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Playing the field in all-pay auctions


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.

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  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.


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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.

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Stephenson, D.G., Brown, A.L. Playing the field in all-pay auctions. Exp Econ 24, 489–514 (2021).

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  • All pay auctions
  • Evolutionary game theory
  • Experiment

JEL Classification

  • D44
  • C73
  • C92