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.