Experienced Bidders in Online Second-Price Auctions

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1 Experienced Bidders in Online Second-Price Auctions Rod Garratt Mark Walker John Wooders November 2002 Abstract When second-price auctions have been conducted in the laboratory, most of the observed bids have been overbids (bids that exceed the bidder s value) and there are very few underbids. Few if any of the subjects in those experiments had any prior experience bidding in auctions. We report on an experiment in which the subjects had substantial prior experience: they were highly experienced participants in ebay auctions. We recruited subjects directly from ebay, recruiting only people who had participated in at least 50 ebay auctions. The subjects then participated in sealed-bid second-price auctions that we conducted on the Internet. Unlike the inexperienced bidders in previous (laboratory) experiments, the experienced bidders exhibited no greater tendency to overbid than to underbid. We thank Greg Crawford, Ron Harstad, and Dan Levin, for helpful comments. We are grateful to John Kagel for providing us with the data sets for the experiments reported in Kagel and Levin. Part of this work was completed while Wooders was a visitor at Hong Kong University of Science and Technology. He is grateful for their hospitality. Department of Economics, University of California, Santa Barbara, California (garratt@econ.ucsb.edu). Department of Economics, Eller College of Business & Public Administration, University of Arizona, Tucson, AZ (mwalker@arizona.edu). Department of Economics, Eller College of Business & Public Administration, University of Arizona, Tucson, AZ (jwooders@bpa.arizona.edu). 1

2 1 Introduction When second-price auctions (SPAs) are conducted in the laboratory, a substantial proportion of the subjects overbid (i.e., they submit bids that exceed their values). There are very few underbids. 1 Kagel and Levin (1993) (henceforth K&L) conducted an experiment with SPAs in which bidders were assigned private values for the item being auctioned. K&L report that in SPAs with five bidders 67 percent of the bids exceeded the bidder s value; in SPAs with ten bidders 58 percent of the bids exceeded values. Kagel, Harstad, and Levin (1987) and Harstad (2000) both report additional evidence of overbidding in SPAs with affiliated private values. Since bidding one s value is always a dominant strategy in SPAs (and overbidding is always weakly dominated by bidding one s value), the overbidding behavior that has been observed in laboratory experiments calls for an explanation. Kagel, Harstad, and Levin suggest that overbidding in SPAs is due to subjects illusion that [bidding in excess of value] improves the probability of winning with no real cost to the bidder, as the second-high-bid price is paid. (pp. 1299) Moreover, they argue that the reason this behavior does not go away with repeated play is that punishment probabilities are weak, given that bidders start with the illusion that bids in excess of [value] increase the probability of winning without materially affecting the price paid, and the majority of the time the auction supports this supposition (pp. 1299). In short, subjects who overbid in SPAs are rarely confronted with the consequences of their mistake. Hence, the learning that often occurs when laboratory subjects participate repeatedly in the same institution may not eliminate overbidding in second-price auctions. In fact, Harstad (2000) reports subjects participating in second-price auctions bid more aggressively over time, quickly breaking through the threshold of bidding equal to value when this is not imposed as a barrier. The persistent failure of subjects to recognize that value-bidding is a weakly dominant strategy is a puzzle. Perhaps it is because laboratory experiments provide subjects only a limited time to consider their decisions and generally involve small stakes. This may be particularly problematic in SPAs because of the obstacles to 1 Previous studies by Coppinger, Smith, and Titus (1980) and Cox, Roberson, and Smith (1982) report underbidding, but in these experiments subjects were not permitted to bid above their private values. 2

3 learning inherent in this mechanism. In contrast, bidders in real-world auctions have time to consider their decisions, they often have participated in many auctions, and the stakes involved are often substantial. Hence, bidders with real-world auction experience might well behave in a way which conforms more closely to the theory. In this paper, we report on an experiment in which the subjects had substantial prior experience in auctions. Our subjects are highly experienced participants in ebay auctions, recruited directly from ebay. We recruited only people whose feedback profiles contained at least 50 entries. Even our least experienced subject likely participated in many more than 50 ebay auctions, since an ebay participant receives feedback from the other party to a transaction only if he or she is the seller or the winning bidder, and it is furthermore not mandatory that either provide feedback about the other. An advantage of using ebay subjects is that we can use bidder characteristics that are external to our experiment to help explain bidding behavior. ebay s bidding system is a dynamic version of a second-price sealed bid auction, and hence ebay experience would seem to be especially relevant to bidding behavior in second-price sealed-bid auctions. On ebay, auctions of a single item are conducted as ascending-price auctions in which bidders submit proxy bids. At any point in time, the high bidder is the bidder who has submitted the highest proxy bid and the current bid equals the second highest proxy bid (plus the bid increment). The high bidder at the time the auction closes wins the auction and pays the current bid. Except that it is dynamic (bidders observe who is the high bidder, the amount of the current bid, and may increase their proxy bids), the auction is essentially a second price auction. In fact, ebay advises bidders to bid their to Decide the maximum you re willing to pay and enter this amount. In effect, ebay is advising bidders to follow their weakly dominant strategy of bidding their value. 2 The subjects we recruited from ebay participated in sealed-bid second-price auctions that we conducted on the Internet. Subjects were sent an invitation, by , that provided a link to a their auction web page. The web page described the rules of the auction, it provided the subject with his or her private value for the item being auctioned, and it provided a form for submitting a bid. 2 Roth et al. point out that uncertain processing of last-second bids can make ebay auctions strategically different from SPAs. 3

4 We find that even highly experienced bidders do not consistently bid their value. However, in contrast to the tendency to overbid displayed by the inexperienced bidders in laboratory experiments, the experienced bidders in our auctions exhibited no greater tendency to overbid than to underbid. The number of subjects who underbid was almost exactly the same as the number who overbid. We find that subjects who have experience selling items on ebay bid significantly less on average than subjects who have only bought. The average bid of subjects with no experience selling on ebay was not significantly different from zero. Subjects with experience selling on average bid significantly below their value. This suggests that experience as a seller changes how one bids in auctions. 2 Experimental Procedures Our goal was to recruit subjects who are highly experienced auction participants, and we use a subject s ebay feedback score as a measure of his experience (i.e., the number of auctions he has participated in). On ebay, a user s feedback score is the number of positive comments minus the number of negative comments, from (unique) users. 3 For several reasons a feedback score is an imperfect measure of experience. When ebay first began operating, a user could leave a positive comment about another user at anytime, while negative or neutral comments had to be transaction related. Hence, a feedback score could be inflated by positive feedback which was not transaction related. ebay now requires all feedback to be transaction related after the close of an auction only the winning bidder can leave feedback for the seller and the seller can leave feedback for only the winning bidder. 4 It is more plausible, however, that feedback scores understate a user s experience since (i) feedback is often not left following a transaction, (ii) non-winning bidders do not receive feedback, and 3 From July 2001 a user s new feedback indicates whether he was the winning bidder or the seller in a transaction. However, up to that time, it is not possible to determine from a feedback profile whether the user s experience came from buying or selling. 4 It is possible, although tedious, to calculate a user s feedback score based only on the feedback left since ebay began to require feedback to be transaction related. We believe this would have only asmalleffect on the feedback scores. 4

5 (iii) no feedback is left for a user who lists an item which does not sell. 5 In addition, obtaining a negative feedback after winning an auction reduces the bidder s feedback score, even though he has more experience. (The difference between a user s total number of feedbacks and his feedback score is generally small since negative feedback is a small fraction of all feedback.) For these reasons, an ebay user is likely to have listed or bid in many more auctions than the number provided by his feedback score. 6 The Auction Each auction had 5 bidders whose values where randomly drawn from the uniform distribution on the interval [$25,$125]. In addition to their profits or losses from bidding in the auction, subjects received a $15 reward for participating. Hence, for the highest bidder, his total earnings were $15 plus his value minus the second highest bid. (If this total was negative, the loss was forgiven and the subject was paid $0.) The other four bidders earned $15. Subjects were fully informed of how their earnings were determined. It s easy to verify that value bidding remains a weakly dominant strategy in a second-price auction with a $15 limit on losses. Subject Recruiting Our recruiting process began by going to ebay s web site and downloading, in the order they appeared, a collection of auction main pages of Morgan silver dollars listed as Ending On The Current Day. The following day, after these auctions had closed, we examined the bid history of each auction sequentially. For every bidder listed in the bid history with a feedback score of 50 or higher we recorded (i) the bidder s ebay ID, (ii) his maximum bid, (iii) the number of times he bid, and (iv) his feedback score. 7 We continued this process until 50 unique IDs had been obtained. The process was then repeated to obtain 50 additional IDs from auctions of Golden Age collectable comics. We recruited subjects from these two categories since these auctions typically had many bidders, thereby reducing the difficulty of obtaining ebay 5 Houser and Wooders (2000) find, for example, the winning bidder leaves feedback for the seller in about one-third of the auctions. 6 The median feedback score at the first round was 134 and the average feedback score was 238. Hence, all the users who bid in our auction had extensive ebay experience. 7 For every bidder other than the winning bidder, his maximum bid is equal to the highest proxy bid he made. The winning bidder s highest bid, however, is the second highest proxy bid plus the bid increment. 5

6 IDs. Moreover, bids in these auctions were roughly in the same range as the subjects values in the experimental auction. Recruiting subjects from categories where, for example, most bids were below $15, say, might introduce a bidding bias in our own auction. In the first round of experiments it s possible that subjects might have been skeptical that they would actually get paid for their participation. To deal with this possibility, several months after subjects were paid their first-round earnings, we invited the participants to a second auction. In the second round the auction rules were the same, but new values were randomly drawn from the same uniform distribution. The first round consisted of several sessions where, in each session, an invitation to participate in our experimental auction was sent via to each one of the 100 ebay IDs. We received a total of 67 bids over six sessions; 37 of these subjects bid again at the second round. Figure 1 shows (i) a c.d.f. of the 100 values (drawn from the uniform distribution on [$25,$125]) usedwheninvitingsubjectsineachsession of round 1, and (ii) the c.d.f. of the values of the subjects who actually chose to participate at round 1. Given the similarity of the two c.d.f. s, it appears that the value assigned to an invitee did not influence his decision whether to participate. Figure 1 goes here. Each subject s invitation gave the deadline for submitting a bid, and contained a link to a personalized web page which described the rules of the auction and provided the subject s value. 8 The personalized web page had three quiz questions; each question required the subject to calculated a hypothetical bidder s earnings given the bidder s value and the highest bid placed by other bidders. Subjects had no direct monetary incentive to answer the questions correctly, but they were required to answer all three questions before they are allowed to submit a bid. The answers to these questions provided us will some indication of whether a subject understood the rules of the auction. The subjects generally understood the instructions as 87% of all questions were answered correctly. After a subject submitted a bid, he was taken to a new page here he was asked to describe his bidding strategy. At the end of each session the bids where placed into groups of five in the order in 8 A sample webpage is available at garratt/auction/sample.html. 6

7 which they arrived. 9 The subjects earnings were calculated, and the subjects were sent an describing the bid and value of each bidder in their auction, as well as their own earnings. Each subject was then mailed a money order for his earnings. Our experiments were conducted between June 2001 and January Beginning in July 2001 new feedback added to a user s feedback profile indicated whether he wasabuyerorasellerintheassociatedtransaction. InSeptember2002,foreach subject who participated in our auction, we counted the number of buyer and seller feedbacks he had in his feedback profile. We assume that this ex-post measure of the subject s type of ebay experience (buyer or seller) is representative of his experience at the time he participated in our auction. 3 Data Analysis Table 1 provide summary statistics for the data used in our analysis. The subjects who participated in our experiment were highly experienced, with a mean feedback score of 265. About 58% of the subjects who participated in our experiment were recruited from Morgan dollar auctions. The subjects submitted an average of 1.45 bids in the auctions there were recruited from. In fact, our subjects had predominately bid only once in auction that they were recruited from, with 50 of the 67 subjects placing only one bid. The sometimes seller variable is a dummy variable which takes the value of 1 for an ebay subject whose feedbacks indicate that he was a seller in some transaction (since ebay began reporting this information) and took the value zero otherwise. Our subjects are roughly evening divided between having always been buyers (46%) and having sometimes been sellers (54%). The subjects who were sometimes sellers had, on average, 130 feedbacks from selling, and a median of 57 feedbacks from selling. Several bidders made the maximum possible bid that could be entered into the bidbox,whichallowed9digits. TheaveragebidandaveragevaluereportedinTable 1 is based on the bids and values of the 94 bids which were of $1000 or less. These 9 Each auction had 5 bidders, but the number of bids received in a session was typically not a multiple of 5. We solved this problem by using some of the subjects bids a second time in order to fill out an incomplete group of 5. For example, if 7 bids were received, then bids 1-5 formed one group which determined the payoffs of bidders 1-5. Bids 3-7 formed a second group, which determined the payoffs of only bidders 6 and 7. 7

8 subjectshadanaveragevalueof$73.48and on average bid $68.23, and hence there was an average overbid of -$5.25. We also constructed a variable giving the absolute value of the difference between a subject s bid and value. This measures the size of a subject s bidding error without discriminating between underbids and overbids. The mean bidding error was $ Figures 2 and 3 plot bids against values for rounds 1 and 2 of the experiment, respectively. Bids above $1000 are high bids and appear as triangles along the top of each figure. These figures visually suggest that the relationship between bids and values is the same from round 1 to round 2. Column (a) of Table 2 reports the results of regressing bids against values with a dummy variable for the round, and with the dummy variable interacted with values. Using a F test, one cannot reject the hypothesis that the coefficients for the round dummy and the interaction variable are jointly zero. 10 In other words, one cannot reject the hypothesis that there is no change in the (linear) relationship of bids to value from round 1 to round 2. Figures 2 and 3 go here. Bidding behavior observed in our experiment may be related to the bidders characteristics or the bidding strategies they use in ebay auctions. Subjects who bid more than once in ebay auctions may follow a strategy of initially bidding low, and later raising their bid (if necessary); in our experimental such subjects might tend to bid lower that other subjects if they don t account for the fact, unlike in an ebay auction, they can bid only once in our auction. The typical amount a subject bids in ebay auctions might influence the amount he bids in our auction. For example, a subject who has never bid more than a few dollars in an ebay auction may be uncomfortable making a large bid in our auction. Column (b) of Table 2 shows the results of regressing bids against values and the subjects ebay characteristics after pooling data for the two rounds. We cannot reject the joint hypothesis that the constant is zero and the coefficient on values is 10 The F statistic.466 is distributed F (2, 89) and the 5% critical value is???. We have F = (SSEc SSE u )/J SSE u /(T K) = ( )/ /(93 4) 8

9 one, which is consistent with the hypothesis of value bidding. 11 However, subjects in ourexperimentwhohaveexperience selling on ebay tend to bid significantly less than subjects who have only bought. The coefficient estimates suggest that, all else equal, subjects who were sometimes sellers bid $15.20 less on average than subjects who were always buyers. These subjects may view their value as a resale value and hence shade their bids to ensure that the price at which they buy is less than the value. 12 In contrast, bidders who have only bought may be more likely to (more accurately) view their value as their own intrinsic value. None of the other bidder characteristics are significant. Column (c) shows that seller experience remains significant when the other bidder-characteristic variables are removed. 13 The dependent variable in the column (d) of Table 2 is the absolute value of the difference between a subject s bid and his value. The dependent variables are the constant and ebay feedback score. If subjects with more experience make smaller bidding errors, then we would expect the coefficient on feedback score to be negative. This estimated coefficient, however, is not significantly different from zero. It is, perhaps, not surprising that a bidder s experience has no effect on the size of his bidding error, given that all of our subjects were highly experienced. The effect of seller experience on bidding behavior can be seen directly simply by comparing the mean overbid in Table 3 for subjects who were sometime sellers and subjects who were always buyers. The mean overbid for subjects who were always buyers is $2.96 which, using a t test, is not significantly different from zero. Subjects with experience selling on average underbid by $12.73, which is significantly different 11 The F statistic is distributed F (2, 80) and the 5% critical value is???. We have F = (SSEc SSE u )/J SSE u /(T K) = ( )/ /(86 6) 12 By bidding 10%, say, below his value a subject ensures himself at least a 10% markup if he interprets his value as a resale price. 13 When there bidder characteristics are removed, one can still not reject the hypothesis that the constant is zero and the coefficient on values is 1. The F statistic is distributed F (2, 80) and the 5% critical value is???. We have F = (SSEc SSE u )/J SSE u /(T K) = ( )/ /(86 3) 9

10 from zero (p value of.014) and also significantly different from the mean overbid of subjects who were only buyers (p value of.031). The mean overbid of subjects who were always buyers rose from round 1 to round 2. This is consistent with the more aggressive bidding that Harstad (2000) reports in later rounds. However, for subjects with experience selling, the mean underbid is virtually unchanged from one round to the next. The difference in the bidding behavior of subjects who were sometimes sellers and subjects who were always buyers is also evident in the frequencies of under-, value-, and over-bids. For subjects who were always buyers these frequencies are 14, 6, and 24, compared to 29, 6, and 18 for subjects who were sometimes sellers. 14 We can reject at the 10% significance level (p value of.071) the hypothesis that these multinomial distribution are the same. Overall the evidence suggests that subjects with seller experience bid differently than subjects with only buyer experience. These subjects bid significantly less than subjects who have only been buyers (and also significantly less than their value); subjects with seller experience also underbid significantly more frequently than subjects whoonlyhaveexperienceasbuyers. Both types of ebay subjects bid differently than the student subjects in K&L. Table 4 shows the frequency of under-, value-, and overbids when value bids are defined as in K&L, i.e., a bid is a value bid if it differs from the subject s value by 5 cents or less. 15 The distribution over the three types of bids is strikingly different forourdata(whetherforalwaysbuyersorsometimessellers)andk&l sdata. Using Pearson s chi-square goodness-of-fit test, one can reject at the 1% level the hypothesis that the distribution over the three types of bids is the same for the ebay subjects whowerealwaysbuyersasitisforthek&lsubjects. (ThevalueofthePearsonQis and the 1% critical value, with two degrees of freedom, is 7.38.) This hypothesis is rejected even more decisively when comparing the distribution over types of bids 14 These frequencies are based on the 97 bids made by subjects who were still registered ebay users when we counted buyer/seller feedback, and includes the bids over $ Given that the stakes in each round of our experiment were about 3.5 times greater than in K&L s experiment, it might make sense to define a value bid as one within $.175 = 3.5($.05) of the subject s value. Doing so leaves the frequencies reported in Table 4 unchanged for the ebay subjects. 10

11 for sometime sellers (or the overall distribution of bids) to the distribution of types of bids made by the K&L subjects. As Table 4 shows, the ebay subjects overall were roughly equally likely to underbid as to overbid, while the subjects in K&L predominately overbid. Experiment Under bids Value bids Over bids Total ebay always buyer 13 (29.5%) 11 (25.0%) 20 (45.5%) 44 (100%) ebay sometime seller 27 (50.9%) 9 (17.0%) 17 (32.1%) 53 (100%) Overall 1 43 (41.3%) 22 (21.2%) 39 (37.5%) 104 (100%) K&L 27 (5.7%) 127 (27.0%) 316 (67.2%) 470 (100%) Table 4 1. The overall count includes the 7 bids for which we could not obtain buyer/seller feedbacks. Revenue and Efficiency Our results, K&Ls results, and the results of other second price auction experiments have established that bidders (even experienced ebay bidders) generally do not bid their value as the theory suggests they should. We now consider the effect suboptimal bidding on seller revenue and auction efficiency. The left-most column of Table 5 shows the seller s ex-ante (i.e., before values are drawn) expected revenue in our auction and in K&L s auction under value bidding. Given the empirical distribution of values drawn, the seller s expected revenue if subjects bid their value is reported in the middle column. Expected seller revenue given the empirical distribution of actual bids (possibly different from the subjects values) is reported in the right-most column. ex-ante Conditional on Value Draw Theoretical Value Bidding Actual Bids ebay $91.67 $92.09 $90.97 K&L $18.87 $18.28 $19.26 Table 5: Seller Revenue 11

12 While there was nearly equal amounts of under bidding and overbidding by the ebay subjects, the net effect on seller revenue is negative expected revenue is only $ (This calculation takes account of the bidders $15.00 limited liability. In particular, if the winning bidder s bid is more than $15 above his value, then seller revenue is the bidder s value plus $15.) In K&L s experiment, the seller s expected revenue given the empirical distribution is $.98 higher given the empirical distribution of bids, than wouldbeexpectedhadsubjectsbidtheirvalue. 16 Table 5 shows that suboptimal bidding has a modest effect overall on seller revenue. The effect of suboptimal bidding on auction efficiency is more striking. Given the empirical distribution of bid-value combinations in our experiment, the high bidder in a randomly selected group of 5 bids has the highest value with only probability 48.5%. If we exclude from the empirical distribution the bid-value combinations in which the bid is above $1000, this probability rises to 55.9%. In contrast, the student subjects in K&L s experiment achieved a far higher level of efficiency; for the empirical distribution of bid-value combinations, 87.3% of the time the high bidder has the highest value. 4 Concluding Remarks ebay bidders participate in an auction institution that is very similar to a secondprice sealed-bid auction. We find that even highly experienced ebay bidders largely do not bid their values, as the theory suggests they should. This result shows that the suboptimal bidding discovered for student subjects in laboratory is robust to even significant experience in non-experimental auctions. There are, however, significant differences in the bidding behavior of ebay subjects with different experience buying and selling, and also significant differences in behavior of ebay subjects overall and student subjects. Subjects with experience selling bid significantly less than subjects who have only bought on ebay. This suggests that the revenue and efficiency properties of auctions might depend in a systematic way on the backgrounds and prior experiences of the bidders. Further, subjects in our experiment are equally likely to underbid as to overbid, in contrast to laboratory 16 This expected revenue calculation for K&L assumes a $10 limited liability. 12

13 experiments which have documented a predominance of overbids. This last difference may be a result of differences in the experimental design, rather than to differences in the subjects experience. For example, our experiment was based on different instructions than K&L s; our experiment was run over the Internet rather than in the laboratory; our experimentinvolvedonlytworounds. 17 References [1] Cooper, D., Kagel, J., Lo, Wei and Qing Liang Gu (1999). Gaming Against Managers in Incentive Systems: Experimental Results with Chinese Students and Chinese Managers, American Economic Review 89, [2] Dyer, D., Kagel, J. and D. Levin. (1989): The Behavior of Naive and Experienced BiddersinCommonValueOffer Auctions: A Laboratory Analysis, Economic Journal 99, [3] Harstad, R. (1990). Dominant Strategy Adoption, Efficiency and Bidders Experience with Pricing Rules, Experimental Economics 3, [4] Kagel, J. and D. Levin (1993). Independent Private Value Auctions: Bidder Behavior in First-, Second- and Third Price Auctions with Varying Numbers of Bidders, Economic Journal 103, Repetition may explains one difference between our data and K&L s. In K&L s experiment subjects participated in more than 20 experiments, which is probably why their subjects tended to bid in whole dollar amounts (40% of all bids). In contrast, in our experiment subjects were less likely to round their bid to a whole dollar amount. 13

14 Table 1: Data characteristics. Variable Obs Mean Std. Dev. Min Max Bid Value Overbid = Bid - Value Absolute value of Overbid Feedback score Auction type: 1 if Morgan, 0 if Comic Number of bids a subject placed in the ebay auction from which he/she was recruited Seller-feedback dummy: 1 if some seller feedback, 0 if only buyer feedback 2 1. Feedback score is the amount of feedback (good, neutral, or bad) in a subject's ebay feedback profile at the time he was invited to participate. Round 2 feedback scores reflect new feedback the subject obtained since round Our experiments were conducted before ebay began to differentiate between buyer and seller feedback. Hence the data used to construct the Seller-feedback dummy was obtained from ebay auctions that occurred after subjects bid in our experiment.

15 Table 2: Regression results for dependent variable "Overbid." Dependent variable Bid Abs. Value of Overbid (a) (b) (c) (d) Constant (10.573) (14.272) (9.560) (3.754)** Value (0.131)** (0.118)** (0.118)** Round dummy: 1 if round 1, 0 if round (19.202) Round dummy times value (0.246) Feedback score (.010) Log of highest bid subject placed in ebay auction (2.830) Auction type: 1 if Morgan, 0 if Comic (7.127) Number of bids a subject placed in the ebay auction from which he/she was recruited (3.757) Seller dummy: 1 if some seller feedback, 0 if only buyer feedback (7.458)* (7.067)* Observations R-squared (adj.) Standard errors are in parentheses: * significant at %5 level, ** significant at the 1% level. All regressions exclude subjects that bid over $1000 (11 obs.). (b) and (c) exclude subjects with no feedback since ebay started to differentiate buyer and seller feedback (7 obs.).

16 Table 3: Mean overbid by round, sometimes sellers and always buyers Category Round 1 Means Round 2 Means Overall Means bid value overbid bid value overbid bid value overbid Always buyer $68.25 $67.78 $0.47 $76.63 $69.05 $7.58 $71.18 $68.22 $2.96 Sometimes seller $67.57 $ $13.05 $59.33 $ $12.13 $64.70 $ $12.73 All subjects $67.52 $ $6.79 $69.53 $ $2.45 $68.23 $ $5.43 All means are calculated excluding bids over $1,000 (11 obs.).

17 CDF of Invitee/Participant Values Participant Values Invitee Values

18 Round Bid Value Median Bid over $10 intervals High bids Bids

19 Round Bid Value Median Bid over $10 intervals High bids Bids