Individual behavior and bidding heterogeneity in sealed bid auctions where the number of bidders is unknown.

Isaac, Mark ; Pevnitskaya, Svetlana ; Schnier, Kurt S. 等

I. INTRODUCTION

Most of the work in sealed bid auction theory and related experimental work has focused on the case in which the number of bidders is known. This is, of course, not always the case in naturally occurring environments. For instance, construction firms submitting bids in a sealed bid procurement auction may not know exactly how many other firms have completed the costly process of preparing and submitting a bid until bidding has closed. There is a body of theory to address environments with an unknown number of bidders (McAfee and McMillan 1987; Matthews 1987; Harstad, Kagel, and Levin 1990; Krishna 2002). (1) The purpose of this article is to provide theoretical considerations and analysis of a large data set of individual bids from laboratory sealed bid auction markets in which buyers have some information about the number of rivals, but not certainty. (2)

According to the received theory, the availability of information about the number of rivals, as expected, directly affects bidding and therefore seller revenue in first-price (FP) auctions but should not affect bidding in the second-price (SP) auctions. Although bidding own valuation is optimal in SP auctions (regardless of the number of bidders), the majority of previous experimental work on auctions with known number of bidders has shown systematic deviation (Kagel 1995) while a few studies report coincidence with theory (see Isaac and James 2000b for an extended discussion). Earlier work on FP auctions with known number of bidders indicated systematic overbidding compared to risk neutral equilibrium. Risk aversion has been suggested as one of the possible explanations for observed overbidding (Cox, Roberson, and Smith 1982).

Dyer, Kagel, and Levin (1989) is the only previous experimental study looking at individual bidding in auctions where the number of bidders is unknown and their design is different from this study. They look at FP auctions with restricted auction size of either six or three participants and ask subjects to submit three bids (two bids for contingent bidding procedure and one bid for noncontingent bidding procedure). Subjects submit all three bids simultaneously and do not know at the time which procedure will be used. They find overbidding in this environment compared to risk neutral Nash equilibrium. Although we test for deviation from risk neutral bidding strategy in a more general setting, our study also addresses a broader set of questions. Previous studies of open-bid ascending "silent" auctions suggested that there may have been revenue loss to the sellers by bidders "guarding" their higher valued items, that is, remaining physically proximate to the bidding stations (Isaac and Schnier 2005). The possibility that "guarding" was at the root of the revenue reductions was investigated by Isaac and Schnier (2006), who replaced the open-bid ascending silent auctions with both FP and SP sealed bid auctions, which is the environment studied in this article. Their paper reported on the aggregate revenue effects of the switch to sealed bid auctions but did not look at individual behavior. It is clear that such analysis of individual bidding behavior must be informed by theories of bidding with an unknown number of rivals, because such is the reality facing bidders in a multiple sealed bid silent auction context. In addition, data from a noncomputerized experiment closely approximating field auctions allows the analysis of possible behavioral adjustments in the field, for example, the attempt to "guard" the item and count the number of bidders. Such an analysis is presented in this article. (3)

Our analysis of FP auctions contributes to the discussion about the causes of heterogeneous deviation of individual bidding from risk neutral theory. Cox, Roberson, and Smith (1982) and Cox, Smith, and Walker (1988) reported heterogeneous bidding above risk neutral equilibrium in FP auctions for mechanisms with fixed number of bidders and Palfrey and Pevnitskaya (2008) for auctions with endogenous entry, where the number of bidders was known at the time of submitting a bid. Those papers show that such deviations are consistent with heterogeneous risk preferences. Risk aversion, however, is not the only suggested explanation for deviations from risk neural Nash equilibrium in FP auctions. Proposed explanations of overbidding include social comparisons, learning, and regret, which have been addressed by recent experimental studies (Engelbrecht-Wiggans and Katok 2007, 2008; Filiz-Ozbay and Ozbay 2007). Risk aversion remains one of the plausible options and we investigate it in the new environment where the number of bidders is unknown. We show theoretically that in the environment with additional uncertainty about the number of bidders, bidding above risk neutral equilibrium remains consistent with risk averse preferences. We find a substantial amount of coincidence with theory in SP auctions but observe systematic deviations from risk neutral bidding in FP auctions in terms of level effect. We test for heterogeneity in two separate ways. First, we use latent class estimation to test for heterogeneity without imposing any structural form on the behavior, that is, not restricting our focus by any conjecture on what may drive deviations. We find evidence of heterogeneity in FP but not SP auctions; furthermore, in FP auctions heterogeneity persists with experience. This is consistent with theory as the number of bidders and risk preferences do not affect optimal bid in the SP auctions but are part of the optimal bidding strategy in the FP auctions. Second, we apply a structural form of nonlinear utility (as a result of risk preferences) and estimate risk preference parameter of each subject and also of groups of subjects allocated into segments by the latent estimation technique. We then check for consistency between the two estimation approaches. We find that heterogeneity in bidding in the FP auctions is consistent with heterogeneity in risk preferences, the attempt to count the number of bidders in the auction, and bidder specific noise.

The rest of the article is organized as follows. In Section II, we present theoretical results on bidding in silent auctions. Section III offers the experimental design and hypotheses. Estimation methods are described in Section IV. The experimental results appear in Section V. Section VI offers our conclusions.

II. A THEORY OF BIDDING WHERE "n" IS UNKNOWN

In the sealed bid silent auctions bidders do not know at the time they are bidding on any specific item how many of the other bidders will be active rivals. Specifically, at the time of submitting a bid, each bidder knows the auction mechanism (here FP or SP), their own valuation of the item, the distribution of valuations of other bidders, and the distribution of the number of bidders in the auction (here over [1, N]). We start with presenting risk neutral theory of bidding in this environment drawing upon McAfee and McMillan (1987), Harstad, Kagel, and Levin (1990), and Krishna (2002).

Not surprisingly, having an unknown number of active bidders, n, out of the N total bidders does not affect the optimal bidding strategy, B(.), for SP sealed bid auctions, namely

(1) B([[upsilon].sub.i]) = [[upsilon].sub.i]

where [[upsilon].sub.i] is bidder i's valuation.

For FP sealed bid auctions, however, the situation is much different. The Nash equilibrium bid function for risk neutral bidders is

(2) [[beta].sub.RN]([[upsilon].sub.i] = [[summation].sub.n] [[omega].sub.n]([[upsilon].sub.i])[[beta].sub.n]([[upsilon].sub.i])

where [[beta].sub.RN]([[upsilon].sub.i]) is a risk neutral bid in an auction with n bidders and the weights, [[omega].sub.n]([[upsilon].sub.i]), are functions of the bidders expectations {[[rho].sub.k]}, which are the probabilities that the bidder is in an auction with k bidders, [[summation].sup.N.sub.k=1] = [[rho].sub.k] = 1. (4) That is, the equilibrium bid function when the bidder "is unsure about the number of rivals he/she faces is a weighted average of the equilibrium bids in auctions when the number of bidders is known by all" (Krishna 2002, 36).

Overbidding in FP single-unit auctions with known number of bidders has been observed in multiple studies with risk aversion being one successful, but not the only, explanation (see Cox, Roberson, and Smith 1982 and Kagel 1995 for the discussion). However, the recent models of bidding in FP auctions where the number of bidders is uncertain focused only on risk neutral participants. (5) In this section, we derive properties of the bidding functions when bidders are risk averse.

Suppose that subjects have income utility U(x), a concave, increasing, differentiable function satisfying U(0) = 0. The symmetric Nash equilibrium bid maximizes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [p.sub.j] is the probability of participating in an auction with j bidders and F([upsilon]) is the cumulative distribution function of bidders' valuations. The optimal bid satisfies the first-order conditions for maximization resulting in the following differential equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the weights [w.sub.j]([upsilon]) = ([F.sup.j-1]([[upsilon].sub.i])[p.sub.j])/[[summation].sub.j] [p.sub.j][F.sup.j-1]([[upsilon].sub.i]). The equilibrium bid function, however, is no longer a weighted combination of bids in corresponding j-bidder auctions. Krishna's proof breaks as it relies on revenue equivalence which no longer holds. The Harsted, Kagel, and Levin derivation relies on linear utility and also no longer holds. Despite this, we can compare bidding functions of risk averse, [beta]([upsilon]), and risk neutral, [[beta].sub.RN]([upsilon]), bidders.

PROPOSITION 1. [beta]([upsilon]) > [[beta].sub.RN]([upsilon])

Proof Following Harsted, Kagel, and Levin, the first-order conditions of the maximization problem of risk neutral bidders results in the following differential equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore inequality [beta]'([[upsilon].sub.i]) [greater than or equal to] [[beta]'.sub.RN]([[upsilon].sub.i]) follows from the concavity of U(x). Following Milgrom and Weber (1982) we conclude that whenever [beta]([[upsilon].sub.i]) [less than or equal to] [[beta].sub.RN]([[upsilon].sub.i]), [beta]'([[upsilon].sub.i]) > [[beta]'.sub.RN]([[upsilon].sub.i]); the equilibrium boundary condition is [[beta].sub.RN]([[upsilon].bar]) = [beta]([[upsilon].bar]) = [[upsilon].bar]. It then follows (Milgrom and Weber, lemma 2) that for [upsilon] > [[upsilon].bar], [beta]([upsilon]) > [[beta].sub.RN]([upsilon]). []

That is, risk averse preferences result in overbidding compared to risk neutral behavior in FP auctions when the number of bidders is unknown.

III. EXPERIMENTAL DESIGN AND HYPOTHESES

We use data from a total of six experimental sessions, each consisting of five auction periods. The two bidding institutions were alternated to allow both within-subject and across-subject comparisons. The five bidding periods of three experimental sessions were sequenced in an FFSSF format; the other three sessions were sequenced SSFFS. Table 1 is a "road map" to the features of this experimental design.

In each period, there were 16 items offered for sale and 8 potential bidders. For each item in each period, the number of active (nonzero value) bidders, between two and eight inclusive, was randomly chosen with a uniform distribution. Then, the subject IDs to receive the positive values were randomly chosen. Finally, each bidder with a positive value for a given auction had their value randomly chosen over the interval (0.00, 20.00] using a uniform distribution. The subjects were informed of these random processes.

Each item was auctioned at a separate station and a physical separation of the bidding stations for different items, just as one would find in naturally occurring silent auctions, was included in the laboratory design. At each bidding station was placed a "supersized" foam coffee mug with an opaque, slotted top. The letter of the associated item was clearly marked on each mug. Each bidder had a pad of bidding slips on which they were required to write their bidder ID, the item being bid on, their bid, and the time (of 7 minutes) remaining on the clock. (6) They inserted the bidding slip into the mug when they wished to bid for an item. (7) The instructions of the experiment can be found in the Appendix. (8)

For the SP auctions, the formal baseline hypothesis is simple: bidders will follow their dominant strategy and bid their value. The experimental auction literature provides a well-known counter-hypothesis, that bidders will generally over-bid their values in a SP sealed bid auction (Cooper 2008).

Figures 1A and 1B display graphs of the risk neutral equilibrium bidding function in the FP auction, [[beta].sub.RN]([[upsilon].sub.i]), for the parameters of our experimental design (r = 1). (9) It should be noted that the bid function here, unlike the standard case of known n, is nonlinear in [[upsilon].sub.i]. As shown theoretically in the previous section, risk averse bidders (0 < r < 1) are expected to bid above [[beta].sub.RN]([[upsilon].sub.i]).

There is a plausible alternative to this theory for the silent auction version of the FP sealed bid auction, what we will call the counting conjecture. It is at least possible that bidders will try to watch other bidders and discern the number of bidders in each of the individual auctions. If so, then the silent auctions are not a part of the standard theoretical framework but simply devolve to a series of simultaneous FP auctions, with bidders bidding according to the formula based on known n. Of course there are many reasons to believe that counting is difficult: bidders must attend to their own decisions and move between different stations; by construction some bids will be placed after others, requiring extensive updating by counters; watching all of the possible rivals may be difficult as they span out across the room and each bidder may place a new bid before the auction ends. (10)

[FIGURE 1 OMITTED]

IV. ESTIMATION METHODS

A. General Analysis of Bidding

For each auction institution we have a relatively robust number of observations. Because the valuations for the items are independently drawn private values, we treat auctions independently. To investigate whether or not bidder behavior is in accordance with the theoretical predictions we estimate a reduced form bid function under both homogeneity and heterogeneity assumptions. The homogeneity assumption implies that all bidders possess the same marginal bidding propensities (regression coefficients), whereas the heterogeneous model allows for there to exist multiple bidding functions within the population. This was achieved utilizing latent class regressions, which have proven to be useful in public good (Anderson and Putterman 2006) and resource extraction experiments (Schnier and Anderson 2006). This method will be further outlined following a general discussion of the bid function estimated.

The bid function we estimate for each auction institution is

(3)

[b.sub.ikt] = [[gamma].sub.0] + [[gamma].sub.1] x [Predicted.sub.ikt] + [[gamma].sub.2] x [Number.sub.kt] + [[gamma].sub.3] x New + [[gamma].sub.4] x New x [Predicted.sub.ikt] + [[gamma].sub.5] x New x [Number.sub.kt]

where [b.sub.ikt] represents participant i's bid on an item with value k in time period t, within the panel for either the FP or the SP auction. Within the SP auction the variable [Predicted.sub.ikt] is bidder i's value for item k in period t, [Value.sub.ikt], which in theory is equal to the bid. In the FP auction we use bidder i's predicted risk neutral equilibrium bid for item k in period t. The variable, [Number.sub.kt] indicates the number of active bidders for item k in period t and is included to investigate the counting conjecture discussed earlier. New is a dummy variable indicating whether or not participant i's bid occurred during their first exposure to either auction institution. (11) Should bidder behavior follow the theoretical predictions we would expect [[gamma].sub.0] = 0, [[gamma].sub.1] = 1, and [[gamma].sub.2] = 0. If the counting conjecture is to be supported we would require a nonzero and significant coefficient [[gamma].sub.2]. If individual i's behavior requires some learning with the institution, we expect some of [[gamma].sub.3], [[gamma].sub.4], or [[gamma].sub.5] to be nonzero. To estimate this bidding function we utilize a double censored tobit regression with an upper bound of $20.00, the maximum induced value in the experiment, and a lower bound of $0.00, the minimum induced value. (12)

For the subset of "experienced" bids Equation (3) degenerates to

(4)

[b.sub.ikt] = [[gamma].sub.0] + [[gamma].sub.1] x [Predicted.sub.ikt] + [[gamma].sub.2] x [Number.sub.kt]

as New = 0 for all "experienced" bids. We estimate Equation (4) to investigate the effect of experience on bidding and heterogeneity by comparing the results to the estimation of Equation (3). Regression results for Equation (3) where we use all data are denoted New = 0, 1 results, indicating the presence of the New dummy variable, and for experienced bidding when New = 0, used in Equation (4), the results are denoted Experienced.

B. Presence and Degree of Heterogeneity in Bidding

We next test for the presence and degree of heterogeneity in bidding. To conduct the heterogeneous estimation we use El-Gamal and Grether's (1995, 2000) estimation classification (EC) algorithm to endogenously group subjects into a prespecified number of bidder types. (13) The EC algorithm assumes that each subject's bids can be described by a bidding function, b([gamma]), as in Equation (3), where [gamma] is an unknown parameter vector. Heterogeneity is introduced by allowing for there to exist a prespecified number of different "types," or segments, H, with each "type" possessing their own parameter vector [[gamma].sub.h]. The estimation of the parameter vectors of each type/segment h = 1, ..., H is conducted simultaneously with the determination of each subject's type (or segment assignment). This is achieved by having each subject's contribution to the likelihood function, [GAMMA] = ([[gamma].sub.1], ..., [[gamma].sub.h]), be the maximum of the joint likelihood of all their bid observations, n, across the H types. (14) In essence, each of the H different parameter vectors is "tested" for all individuals and only that parameter vector which best fits their contribution to the likelihood function is selected. Furthermore, each of the H different parameter vectors is endogenously determined within the maximum likelihood operator and for each iteration of the maximization the parameter vector "tests" are conducted. The log-likelihood function utilizing the EC algorithm and the log-likelihood for the tobit regression, denoted by

L(x), is expressed as

ln[L(b; X|[GAMMA], H)]

= [[summation].sup.m.sub.j=1] arg [max.sub.h] [[[summation].sup.T.sub.t=1] [[summation].sup.n.sub.i=1] x ln[L([b.sub.itj]; [X.sub.itj]|[[gamma].sub.h])]]

where m is the number of subjects in the experiment (48) and [X.sub.itj] is a matrix of independent variables captured in the bidding function (1). The estimates of [GAMMA] are subject to the assumptions regarding the number of "types" within the population. Estimation proceeds by first setting H = 1, the homogeneous model, and then increasing H until the test statistics indicate that a sufficient number of "types" have been specified. The test statistics used to determine the appropriate number of "types" or segments were the Bayesian information criterion (BIC), the Akaike Information Criteria (AIC), the Consistent Akaike Information Criteria (cnAIC), the corrected Akaike Information Criterion (crAIC) and likelihood ratio (LR) test.

SP auction test statistics were not necessary because the models continuously degenerated to the homogeneous case (H = 1). For the FP auctions, we start with conducting the estimation for all data (New = 0, 1) and follow with estimation using only Experienced data as described below. We note that the LR and BIC test statistics generally decrease so we focus on the dynamics of the AIC criteria as the number of segments increases. AIC criteria also decrease but tend to flatten for the four- to five-tier segmentation, indicating the latter. For both data sets we were unable to reach convergence for the six-segment models. Estimations for the New = 0, 1 data show that increasing heterogeneity up to three segments captures heterogeneity in bidding in terms of differences in coefficients on core variables; however, going beyond three segments just subdivides the existing tiers according to the effect of experience while leaving all of the key results intact. Therefore for the New = 0, 1 data additional tiers beyond three segments are just telling us that experience matters for heterogeneity, supporting a separate estimation on the Experienced data which we conduct next. For the experienced data, going from four to five segments assigns just one subject (who is clearly an outlier) to the fifth segment. The lineup of the general bidding propensities of segments remains unchanged. The creation of a one-person segment also raises concerns about beginning to "overfit" the data. If we push this model to its extreme we could end up with a separate segment for each subject. Considering the economic implications and statistical considerations described above and also choosing the most conservative approach for describing the heterogeneity results, we report estimations for three segments for all data and four segments for data with experienced bidding. (15)

The results of these estimations are presented in Tables 2-4. In each table we show homogeneous estimates (H = 1) as a reference point and then report results from the best fitted number of segments.

C. Risk Preferences

We separately conduct a structural estimation of the bidding data and estimate risk preference parameters of individual subjects as well as groups of subjects assigned to the same segment by the latent technique. Our objective is to evaluate the consistency of the risk aversion hypothesis with observed behavior and the heterogeneity classification of the previous part as well as compare our results to the previous studies.

For estimation purposes we adopt CRRA utility function u(x)= [x.sub.r], where r is the risk parameter such that r = 1 for risk neutral bidders, 0 < r < 1 for risk averse bidders, and

r > 1 for risk loving bidders. For CRRA utility function and our experimental design, the differential equation for the optimal bid becomes:

[beta]([[upsilon].sub.i]) = (1/r[[upsilon].sub.i])([[upsilon].sub.i] - [beta]([[upsilon].sub.i])) ([[summation].sup.8.sub.j=2] (j - 1)[([[upsilon].sub.i]/2000).sup.j-1] [p.sub.j]) / [[summation].sup.8.sub.j=2] [([[upsilon].sub.i]/2000).sup.j-1] [p.sub.j]).

The closed form solution for optimal bid cannot be obtained; however, we are able to solve the above differential equation numerically given specific valuation and risk parameter and obtain a theoretical bid. For every valuation, theoretical bid for a given r can be compared to the observed bid. We then estimate risk parameters in the following way. For each experimental observation, we obtain a numerical solution for the optimal theoretical bid based on r. The accuracy of the theoretical prediction is reflected by the difference between theoretical and actual bids. Our estimate of r minimizes the sum of absolute deviations between theoretical and observed bids as illustrated below.

The estimate of the individual risk preference parameter for each subject i is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[upsilon].sub.ij] is the drawn valuation of bidder i in their auction j, [n.sub.i] is the number of auctions that bidder i participated in, [[beta].sup.*] ([[upsilon].sub.ij], r) the optimal theoretical bid (given r and [[upsilon].sub.ij]), and [[beta].sub.ij] the observed bid for value [[upsilon].sub.ij].

The estimator for the whole population of bidders is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For the whole population the estimate r = 0.375 minimizes the above function. This value is similar to those observed in many previous studies (Isaac and James 2000a). The estimates of risk parameters for each segment, [G.sub.k], are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The results of these estimations are presented in Tables 6 and 7.

V. RESULTS

We have 1,050 and 1,013 observations on bidder's value and their corresponding bid in the SP and FP auctions, respectively. (16) Essentially all positive valuations resulted in bids, thus not creating an endogenous entry setup. Only about 6% of positive valuations did not result in submitted bids and this number would be reduced further with less conservative estimation (e.g., disqualified duplicate bids and bids with missing information could indicate the desire to bid on all active units but making errors in the bidding process). The number is comparable to a robust phenomenon of "throw away" zero bids in computerized experimental auctions (where bidders are required to submit a bid) for lower values. The frequency of bids in all treatments decreases over time with essentially no bids right at the closing time. In a sealed bid setup, waiting does not provide information on the amounts of bids of other subjects. Our design makes counting difficult because subjects cannot see the number of already submitted bids in a jar and, given that subjects are active on multiple items, spend substantial time at a particular station to observe the bid submission process. (17)

We next describe our specific findings for SP and FP auctions.

A. SP Auctions

From a purely descriptive point of view, we note that 547 (52.09%) of the bids are equal to value, and that 740 (70.48%) were within 25 cents of value. (18) This compares favorably to other research into dominant strategy mechanisms and is consistent with Isaac and Schnier (2006) who report that average aggregate revenue was very close to the theoretical predictions for these auctions. (19) This result seems robust in our study and is not driven by previous experience in FP auctions.

Turning to the estimation results, we note that the econometric firepower of the latent class process was essentially wasted on the SP auctions, as both the full data set and the data restricted to experienced periods failed to converge beyond a single class. (20) Table 2 shows that for the New = 0, 1 case subjects appear to be following the theoretical predictions except for when they are initially exposed to the SP auction institution. The negative coefficient on the New x Value indicates that subjects bid more than 9% below their value when first exposed to new auction institution. The Value coefficient is not statistically significant from its hypothesized theoretical value (using a t test). (21) The bidding behavior does not depend on the number of bidders in the auction. This is not surprising because in the SP auctions the optimal bid does not depend on n and the counting hypothesis is irrelevant in this case.

Given the statistical significance of some of the "New" variables, we re-estimate the results with only the experienced bidding. The heterogeneous estimation again collapses to a single class outcome. As shown in the right-hand column (Experienced) of Table 2, we find that experienced subjects bid almost exactly like the dominant strategy prediction. A t test on the Value coefficient, the only statistically significant coefficient in the experienced model, confirmed this hypothesis. (22) What chiefly distinguishes the full data set from that restricted to the experienced periods is the much smaller variance around the estimates in the latter model.

B. FP Auctions

As described in Section 4B, we report below the results for the three-segment model for the New = 0, 1 data and the four-segment model for the Experienced data.

Table 3 presents estimation results for homogenous (H = 1) and heterogeneous (H = 3) specifications of the model for all New = 0, 1 data. The results from the homogeneous estimation (H = 1) indicate that subjects, treated as one group, do not behave according to the equilibrium theoretical predictions. Three of the coefficients are statistically different from their hypothesized values. The constant term is positive and statistically significant. Subjects, as an aggregate, are influenced by initial exposure to the auction institution; the significant coefficients on the New and the New x Predicted variables indicate a structural shift in their bid function. The coefficient on the Predicted variable, however, is not significantly different from 1.00 as suggested by theory. These results provide a preliminary look at the data. We next describe the results of the heterogeneous model.

The heterogeneous results for the New = 0, 1 model differ from those obtained with a homogenous model. A majority of the subjects (35), those in Segments 1 and 2, tend to bid above the equilibrium theoretical predictions. t tests on the Predicted coefficient indicate that this coefficient is significantly greater than 1.00 in both cases. (23) For all three segments the estimated constant is positive, but it is statistically significant (with 95% confidence) only for Segment 1. Segments 2 and 3 are distinguished from Segment 1 also by the fact that the former show stronger structural shifts from the New condition. In no case is there any significant evidence of the counting conjecture.

The 12 bidders in Segment 1 have a propensity to bid a large predetermined amount, as reflected in the large and statistically significant constant term (1.2590) and also depart from risk neutral theory by having a coefficient on Predicted (1.1400) that is significantly greater than 1. The 23 bidders in Segment 2 have a much smaller constant term (0.4552) and they come closest to the coefficient of 1.00 on the Predicted variable (1.0800). The 13 bidders in Segment 3 are particularly interesting. They have an intermediate constant of 0.8617 (which is not significant for any standard confidence level) and their coefficient on Predicted is only 0.9076. The coefficients for this segment tend to have larger standard errors than the other two groups. (24)

Table 4 presents the estimations for the Experienced data and model specification (4). The homogenous model (H = 1) yields nearly identical parameter estimates for the Constant, Predicted, and Number coefficients as those depicted for the homogenous case in Table 3 for all data. Heterogeneity in bidding in the FP auctions persists with experience, suggesting the stable cause of deviations from the RNNE that becomes more refined when the initial noise settles down with experience. Segments 1 and 4 (with 17 and 12 bidders, respectively) have a positive (and significant) constant, and slightly greater than 1 coefficient on the predicted bid. For Segments 2 and 3 (with 15 and 4 bidders, respectively), the coefficient on the constant term is not significantly different from zero. Segment 2's Predicted coefficient is very close to 1, while for Segment 3 it is below 1 and equal to 0.6359. Furthermore, Segment 2 has a Number coefficient that is significantly different from zero (p = .02), the only time the counting hypothesis is supported in this study. (25) This means that a small subset of subjects attempt to access the number of bidders in the auction and slightly adjust their bids for auctions with higher number of participants.

Unlike the SP auctions where the model always converged to 1 segment (homogeneity), in the FP auctions heterogeneity persisted with experience resulting in four-segment specification for experienced data and a three-segment specification for all data. Such difference in heterogeneity between mechanisms is consistent with heterogeneity in risk preferences which should play no role in the SP auction (where the solution concept is dominance) and directly affect the bidding in the FP auctions. We study these considerations in detail in the next section.

C. Individual Characteristics of Heterogeneity

We next investigate if there is any regularity in how specific subjects are assigned to segments. Following methodology of Section 4C we study whether the pattern of bidding consistent with risk aversion plays a role in segment assignments.

First, consider the SP auctions, where risk preferences play no role for the bidding strategy and the model collapsed to one segment. Using all of the data (New = 0, 1) the point predictions of the coefficients are similar but noisier than estimates for Experienced bidding. This suggests the presence of noise in behavior in the face of a new institution, a propensity that does not qualify as risk aversion.

The analysis of the FP auctions, however, indicates that heterogeneity persists with experience. We explore below whether there is a common factor in the segment assignments for the FP auction that is consistent with heterogeneity in risk preferences. (26)

We report the estimated risk parameter (as well as the standard error and the number of observations) for each individual bidder in Table 6. Estimation results of risk parameters for the segments formed by the EC algorithm are presented in Table 7. As the segment numbers are arbitrary, we order the results in Table 7 in increasing risk preference parameter, r. We perform these estimations for all data (New = 0, 1) and for only experienced bidders' data (Experienced). Bidders risk parameters, r, range from 0.035 (subject 27) to 34.9 (subject 7) for all data (New = 0, 1) and from 0.045 (subject 27) to 27.30 (subject 7) for experienced data. Subject 7's bidding is an outlier and is also characterized by the largest noise. (27) The second highest risk preference parameter is 2.55 for all data and 2.38 for experienced data (subject 3). The ranking of individual risk parameters is preserved fairly well with experience as shown in Table 6. The distributions of the individual risk parameters are displayed in Figure 2. These distributions are obtained by counting the frequency of individual [r.sub.i]s in the intervals with length 0.1, that is, from 0 to 0.1, from 0.1+ to 0.2, and so forth. The arrows on the graph indicate the estimated risk aversion parameters of segments from Table 7. The distribution of individual risk parameters for all data is somewhat uniform with no distinct peaks. The estimated risk parameters of three optimal segments formed by EC algorithm (0.21, 0.52, and 0.99) are spread about evenly among risk averse to risk neutral range of risk parameters. The distribution of individual risk parameters changes dramatically when we look at only experienced bidders and has two very distinct peaks. Segments 1 and 4 (containing a total of 29 of the 48 bidders) line up at the first peak in the distribution and Segment 2 (containing 15 out of 48 bidders) lines up near the second peak. Segment 3 is at the right tail and has the highest r = 1.505 (least risk averse). Bidding in Segments 1, 2, and 4 is consistent with risk averse preferences (r = 0.375, 0.505, and 0.205, respectively). Segment 2 is the only group to show significant counting behavior. Therefore, although a large number of bidders exhibited risk averse risk preferences (if we were to use this hypothesis) the behavioral phenomenon of counting also affects heterogeneity and segment assignments.

[FIGURE 2 OMITTED]

Figures 1A and 1B show bidding data by segments for Experienced data and also two theoretical benchmarks: risk neutral (r = 1) bid and bid function based on the optimal r estimate for each segment. The graphs show that different segments have different levels of bidding noise. Data in Segments 1 and 4 cluster tightly around theoretical predictions, while data in Segment 2 has larger noise. Segment 3 is associated with particularly large bidding noise. Some subjects, for example, 3 and 7, bid all over the nondominated range of bids ([b.sub.ij] [less than or equal to] [[upsilon].sub.ij]). It thus appears that the estimation technique is sorting the bidders in segments based also on the intrinsic noise.

Table 5 presents the contingency table for the segment assignments in all data and only in experienced periods. If there is a stable cause of heterogeneity, the assignment of subjects to segments should be consistent. The chi-squared statistic for the associated contingency test is significant, indicating there is a common element to the heterogeneity in the bidders' behavior that survives the effect of experience. If risk preferences are stable, there would be higher numbers along the main diagonal and lower numbers or zeros in the lower left and upper right corners. This tendency is observed in the table with a slight deviation, which is because of other factors affecting heterogeneity besides risk preferences. The majority of subjects (23) from the "middle" risk aversion Segment 2 (r = 0.515) based on New = 0, 1 data remained in "middle" Segments 1 and 2 (r = 0.375 and 0.505, respectively) for Experienced data. Ten out of 12 subjects in the most risk averse (lowest r) segment of all data (r = 0.205) remain in the most risk averse segment of experienced data (r = 0.205). There are only two subjects, 9 and 48, who were in Segment 3 (r = 0.985) for all data and joined Segment 1 (r = 0.205) for Experienced data. As follows from Table 6, the estimated risk parameters for New = 0, 1 and Experienced data were, respectively, 0.36 and 0.29 for subject 9 and 0.26 and 0.24 for subject 48. Although the estimated risk parameters of these subjects did not change much with experience, the bidding noise (standard deviation from theoretical bid) dropped significantly: from 2.84 to 0.28 for subject 9 and from 2.00 to 0.26 for subject 48. Table 7 indicates that Segment 3 for all data is characterized by the largest bidding noise (standard deviation from theoretical bid is 4.16). This is yet another indication that bidder specific noise is utilized by the EC algorithm to identify heterogeneity among subjects.

Our results demonstrate that heterogeneity in bidding captured by the latent class estimation technique is consistent with three characteristics: risk preferences, counting the number of bidders, and the intrinsic bidding noise of a given subject.

VI. CONCLUSIONS

Previous studies of individual bidding data in sealed bid auctions with known number of bidders indicated that bidders deviate from theoretical predictions and exhibit heterogeneity (Cox, Roberson, and Smith 1982; Kagel 1995). In this article we study individual bidding behavior in sealed bid auctions with unknown number of bidders (modeling uncertainty in a Bayesian way) and look for the consistency of individual bidding decisions with models of individual bidding behavior. Previous theoretical work in this environment identified the dominant strategy in the SP auctions and risk neutral equilibrium model in the FP auctions, which we adapted to our experimental design. We then extended existing risk neutral theory of bidding in FP auctions with unknown number of bidders to account for risk preferences. The permutations on the basic models that we considered were, first, whether there was any indication that bidders could discern (count) the number of bidders in each of the auctions; second, an experience effect; and third, whether there might be individual heterogeneity and the possibility that it could be explained by heterogeneous risk preferences.

We found solid support for the proposition that bidders come close to the dominant strategy of truthful revelation in SP sealed bid auctions, especially with a modest amount of repetition.

To the extent that these results differ from other studies, further experimental examination of the causes should provide fertile ground for exploration. We propose a few conjectures for consistency with theory in SP auctions. Our design is slightly different from previous studies where the number of bidders was known at the time of submitting a bid. Not providing the information on the number of bidders may make the dominant strategy more transparent in our setting and facilitate subjects to figure out the optimal strategy. There is evidence that not providing irrelevant information with respect to the decision process helps subjects concentrate on the essential in SP auctions. For example, Guth and Ivanova-Stenzel (2003) demonstrate that the lack of commonly known beliefs about certain features of the game leads to crowding out of the overbidding in SP auctions. Learning about the number of rivals may trigger competitiveness in subjects even if this information is irrelevant for the optimal bidding. The noncomputerized design with multiple stations in the same round might expedite learning within each round. Several studies report convergence to a dominant strategy over time (see Kagel 1995 for a discussion). Cox, Smith, and Walker (1985) report that in multiple unit, uniform price, sealed bid auctions, only a minority of the bids were above values. (28)

In the FP auctions the majority of bidders demonstrate a modest amount of overbidding compared to the risk neutral model and we show theoretically that this over-bidding is consistent with risk aversion. The latent class estimation process, which does not make any structural assumptions on the cause of heterogeneity in bidding, indicates that heterogeneity persists with experience and yields segment assignments that are in general consistent with the patterns of estimated individual and segment risk aversion, especially for experienced bidders. There is evidence, however, that heterogeneity in bidding is driven by components beyond the standard model of risk aversion. The segment assignments in the New = 0, 1 case and the Experienced case are generally consistent but not identical, indicating heterogeneity that changes with experience in the institution.

Bidding noise differs across segments and tends to decrease with experience in both FP and SP auctions. We find that the model converged to relatively homogeneous behavior in the SP auctions for both the full and experienced data sets, New = 0, 1 and Experienced, respectively. In the FP auctions, heterogeneity persisted, while the bidding noise decreased with experience. One of the characteristics affecting bidding in the FP auctions is the phenomena of counting the number of bidders, which is significant in one segment. Segment assignments are also influenced by the magnitude of bidder specific noise (which is heterogeneous).

Our results indicate that in the field settings where the number of bidders is typically unknown, theory provides a good benchmark for predictions. In the SP auctions bidding coincides with theory while overbidding remains in the FP auctions. To the extent this phenomenon is robust, FP auctions would generate higher revenue; in the SP auctions, the auctioneer may reveal the number of rivals to participants to increase the level of bidding and therefore expected revenue. These results provide an interesting menu for further research on auctions with an unknown number of bidders. Two obvious areas are (1) a switch to common and affiliated as opposed to independent private values; and (2) allowing bidders to choose bidding in FP, SP, or ascending auctions (Ivanova-Stenzel and Salmon 2004) and analyzing comparative performance. The results presented here also inform the longstanding debate on the role of risk aversion in FP auction overbidding and bidding heterogeneity. A study determining the extent to which the factors that we identified explain heterogeneity would provide a further contribution to the theory of individual behavior in auctions.

ABBREVIATIONS

AIC: Akaike Information Criteria
BIC: Bayesian Information Criterion
cnAIC: Consistent Akaike Information Criteria
crAIC: Corrected Akaike Information Criterion
EC: Estimation Classification
FP: First Price
LR: Likelihood Ratio
SP: Second Price

doi:10.1111/j. 1465-7295.2011.00393.x

APPENDIX: EXPERIMENTAL INSTRUCTIONS (vSFL.4)

Introduction

This is an experiment in the economics of market decision making. Depending upon the decisions that you and other participants make, you have the opportunity to earn a considerable amount of money which will be paid to you in a check on a research account at the end of the experiment. Under no circumstances will you leave the experiment owing money to the experimenters. Your rights as a subject in an experiment have been explained to you in your sign-in document. Except as you might be directed in these instructions or by the experimenter, please do not talk or otherwise communicate with any other participant during the course of the experiment.

Today's Markets

First, we must mention that from this point on, all dollar amounts, unless otherwise specified, are "experimental dollars" (E$). At the end of the experiment, you will earn 25 U.S. cents for each one experimental dollar that you earn. Thus, if your payoff earnings say $5.00 (E$), you will actually earn $1.25. If your earnings say $8.00 (E$), you will actually earn $2.00 and so forth. The only amount that does not get adjusted by this exchange rate is your $7.00 for showing up. That $7.00 is already guaranteed in U.S. dollars. Therefore, from this point on, all costs, values, payments profits, and so forth that we use will be experimental dollars.

Around this room you will see 16 "auction stations." At each station there is a different hypothetical item for sale by auction. Each item is known by a letter, which has no other meaning other than to identify each item. There is exactly one item for sale at each auction station.

The process for selling each item is as follows. At each station, there is a foam mug marked with the letter designation for the item for sale. You have been given your own pad of sticky notes. If you wish to make a bid to purchase that item, please take a sticky note and write on it all of the following four items:

* The letter designation of the item, for example, A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P

* Your bidder number, for example, 1, 2, 3, 4, 5, 6, 7, or 8

* The time as shown on the clock

* Your bid in experimental dollars and cents

Your sticky note bid must have all four items on it to be a valid bid, and all decisions by the experimenters disqualifying incomplete bids are final. If you find that you submitted a bid with missing information, you can at any time submit a complete, second bid. It is recommended that you fold your completed bid form securely after you finish writing on it. This will provide a greater degree of privacy.

You may make bids on as many or as few of the 16 items as you like. You may submit more than one bid on an item. All auctions close at the same time (7 minutes elapsed time on the official timer). The winning bidder for each item is the highest bid in the coffee can when time has expired. The winning bidder pays his/her own bid (that is, the highest or winning bid) to the experimenter. If there is a tie for highest bid, one of the experimenters will break the tie using a random number table. Tie breaking decisions are final and nonnegotiable. Different bidders may very well bid on different items. A single bidder may win zero, one, two, several, or even all of the items.

What Are My Values for the Items? How Are They Chosen?

First, it is important to note that each of you begins with a $5.00 initial reserve simply for participating. This is in addition to your "show up" fee. All earnings from the auction markets are on top of this $5.00 (remember, all amounts are in experimental dollars).

At the beginning of each auction period, you will be given a "payoff chart" so that you can keep track of your earnings. There is a column for each of the 16 items. The next row is labeled "my value for this item." The number in that row represents how much we will pay to you if you win the auction for that item. Some of these entries may be "zero."

If you win the auction for that item, then the following is how your earnings are calculated:

(MY VALUE - MY WINNING BID)/MY EARNINGS.

If you do not win an item, your earnings at that auction station are zero.

Suppose, for example, your value for an item was $24.00, and you won the auction with a bid of $21.40. Then, your earnings would be: $24.00 - $21.40 = $2.60 (remember, experimental dollars). (These numbers are for illustrative purposes only and have no meaning for the actual experiment).

Where do these values come from? That is a very good question. The values for each item generally differ from one bidder to another. That is why we used a random process to hand out the folders when you arrived. Furthermore, each bidder will generally have different values for different items, and different values for the same item between periods.

In choosing the values, we first decided for each item in each period how many bidders would have positive values. We used a random number process to choose, independently, a number of 2, 3, 4, 5, 6, 7, or 8 bidders having positive values. Each number was equally likely. Then, we used a similar random number process to decide which bidder IDs would be the bidders with the positive values. Each bidder ID was equally likely to be chosen for each item. Finally, for those bidder IDs with positive values on a particular item in a particular period, we drew a different value number from the set of number $00.01, 00.02, ..., 19.99, 20.00. Each value was equally likely.

Notice that this means there will always be at least two bidders with positive values for each item.

In other words, consider the G object in round one. First, a random number process might tell us that three bidders will have positive values. Then, a second random number process might tell us that the three bidder IDs to receive the positive values for the G object are 1, 3, and 4. Finally, we would then draw three random values between 00.00 (29) and 20.00, say $0.78 for 1, $2.08 for 3, and $18.74 for 4. All other bidders would have value $00.00 for G for round one. Different numbers, IDs, and values would be drawn for each object in each round. These numbers are hypothetical and have no relation to the actual numbers drawn, but they illustrate the process we went through for each object, and each bidder, in every round.

Your earnings in any one period are the sum of all of the earnings on items you won. Your earnings (in experimental dollars) in the entire experiment are the sum of your earnings in each period.

Closing Thoughts

As described above the values for the items are likely to be different from one bidder to another, from one item to another for a particular bidder, and from one period to another. Recall that these are chosen independently, so that if you get several very low or very high values it means nothing for what might happen in the future.

All bidders are expected to cooperate with the rules of the experiment. By agreeing to participate, you agree to follow the rules, and you understand that anyone who does not do so may, at the discretion of the experimenter, be asked to leave the experiment with only the "show up" fee of $7.00.

It is possible to lose money in an auction. This will occur if you bid higher than your value on an item and you win the auction. In such a case, YOUR VALUE - YOUR WINNING BID would be negative. We do not stop anyone from doing this, however, if you bid less than or equal to your value, you will never be in a position to lose money. If you do make decisions such that you lose money, we will subtract it first from your accumulated earnings, and second from your $5 initial reserve. Any bidder who loses so much money that he/she has eliminated their accumulated earnings and their $5.00 initial reserve will be allowed to remain in the experiment only under a rule that they bid less than or equal to their value on each item from then on out.

CHANGE IN MARKET PRICE RULE

Until we announce otherwise, we will be using a different rule to determine the market price (the payment) for the winning bidder in the auctions, as follows:

The winning bidder for each item remains the highest bid in the foam cup when time has expired. However, the winning bidder pays to the experimenter the highest bid submitted by a losing bidder. If there is a tie for highest bid, one of the experimenters will break the tie using a random number table. Tie breaking decisions are final and nonnegotiable.

All other aspects of the auctions remain exactly the same as in the original instructions.

It is still possible to lose money in an auction. This will occur if three things happen: (1) you bid higher than your value on an item and (2) you win the auction, and (3) your payment (the highest bid submitted by a losing bidder) is greater than your value. In such a case, YOUR VALUE - YOUR PAYMENT would be negative. We do not stop anyone from doing this, however, if you bid less than or equal to your value, you will never be in a position to lose money. If you do make decisions such that you lose money, we will subtract it first from your accumulated earnings, and second from your $5 initial reserve. Any bidder who loses so much money that he/she has eliminated their accumulated earnings and their $5.00 initial reserve will be allowed to remain in the experiment only under a rule that they bid less than or equal to their value on each item from then on out.

REFERENCES

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(1.) A recent paper by Levin and Ozdenoren (2004) looks at how bidders and seller respond to ambiguity about the number of bidders.

(2.) In this article, we model the uncertainty about the number of bidders in a Bayesian way (i.e., as risk and not as ambiguity). This is a similar (but more general) framework as in Dyer, Kagel, and Levin (1989).

(3.) One of our main objectives is therefore to explore the usefulness of the received theory in an environment that is a closest approximation of the field; our research is not specifically designed to test the theory in the sense of giving it its "best shot."

(4.) Krishna's notation is stated in terms of "'number of other bidders."

(5.) Matthews compares two auction mechanisms for buyers with CARA, DARA, and IARA preferences.

(6.) Only one winning bidding slip (out of 480) was invalidated of absent information. In addition, there were six nonwinning bidding slips invalidated for various reasons of absent or incorrect information whose status cannot be determined (out of a total of 2,205).

(7.) In order to keep bidders from wanting to pry open the slots in order to see if they were the only bidder, each mug always contained at least one yellow bid slip marked "blank." Subjects knew this, and in fact a volunteer monitor from the population of subjects (undergraduate students) saw us put one in each mug.

(8.) Isaac and Schnier (2006) used this data to look solely at the aggregate market revenue and efficiency properties of auctions. The primary result of Isaac and Schnier (2006) was that the sealed bid mechanisms yielded both greater revenues and higher aggregate efficiencies when compared with the traditional ascending silent auctions. Specifically they find that efficiency was 0.97 for FP and 0.98 for SP auctions and the revenue index (based on risk neutral theory) was 1.08 for FP auctions and 0.95 for SP auctions. The Isaac and Schnier paper contains extended descriptions of some of the physical attributes of the auctions such as geographical distribution of stations around the room and the construction of bidding mugs.

(9.) Theoretical bid functions were obtained using CRRA utility function of the form U = [x.sub.r], so r = 1 is risk neutrality.

(10.) A second alternative is that bidders will use a rule of thumb rather than the nonlinear bid function implied by the theory. The most obvious candidate is that they will work with the median numbers of bidders, n = 5, and hence will bid 80% of their value ((n - 1)/n). We ran a parallel series of estimations to those reported here using this behavioral benchmark. A later footnote explains their similarity to the main results.

(11.) New = 1 for the first round decisions with the particular auction mechanism.

(12.) The likelihood function for the tobit estimation was programmed in GAUSS and we utilized the MAXLIK toolbox to obtain our parameter estimates.

(13.) Bidder heterogeneity could be controlled for using either a fixed-effects or a random-effects tobit regression. However, this estimation technique would not allow us to determine whether or not individual subjects (or groups of subjects) possessed alternative bidding functions which are statistically different from others within the experiment.

(14.) This method may generate a likelihood function which possesses a number of local maxima. To ensure that our parameter estimates are the global maxima we estimate the FP model 2,000 times and the SP model 500 times using different random starting parameters.

(15.) Detailed estimation results for up to five-segment models for both data sets are available from the authors upon request.

(16.) The following data were excluded relative to the initial valid data set. First, we excluded all bids from the second period of the second FFSSF experiment as one bidder clearly acted upon a misunderstanding of the instructions. Also we removed 62 bids that appear to be double bids. These bids are relatively equally split between the FP (33 out of 1,013) and SP (29 out of 1,050) auctions, and appear to be mainly mistakes rather than re-bids as about half of them are decreasing. We have conducted all the tests reported here including these bids and all of the main conclusions reported here remain unchanged.

(17.) We also conducted analysis of the relation between the timing of the bid and valuation and estimated 12 different models. The best statistical fit came from panel estimations (random effects and fixed effects for individuals as well as dummy variables for periods) with bid value entering both directly and in a squared term. In these models the coefficients on bid value were not statistically significant. In most of the remaining models (linear time remaining only and/or not accounting for panel structure) the coefficients on bidder value were statistically significant but with little in the way of quantitative effect. Specifically, the results indicate that a $1 increase in bidder valuation increases the time remaining in the auction by between 0.73 and 1.56 seconds (the maximum possible acceleration for the whole interval of valuations therefore being on the order of 30 seconds in a 7-minute period). This indicates that most subjects do not attempt to wait and count the number of bidders on their high value items. If anything, they are submitting bids on their higher valued items a few seconds earlier.

(18.) We report results in terms of experimental dollars (one experimental dollar was .25). The bidder who had misunderstood the instructions in an early round was subsequently in the first stage of our bankruptcy rule. which restricted bidding over value. This was the only bidder to enter any stage of the bankruptcy rule. Removing all the data for this bidder yields revelation percentages that are actually higher, 53.26% (exact) and 71.86% (within 25 cents), respectively. This bidder persistently bid below value in the SP auctions.

(19.) This result is robust in our study and is not driven by previous experience in FP auctions. When we separately analyze bidding in SP auctions prior to exposure to FP auction in the SSFFS treatment (dropping New data) we still find that 82.79% of bids are within 25 cents of valuation.

(20.) In fact, the tight fit of the data to the theoretical model apparently caused problematic convergence in GAUSS for even the single-segment model. The estimates we report in that one case are instead from a STATA estimation.

(21.) The t test yielded a value of 0.702.

(22.) The t test yielded a value of 0.692.

(23.) The t tests yielded values of 10.294 and 5.161 for h = 1 and h = 2, respectively.

(24.) We bad originally considered what we termed a "rule of thumb" or "behavioral" alternative model for bidding in the FP auctions, namely that bidders would bid the Nash equilibrium bid as though they were in the median size group (N = 5). The risk neutral equilibrium version would be a bid of 0.8 times value. Overbidding would be transparently consistent with risk aversion just as in the standard literature. As it turns out, the results using this behavioral alternative are qualitatively very similar to those we presented here. The results with the linear approximation do indicate a better fit in that the constant terms are smaller and less significant, and they have slightly larger log-likelihood. However, the change does nothing to eliminate the apparent overbidding with respect to the prediction, in fact, with respect to the behavioral benchmark, the estimated overbidding is even worse. Furthermore the classification schemes are very close.

(25.) The next smallest p value is 0.31 in Segment 1.

(26.) Harstad (2000) reports that subjects may transfer the experience from one auction mechanism to the other. To check for the order effect, we test for possible differences in heterogeneity along with the level of bidding in FP auctions between the two experimental design sequences. First, using the structural estimation of individual risk preference parameters (that are indicative of the level of overbidding) we construct a distribution of risk parameters (bidding propensities) for each sequence using 11 bins range (ten risk averse bins with the step 0.1 and one nonrisk averse bin). A chi-square test cannot reject the hypothesis that two samples come from a common distribution. A second test relates the possible effect of sequencing to our core results, the latent class estimation. Because of the known effect of experience and the more robust latent classification in the FP results, we use the experienced FP (four-segment) classification for analysis. We conducted a two-sample chi-squared test of observed frequencies. The null hypothesis is that assignments to the four categories are the same for the FFSSF individuals as for the SSFFS individuals. The difference is not significant with the Yates correction as a result of the fact that we only observe integral values.

(27.) As discussed earlier the five-segment model created a fifth segment consisting of just subject 7, the "outlier." We have elected to focus on the four-segment model to provide a conservative profile of segmentation.

(28.) Specifically, 33% of bids of inexperienced subjects and 15% of bids of experienced subjects were above value.

(29.) We thank an anonymous referee for pointing out the typo in the Instructions. Note that unlike the statement in the previous paragraph, the lowest possible value is listed as 00.00, not 00.01. A value of 00.00 has never been drawn for the active bidder.

Isaac: Department of Economics, Florida State University, Tallahassee, FL 32306. Phone (850) 644-7081, Fax (850) 644-4535, E-mail misaac@fsu.edu

Pevnitskaya: Department of Economics, Florida State University, Tallahassee, FL 32306. Phone (850) 645-1525, Fax (850) 644-4535, E-mail spevnitskaya@fsu.edu

Schnier: Department of Economics, Andrew Young School of Policy Studies, Georgia State University, Atlanta, GA 30303. Phone (404) 413-0159, E-mail kschnier@gsu.edu

 TABLE 1
 Experimental Design

 Number of
Institution Sessions Period 1 Period 2 Period 3

FFSSF 3 First price First price Second price
SSFFS 3 Second price Second price First price

Institution Period 4 Period 5

FFSSF Second price First price
SSFFS First price Second price

 TABLE 2
 Latent Class Regressions-Second-Price
 Auction (a)

 New = (0,1) Experienced
Variable H = 1 H = 1

Constant 0.0292 (0.14) -0.0791 (-0.981)
Value 1.0071 ** (99.01) 1.0027 ** (259.677)
Number -0.0188 (-0.61) 0.0029 (0.242)
New -0.1746 (-0.52)
New x Value -0.0939 ** (-5.91)
New x Number 0.0818 (1.61)
[[sigma].sup.2] 1.4789 0.5617
Number in segment 48 48
Log-Likelihood -1.887.89 -604.33

(a) t-Statistics for the regression coefficients are shown in
parentheses.

* Statistically significant at 90% confidence, t test.

** Statistically significant at 95% confidence, t test.

 TABLE 3
Latent Class Regressions-First-Price Auction: Three-Segment Model
 (New = 0,1) (a)

 Homogeneous
Variable H=1 H=3;h=1

Constant 0.9389 ** (2.52) 1.2590 ** (6.13)
Predicted 1.0351 ** (46.242) 1.1400 ** (83.73)
Number 0.0125 (-0.22) -0.0066 (-0.22)
New 1.1010 ** (2.00) -0.0518 (-0.17)
New x Predicted -0.1834 ** (-5.34) 0.0175 (0.92)
New x Number -0.1153 (-1.37) 0.0005 (0.01)
[[sigma].sup.2] 2.5762 0.6613
Number in segment 48 12
Log-likelihood -2,393.01

 Heterogenous
Variable H=3;h=2 H=3;h=3

Constant 0.4552 * (1.733) 0.8617 (1.733)
Predicted 1.0800 ** (69.66) 0.9076 ** (18.43)
Number 0.0605 (1.55) 0.0524 (0.37)
New 0.3452 (2.19) 1.4123 (0.91)
New x Predicted -0.0819 ** (-3.45) -0.5219 ** (-6.14
New x Number -0.0685 (-1.19) -0.0011 (-0.00)
[[sigma].sup.2] 1.2342 3.3138
Number in segment 23 13
Log-likelihood -1,795.94

(a) t-Statistics for the regression coefficients are shown in
parentheses.

* Statistically significant at 90% confidence, t test.
** Statistically significant at 95% confidence, t test.

 TABLE 4
Latent Class Regressions-First-Price Auction: Four-Segment
 Model (Experienced) (a)

 Homogeneous
Variable H=1 H=4;h=1

Constant 0.9438 ** (3.15) 0.9664 ** (5.06)
Predicted 1.0349 ** (56.987) 1.1102 ** (100.94)
Number 0.0120 (0.270) -0.0292 (-1.03)
[[sigma].sup.2] 2.0899 0.7113
Number in segment 48 17
Log-likelihood -1,223.38

 Heterogeneous
Variable H=4;h=2 H=4;h=3

Constant 0.2213 (0.68) 0.6209 (1.33)
Predicted 1.0501 ** (53.08) 0.6359 ** (8.34)
Number 0.1069 ** (2.25) 0.1252 (0.62)
[[sigma].sup.2] 1.3221 3.4998
Number in segment 15 4
Log-likelihood -827.38

Variable H=4;h=4

Constant 1.2066 ** (7.16)
Predicted 1.1547 ** (99.67)
Number -0.0043 (-0.17)
[[sigma].sup.2] 0.5690
Number in segment 12
Log-likelihood

(a) t-Statistics for the regression coefficients are shown in
parentheses.

* Statistically significant at 90% confidence, t test.
** Statistically significant at 95% confidence, t test.

 TABLE 5
Contingency Table of Assignments (As a Result of Arbitrary Assignment
 of Segment Numbers We Order the Segments in Increasing r)

 First-Price Auction
 (Experienced)

 Segment 4 Segment 1
 (r- = 0.205) (r = 0.375)

First-Price Segment 1 (r = 0.205) 10 2
Auction Segment 2 (r = 0.515) 0 12
(New = 0, 1) Segment 3 (r = 0.985) 2 3
 Total 12 17

 First-Price Auction
 (Experienced)

 Segment 2 Segment 3
 (r = 0.505) (r = 1.505)

First-Price Segment 1 (r = 0.205) 0 0
Auction Segment 2 (r = 0.515) 11 0
(New = 0, 1) Segment 3 (r = 0.985) 4 4
 Total 15 4

 Total

First-Price Segment 1 (r = 0.205) 12
Auction Segment 2 (r = 0.515) 23
(New = 0, 1) Segment 3 (r = 0.985) 13
 Total 48

 TABLE 6
 Estimates of Individual Risk Preference
 Parameters

 Experienced New = 0 and 1

Bid ID r SD N r SD N

1 1.125 2.223 20 1.515 2.854 27
2 0.805 0.622 14 0.965 0.885 23
3 2.375 1.957 19 2.545 1.765 27
4 0.425 0.973 19 0.425 4.546 30
5 0.205 0.718 20 0.195 0.722 28
6 0.515 1.173 15 0.565 2.001 24
7 27.3 4.370 18 34.90 3.731 25
8 0.295 2.372 19 0.865 2.242 27
9 0.285 0.284 11 0.355 2.836 17
10 0.295 0.524 7 1.395 2.582 16
11 0.185 0.567 9 1.025 1.428 18
12 0.615 0.419 10 0.705 2.589 18
13 0.385 1.118 10 0.455 0.962 23
14 0.425 1.191 9 1.345 3.245 20
15 0.285 0.370 9 0.535 0.769 19
16 0.385 0.683 10 0.675 1.581 17
17 0.505 0.432 21 0.515 1.854 27
18 0.595 0.923 13 0.575 0.898 21
19 0.265 0.581 18 0.335 5.681 26
20 0.235 1.310 19 0.315 2.151 30
21 0.505 0.731 16 0.515 0.987 23
22 0.785 0.748 17 0.785 0.744 27
23 0.895 2.413 17 0.995 2.441 27
24 0.255 1.577 21 0.415 1.409 32
25 0.135 0.213 7 0.135 0.233 16
26 0.345 0.601 10 0.335 0.607 20
27 0.045 0.152 7 0.035 0.144 19
28 0.455 0.741 6 0.455 0.753 7
29 0.515 0.573 7 0.515 0.548 15
30 0.565 1.129 9 0.725 0.910 23
31 0.355 0.418 9 0.265 0.988 20
32 0.305 0.461 11 0.275 0.403 19
33 0.515 1.273 7 0.465 0.895 16
34 0.265 1.885 11 0.265 1.373 21
35 0.115 0.246 7 0.105 0.193 19
36 0.625 2.751 9 0.865 5.797 19
37 0.105 1.074 7 0.125 0.727 15
38 0.235 0.134 9 0.205 0.267 22
39 0.265 0.792 9 0.315 1.389 20
40 0.505 1.552 10 0.505 1.306 17
41 0.145 0.406 7 0.185 0.410 15
42 0.255 0.404 11 0.215 0.362 21
43 0.325 0.861 6 0.515 0.864 18
44 0.295 0.821 12 0.315 0.681 24
45 0.525 0.601 7 0.515 0.459 13
46 0.235 0.274 9 0.235 0.341 23
47 0.985 1.188 9 0.695 1.387 19
48 0.235 0.264 11 0.255 2.004 20
All 0.375 2.205 568 0.455 2.846 1,013

 TABLE 7
Risk Preference Parameters by Segments (Segments Are Ordered in
 Increasing r for Presentation; N, Number of Observations)

 Experienced New = 0 and 1

Bid ID r SD N r SD N

All 0.375 2.205 568 All 0.455 2.846 1,013
Segment 4 0.205 0.738 117 Segment 1 0.205 0.652 238
Segment 1 0.375 0.636 177 Segment 2 0.515 1.316 476
Segment 2 0.505 1.423 200 Segment 3 0.985 4.162 299
Segment 3 1.505 3.824 74