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Indefinitely repeated contests: An experimental study

Abstract

We experimentally explore indefinitely repeated contests. Theory predicts more cooperation, in the form of lower expenditures, in indefinitely repeated contests with a longer expected time horizon. Our data support this prediction, although this result attenuates with contest experience. Theory also predicts more cooperation in indefinitely repeated contests compared to finitely repeated contests of the same expected length, and we find empirical support for this. Finally, theory predicts no difference in cooperation across indefinitely repeated winner-take-all and proportional-prize contests, yet we find evidence of less cooperation in the latter, though only in longer treatments with more contests played. Our paper extends the experimental literature on indefinitely repeated games to contests and, more generally, contributes to an infant empirical literature on behavior in indefinitely repeated games with “large” strategy spaces.

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Notes

  1. Games that are repeated a known number of times are termed finite supergames or finitely repeated games, while games with an unknown time horizon are indefinite supergames or indefinitely repeated games (Friedman 1971).

  2. Dechenaux et al. (2015) survey experimental contest research while Dal Bó and Fréchette (2018) review the experimental supergame literature. The experimental indefinite supergame literature has largely focused on the Prisoner’s Dilemma. For example, see Murnighan and Roth (1983); Dal Bó (2005); Duffy and Ochs (2009); Dal Bó and Fréchette (2011); Dal Bó and Fréchette (2018). Non-Prisoner’s Dilemma indefinite supergame experiments include Palfrey and Rosenthal (1994); Sell and Wilson (1999); Tan and Wei (2014); Lugovskyy et al. (2017) (public goods), Holt (1985); Feinberg and Husted (1993) (oligopoly), Camera and Casari (2014); Duffy and Puzzello (2014) (monetary exchange), Engle-Warnick and Slonim (2006a, b) (trust), McBride and Skaperdas (2014) (conflict), and Kloosterman (2020) (coordination).

  3. Some studies mostly confirm theory (Dal Bó 2005; Duffy and Ochs 2009; Dal Bó and Fréchette 2011; Fréchette and Yuksel 2017), while others report more mixed support for theory (Roth and Murnighan 1978; Murnighan and Roth 1983; Normann and Wallace 2012; Lugovskyy et al. 2017; Kloosterman 2020).

  4. See Roth and Murnighan (1978). For comparisons of supergame termination rules, see Normann and Wallace (2012) and Fréchette and Yuksel (2017).

  5. Examining cooperation across continuation probabilities is not the main concern of Holt (1985).

  6. Kloosterman (2020) reports low cooperation in an indefinite \(3\times 3\) coordination game—a game with a slightly larger strategy space than the PD.

  7. Relaxing the risk-neutrality assumption and considering, say, symmetrically risk-averse players exhibiting CARA preferences changes predictions quantitatively, but preserves our comparative statics, with the exception that expenditure levels in PP and WTA contests no longer coincide. In Sect. 4, we elaborate on the behavior of non-risk-neutral players in each contest setting.

  8. Contests are typically defined with continuous, unbounded strategy spaces. In this paper, we define strategy spaces as discrete and bounded because that is the way they are implemented in our experiment. This discrete expenditure level restriction is also useful because it allows for the existence of the best deviation from socially optimal play. For a sufficiently large V, this modified, discrete game is a good approximation of the original, continuous game and has the same equilibrium, socially optimal expenditures, and payoffs as the original game, provided that V is divisible by 4.

  9. This is the unique NE of the contest game with continuous strategy spaces (see, e.g., Szidarovszky and Okuguchi 1997). For our parameterization, we verify numerically that it is also the unique NE in the discrete game.

  10. With this sample size, average expenditure comparisons between any two treatments have power 29%, 86%, and 99% for effect sizes (Cohen’s d) 0.2, 0.5 and 0.8, respectively, at \(\alpha =0.10\). Thus, the conventional 80% power level would be reached for medium effect sizes and beyond.

  11. While the grim trigger strategy involving \(x^\mathrm{so}=0\) cannot be supported under \(\delta =0.5\), other symmetric strategy pairs \((\bar{x},\bar{x})\), with \(\bar{x} \in (0,x^*]\), can be supported. The most cooperative outcome that can be supported is the grim trigger strategy involving cooperation at \(\bar{x}=4\) until defection.

  12. The zipper (turnpike) matching protocol was implemented with 20 participants over 10 supergames. Before the first supergame, participants were divided into two groups (A and B) of 10 participants. Within each group, participants were assigned unique ID numbers, \(i_A,i_B\in \{1,\dots ,10\}\). For the first supergame, participant \(i_A\) from group A was matched with participant \(i_B\) from group B so that \(i_A=i_B\). In subsequent supergames, group B participants were rotated into novel matches with group A participants. Specifically, in supergame S, participant \(i_A\) was always matched with participant \(i_B=10\cdot \mathbb {1}{\{S>i_A\}}+(i_A+1)-S\).

  13. Many indefinite supergame experiments use this procedure (see, e.g., Dal Bó 2005; McBride and Skaperdas 2014).

  14. See Fréchette and Yuksel (2017) for additional ways to implement discounting in indefinitely repeated games and for comparisons between them. We briefly considered, but ultimately rejected having participants play for infinite time due to IRB concerns.

  15. Please see Table 2 in the supplementary material for the specific seeds. For Low \(\delta\) sequences we used the years 1985, 1995, and 2005; for the High \(\delta\) sequences we used the years 1990, 2000, and 2010.

  16. We did not use the words “supergame” or “round” in the experimental instructions. Instead, we referred to supergames as “periods” and to rounds within the supergame as “decisions.”

  17. Please see Table 1 in the supplementary material for treatment-specific mean earnings. Following the main portion of the experiment, participants completed an incentivized risk task where they chose a single “switch point” from a menu of 21 choices between the same lottery—\((\$2.00,\$0.00;0.5,0.5)\)—and different sure amounts of money—from $0.00 to $2.00 in increments of $0.10. They then answered a qualitative risk question [“risk_qual” from the Preference Survey Module introduced by Falk et al. (2016)] and a few demographic questions before being paid. Instructions for the incentivized risk task are in the supplementary material.

  18. The analysis of risk-dominant equilibria has shed light on the role of expectations in cooperative behavior in indefinitely repeated PD experiments (Blonski et al. 2011; Dal Bó and Fréchette 2011; Kloosterman 2020).

  19. We consider these two strategies to succinctly highlight the trade-off between risk and cooperation.

  20. Formally, the stage game best response to expenditure \(\bar{x}<V/4\) is \(\hat{x}=\sqrt{V\bar{x}}-\bar{x}\) (ignoring the integer problem) and the payoff from optimal deviation is \(\hat{\pi } = (\sqrt{V}-\sqrt{\bar{x}})^2\), whereas the payoff from the cooperative strategy profile \((\bar{x},\bar{x})\) is \(V/2-\bar{x}\). A derivation similar to the one in Sect. 2.2 produces a threshold value of the discount factor \(\tilde{\delta }=\frac{(\sqrt{V}-\sqrt{\bar{x}})^2-\frac{V}{2}+\bar{x}}{(\sqrt{V}-\sqrt{\bar{x}})^2-\frac{V}{4}}\). It can be shown that \(\tilde{\delta }\) decreases monotonically in \(\bar{x}\) for \(\bar{x}\in [0,\frac{V}{4}]\).

  21. Tables 5, 6, and 7 in the supplementary material contain average expenditures for every five contests, by supergame using data from All Rounds, and by supergame using data from Round 1 only, respectively.

  22. Figure 3 in the supplementary material shows mean expenditure by round, by treatment and session, for each of the empirical comparisons we make in this section.

  23. Tables 26 and 27 in the supplementary material indicate that our analysis with quadratic specifications is robust to using linear specifications instead. Table 8 in the supplementary material has regression estimates from a specification with only session fixed effects. As we discuss in detail in this section, this very simple specification fails to capture important experience effects.

  24. We do present regression results with Supergame instead of Contests in the supplementary material. Our results are robust to using supergame (Supergame) instead of cumulative round (Contests).

  25. The partial effect of an additional contest (round) of experience on expenditure is, itself, a function of Contests. For example, the estimated difference in the effect of an additional contest (round) of experience in the HighDelta treatment compared to the LowDelta treatment is \(\hat{\beta }_2+2\hat{\beta }_4Contests\).

  26. These results are robust to the inclusion of regressors capturing participant age, gender, and risk preferences. See the supplementary material for these robustness results.

  27. They also match empirical cumulative density functions (CDFs) for expenditure. See Fig. 5 in the supplementary material.

  28. Figure 4 in the supplementary material plots “overbidding” (deviations from the stage game NE) directly. There is always significant overbidding in WTA-Low \(\delta\)-Indefinite and PP-Low \(\delta\)-Indefinite. After the first few rounds, there is no significant overbidding in WTA-High \(\delta\)-Indefinite. In PP-High \(\delta\)-Indefinite, overbidding converges to a level slightly above 0 ECUs.

  29. Tables 24 and 35 in the supplementary material show that the results from Tables 6 and 7 are robust to including participant-level controls for age, gender, and risk.

  30. The interested reader can find analysis comparing High \(\delta\)-Finite contests to Low \(\delta\)-Finite contests in the supplementary material, in Fig. 9 and in Tables 19 and 20.

  31. While we focus on Round 1 results in the main text, the reader may be curious about Round 5 expenditures—the last round in the High \(\delta\)-Finite treatments. Figure 7a in the supplementary material shows there is no significant final round effect (increase in expenditure) in either High \(\delta\)-Finite treatment. However, Figure 7b, which compares mean expenditure in Round 5 across Indefinite and Finite treatments, reveals that Round 5 expenditure was significantly lower in the Indefinite treatments relative to their Finite counterparts. This latter result is consistent with theory.

  32. Our conclusions are robust to participant-level control variables or to using Supergame instead of Round to account for experience. Please see the supplementary material for robustness analysis.

  33. These conclusions are robust to the inclusion of participant-level control variables or to using Supergame to account for experience. Please see the supplementary material. Ex post power analysis indicates an effect size of \(d<0.01\) (very small) for the Low-\(\delta\)-Finite comparison. A sample size of 1,445,116 participants per treatment would be needed for 80% power (assuming \(\alpha =0.10\)).

  34. Ex post power analysis indicates an effect size of \(d=0.14\) (small), and that 592 participants per treatment would be needed for 80% power (with \(\alpha =0.10\)).

  35. They are also robust to controlling for participant-level variables or to using Supergame to account for experience. Please see the supplementary material.

  36. We came to this idea independently, but we wish to acknowledge discussions with Gabriele Camera and Guillaume Fréchette who expressed similar ideas related to this topic.

  37. Table 28 in the supplementary material contains data on equal expenditure choices. In addition to PP contests having a greater percentage of equal expenditure than WTA contests, the percentage of equal expenditure is always greater in High \(\delta\) contests than in Low \(\delta\) contests (as would be expected).

  38. See Fig. 6 in the supplementary material.

  39. This contrasts somewhat with Cournot duopoly experiments (without communication). For example, Holt (1985) finds some evidence of collusion in indefinite Cournot duopolies, while Huck et al. (2004) observe some collusion in finitely repeated, one-shot Cournot duopolies.

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Correspondence to Philip Brookins.

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

We thank the Economic Science Institute, the Marquette University College of Business Administration, and the Max Planck Institute for Research on Collective Goods for funding. Megan Luetje and Arthur Nelson provided excellent assistance conducting the experiments. For helpful feedback, we thank seminar participants at Chapman University, Marquette University, the 2017 Contests: Theory and Evidence Conference (University of East Anglia), and Werner Güth and participants of the Experimental and Behavioral Economics Workshop (LUISS Guido Carli University, Rome). This paper supersedes an earlier manuscript that circulated online under the same title. Relative to that earlier manuscript, this paper features improved experimental procedures, less ambiguous instructions, and wholly new data (a 50% increase in sample size over the older manuscript). We thank Roberto Weber and two anonymous referees for comments that have greatly improved the paper. Any errors are ours alone.

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Brookins, P., Ryvkin, D. & Smyth, A. Indefinitely repeated contests: An experimental study. Exp Econ 24, 1390–1419 (2021). https://doi.org/10.1007/s10683-021-09703-0

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Keywords

  • Contest
  • Indefinitely repeated game
  • Cooperation
  • Experiment

JEL classification codes

  • C72
  • C73
  • C91
  • D72