Skip to main content
SpringerLink
Log in

Varying the number of bidders in the first-price sealed-bid auction: experimental evidence for the one-shot game

  • Published:
Theory and Decision Aims and scope Submit manuscript

Abstract

The paper reports experimental data on the behavior in the first-price sealed-bid auction for a varying number of bidders when values and bids are private information. This feedback-free design is proposed for the experimental test of the one-shot game situation. We consider both within-subjects and between-subjects variations. In line with the qualitative risk neutral Nash equilibrium prediction, the data show that bids increase in the number of bidders. However, in auctions involving a small number of bidders, average bids are above, and in auctions involving a larger number of bidders, average bids are below the risk neutral equilibrium prediction. The quartile analysis reveals that bidding behavior is not constant across the full value range for a given number of bidders. On the high value quartiles, however, the average bid–value ratio is not different from the risk neutral prediction. The behavior is different when the winning bid is revealed after each repetition.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Engelbrecht-Wiggans (1980) surveys the theoretical literature on bidding models.

  2. Grimm and Schmidt (2000) more generally show that fanning out or fanning in of bidding strategies can result from quasiconcave or quasiconvex preferences.

  3. There is evidence from various experimental environments that repeated interaction with feedback influences decision-making. The feedback-free approach is the state-of-the-art approach in the experimental literature on individual decision making under risk and uncertainty (Hey 1991; Camerer 1995). A treatment effect between feedback-free and repeated-game approach was reported for different experimental environments including public goods (Neugebauer et al. 2009), guessing games (Weber 2003; Grosskopf and Nagel 2008), sequential first-price auctions (Neugebauer 2004), and single-unit first-price auctions (Neugebauer and Perote 2008).

  4. One contribution of our study is that we consider comparatively large number of bidders, including \(N = 14\) and \(N = 21\). Most studies on the first-price auction focused on a rather small number of bidders, \(N \le 4\), exceptions are reported below.

  5. Under the assumption that the bidders exhibit constant relative risk aversion and the risk-aversion measures are independently drawn from a commonly known distribution, the existence of a Nash equilibrium has been shown (Cox et al. 1982a). In equilibrium, bids are above the risk neutral Nash equilibrium and are positively correlated with the degree of individual risk aversion.

  6. In Neugebauer and Selten’s work, the subject’s value was fixed at unity and computerized competitors’ bids were independently drawn from the uniform distribution over the unit interval. In between-subjects variation three feedback conditions were examined in different market sizes \(N = \{3, 4, 5, 6, 9\};\) (1) winning bid and competitors’ highest bid, (2) winning bid, (3) winning bid only if subject wins. Above-equilibrium bidding was the average action in condition (2) for all \(N\), and in the conditions (1) and (3) for \(N\) = 4.

  7. Related experimental studies of oligopolistic markets have also shown that competitive pressures increase with the number of oligopolists (Huck et al. 2000, 2004; Dufwenberg and Gneezy 2000; Abbink and Brandts 2008; Brandts and Guillen 2007). In a remotely related theoretical and now classical paper, Selten (1973) suggested that quite extreme competition-effects can occur in oligopolies.

  8. For the high value quartile in the small market size, \(N = 3,\) the bid–value ratios are higher in within-subjects variation. This evidence seems to suggest that behavior in small markets is quite sensitive to variations.

  9. This way of presenting the problem is theoretically equivalent to having values and bids over the unit interval rounded to the second decimal. As reported below, a finer scale has been applied in the experiment for within-subjects variation.

  10. The instructions are appended to the paper.

  11. The reported task was the first one of several tasks in an experimental session.

  12. The experiment conducted in York involved market size \(N = 7\) only. In York, the participants were from different fields of study, while in Hannover all participants were economics students. For the market size \(N = 7\) (the data including some replies to the debriefings are detailed in Neugebauer and Perote 2008), there were no significant behavioral differences between the samples from Hannover and York. Therefore, we include the data from York in the sample. As a matter of fact, the stated observations do not change if these data are excluded.

  13. In experiment 1, in 58 of 5,500 random draws the outcome was a zero private value; in experiment 2: we had 4 zero draws of 12,600 draws. In experiment 3, the private value was always positive.

  14. Exact \(p\) values are 0.071, 0.211, 0.065, 0.005, and 0.000, for \(N = \{3, 5, 7, 9, 14\}\) respectively.

  15. The null hypothesis that all samples come from the same distribution is tested against the ordered alternative that bids weakly increase with \(N\) with at least one inequality. The test is conducted one-tailed as the prediction of the risk-neutral Nash equilibrium indicates an increase of the bid–value ratio with \(N\).

  16. In the repeated first-price auction experimental design that reveals at least the winning bid after each period, average bidding above the RNNE results for all market sizes \(N \le 10\). Experimental markets with more than 10 participants have not been investigated before.

  17. The individual average bid–value ratios are recorded in Table 5 in Appendix for each quartile and overall. Confirmatory results to the reported ones are obtained if one examines the behavior conditional on being assigned the high value that is expected to win in an efficient market (see Neugebauer 2007), or if one considers only bids of values equal or above 90, or if one considers only values that in the RNNE have at least a probability of winning of 0.25 (this would involve values of at least 50 in the market with \(N = 3;\) it would involve values of at least 71 in the market with \(N = 5;\) etc.).

  18. Splitting the data by group size yields significant differences between the first and the second quartile for \(N > 3\) (\(p < 0.015\)), and between the second and the third quartile for \(N > 5\) (\(p < 0.015\)). None of the sessions shows a significant effect on the 5 % significant level between the third and the fourth quartile.

  19. \(p\) values are 0.722, 0.017, 0.000, and 0.000, for \(N = \{3, 5, 7, 9, 14\}\), respectively.

  20. To avoid unintended bidding above value, each such bid requires an extra confirmation by the subject.

  21. The software implemented the following matching protocol which was not explained in detail to subjects. Subjects are randomly assigned numbers {1, 2,..., 21}. The subjects of the first three numbers are matched in the first market of size \(N = 3,\) the second three numbers are assigned to the second market, etc. Similarly, for the market size \(N = 7;\) the first seven numbers are assigned to the first market, etc. For market size \(N = 14,\) the first fourteen numbers are assigned to the first market. The numbers {15,..., 21} are assigned to the second. This market of size \(N = 14\) is completed with the bids of the subjects numbered {8,..., 14}, whose bids are relevant for the price determination in that second market, but whose payoffs are exclusively determined in the first market of size \(N = 14.\) Finally, each subject submits a bid to the market of size \(N = 21.\)

  22. Comparing the bid–value ratios for the within-subject experiment between conditions, we find that the increasing-pay treatment generates higher averages than the constant-pay treatment. However, only for the low-value quartile are these differences significant at the 5 % significance level. We conduct a two-tailed Mann–Whitney test for the comparison of the bid–value ratio across treatment conditions. The \(p\) values of the test conducted on the low value quartile are {0.046, 0.044, 0.024, 0.031} for \(N = \{3, 7, 14, 21\}\).

  23. The \(p\) values of the two-tailed Mann–Whitney test are {0.901, 0.349, 0.183, 0.023, 0.261} for \(N = 3,\) {0.737, 0.240, 0.166, 0.401, 0.323} for \(N = 7,\) and {0.068, 0.420, 0.590, 0.692, 0.181} for \(N = 14,\) where the first \(p\) value in the curly brackets represents the first quartile, the second quartile, ..., the fourth quartile; finally, the last \(p\) value represents the two sample test of bid–value ratios for the corresponding number of bidders.

  24. Unfortunately there was a high no-show rate, such that we were unable to obtain data from a third group. However, as the results are quite clear, we refrained from running a further session.

  25. In line with earlier results on the repeated-game design, this observation suggests that the difference of the average bid–value ratio from the RNNE may decrease with an increasing number of bidders.

  26. In between-subjects variation only four observations of market size \(N = 7\) are independent from the feedback-free approach (INFO1). So, the Fisher test involves the first six observations of Table 4.

  27. The strategy method has been applied to repeated first-price auctions with feedback information in Selten and Buchta (1999), Güth et al. (2003), Pezanis-Christou and Sadrieh (2003), Kirchkamp et al. (2008), Kirchkamp et al. (2009), and Kirchkamp and Reiss (2011).

  28. In the sessions, \(N\) was substituted by the number of participants \(N = \{3, 5, 7, 9, 14\}\).

References

  • Abbink, K., & Brandts, J. (2008). Pricing in Bertrand competition with increasing marginal costs. Games and Economic Behavior, 63, 1–31.

    Article  Google Scholar 

  • Andreoni, J., Che, Y.-K., Kim, J. (2007). Asymmetric information about rivals’ types in standard auctions: An experiment. Games and Economic Behavior, 59(2), 240–259.

    Google Scholar 

  • Armantier, O., & Treich, N. (2009). Subjective probabilities in games: An application to the overbidding puzzle. International Economic Review, 50(4), 1079–1102.

    Article  Google Scholar 

  • Battalio, R. C., Kogut, J., & Meyer, J. (1990). Individual and market bidding in a Vickrey first-price auction: Varying market size and information. In L. Green & J. H. Kagel (Eds.), Advances in behavioral economics (Vol. 2). Norwood: Ablex Publishing.

    Google Scholar 

  • Brandts, J., & Guillen, P. (2007). Collusion and fights in an experiment with price setting firms and advanced production. The Journal of Industrial Economics, 55, 453–473.

    Article  Google Scholar 

  • Camerer, C. (1995). Individual decision making. In J. H. Kagel & A. E. Roth (Eds.), The handbook of experimental economics. Princeton: Princeton University Press.

    Google Scholar 

  • Chen, K.-Y., Plott, C. R. (1998). Nonlinear behavior in sealed bid first-price auctions. Games and Economic Behavior, 25, 34–78.

    Google Scholar 

  • Conover, W. J. (1999). Practical nonparametric statistics. New York: Wiley.

    Google Scholar 

  • Cox, J. C., Roberson, B., & Smith, V. L. (1982a). Theory and behavior of single object auctions. Research in Experimental Economics, 2, 1–43.

    Google Scholar 

  • Cox, J. C., Smith, V. L., & Walker, J. M. (1982b). Auction market theory of heterogeneous bidders. Economics Letters, 9, 319–325.

    Article  Google Scholar 

  • Cox, J. C., Smith, V. L., & Walker, J. M. (1988). Theory and individual behavior of first-price auctions. Journal of Risk and Uncertainty, 1, 61–99.

    Article  Google Scholar 

  • Crawford, V. P., & Iriberri, N. (2007). Level k auctions: Can a nonequilibrium model of strategic thinking explain the Winner’s curse and overbidding in private value auctions? Econometrica, 75, 1721–1770.

    Article  Google Scholar 

  • Dorsey, R., & Razzolini, L. (2003). Explaining overbidding in first price auctions using controlled lotteries. Experimental Economics, 6(2), 123–40.

    Article  Google Scholar 

  • Dufwenberg, M., & Gneezy, U. (2000). Price competition and market concentration: An experimental study. International Journal of Industrial Organization, 18, 7–22.

    Article  Google Scholar 

  • Dufwenberg, M., & Gneezy, U. (2002). Information disclosure in auctions: An experiment. Journal of Economic Behavior & Organization, 48, 431–444.

    Article  Google Scholar 

  • Dyer, D., Kagel, J. H., & Levin, D. (1989). Resolving uncertainty about the number of bidders in independent private-value auctions: An experimental analysis. The RAND Journal of Economics, 20, 268–279.

    Article  Google Scholar 

  • Engelbrecht-Wiggans, R. (1980). Auctions and bidding models: A survey. Management Science, 26, 119–142.

    Google Scholar 

  • Engelbrecht-Wiggans, R., & Katok, E. (2007). Regret in auctions: Theory and evidence. Economic Theory, 33, 81–101.

    Article  Google Scholar 

  • Fischbacher, U. (2007). z-Tree: Zurich toolbox for ready-made economic experiments. Experimental Economics, 10, 171–178.

    Article  Google Scholar 

  • Goeree, J. K., Holt, C. A., & Palfrey, T. R. (2002). Quantal response equilibrium and overbidding in private-value auctions. Journal of Economic Theory, 104, 247–272.

    Article  Google Scholar 

  • Grimm, V., & Schmidt, U. (2000). Equilibrium bidding without the independence axiom: A graphical analysis. Theory and Decision, 49(4), 361–374.

    Article  Google Scholar 

  • Grosskopf, B., & Nagel, R. (2008). The two-person beauty contest. Games and Economic Behavior, 62, 93–99.

    Article  Google Scholar 

  • Güth, W., & Ivanova, Stenzel R. (2003). Learning to bid-an experimental study of bid function adjustments in auctions and fair division games. The Economic Journal, 113, 477–494.

    Article  Google Scholar 

  • Hey, J. D. (1991). Experiments in economics. Oxford: Blackwell Pulishers.

    Google Scholar 

  • Huck, S., Normann, H. T., & Oechssler, J. (2000). Does information about competitors’ actions increase or decrease competition in experimental oligopoly markets? International Journal of Industrial Organization, 18, 39–57.

    Article  Google Scholar 

  • Huck, S., Normann, H. T., & Oechssler, J. (2004). Two are few and four are many: Number effects in experimental oligopolies. Journal of Economic Behavior & Organization, 53, 435–446.

    Article  Google Scholar 

  • Kagel, J. H. (1995). Auctions: A survey of experimental research. In J. H. Kagel & A. E. Roth (Eds.), The handbook of experimental economics (pp. 501–586). Princeton: Princeton University Press.

    Google Scholar 

  • Kagel, J. H., & Levin, D. (1993). Independent private value auctions: Bidder behaviour in first-, second-and third-price auctions with varying numbers of bidders. The Economic Journal, 103, 868–879.

    Article  Google Scholar 

  • Kagel, J. H., & Levin, D. (2008). Auctions: A survey of experimental research, 1995–2008. In J. H. Kagel & A. E. Roth (Eds.), Handbook of experimental economics (Vol. 2). Princeton: Princeton University Press.

    Google Scholar 

  • Kirchkamp, O., & Reiss, J. P. (2011). Out-of-equilibrium bids in first-price auctions: Wrong expectations or wrong bids. The Economic Journal, 121, 1361–1397.

    Article  Google Scholar 

  • Kirchkamp, O., Reiß, J. P., & Sadrieh, A. (2008). A pure variation of risk in firstprice auctions, Technical Report No. 058, METEOR Research Memorandum, Maastricht University.

  • Kirchkamp, O., Poen, E., & Reiß, J. P. (2009). Outside options: Another reason to choose the first-price auction. European Economic Review, 53(2), 153–169.

    Article  Google Scholar 

  • Neri, C. (2012). Eliciting beliefs in continuous-choice games: A double auction experiment. University of St. Gallen, Working Paper.

  • Neugebauer, T. (2004). Bidding strategies of sequential first price auctions programmed by experienced bidders. Cuadernos de Economía, 27, 153–184.

    Google Scholar 

  • Neugebauer, T. (2007). Bid and price effects of increased competition in the first-price auction: experimental evidence. CREFI-LSF Working Paper Series, 7–17.

  • Neugebauer, T., & Selten, R. (2006). Individual behavior of first-price auctions: The importance of information feedback in computerized experimental markets. Games and Economic Behavior, 54, 183–204.

    Article  Google Scholar 

  • Neugebauer, T., & Pezanis-Christou, P. (2007). Bidding behavior at sequential first-price auctions with(out) supply uncertainty: A laboratory analysis. Journal of Economic Behavior & Organization, 63(1), 55–72.

    Article  Google Scholar 

  • Neugebauer, T., & Perote, J. (2008). Bidding ‘as if’ risk neutral in experimental first price auctions without information feedback. Experimental Economics, 11, 190–202.

    Article  Google Scholar 

  • Neugebauer, T., Perote, J., Schmidt, U., & Loos, M. (2009). Selfish-biased conditional cooperation: On the decline of contributions in repeated public goods experiments. Journal of Economic Psychology, 30, 52–60.

    Article  Google Scholar 

  • Ockenfels, A., & Selten, R. (2005). Impulse balance equilibrium and feedback in first price auctions. Games and Economic Behavior, 51, 155–170.

    Article  Google Scholar 

  • Pezanis-Christou, P., & Sadrieh, A. (2003). Elicited bid functions in (a)symmetric firstprice auctions. Discussion Paper No. 2003-58, CentER, Tilburg University.

  • Selten, R. (1967). Die Strategiemethode zur Erforschung des eingeschränkt rationalen Verhaltens im Rahmen eines Oligopolexperiments. Beiträge zur experimentellen Wirtschaftsforschung, 1, 136–168.

    Google Scholar 

  • Selten, R. (1973). A simple model of imperfect competition, where 4 are few and 6 are many. International Journal of Game Theory, 2, 141–201.

    Article  Google Scholar 

  • Selten, R., & Buchta, J. (1999). Experimental sealed bid first price auctions with directly observed bid functions. In D. Budescu, I. Erev, & R. Zwick (Eds.), Games and human behavior: Essays in honor of Amnon Rapoport (pp. 101–116). Wien: Physica Verlag.

    Google Scholar 

  • Selten, R., Mitzkewitz, M., & Uhlich, G. R. (1997). Duopoly strategies programmed by experienced players. Econometrica, 65, 517–555.

    Article  Google Scholar 

  • Vickrey, W. (1961). Counterspeculation, auctions, and competitive sealed tenders. The Journal of Finance, 16, 8–37.

    Google Scholar 

  • Weber, R. A. (2003). Learning with no feedback in a competitive guessing game. Games and Economic Behavior, 44, 134–144.

    Article  Google Scholar 

Download references

Acknowledgments

We acknowledge helpful comments from Utz Weizel, Olivier Armantier, Jordi Brandts, James Cox, Vince Crawford, Jacob Goeree, Veronika Grimm, Charles Holt, Heidrun Hoppe, Rudi Kerschbamer, Paul Pezanis-Christou, Amnon Rapoport, Karim Sadrieh, Reinhard Selten and other participants at the GfeW meeting in Goslar, the international ESA meeting in Rome and the local ESA meeting in Tucson. Financial support through the EU-TMR Research Network ENDEAR (FMRX-CT98-0238), Recherche-UL (F2R-LSF-PUL-09BFAM), and the Department of Economics at Radboud University Nijmegen is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Department of Economics, Radboud University Nijmegen, Thomas van Aquinostraat 5, 6525 GD,  Nijmegen, The Netherlands

    Sascha Füllbrunn

  2. Luxembourg School of Finance, Université de Luxembourg, Campus Kirchberg 4, rue Albert Borschette, 1240,  Luxembourg, Luxembourg

    Tibor Neugebauer

Authors
  1. Sascha Füllbrunn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tibor Neugebauer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tibor Neugebauer.

Appendices

Data on the repeated game design are available upon request.

Appendix

1.1 Instructions (between-subjects experiment)

General information

  1. 1.

    You are about to participate in 50 rounds of an auction experiment. In each of these rounds, you will be assigned to a group of \(N\) bidders:Footnote 28 yourself and 6 other participants. Your group will stay the same throughout the experiment. However, you will not receive any information about the identity of the other group members.

  2. 2.

    In each of the 100 rounds, one fictitious item will be sold for which you have to submit a bid. A bid consists in proposing a price of purchase (i.e., an integer number between 0 and 100).

The auction rule

  1. 3.

    Your bid must be always a number between 0 and 10,000. In each auction round, the bidder who submits the highest bid wins the auction.

  2. 4.

    If ever the highest bid is submitted by more than one bidder, the winner will be determined randomly. (There will be an equal chance for each of them to be selected as the winner.)

  3. 5.

    The winner of the auction round is awarded the item and pays a price equal to her/his bid.

Your payoff in an auction round

  1. 5.

    At the outset of each auction round, the computer draws integer numbers between 0 and 10,000 at random, one for each bidder. (These numbers are independent of each other.)

  2. 6.

    One of these numbers will be assigned to you. The number represents your resale value for the item for sale.

  3. 7.

    Your resale value is the amount the experimenter is going to pay you if you win the item in the auction round.

  4. 8.

    Therefore, if you win the item in the market to which you participate, your round payoff will be equal to the difference between your resale value and your bid. If you don’t win the item, your round payoff will be zero.

  5. 9.

    Note: In order to prevent negative payoffs, you will NOT be allowed to submit a bid above your resale value.

Your payoff in the experiment

  1. 10.

    Round payoffs, bids, prices and resale values are expressed in the Experimental Currency Unit ECU.

  2. 11.

    At the end of the experiment you will be paid your accumulated payoff of the experiment privately in the adjacent office. The exchange rate will be 1 ECU = 0.0015 (constant pay treatment). The exchange rate differs between markets. In the 3-bidders market, 100 ECU = €0.05; in the 7-bidders market, 100 ECU = €0.233; in the 14-bidders market, 100 ECU = €0.875; and in the 21-bidders market, 100 ECU = €1.925.

Information feedback

  1. 12.

    You will not receive any information about prices or payoffs. Throughout the experiment you will be given an on-screen record of all information you have received in the previous auction rounds including values and bids.

  2. 13.

    After 50 rounds, you will receive full information on prices and payoffs per period and overall.

1.2 Instructions (within-subjects experiment)

General information

  1. 1.

    You are about to participate in 50 rounds of an auction experiment. In each of these rounds, you will simultaneously propose a price (submit a bid) in four auction markets. You participate to each of the four markets with equal probability, but the actual market to which you participate is revealed to you only in hindsight.

  2. 2.

    The four markets in which you simultaneously bid differ in the number of bidders. The first auction market has 3 participants (3-bidder market), the second has 7 participants (7-bidder market), the third 14 (14-bidder market), and the fourth 21 (21-bidder market). The participants in each of these groups stay the same. Unless you bid in the 21-bidder market, however, you will not know the identity of the other group members.

  3. 2.

    In each of the 100 rounds, one fictitious item will be sold for which you have to submit a bid. A bid consists in proposing a price of purchase (i.e., an integer number between 0 and 100).

The auction rule

  1. 3.

    In each auction round, the bidder who submits the highest bid wins the auction.

  2. 4.

    If ever the highest bid is submitted by more than one bidder, the winner will be determined randomly. (There will be an equal chance for each of them to be selected as the winner.)

  3. 5.

    The winner of the auction round is awarded the item and pays a price equal to her/his bid.

Your payoff in an auction round

  1. 5.

    At the outset of each auction round, the computer draws integer numbers between 0 and 100 at random, one for each bidder. (These numbers are independent of each other.)

  2. 6.

    One of these numbers will be assigned to you. The number represents your resale value for the item for sale.

  3. 7.

    Your resale value is the amount the experimenter is going to pay you if you win the item in the auction round.

  4. 8.

    Therefore, if you win the item, your round payoff will be equal to the difference between your resale value and your bid. If you don’t win the item, your round payoff will be zero.

  5. 9.

    Note: In order to prevent negative payoffs, you should NOT submit a bid above your resale value.

Your payoff in the experiment

  1. 10.

    Round payoffs, bids, prices and resale values are expressed in the Experimental Currency Unit ECU.

  2. 11.

    At the end of the experiment you will be paid your accumulated payoff of the experiment privately in the adjacent office. The exchange rate will be 1 ECU = \({\pounds }\)0.06 (UK, \(N = 7\)); 1 ECU = € {0.05, 0.05, 0.10, 0.10, 0.20} (Germany, \(N = \{3, 5, 7, 9, 14\}\)).

Information feedback

  1. 12.

    You will not receive any information about prices or payoffs.

  2. 13.

    Throughout the experiment you will be given an on-screen record of all information you have received in the previous auction rounds including values and bids.

Tables

Data on the repeated game design are available upon request.

See Tables 5, 6, and 7.

Table 5 Individual average bid–value ratio by segment in between-subjects experiment
Full size table
Table 6 Individual average bid–value ratio by segment in within-subjects experiment—constant-pay condition
Full size table
Table 7 Individual average bid–value ratio by segment in within-subjects experiment—increasing-pay condition
Full size table

Rights and permissions

Reprints and permissions

About this article

Cite this article

Füllbrunn, S., Neugebauer, T. Varying the number of bidders in the first-price sealed-bid auction: experimental evidence for the one-shot game. Theory Decis 75, 421–447 (2013). https://doi.org/10.1007/s11238-013-9378-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11238-013-9378-z

Keywords

JEL codes

Search

Navigation