Skip to main content

Playing the field in all-pay auctions

Abstract

We provide the first examination of all-pay auctions using continuous-time protocols, allowing subjects to adjust their bid at will, observe payoffs almost instantaneously, and gain more experience through repeated-play than in previous, discrete-time, implementations. Unlike our predecessors—who generally find overbidding—we observe underbidding relative to Nash equilibrium. To test the predictions of evolutionary models, we vary the number of bidders and prizes across treatments. If two bidders compete for a single prize, evolutionary models predict convergence to equilibrium. If three bidders compete for two prizes, evolutionary models predict non-convergent cyclical behavior. Consistent with evolutionary predictions, we observe cyclical behavior in both auctions and greater instability in two-prize auctions. These results suggest that evolutionary models can provide practitioners in the field with additional information about long-run aggregate behavior that is absent from conventional models.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Notes

  1. Inexperience is one of many explanations for overbidding relative to Nash equilibrium in all-pay auctions (see Dechenaux et al. 2014). Preference-based explanations (e.g., joy of winning an item) may cause overbidding to continue in the long-run. We explore this issue in more detail in our concluding section.

  2. Time-averaged evolutionary predictions are similar to the Nash predictions, but dynamic evolutionary predictions are very different from the Nash predictions.

  3. Fixed-point models, including both Nash equilibrium and quantal response equilibrium (McKelvey and Palfrey 1995), identify invariant points of an operator on the strategy space. Nash equilibria are fixed points of the best response and quantal response equilibria are fixed points of the quantal response. Consequently, fixed-point models never predict instability in the distribution of strategies. Equilibrium selection models can address relative stability across multiple equilibria, but are less useful in settings with a unique equilibrium. In contrast, evolutionary models describe a dynamic adjustment process that can explain persistently non-convergent cyclical patterns in the distribution of strategies across a population.

  4. For example, consider the design of a contest structure to allocate grant funding. Several distinct contest structures may yield identical equilibrium predictions. Evolutionary models can help policymakers identify which of these structures is most likely to induce convergence on the desired equilibrium. Conversely, a policymaker who fails to consider evolutionary models may select a contest structure that has desirable equilibrium properties but induces undesirable non-convergent behavior. We elaborate on this idea in our concluding section.

  5. See Dechenaux et al. (2014) and Brown and Stephenson (2019) for a survey on each topic, respectively.

  6. The all-pay auction has a continuous strategy space which we discretize into 1001 actions (see Sect. 4).

  7. Since the expected payoff from bidding zero is zero, the expected payoff from other bids in the support of the Nash equilibrium must also be zero.

  8. Since \(\sqrt{x}>x\) for all \(x\in (0,1)\), we have \(\frac{b_i}{v}>1-\sqrt{1-\frac{b_i}{v}}\) for all \(b_i\in (0,v)\).

  9. This instantaneous change is known as a “jump-adjustment:” the alternative “continuous-adjustment” has strategies gradually change upon subject input. Cason et al. (2014) employs each method in a separate treatment. Stephenson (2019) employs the latter method.

  10. Because payoffs are calculated ten times per second, one could interpret this as a finitely repeated game. This approximation of continuous time is common in the literature (see Cason et al. 2014; Oprea et al. 2011; Stephenson 2019). Further, it is unlikely subjects had the cognitive ability or physical reflexes to make adjustments ten times per second, making the game effectively continuous from the subjects’ standpoint.

  11. None of the five outcome variables (see Table 2) differ substantially between information treatments, holding the auction constant (see Table A.1 in electronic supplementary material). Regression analysis (not provided) confirms the main results of this paper even in specifications that use information treatment as an explanatory variable.

  12. Subjects in our experiment could instantaneously adjust their action. In Stephenson (2019) and some treatments of Cason et al. (2014), subjects made continuous adjustments to their action over time, so they lack comparable jumps that can be counted (see footnote 9). Of the studies with jump adjustments, Cason et al. (2014) and Oprea et al. (2011) do not mention the frequency of adjustments they observed. Stephenson (2020) observed an adjustment every 6.67 seconds in a 24-action, dominant strategy, school choice game.

  13. Unless otherwise noted, we provide comparisons of session level averages in our analysis. Using period-level averages instead would not qualitatively affect our results. If anything, using period-level treatment comparisons, whether parametric or non-parametric, would make the differences between treatments significant at lower thresholds.

  14. See Table A.2 for a detailed description of the maximum likelihood estimates.

  15. If it had been feasible to clearly depict a payoff distribution over the 1000 dimensional space of mixed strategies, then we might have asked subjects to directly select mixed strategies. Alternatively, if each session involved thousands of subjects, then we might have obtained a closer approximation of the large population limit. In such experiments, time-averaged bids might have been even closer to the Nash predictions.

References

  • Abreu, D., & Sethi, R. (2003). Evolutionary stability in a reputational model of bargaining. Games and Economic Behavior, 44(2), 195–216.

    Article  Google Scholar 

  • Alós-Ferrer, C., Ania, A. B., & Schenk-Hoppé, K. R. (2000). An evolutionary model of bertrand oligopoly. Games and Economic Behavior, 33(1), 1–19.

    Article  Google Scholar 

  • Anderson, S. P., Goeree, J. K., & Holt, C. A. (1998). Rent seeking with bounded rationality: An analysis of the all-pay auction. Journal of Political Economy, 106(4), 828–853.

    Article  Google Scholar 

  • Barut, Y., & Kovenock, D. (1998). The symmetric multiple prize all-pay auction with complete information. European Journal of Political Economy, 14(4), 627–644.

    Article  Google Scholar 

  • Baye, M. R., Kovenock, D., & De Vries, C. G. (1993). Rigging the lobbying process: An application of the all-pay auction. American Economic Review, 83(1), 289–294.

    Google Scholar 

  • Baye, M. R., Kovenock, D., & De Vries, C. G. (1996). The all-pay auction with complete information. Economic Theory, 8(2), 291–305.

    Article  Google Scholar 

  • Benndorf, V., Martinez-Martinez, I., & Normann, H.-T. (2016). Equilibrium selection with coupled populations in hawk-dove games: Theory and experiment in continuous time. Journal of Economic Theory, 165, 472–486.

    Article  Google Scholar 

  • Bigoni, M., Casari, M., Skrzypacz, A., & Spagnolo, G. (2015). Time horizon and cooperation in continuous time. Econometrica, 83, 587–616.

    Article  Google Scholar 

  • Brown, A., & Stephenson, D. (2019). Handbook of experimental game theory, chapter games with continuous-time experimental protocols. (Forthcoming)

  • Cason, T. N., Friedman, D., & Hopkins, E. (2014). Cycles and instability in a rock-paper-scissors population game: A continuous time experiment. Review of Economic Studies, 81(1), 112–136.

    Article  Google Scholar 

  • Cason, T. N., Friedman, D., & Hopkins, E. (2020). An experimental investigation of price dispersion and cycles. https://krannert.purdue.edu/faculty/cason/papers/price-disp-cycles.pdf.

  • Chatterjee, K., Reiter, J. G., & Nowak, M. A. (2012). Evolutionary dynamics of biological auctions. Theoretical Population Biology, 81(1), 69–80.

    Article  Google Scholar 

  • Davis, D. D., & Holt, C. A. (1993). Experimental economics. Princeton: Princeton University Press.

    Google Scholar 

  • Davis, D. D., & Reilly, R. J. (1998). Do too many cooks always spoil the stew? An experimental analysis of rent-seeking and the role of a strategic buyer. Public Choice, 95(1–2), 89–115.

    Article  Google Scholar 

  • Dechenaux, E., Kovenock, D., & Sheremeta, R. M. (2014). A survey of experimental research on contests, all-pay auctions and tournaments. Experimental Economics, 18(4), 609–669.

    Article  Google Scholar 

  • Ernst, C., & Thöni, C. (2013). Bimodal bidding in experimental all-pay auctions. Games, 4(4), 608–623.

    Article  Google Scholar 

  • Fudenberg, D., & Levine, D. K. (1998). The theory of learning in games (Vol. 2). Cambridge: MIT Press.

    Google Scholar 

  • Gneezy, U., & Smorodinsky, R. (2006). All-pay auctions: An experimental study. Journal of Economic Behavior & Organization, 61(2), 255–275.

    Article  Google Scholar 

  • Greiner, B. (2015). Subject pool recruitment procedures: Organizing experiments with orsee. Journal of the Economic Science Association, 1(1), 114–125.

    Article  Google Scholar 

  • Hens, T., & Schenk-Hoppé, K. R. (2005). Evolutionary stability of portfolio rules in incomplete markets. Journal of Mathematical Economics, 41(1–2), 43–66.

    Article  Google Scholar 

  • Hodler, R., & Yektaş, H. (2012). All-pay war. Games and Economic Behavior, 74(2), 526–540.

    Article  Google Scholar 

  • Hopkins, E. (1999). A note on best response dynamics. Games and Economic Behavior, 29(1), 138–150.

    Article  Google Scholar 

  • Hopkins, E., & Seymour, R. M. (2002). The stability of price dispersion under seller and consumer learning. International Economic Review, 43(4), 1157–1190.

    Article  Google Scholar 

  • Lugovskyy, V., Puzzello, D., & Tucker, S. (2010). An experimental investigation of overdissipation in the all pay auction. European Economic Review, 54(8), 974–997.

    Article  Google Scholar 

  • Marinucci, M., & Vergote, W. (2011). Endogenous network formation in patent contests and its role as a barrier to entry. Journal of Industrial Economics, 59(4), 529–551.

    Article  Google Scholar 

  • McKelvey, R. D., & Palfrey, T. R. (1995). Quantal response equilibria for normal form games. Games and Economic Behavior, 10(1), 6–38.

    Article  Google Scholar 

  • Oprea, R., Henwood, K., & Friedman, D. (2011). Separating the hawks from the doves: Evidence from continuous time laboratory games. Journal of Economic Theory, 146(6), 2206–2225.

    Article  Google Scholar 

  • Potters, J., De Vries, C. G., & Van Winden, F. (1998). An experimental examination of rational rent-seeking. European Journal of Political Economy, 14(4), 783–800.

    Article  Google Scholar 

  • Sandholm, W. H. (2010). Population games and evolutionary dynamics. Cambridge: MIT Press.

    Google Scholar 

  • Smith, J. M. (1982). Evolution and the theory of games. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Stephenson, D. (2019). Coordination and evolutionary dynamics: When are evolutionary models reliable? Games and Economic Behavior, 113, 381–395.

    Article  Google Scholar 

  • Stephenson, D. (2020). Assignment feedback in school choice mechanisms. In Working paper.

  • Stephenson, D. G., & Brown, A. L. (2020). Characterizing persistent disequilibrium dynamics: Imitation or optimization?

  • Taylor, P. D., & Jonker, L. B. (1978). Evolutionary stable strategies and game dynamics. Mathematical Biosciences, 40(1), 145–156.

    Article  Google Scholar 

Download references

Acknowledgements

We thank the Texas A&M Humanities and Social Science Enhancement of Research Capacity Program and the College of Liberal Arts at Texas A&M for their generous financial support of this research. This paper benefitted from helpful comments by Tim Cason, Catherine C. Eckel, Daniel Fragiadakis, Dan Friedman, Dan Kovenock, Ryan Oprea, as well as the continual support and feedback the experimental research team at Texas A&M. We are also grateful for the insights given by attendees of the 2013 North American and World Meetings of the Economic Science Association.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Daniel G. Stephenson.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material (PDF 20261 KB)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Stephenson, D.G., Brown, A.L. Playing the field in all-pay auctions. Exp Econ 24, 489–514 (2021). https://doi.org/10.1007/s10683-020-09669-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10683-020-09669-5

Keywords

  • All pay auctions
  • Evolutionary game theory
  • Experiment

JEL Classification

  • D44
  • C73
  • C92