Overcoming unfairness via repeated interactions in mini-ultimatum game

TL;DR

This paper models the evolution of fairness in a repeated mini-ultimatum game, showing how reactive strategies can promote and sustain fairness depending on the game length and population dynamics.

Contribution

It introduces a theoretical framework analyzing fairness evolution in repeated interactions using reactive strategies and a two-population stochastic dynamics model.

Findings

Fair reactive strategies promote fairness below a critical game length.

Fairness is maintained by compliant offerers and accepters above the critical length.

Specific reactive strategies facilitate long-term fairness in mutation-selection dynamics.

Abstract

Repeated interactions are ubiquitous and known to promote social behaviour. While research often focuses on cooperation in the Prisoner's Dilemma, experimental evidence suggests repeated interactions also foster fairness. This study addresses a gap in the literature by theoretically modelling the evolution of fairness within a repeated mini-ultimatum game. Specifically, we construct a repeated-game framework where offerers and accepters interact using reactive strategies. We then investigate whether fair reactive strategy pairs are resilient against unfair mutants in a two-species population. By analyzing short-term evolutionary stability via the concept of two-species evolutionary stable strategy, we identify a critical effective game length: below this value, fairness is promoted by offerers and accepters who comply with their partner's past actions. Above this critical value, fairness is maintained by `complier' offerers and fair accepters. We also show that specific reactive strategies effectively facilitate the emergence and sustenance of fairness in long-term mutation-selection dynamics. To this end, we develop a two-population stochastic dynamics model -- a generalization of classical adaptive dynamics -- that accounts for finite population sizes and non-local mutants in the reactive strategy space.