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Arbitration supports reciprocity when there are frequent perception errors


Reciprocity is undermined by perception errors, mistakes that cause disagreement between interacting individuals about past behaviour. Strategies such as win–stay–lose–shift and generous tit-for-tat can re-establish cooperation following a perception error, but only when errors arise infrequently. We introduce arbitration tit-for-tat (ATFT), a strategy that uses third-party arbitration to align players’ beliefs about what transpired when they disagree. We show that, when arbitration is moderately accurate, ATFT is a strong subgame-perfect equilibrium and is evolutionarily stable against a range of strategies that defect, cooperate, ignore arbitration or invoke arbitration unnecessarily. ATFT can persist when perception errors are frequent, arbitration is costly or arbitration is biased. The need for third parties to resolve perception errors could explain why reciprocity is rare in other animals despite opportunities for repeated interactions and why human reciprocity is embedded within culturally transmitted moral norms in which community monitoring plays a role.

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Fig. 1: The minimum arbitration accuracy necessary for ATFT to have higher fitness than a deviant who defects at the gg node, as a function of error rate.
Fig. 2: Minimum arbitration accuracy necessary for ATFT to have higher expected fitness than an individual who does not call the arbitrator after the partner mistakenly defects and who cooperates on the next interaction because the partner remains in good standing.
Fig. 3: ATFT resists invasion by the best memory-1 strategy.
Fig. 4: The minimum arbitration accuracy necessary to resist invasion by Cheater or Tolerant as a function of error rate for three levels of arbitration cost.
Fig. 5: Invasion criteria for ATFT.

Data availability

There is no empirical data associated with this paper.

Code availability

Mathematica and Matlab scripts used to solve the fitness equations, perform Monte Carlo simulations and create the plots are publicly available at


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We thank M. Hoffman for help with proving that ATFT is subgame perfect. We also thank J. Silk, P. Richerson and J. Henrich for useful comments. This research was funded by the John Templeton Foundation (grant no. 48952). The views expressed in this publication are those of the authors and do not necessarily reflect the views of the funding agency.

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S.M. and R.B. conceived the study, developed the model and wrote the paper.

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Correspondence to Sarah Mathew.

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Supplementary Notes 1–7, Supplementary Figs. 1–22 and Supplementary Tables 1–58.

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Boyd, R., Mathew, S. Arbitration supports reciprocity when there are frequent perception errors. Nat Hum Behav 5, 596–603 (2021).

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