alpha]'s, each of the four auction mechanisms generates the same expected quantity conditional on payment. Further, the expected payment in an auction where the quantity is fixed at the level expected in a fixed revenue auction with payment [p.sup.*] is [p.sup.*]. That is E(p|q = E(q|[p.sup.*])) = [p.sup.*]. Similarly, E(q|p = E(p|[q.sup.*]))=[q.sup.*].

III. EXPERIMENTAL DESIGN AND PROCEDURES

Given that observed behavior in laboratory experiments with fixed quantity auctions sometimes differs from the theoretical predictions, we conducted a series of laboratory auctions to explore how people behave in fixed revenue auctions. We designed this experiment to determine what, if any, behavioral differences arise when the dimension of the auction is changed. Because the ordering of observed prices in standard auctions is well established, our experiment seeks to determine if this ordering is maintained when it is the revenue that is prespecified.

As detailed by Cox, Roberson, and Smith (1982), maintaining similar message spaces and expected payoffs across treatments is imperative in an auction experiment. Here, this entails comparability not only across institutions but also across the dimension of the auction, fixed revenue versus fixed quantity dimension. It is straightforward to calculate that the expected profit of a bidder in the fixed quantity auction is ([bar.[alpha]] - [[alpha].bar] /[n(n + 1)], while the expected profit to a bidder in the fixed revenue auction is ([bar.[alpha]] - [[alpha].bar])/[[beta]n(n+ 1)]. To maintain similarity, the slope parameter 13 is set equal to 1. With this slope for the value functions, [bar.v] - [v.bar] = [bar.q] - [q.bar] so that the vertical distance between value-curves is the same as the horizontal distance.

The other parameters in the experiments were as follows. Each auction had n = 5 bidders. The random components of the values functions, the [alpha]'s, were distributed U[-10, 5], which was public information among the bidders. With these parameters, the risk-neutral expected profit of the winner was $2.50 per auction. This choice of distribution also gives symmetry in the expected price and quantity. In the standard price auctions, the quantity is fixed at 13, thus yielding an expected price ([[alpha].bar]+bq) + [(n - 1)/(n + 1)]([bar.[alpha]] - [[alpha].bar]) = 13. Similarly, the price is set at 13 in the fixed revenue auction, which according to Equation (6) gives an expected quantity of 13. (11) With these parameters, maximum willingness to pay is distributed U[3, 18] and minimum acceptable quantity is distributed U[8, 23].

The English and Dutch auctions require additional parameterization in the form of clock increments and starting and stopping amounts. All clock increments were set at 0.1, which was also the discreteness allowed for bids in the sealed auctions, (12) thereby maintaining an identical message space in each treatment. In a fixed quantity auction, the natural starting price for the English auction and natural stopping price for the Dutch auction are zero. (13) However, there does not appear to be a natural starting price for the Dutch auction or stopping price for the English clock. This ambiguity problem does not occur in the quantity dimension. When it is the revenue that is fixed, zero serves as a natural starting quantity for the Dutch auction and a natural stopping quantity for the English auction. The seller's total inventory is a natural starting quantity for the English clock and stopping quantity for the Dutch clock. (14) To maintain parity between the dimensions of the auction, it is important that the starting point be as far from the expected termination point in both cases. That is, the Dutch price should start as far above the expected price as the Dutch quantity starts below the expected quantity. Since the natural starting place of 0 for the Dutch quantity auction is 13 units below the expected quantity of 13, the appropriate starting price for the Dutch auction is 13 units above the expected price of 13 which is 26. The natural stopping point for the Dutch price is 0, which is 13 units below the expected price, and thus, the appropriate stopping point for the Dutch quantity is 13 units above the expected quantity; hence, the lot size is set at 26. For the English auction, the natural starting price is 0, which is 13 units below the expected price of 13. Therefore, in the English quantity auction, the appropriate starting point is 13 units above the expected quantity of 13 which is 26. Similarly, the natural ending point for the English quantity auction is 0, and thus, for the English price auction, it is 26. See Figure 2 for a graphic representation of this parallelism in the experimental design. These parameters also have the desirable feature that all four clock auctions should on average take the same amount of time to determine a winner as the expected price and quantity are 13. For consistency, sealed bids also had to be between 0 and 26.

We conducted experimental sessions for each of the four mechanisms in a fixed revenue setting. For calibration, we also conducted auctions with a fixed quantity. But given the large literature on sealed bid, single-unit (lot) auctions, we investigated only the English and Dutch price clock auctions. The total number of laboratory sessions was 24: four replications of each treatment in the 2 x 2 design of {FixedRevenue, FixedQuantity} x {Dutch, English} and four replications of each of the two sealed bid fixed revenue institutions.

[FIGURE 2 OMITTED]

In each treatment, subjects were shown their random a. In the fixed quantity auctions, a subject's screen indicated that her value equaled her random component plus 13. A subject's screen also displayed the current clock price and the profit that the subject would receive if she bought at the current clock price. Subjects in the fixed revenue auctions were shown that the price was always 13. In the fixed revenue clock auctions, the subjects were shown the component of their value that came from the clock and what their total value would be if they were to buy at the current clock quantity. In the sealed bid auctions, subjects were told that they could type a quantity component bid. After a subject entered such a bid, his value and profit conditional on winning were updated on the screen. As explained to the subjects, these numbers were a lower bound on profit and value in the penultimate-quantity auction. Unlike in the clock auctions, subjects confirmed their bids in the sealed auctions. Subjects received feedback after each period in terms of the market price or quantity. This information along with their private information and bids was displayed in a table on the subject's screen.

In each session, subjects first read written instructions and then participated in eight unpaid practice periods. (15) After the practice rounds, the experiments continued for an additional 15 periods. (16) Hence, the data set includes 120 subjects and 360 auctions. Subjects were randomly recruited from classes at the University of Arkansas and only participated in one auction mechanism with either the fixed revenue or the fixed quantity dimension. The laboratory sessions lasted less than 1 h, and subjects were paid $5.00 for showing up on time plus their salient earnings which averaged approximately $5.01 across all treatments.

IV. RESULTS

In what follows, we report our results as a series of six findings. The careful construction of the experimental environment allows for a direct comparison between dimensions even though the participants bid prices in one case and quantities in the other. Given the random draw, [alpha], a bidder's value curve was a line with slope [beta] = 1 that is parallel to the extreme cases shown in Figure 2. For each subject, one observes a bid for the fixed quantity or payment and how much this deviates from the point on the bidder's value curve associated with the fixed quantity or payment. Being further below the curve results from a lower bid in the fixed quantity auction, but in the fixed revenue auctions, it is the consequence of a higher bid. Comparisons across dimensions are thus measured relative to a line with slope 1. While any such line would work, it is natural to think of the fixed quantity auction as a standard and bids in such auctions as a distance above 0, the minimum possible bid. For fixed revenue auctions, one needs the distance below the maximum bid of 26. We define the variable [Transaction.sub.ijs] as the standardized transaction of subject s in session i and auction j. For the fixed quantity auction, [Transaction.sub.ijs] = [Price.sub.ijs], which is the observed price. If the session is a fixed revenue auction, then [Transaction.sub.ijs] = 26 - [Quantity.sub.ijs], where [Quantity.sub.ijs] is the observed transaction quantity for a fixed revenue of 13. (18)

We begin by comparing the transactions in the fixed revenue and fixed quantity versions of the Dutch and English clock auctions. We employ a linear mixed-effects model as the basis for the quantitative support for this and other findings. The treatment effects (Dutch vs. English auctions and FixedRevenue vs. FixedQuantity) and an interaction effect from the 2 x 2 design are modeled as (0-1) fixed effects, while the 16 independent sessions and winning bidders within the sessions are modeled as random effects [e.sub.i] and [[zeta].sub.is] respectively. As a control for the across-auction variation of the realizations of the [alpha]'s, we include deviations of the relevant kth highest realization from their theoretical expected values, denoted by [a.sub.k]. (19) We do this because the predicted standardized transactions in each round are conditioned on the observed [alpha] realizations; so, the location of the second highest [alpha] should not matter in a first-price or last-quantity auction but should identify the transaction amount in a second-price or penultimate-quantity auction. For [alpha] ~ U[-10, 5], the expected values of the highest and second highest realization of five draws are 2.5 and 0, respectively. Specifically, the model that we estimate via maximum likelihood is:

[Transaction.sub.ijs] = [mu] + [e.sub.i] + [[zeta].sub.is] + [[beta].sub.1][Dutch.sub.i] + [[beta].sub.2][FixedRevenue.sub.i] + [[beta].sub.3][Dutch.sub.i][FixedRevenue.sub.i] + [[phi].sub.1][a.sub.1,ij] + [[phi].sub.2][a.sub.2,ij] + [[gamma].sub.1][a.sub.1,ij][Dutch.sub.i] + [[gamma].sub.2][a.sub.2,ij][Dutch.sub.i] + [[delta].sub.1][a.sub.1,ij][FixedRevenue.sub.i] + [[delta].sub.2][a.sub.2,ij][FixedRevenue.sub.i] + [[eta].sub.1][a.sub.1,ij][Dutch.sub.i][FixedRevenue.sub.i] + [[eta].sub.2][a.sub.2,ij][Dutch.sub.i][FixedRevenue.sub.i] + [[epsilon].sub.ijs],

where [e.sub.i] ~ N(0, [[sigma].sup.2.sub.1]), [[zeta].sub.is] ~ N(0, [[sigma].sup.2.sub.2]) and [[epsilon].sub.ijs] ~ N(0, [[sigma].sup.2.sub.3])

FINDING 1. Consistent with previous work, standardized transactions in a Dutch clock auction are greater than standardized transactions in an English clock auction for both fixed revenue and fixed quantity mechanisms. There is no difference in the standardized transactions in the fixed revenue and fixed quantity mechanisms.

Table 1 reports the estimates for the above model. The benchmark for the treatment effects is the fixed quantity English auction. The point estimate for the average standardized transaction in this treatment, [??] = 13.07, is nearly identical to the expected theoretical standardized transaction of 13. From this, we can infer that the bidders are following their dominant strategy to bid until the price exceeds their value. The Dutch clock institution has significantly higher standardized transactions, increasing the standardized transaction amount by [[??].sub.1] = 0.74 (p value = .0130). This result is consistent with risk-averse bidding and with previous work. However, none of the terms involving FixedRevenue are statistically significant, individually or jointly (Likelihood Ratio statistic = 2.46, p value = .8729). Hence, we conclude that transaction amounts in fixed revenue auctions are equivalent to those in fixed quantity auctions when the settings are directly comparable.

FINDING 2. The dimension of the auction, fixed revenue or fixed quantity, does not affect efficiency in the Dutch clock auction but does (marginally) affect efficiency in the English auction.

All four clock auctions were highly efficient. Average efficiency in the Dutch quantity clock and Dutch price clock auctions was 99.1% and 97.5%, respectively. Average efficiency in the English quantity clock and English price clock auctions was 97.2% and 99.5%, respectively. Figure 3 plots the average efficiency over the 15 periods for each session (by treatment). Efficiency is defined as the winning bidder's surplus divided by the maximum possible surplus. Using the average efficiency in a session as the unit of observation, we cannot reject the null hypothesis that the Dutch quantity clock and Dutch price clock auctions are equally efficient based upon the Wilcoxon rank-sum test ([U.sub.4,4] = 8, p value = 1.0000). There is marginal evidence to reject the null hypothesis that the English quantity clock and English price clock auctions are equally efficient ([U.sub.4,4] = 15,p value = .0571). The magnitude of this difference is relatively small as the "poorer" performing English quantity clock auction was 97.2% efficient.

[FIGURE 3 OMITTED]

FINDING 3. Consistent with previous work, Dutch clock fixed revenue auctions are not behaviorally isomorphic to last-quantity sealed bid auctions.

Cox, Roberson, and Smith (1982) reported the same result for fixed, single-unit auctions. Table 2 reports the estimates of a linear mixed-effects model that tests the theoretical isomorphism between Dutch clock and last-quantity sealed bid auctions. This data set includes four sessions of Dutch clock auctions and four sessions of last-quantity sealed bid auctions. The dependent variable is the transaction quantity, and the treatment effect of interest is the institution. (20) Since we expect ex ante that the bids are a function of the highest [[alpha].sub.i], we also include the [[alpha].sub.1] variable as a control for variation of the [alpha] draws. The average quantity in the Dutch clock fixed revenue auctions is [??] = 12.17 which is less than risk-neutral prediction of 13 and consistent with risk-averse bidding. (Recall that the Dutch quantity starts at 0 and increases until the first bidder accepts the quantity on the clock.) Last-quantity sealed bid auctions have even more risk-averse outcomes, lowering transaction quantities by [[??].sub.1] = -0.47 (p value = .0460). In the Dutch auctions, the estimated slope of the quantity bid function is [[??].sub.1] = -0.87, which is very close to the slope of the bid risk-neutral function of -(n - 1)/ n = -0.80. There is no evidence that last-quantity bid functions are steeper [[??].sub.1] = 0.02 (p value = .6981). Cox, Smith, and Walker (1983) concluded that this difference is the result of bidders improperly updating their priors as opposed to the "excitement" of watching the clock to continuing tick. Panel (A) of Figure 4 plots the winning quantity bids against the winning bidder's value for that quantity along with the risk-neutral prediction. Panel (B) of Figure 4 plots the same information in manner more consistent with previous auction experiments; the x-axis has the bidder's value parameter and the y-axis is the actual bid. For the familiar fixed quantity auction, the risk-neutral prediction would be upward sloping since a higher value for the fixed number of units would lead to a higher price bid and risk-averse bidders would bid above that prediction. For fixed revenue auctions, a higher a leads to a lower quantity bid, so the risk-neutral bid function is downward sloping and risk-averse bidders would bid below this prediction.

[FIGURE 4 OMITTED]

FINDING 4. Dutch clock and last-quantity sealed bid auctions are equally and highly efficient.

There are only 11 auctions out of 120 that are less than 100% efficient: 5 Dutch clock and 6 last-quantity sealed bid. The average efficiency is 99% over the 60 auctions for each institution (see Figure 3). Using a Wilcoxon rank-sum test, we cannot reject the null hypothesis that the two institutions are equally efficient ([U.sub.4,4] = 8.5, p value =.8857).

FINDING 5. Consistent with previous work, English clock fixed revenue auctions are not behaviorally isomorphic to penultimate-quantity sealed bid auctions.

Kagel and Levin (1993) found that in single-unit, second-price sealed bid auctions, bidders consistently bid higher than the dominant strategy prediction, even with experience in the auction mechanism. They speculated that bidders fall to the illusion that bidding higher is a low-cost means of increasing the probability of winning. We also find that bidders in the penultimate-quantity sealed bid auction similarly over-reveal (by submitting quantities less than their dominant strategy prediction). Because English auctions are conducted in real time, they provide immediate and overt feedback as to what a bidder should and should not bid, and so bidders adopt the strategy very quickly. Sealed bid auctions do not offer such feedback.

Table 3 reports the estimates of a linear mixed-effects model that tests the theoretical isomorphism between English clock and penultimate-quantity sealed bid auctions. This data set includes four sessions of English clock auctions and four sessions of penultimate-quantity sealed bid auctions. The model includes the [a.sub.2] variable as a control on the variation of the second highest [alpha]. The average quantity in the English fixed revenue auction is [??] = 13.23, which compares quite favorably to the theoretical prediction of 13. The penultimate-quantity sealed bid auctions have lower transaction quantities by [[??].sub.2] = -0.58 (p value = .0344). As predicted, the relationship between the second highest a and the transaction quantities is almost exactly -1 ([[??].sub.2] = - 0.97,p value <.0001). Figure 5 plots bids against the second lowest [alpha] along with the dominant strategy prediction. Again, as opposed to the more familiar fixed quantity setting in which a higher value curve leads to a higher "price" bid, in the fixed revenue auction a higher [alpha] leads to a lower "quantity" bid. In Panel (A), only the quantity-setting bids are plotted against that bidder's [alpha]. This panel clearly displays the Kagel and Levin observation of over-revelation in the penultimate-quantity sealed bid auctions. Just as Cox, Roberson, and Smith (1982) reported, subjects also "throw away" bids in the English auctions when they receive a very low at and do not expect to win the auction. Hence, they exit nearly immediately.

FINDING 6. We cannot reject the null hypothesis that penultimate-quantity sealed bid auctions are as efficient as English clock auctions.

As Figure 3 indicates, three of the penultimate-quantity sealed bid auctions are rather inefficient; a full 10 percentage points below the minimum efficiency observed in any of the other conditions. The fourth session, however, is highly efficient. As a result, we cannot reject the null hypothesis that the two institutions are equally efficient ([U.sub.4,4] = 13, p value = .2000).

[FIGURE 5 OMITTED]

V. CONCLUSIONS

The Dutch price, English price, first-price sealed bid, and second-price sealed bid auctions are all commonly used institutions that have been studied extensively both theoretically and empirically. This paper establishes the theoretical and behavioral properties of these standard auction mechanisms employed to raise a fixed amount of revenue for the seller. While this represents a shift in how auctions can be used, under one generalization of the standard assumptions regarding values, the predictions are similar to standard theory. Not surprisingly, with this assumption, a bidder in the English quantity or penultimate-quantity sealed bid auction should truthfully reveal her value, while a bidder in the Dutch quantity or the last-quantity auction does not. However, if values are not linear in quantity, the complexity of the auction is dramatically increased. Our choice of value functions is restrictive to enable a direct behavioral comparison with previous research and is not held to be a general description of bidder values. This is an area deserving further study. A related issue pertains to mechanism design: when is a fixed revenue auction optimal? The answer to this can depend on a variety of factors including the cost structure of the seller, the distribution of buyer value functions, and the transactions and contracting costs associated with conducting the auction and the resulting trades. While these are important issues to be resolved, they are beyond the scope of this paper. Our goal is to understand the basic properties of such an auction. We take as given that the seller is attempting to receive a fixed payment for the least quantity (cost) while dealing with a single buyer.

The results of our experiment indicate that auction outcomes are not affected by the dimension of the auction. Bidders truthfully reveal in the English auction but not in the penultimate-quantity sealed bid auction. Bidders attempting to game this sealed bid institution nominally lower efficiency relative to the English auction. As in the price dimension, the last-quantity auction and the Dutch clock auction are not behaviorally isomorphic. The sealed bid institution leads to lower quantities (analogous to higher prices in a fixed quantity auction). However, the two institutions are equally and highly efficient. Quantity equivalence does not hold across institutions, just as revenue equivalence has regularly been found not to hold in the standard setting. In fact, the dimension of the auction does not change the magnitude or ordering of the differences between auction institutions. These findings suggest that auction behavior is extremely robust and thus lends additional credence to previous work.

REFERENCES

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(1.) See Lucking-Reiley (1999) and Hansen (1985, 1986) for tests of revenue equivalence with naturally occurring field data. Maskin and Riley (1985) and Riley (1989) discussed possible explanations for the failure.

(2.) There is also a class of market auctions that serve to determine both price and quantity. The double auction and the Walrasian auction are two examples of this type of auction. McCabe, Rassenti, and Smith (1990) provided a description of other institutions such as the double Dutch auction.

(3.) Transaction costs can explain why a buyer or seller might want to hold a single auction for a set of items even though it might not achieve as much revenue as auctioning items off separately. Auctions for surplus equipment are often structured such that a bidder has to purchase an entire pallet of used items rather than a single item. Vast tracts of land are rarely auctioned off in square foot parcels. Similarly, transaction costs could explain why someone would prefer to hold a fixed revenue auction, thus guaranteeing a single transaction.

(4.) Dastidar (2006) also considered such auctions.

(5.) A framing effect is the observation that people make different decisions for the same available actions over same outcomes when they (presumably) conceive the decision-making problem as being different. For example, the visual electronic interface of an online English auction is a different decision-frame than an oral English auction with a live auctioneer and bidding paddles.

(6.) Given the complexity of the mathematical optimization problem, it seems doubtful that most bidders actually construct the equilibrium bid function.

(7.) There are two different formats of English auctions, an outcry version in which bids come from the floor and jump bidding is possible and a clock version in which the bid price is controlled by the auctioneer. Our paper focuses on the latter, and the interested reader is referred to Kamecke (1998) for a discussion of the theoretical differences between the two.

(8.) While multiunit variations of standard auctions, such as the double Dutch (McCabe, Rassenti, and Smith 1990), endogenously determine quantity, in those auctions each bidder is deciding if she should trade a fixed unit for the bid price.

(9.) See Dastidar (2006) for a more generalized discussion of auctions in which the bids are in terms of quantities.

(10.) At q = [??], v = p. In the standard, fixed quantity auction, a bidder's value is the breakeven price.

(11.) The parameters were selected so that the expected price and quantity were not focal points.

(12.) Note that with n = 5 bidders, the slope of the bid function is 0.8 = (n - 1)/n which can be realized with an increment of 0.1.

(13.) Stopping price refers to a price at which the auction is ended. In a Dutch price auction, the seller's reserve price might also serve as a stopping price.

(14.) Not having a stopping quantity in a Dutch quantity auction exposes the seller (experimenter) to the possibility of an infinite loss. The same would be true in a Dutch price auction if the price were allowed to become negative and fall indefinitely.

(15.) A copy of the instructions is available from the authors upon request.

(16.) The sealed bid auctions ran much more quickly than the clock auctions, so as many as 15 additional auctions were also conducted. The results do not differ in any meaningful way when including these auctions, so for the sake of parsimony, the analysis focuses only on the first 15 auctions in each session.

(17.) Recall that the expected profit to the winner was $2.50 each round. The observed average payoff is the result of aggressive, risk-averse bidding behavior which is detailed in the next section. Risk-averse behavior implies saliency in the rewards, that is, subjects are earnestly engaged in the bidding task.

(18.) For example, a price of 17 which is 4 units above the expected price of 13 is the same deviation as a quantity of 9 which is 4 units below the expected quantity of 13. Formally, we take transaction amount = expected price + (expected quantity--observed quantity), which is 26-observed quantity. In the example, 26-9 = 17. This is related to the over-revelation of Equation (5).

(19.) A priori, we expect that the standardized transaction in the Dutch auction is dependent upon the highest realization of [alpha], while it is dependent upon the second highest realization in the English auction.

(20.) Findings 3 and 5 focus only on fixed revenue auctions and thus rely upon the actual quantity rather than the standardized transaction amount used in Finding 1 for comparing behavior in different dimensions.

CARY A. DECK and BART J. WILSON, We wish to thank seminar participants at CIRANO and the University of Alaska Anchorage as well as two anonymous referees for helpful comments. Financial support from the Walton College of Business is gratefully acknowledged.

Deck: Associate Professor, Department of Economics, University of Arkansas, Fayetteville, AR 72701. Phone 1-479-575-6226, Fax 1-479-575-3241, E-mail cdeck@walton.uark.edu

Wilson: Associate Professor, Interdisciplinary Center for Economic Science, George Mason University, Fairfax, VA 22030. Phone 1-703-993-4845, Fax 1-703-993-4851, E-mail bwilson3@gmu.edu

TABLE 1
Estimates of the Linear Mixed-Effects Model of Standardized
Transactions for Fixed Revenue versus Fixed Quantity
[AuctionsTransaction.sub.ij] = [mu] + [e.sub.i] + [[zeta].sub.is]
+ [[beta].sub.1][Dutch.sub.i] + [[beta].sub.2]
[FixedRevenue.sub.i] + [[beta].sub.3][Dutch.sub.i]
[FixedRevenue.sub.i] + [[phi].sub.1][a.sub.1,ij] +
[[phi].sub.2][a.sub.2,ij] + [[gamma].sub.1][a.sub.1,ij]
[Dutch.sub.i] + [[gamma].sub.2][a.sub.2,ij][Dutch.sub.i]
+ [[delta].sub.1][a.sub.1,ij][FixedRevenue.sub.i] +
[[delta].sub.2][a.sub.2,ij][FixedRevenue.sub.i] + [[eta].sub.1]
[a.sub.1,ij][Dutch.sub.i][FixedRevenue.sub.i] +
[[eta].sub.2][a.sub.2,ij][Dutch.sub.i][FixedRevenue.sub.i]
+ [[epsilon].sub.ijs]

 Standard Degrees of
 Estimate Error Freedom (a)

[mu] 13.07 0.18 151
Dutch 0.74 0.25 12
FixedRevenue -0.26 0.25 12
Dutch x FixedRevenue 0.25 0.36 12
[a.sub.1] -0.06 0.08 151
[a.sub.2] 0.98 0.07 151
[a.sub.1] x Dutch 0.85 0.12 151
[a.sub.2] x Dutch -0.93 0.09 151
[a.sub.1] x FixedRevenue 0.12 0.13 151
[a.sub.2] x FixedRevenue -0.09 0.10 151
[a.sub.1] x Dutch x FixedRevenue -0.16 0.17 151
[a.sub.2] x Dutch x FixedRevenue 0.07 0.13 151
 LR: [[beta].sub.2] = [[beta].sub.3] =
 [[delta].sub.1] = [[delta].sub.2] =
 [[eta].sub.1] = = [[eta].sub.2] = 0
 238 observations (b)

 t Statistic p Value

[mu] 74.18 <.0001
Dutch 2.91 .0130
FixedRevenue -1.05 .3153
Dutch x FixedRevenue 0.71 .4905
[a.sub.1] -0.66 .5093
[a.sub.2] 14.77 <.0001
[a.sub.1] x Dutch 7.13 <.0001
[a.sub.2] x Dutch -10.13 <.0001
[a.sub.1] x FixedRevenue 0.96 .3401
[a.sub.2] x FixedRevenue -0.90 .3706
[a.sub.1] x Dutch x FixedRevenue -0.94 .3487
[a.sub.2] x Dutch x FixedRevenue 0.58 .5659
 2.46 .8729

 238 observations (b)

(a) The linear mixed-effects model for repeated measures treats
each session as of one degree of freedom with respect to the
treatments in the 2 x 2 design: Dutch, FixedRevenue, and Dutch x
FixedRevenue variables. Hence, the degrees of freedom for the
estimates of these fixed effects are 12 = 16 sessions - 4
parameters. The linear mixed-effects model is fit by maximum
likelihood with 16 groups. For brevity, the session random
effects are not included in the table.

(b) In two Dutch clock price auctions, a subject inadvertently
clicked on the Buy button nearly immediately after the auction
started. Omitting any one of 238 included data points does not
change the above estimates in any discernable way. However, these
two outliers exert undue influence on the estimates, i.e., bias
the estimates, and are excluded.

TABLE 2
Test of Dutch Clock and Last-Quantity
[IsomorphismQuantity.sub.ijs] = [mu] + [e.sub.i] +
[[zeta].sub.is] + [[beta].sub.1][Sealed.sub.i] + [[phi].sub.1]
[a.sub.1,ij] + [[gamma].sub.1][a.sub.1,ih][Sealed.sub.i] +
[[epsilon].sub.ijs]

 Degrees
 Estimate Standard Error of Freedom

[mu] 12.17 0.14 79
Sealed -0.47 0.19 6
[a.sub.1] -0.87 0.04 79
[a.sub.1] x Sealed 0.02 0.05 79

 t Statistic p Value

[mu] 87.78 <.0001
Sealed -2.51 .0460
[a.sub.1] -21.32 <.0001
[a.sub.1] x Sealed 0.39 .6981

TABLE 3
Test of English Clock and Penultimate-Quantity
[IsomorphismQuantity.sub.ijs] = [mu] + [e.sub.i] +
[[zeta].sub.is] + [[beta].sub.1][Sealed.sub.i] + [[phi].sub.2]
[a.sub.2,ij] + [[gamma].sub.2][a.sub.2,ih][Sealed.sub.i] +
[[epsilon].sub.ijs]

 Degrees of
 Estimate Standard Error Freedom

[mu] 13.23 0.11 80
Sealed -0.58 0.21 6
[a.sub.2] -0.97 0.04 80
[a.sub.2] x Sealed 0.01 0.08 80
 120 observations

 t Statistic p Value

[mu] 124.96 <.0001
Sealed -2.73 .0344
[a.sub.2] -23.00 <.0001
[a.sub.2] x Sealed 0.13 .8983
 120 observations