Asymmetric voluntary cooperation: a repeated sequential best-shot experiment


This paper tests the robustness of voluntary cooperation in a sequential best shot game, a public good game in which the maximal contribution determines the level of public good provision. Thus, efficiency enhancing voluntary cooperation requires asymmetric behavior whose coordination is more difficult. Nevertheless, we find robust cooperation irrespective of treatment-specific institutional obstacles. To explain this finding, we distinguish three behavioral patterns aiming at both, voluntary cooperation and (immediate) payoff equality.

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  1. 1.

    Further examples are provided, e.g., by Hirshleifer (1983) or Barbieri and Malueg (2014).

  2. 2.

    This narrow definition requires not only efficiency gains but also more or less fair burden sharing to distinguish it from cooperation with exploitation and self-sacrifice.

  3. 3.

    In one-shot experiments with best shot games, equilibrium behavior excludes fair and efficient voluntary cooperation due to the non-convexity of the set of feasible payoff vectors, a finding that is behaviorally confirmed by Prasnikar and Roth (1992). Allowing the low contributor to compensate the high one via transfer payments offers a way to establish fair cooperation already in the one-shot base game.

  4. 4.

    Bjedov et al. (2016) consider a two-stage variant of the asymmetric coordination game and show that communication facilitates coordination of alternating.

  5. 5.

    Whereas for base games with binary choices voluntary cooperation is maintained up to a short termination phase, base games with more continuous choices often trigger a continuous decline, which can be attributed to the coexistence of some initial freeriding and conditional cooperation.

  6. 6.

    This avoids the disadvantage of corner solutions, which allow only one-sided deviations from the respective benchmark.

  7. 7.

    With the transfer option in the last stage our best shot scenario resembles a trust game with two potential trustors determining endogenously who becomes the maximal and possibly only contributor.

  8. 8.

    Note that transfer payments may also be triggered by a reciprocity motive (Falk and Fischbacher 2006). However, our design does not allow to distinguish reciprocal preferences from mutually beneficial efficiency seeking.

  9. 9.

    Duffy and Feltovich (1999) discuss one treatment with partial information about the opponent’s cooperation record resembling our design due to its asymmetric setup.

  10. 10.

    Vicary (1990) provides a theoretical rationale for transfer payments in a weakest-link game.

  11. 11.

    Roy (2012) also reports some sort of turn-taking in a favor exchange game with private information.

  12. 12.

    See also the comment by Kuzmics and Rogers (2012).

  13. 13.

    Participants are informed about the number of rounds they will play in the instructions at the beginning of a session.

  14. 14.

    See Appendix for a translated version of the (German) instructions.

  15. 15.

    In the first trial round, the computer contributed 100 points, in the second zero points. If the computer could make a transfer payment, it was 10 points.

  16. 16.

    The control questions started with some yes-no-questions, ensuring general understanding of the instructions. Then four hypothetical sets of contributions (and transfers) were presented for which subjects had to compute the resulting intermediate and final payoffs. To avoid any specific reference point, we used various combinations of example contributions.

  17. 17.

    Note that such cooperative behavior does not necessarily need social preferences to be established. Also two purely self-interested players may try to coordinate on cooperative behavior to maximize their payoffs.

  18. 18.

    Luhmann et al. (2008) provide neural data suggesting a very similar interpretation.

  19. 19.

    Randomization by player 2 informs player 1 about player 2’s intentions in stage 2, while this information is transmitted in stage 3 when compensating and in stage 2 of the next round when alternating.

  20. 20.

    If nothing else is stated, reported p values refer to two-sided Wilcoxon signed rank tests, treating matching groups as the unit of observation. For tests of differences between treatments, we similarly use two-sided Wilcoxon rank sum tests.

  21. 21.

    The larger noise level below 20 than above 80 can be explained by phases in which \(max\{c_{1},c_{2}\}=0\) is played either during a “war of attrition” (when both want their partner to be the only contributor) or when wanting to punish a noncooperative partner.

  22. 22.

    Average transfers are also (insignificantly) higher in AllPay than in Base: On average, participants in the first condition transfer 26.69 points in AllPay and 20.94 points in Base (\(p = 0.11\)). In the second condition, average transfers are 23.34 points in AllPay and 20.74 points in Base (\(p = 0.66\)).

  23. 23.

    Average profits per player (including efficiency losses in the AllPay treatment due to both players contributing) are again very similar. Profits in the first condition are 132.99 in Base, 134.93 in AllPay, and 131.84 in NoTransfer. In the second condition (without the 100 points endowment) profits are 34.09 in Base, 34.13 in AllPay, and 30.98 in NoTransfer.

  24. 24.

    We classify a few observations as fitting to alternating, when players alternate over larger intervals, such as every second round or even once after ten rounds, indicating considerable tolerance regarding short-term inequality. Alternating may also come along with alternating in transfers.

  25. 25.

    Here we treat each pair, matched in the third supergame, as an independent observation.

  26. 26.

    The remaining pairs play a weaker version of compensating with one player contributing less than the efficient level and the other making strictly positive transfers.

  27. 27.

    Considering all rounds instead of the first round only also yields similar findings in Base and NoTransfer, while first and second movers in AllPay contribute the maximal amount about equally often.


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Corresponding author

Correspondence to Lisa Bruttel.

Additional information

We thank the editor and two anonymous referees of this journal for very helpful suggestions. Frederike Lannte and Christoph Spyra provided valuable research assistance.

Appendix: Instructions for the Base treatment in the first condition

Appendix: Instructions for the Base treatment in the first condition

Welcome and thank you for participating in this experiment. Please read these instructions very carefully. From now on we ask you to remain seated and to stop communicating with other participants. If you have any questions, please raise your hand. We will come to your place and answer your questions in private.

These instructions are the same for all participants.

Your earnings in this experiment will be counted in points. For every 500 points you earn, you will be paid 1 euro in cash directly at the end of the experiment. For showing up you receive an initial endowment of 1,500 points credited to your points account.

You will participate in the following sub-experiment three times. Each sub-experiment consists of 20 rounds. You interact in one sub-experiment repeatedly with the same other participant but in different sub-experiments with different participants. You will not be informed who these other participants are, nor will they learn your identity.

There are two different roles in this experiment. These roles are denoted with 1 and 2. Your role will be assigned randomly at the beginning of the experiment. You then decide only in the role assigned to you. Your role stays the same in all sub-experiments. In the following, the participant who is assigned role 1 or 2, respectively, is called participant 1 or 2, respectively.

In all three sub-experiments, one round consists of three stages. At the beginning of each round, both participants receive 100 points. In the first stage, participant 1 decides how many out of the 100 points he wants to contribute to a joint project (contribution 1). Contributions can only be made in steps of 10. In the second stage, participant 2 is informed about the contribution of participant 1 and decides about his contribution to the project (contribution 2). Here again, contributions can only be made in steps of 10.

The payout from the project is determined depending on the own contribution and that of the other participant. Both participants receive the same payout. The size of the payout depends on the higher of the two contributions, no matter if it was made by participant 1 or 2. The lower of the two contributions does not matter for the payout from the project. The lower contribution is refunded immediately to the participant who made it. If both participants choose the same contribution, a random draw decides who of the two participants (1 or 2) makes his contribution, the other contribution is refunded.

Highest contribution 0 10 20 30 40 50 60 70 80 90 100
Payout for both participants 0 27 40 47 55 61 67 73 80 83 85

At the end of the second stage, both participants are reminded of their own contribution and informed about the contribution of the other participant. Furthermore, you are informed about the payout from the project for both participants.

In the third stage, the participant who did not pay in anything can make a voluntary transfer payment to the other participant (who made the higher contribution to the project). This payment can amount to between 0 and 100 points.

At the end of each round you learn

  • the contribution of participant 1 in the first stage

  • the contribution of participant 2 in the second stage

  • the amount of the transfer payment in the third stage

  • your final profit in the current round

  • your current total profit from this sub-experiment (separately for each sub-experiment).

Your profits from all three sub-experiments are added up at the end of the experiment and paid out to you in cash. The exchange rate is 500 points to 1 euro.

After reading the instructions, you do have the possibility to familiarize yourself with the experiment in two practice rounds. In these practice rounds, you do not interact with another participant but with a computer program. The practice rounds are not relevant for your payoff. Afterwards you will be asked to answer some control questions. Only then will the actual experiment start with the first sub-experiment. After the experiment, we will ask you to answer a short questionnaire.

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Bruttel, L., Güth, W. Asymmetric voluntary cooperation: a repeated sequential best-shot experiment. Int J Game Theory 47, 873–891 (2018).

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  • Best shot game
  • Coordination
  • Transfer
  • Refund
  • Experiment

JEL Classification

  • C71
  • C73
  • C91