Economics Letters 85 (2004) 279 – 286 www.elsevier.com/locate/econbase

Learning to play second-price auctions, an experimental study Jorge G. Aseff * Department of Economics, DePaul University, 1 East Jackson Blvd., Chicago, IL 60604, USA Received 25 March 2003; received in revised form 13 April 2004; accepted 19 April 2004 Available online 24 August 2004

Abstract Existing experimental results in second-price auctions show that subjects bid consistently above their valuations. Experience, gained repeating a second-price auction, does not improve subject’s bidding behavior. When subjects ‘introspect,’ bidding behavior improves. Overbidding and underbidding fall significantly. D 2004 Elsevier B.V. All rights reserved. Keywords: Second-price auctions; Overbidding; Underbidding JEL classification: C7; D44; D81

1. Introduction It is well known [cf. Kagel et al. (1987) and the references therein] that English and Vickrey auctions have quite different experimental characteristics.1 In English auctions, observed behavior rapidly converges to predicted behavior; in Vickrey auctions subjects tend to bid above their valuation even though bidding their valuation is a weakly dominant strategy. Changes in the experimental design such as repetition and the use of experienced subjects has shown little improvement. Cox et al. (1982) analyze single-unit second-price auctions and find that prices converge from below to equilibrium bids. However, in their experiments, subjects were specifically prohibited from bidding above their valuations. The differences in observed behavior have been attributed to differences in the learning environment of both auctions [cf. Kagel and Levin (1993)]. Subjects in English auctions readily observe the price as it surpasses their valuation and immediately realize the consequences of over or underbidding. Specifically, the costs of overbidding are more likely to materialize in English auctions. To the contrary, subjects in * Tel.: +1-312-362-8355; fax: +1-312-362-5451. E-mail address: [email protected] (J.G. Aseff). 1 In the sequel, Vickrey auction and second-price auction will be used interchangeably. 0165-1765/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2004.04.016

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Vickrey auctions may not experience the consequences of overbidding: overbidding generates a loss only when the object is secured and the second highest bid is above the winner’s valuation. Therefore, bidders that are less reluctant to overbid in second-price auctions are not too surprising. We show that learning in Vickrey auctions may be easier to accomplish through introspection. In our experiments, subjects participated in two identical sessions of second-price auctions that were approximately one day apart. Our results suggest that having one night to ‘think’ about the game enhances subjects’ understanding of the auction. Indeed, their performance during the second day improves significantly. Considering all observations, our results indicate that average overbidding falls by 37.15% on the second day, and average underbidding falls by 71.59% on the second day. Nevertheless, even though overbidding and underbidding fall significantly, they remain statistically significant. Therefore, this study indirectly suggests that the overbidding result persists in experimental settings that allow for introspection. Kagel and Levin (1993) analyze second-price auctions with experienced subjects and find that overbidding persists. In their experiments, subjects gain experience in a common values setting, that is to say in a different game. Harstad (1990) conducts second-price auctions with subjects that earned experience in a first-price auction. He finds that on average, bids are below equilibrium. Finally, in a different context, Blume and Gneezy (2000) study cognitive differences in pure coordination games and find that players that have previously played the game against themselves play significantly better when they meet an opponent. Therefore, the ability to describe the game to themselves seems to make a difference in the selection of optimal strategies. In the remainder of the article we describe the experimental sessions (Section 2), explain the experimental design (Section 3), present the main findings (Section 4), and state our concluding remarks (Section 5).

2. The second price auctions Each day, bidders participate in a series of single-unit, second-price auctions. In each auction, one certificate is auctioned off among three potential buyers, bidders A, B, and C. The identity of the bidder changes from round to round; i.e., bidder A in one round may be bidder C in the next and so on. The valuation vi of each bidder is privately observed and drawn from the set {0, 1, 2,. . ., 98, 99, 100} according to a uniform distribution. The bidder’s valuation for the certificate represents the bidder’s type. After bidders observe their types, they submit a bid for the certificate. Bids are restricted to be between 0 and 100 inclusive, and not restricted to be less than or equal to the bidder’s type. Once all bids have been received, the certificate is allocated to the bidder that submitted the highest bid, who then pays a price equal to the second-highest bid. Subjects know that in the event of a tie, the certificate is assigned randomly. It is well known that in a single-unit second-price auction with private values, a weakly dominant strategy is to bid an amount equal to the private valuation.

3. Experimental design Subjects were randomly selected from a pool of undergraduate students at Arizona State University’s college of business. Experiments were conducted in two consecutive days. In each day, subjects

J.G. Aseff / Economics Letters 85 (2004) 279–286

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Fig. 1. Screen observed by subjects.

participate in 20 rounds of second-price auctions, where in each round a single certificate is to be sold to one of three bidders, A, B, and C. Each group of three bidders remains unchanged through all the sessions in a given day. The first day of participation a computer terminal is arbitrarily assigned to every bidder. To minimize possible collusive agreements the experimenter ensures that bidders that interacted in day 1 do not interact in day 2. Subjects did not know about the group changes before day 2. A set of instructions containing examples of the allocation rule and of the calculation of payoffs was provided to each subject. The instructions were read aloud both days and a brief quiz was completed right before the beginning of the actual experiment. There were two rounds of questions, one before the quiz, and one after the answers to the quiz were revealed. The types of all bidders were selected in advance from the set {0, 1, 2,. . ., 98, 99, 100}. These types are allowed to differ from round to round. Subjects were not prevented from bidding an amount greater than the type assigned to them by the program. It is explained to them that a bid can be any whole number between 0 and 100 inclusive. At the beginning of each auction, subjects observe the screen portrayed in Fig. 1. After each auction, the results are displayed. Then, subjects observe the bids of all participants, the points accumulated in the auction, and the points accumulated up to the given auction. The information pertinent to each subject is highlighted. To account for potential losses, each subject was endowed with an initial balance of 50 points. At the end of each session, points were exchanged for cash at a predetermined and publicly known rate of $0.10 per point. A minimum of $5.00 was guaranteed to each subject, although they were not informed of this in advance. Also, at the end of the session of day 1, it was announced that a bonus of $10.00 would be added to the earnings of day 2. This was done to further the incentives of subjects to return the next day given that their participation in day 2 was crucial.

4. Results Our results are based on a total of 24 subjects that participated in two series of Vickrey auctions that were approximately one day apart. Our main finding can be summarized as follows: The bidding behavior of subjects during the second day of the Vickrey auction is significantly closer to equilibrium behavior than the bidding behavior during the first day. See Fig. 2 for an illustration of this finding. The 45j line in each panel of Fig. 2 represents equilibrium behavior. This finding is supported by results 1–3 below.

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Fig. 2. Results from second-price auction.

Let Si;t ¼ ðBi;t  vi;t Þ denote the net gain of subject i in an auction P t that took place on day t, for t = 1, 2. Si;t , when Si;t z 0 ; and average Then, average overbidding is defined by P Overbidt = 1=Rt Ri¼1 Mt underbidding is defined by Underbidt = 1=Mt i¼1 Si;t , when Si; t V 0. Note that Rt and Mt specify the number of auctions with Si;t z 0 and Si;t V 0 respectively. 4.1. Result 1 (Overbidding) Subjects overbid at a higher rate in day 1 than in day 2. The difference between average overbidding in day 1 and day 2 is statistically significant at the level of 5%. The statistical tests regarding overbidding appear in Table 1. To analyze the auctions by the type of the bidder, we introduce a moving threshold x. Thus, vi z x means that we are considering

Table 1 Overbidding vi z

Overbid1

Overbid2

Mean difference

%D

0 10 20 30 40 50 60 70 80 90

11.2649 11.2510 11.6669 11.0398 6.6589 6.7715 6.5955 6.9167 4.1000 3.0833

7.0791 6.9505 6.9991 7.0376 7.0946 7.1085 7.1422 7.1211 5.3805 4.4637

4.0608 4.3005 4.6578 4.0021  0.4356  0.3370  0.5467  0.2144  1.2806  1.3804

 37.1578  38.2177  37.7913  36.2524 6.5430 4.9760 8.2890 2.9550 31.2300 44.7702

(0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0001)

p-Values between parenthesis.

(0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)

(0.0349) (0.0315) (0.0460) (0.0382) (0.3737) (0.4008) (0.3276) (0.4285) (0.1004) (0.0559)

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Table 2 Underbidding vi z

Underbid1

Underbid2

Mean difference

%D

0 10 20 30 40 50 60 70 80 90

7.2050 (0.0000) 7.4397 (0.0010) 7.6644 (0.0010) 7.8552 (0.0011) 8.2936 (0.0020) 8.7012 (0.0020) 9.4280 (0.0028) 11.0135 (0.0050) 12.3765 (0.0089) 11.3333 (0.0260)

2.0469 1.8796 1.9112 2.1523 2.4485 2.4633 2.7445 3.5317 4.3194 0.5000

5.1580 (0.0146) 5.5601 (0.0121) 5.7532 (0.0116) 5.7023 (0.0149) 5.8450 (0.0222) 6.2380 (0.0201) 6.6835 (0.0273) 7.4820 (0.0440) 8.0570 (0.0681) 10.8333 (0.0311)

 71.5905  74.7355  75.0639  72.6003  70.4772  71.6901  70.8899  67.9329  65.0999  95.5882

(0.0155) (0.0364) (0.0346) (0.0330) (0.0227) (0.0221) (0.0368) (0.0483) (0.0553) (0.0458)

p-Values between parenthesis.

subjects with type greater or equal than x; e.g., x = 0 means that subjects of all types are considered. Considering all types, overbidding falls by 37.15% and this reduction is statistically significant. Overall, overbidding remains positive and significant even when subjects introspect. Therefore, the earlier overbidding result is robust to introspection in experimental second-price auctions.2 The reduction in overbidding is significant for types between 0 and 30. However, when vi z 30 the change in overbidding is no longer significant. We contemplate two reasons why overbidding changes less for higher types. First, subjects were not allowed to bid more than 100 points. Then, as vi approaches this upper bound, overbidding must fall. Second, empirical evidence suggests that when stakes are higher, subjects tend to play better. In our context high-type bidders can potentially earn larger amounts, thus if they do indeed play better, the improvement in their behavior will be less noticeable. 4.2. Result 2 (Underbidding) Subjects underbid at a higher rate in day 1 than in day 2. The difference between average underbidding in day 1 and day 2 is statistically significant at the level of 5%. The statistical tests regarding underbidding appear in Table 2. Considering all types, underbidding falls by 71.6%, and this reduction is statistically significant. Furthermore, except for 80 V vi V 90, the reduction in underbidding is significant at the level of 5%. Also, the magnitudes of the changes in underbidding for all different types suggest that observed bids approach equilibrium bids from below faster than from above. The adjustment in underbidding is especially interesting when we consider high values of vi, since these subjects can potentially underbid by more. Indeed, Underbid1 shows that as vi increases, so does underbidding. However, Underbid2 fluctuates less than Underbid1 and moreover, for vi z 90 is not significantly different from 0 at the level of 5%. It is important to note that given a hightype, the expected cost of underbidding is higher than the expected cost of overbidding. Subjects seemed to have realized this in day 2.

2

I thank an anonymous referee for making this remark explicit.

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Table 3 Variance ratio test vi z 0 10 20 30 40 50 60 70 80 90

(1) All Si,t

(2) Si,t z 0

(3) Si,t V 0

Var1

Ratio

Var1

Ratio

334.2290 318.5890 316.4720 311.7830 300.3660 315.3180 332.1530 369.4390 375.2310 414.3330

5.01883 (0.00000) 6.22878 (0.00000) 6.11212 (0.00000) 6.22025 (0.00000) 5.84259 (0.00000) 6.19449 (0.00000) 6.44942 (0.00000) 7.47885 (0.00000) 13.14660 (0.00000) 54.20630 (0.00000)

191.5300 157.6780 148.7350 118.9630 51.0439 52.4622 41.3642 43.1091 16.7525 6.31398

3.01650 3.57906 3.33750 2.82938 1.23820 1.30211 0.97891 1.10431 0.95919 1.00950

(0.00000) (0.00000) (0.00000) (0.00000) (0.05857) (0.03300) (0.44722) (0.27266) (0.43637) (0.48035)

Var1

Ratio

239.2110 254.3460 263.3220 283.1750 344.1570 362.5310 406.7650 459.1870 493.0820 605.1790

29.90270 29.43950 29.11960 28.01470 28.34670 28.43300 31.26650 29.03880 25.32870 2118.13000

(0.00000) (0.00000) (0.00000) (0.00000) (0.00000) (0.00000) (0.00000) (0.00000) (0.00000) (0.00000)

p-Values between parenthesis.

Results 1–2 suggest that subjects play significantly better on the second day. Consequently, one would expect a smaller variance for Si;2. Hence, Table 3 contains Variance Ratio tests for Si; t . Column (1) contains the Variance Ratio test when we consider all subjects’ net gains. Columns (2) and (3) correspond to observations where subjects overbid and underbid respectively. Let Vart = Variance(Si,t), and Ratio = Var1/Var2. The tests reveal that the variance of Si; t : decreases significantly at the 5% level when we consider the full sample. Considering the subset of the sample where subjects underbid, the reduction of the variance is significant except for 40 V vi V 50. Table 1–3 confirm the notion already mentioned above that subjects adjust for underbidding at a faster rate than for overbidding. Consider the following observations. First, the mean difference in overbidding is approximately 4 points, and the mean difference in underbidding is approximately 5 points. Second, the drop in variability is almost 10 times larger in the case of underbidding than in the case of overbidding. Therefore, our results indicate that the convergence

Table 4 Linear fit by type vi z

a1

b1

0 10 20 30 40 50 60 70 80 90

12.5277 (0.0000) 12.6852 (0.0000) 12.7710 (0.0000) 13.1097 (0.0000) 11.0428 (0.0000) 14.5202 (0.0243) 22.3597 (0.0175) 31.5469 (0.0185) 36.7778 (0.3381) 114.9400 (0.4255)

0.8167 0.8143 0.8132 0.8081 0.8338 0.7910 0.6972 0.5916 0.5322  0.2888

p-Values between parenthesis.

(0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0002) (0.2116) (0.8501)

R21

a2

b2

0.6354 0.5866 0.5730 0.5226 0.3489 0.2542 0.1344 0.0675 0.0155 0.0007

6.3507 (0.0000) 6.1725 (0.0000) 6.5155 (0.0000) 6.4250 (0.0000) 7.3425 (0.0007) 6.7912 (0.0083) 17.5312 (0.0000) 24.7633 (0.0000) 25.3667 (0.0204) 32.3777 (0.0943)

0.9933 0.9958 0.9912 0.9924 0.9809 0.9877 0.8594 0.7768 0.7695 0.6989

R22 (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0011)

0.9226 0.9245 0.9185 0.9063 0.8077 0.7675 0.5975 0.4828 0.2886 0.2065

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to equilibrium found by Cox et al. (1982) is a consequence of prohibiting subjects from bidding above their valuations. 4.3. Result 3 Bids are better explained by a linear relationship in day 2 than in day 1. If subjects were playing according to their equilibrium strategies, then regressing their bids against their types would yield an intercept coefficient of zero, and a slope coefficient of one. We estimated the relationships Bi; t ¼ at þ bt vi; t , for i = 1, . . ., 24 with the data collected in t = 1, 2. The results of both estimations are presented in Table 4. Considering all types, one can see that both parameters of the regression are closer to its theoretical counterparts in the regression with data from day two. An intercept strictly greater than zero suggests that subjects determine their bids adding a markup to their valuations. This markup decreases significantly in day two but it remains significant. As we consider higher types, the fit is consistently better in day 2. Within day 2, the regression lines shift up and become flatter, this in turn deteriorates the goodness of fit. Fig. 2-b shows that as the minimum type considered increases, the regressions capture the cluster of bids above the 45j line.

5. Conclusions Repetition may be a difficult method to understand the workings of second-price auctions. We have analyzed the results of two experimental sessions involving single-unit second-price auctions with private values. The experimental sessions were approximately one day apart and thus subjects were given time to introspect. The results of the experiments allow us to conclude that bids are significantly closer to equilibrium behavior after the time for introspection. However, overbidding remains statistically significant, suggesting that the proclivity of subjects to overbid in second-price auctions is robust to changes in the experimental setting. The improvement in underbidding is more noticeable, as indicated by the larger reduction in average underbidding. Acknowledgements I am grateful to Alejandro Manelli and Martin Sefton for helpful conversations and encouragement, and to an anonymous referee for very useful remarks. Financial support of the Association of Students of the Arizona State University research grant is also acknowledged. I also benefited from participating in NSF project SBR-9810840. Any errors are solely my responsibility.

References Blume A., Gneezy U., 2000. Cognitive Forward Induction and Coordination without Common Knowledge: theory and Evidence Mimeo. University of Pittsburgh. Cox, J., Roberson, R., Smith, V.L., 1982. Theory and behavior of single object auctions. Smith, V.L. (Ed.), Research in Experimental Economics. vol. 2. JAI Press, Smith.

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Harstad R.M., 1990. Dominant Strategy Adoption, Efficiency, and Bidders’ Experience with Pricing Rules, Mimeo. Virginia Commonwealth University. Kagel, J.H., Levin, D., 1993. Independent private value auctions: bidder behaviour in first-, second- and third-price auctions with varying number of bidders. The Economic Journal 419, 868 – 879. Kagel, J.H., Harstad, R.M., Levin, D., 1987. Information impact and allocation rules in auctions with affiliated private values: a laboratory study. Econometrica 55, 1275 – 1304.