Evolution of fairness in the one-shot anonymous Ultimatum Game

Abstract

Classical economic models assume that people are fully rational and selfish, while experiments often point to different conclusions. A canonical example is the Ultimatum Game: one player proposes a division of a sum of money between herself and a second player, who either accepts or rejects. Based on rational self-interest, responders should accept any nonzero offer and proposers should offer the smallest possible amount. Traditional, deterministic models of evolutionary game theory agree: in the one-shot anonymous Ultimatum Game, natural selection favors low offers and demands. Experiments instead show a preference for fairness: often responders reject low offers and proposers make higher offers than needed to avoid rejection. Here we show that using stochastic evolutionary game theory, where agents make mistakes when judging the payoffs and strategies of others, natural selection favors fairness. Across a range of parameters, the average strategy matches the observed behavior: proposers offer between 30% and 50%, and responders demand between 25% and 40%. Rejecting low offers increases relative payoff in pairwise competition between two strategies and is favored when selection is sufficiently weak. Offering more than you demand increases payoff when many strategies are present simultaneously and is favored when mutation is sufficiently high. We also perform a behavioral experiment and find empirical support for these theoretical findings: uncertainty about the success of others is associated with higher demands and offers; and inconsistency in the behavior of others is associated with higher offers but not predictive of demands. In an uncertain world, fairness finishes first.

Fig. 1.

With intermediate selection and mutation, the most common strategy is fair, having p > q > 0. Shown are the frequencies of [p,q] pairs averaged over 10⁸ generations. To aid convergence, the p and q values of agents in the simulations in the figure are discretized in increments of 1/12 (all other simulations use a continuous strategy space). Red indicates high frequency; yellow indicates lower frequency. The most common strategy is indicated with a black ×. Simulations use n = 100 and u = 10^−1.25, with w varying across 10^−1.5 (A), 10⁻¹ (B), 10^−0.5 (C), and 10² (D). Similar results are obtained using other mutation rates (Fig. S1). Note that strong selection drives the population to the smallest possible nonzero value of p = q = 1/12 in D (rather than p = q = 0), for the following reason: although p = q = 1/12 is neutral with p = 1/12, q = 0, the latter strategy can be invaded by p = q = 0, which in turn is risk-dominated by p = 1/12, q = 1/12; due to the discretized strategy space, no intermediate strategies exist.

Fig. 2.

Across a wide range of selection strengths and mutation rates, evolution results in fairness on average: the mean minimum amount demanded has q > 0, and the mean offer has p > q. Shown are time-averaged values of p and q over 10⁸ generations, using the population size n = 100 and mutation rate (A) u = 10⁻³, (B) u = 10⁻², and (C) u = 10⁻¹. Shown in yellow are the parameter regions which agree with experimental data, 0.3 < p < 0.5 and 0.2 < q < 0.35, based on additional simulations examining selection strengths in increments of 0.1.

Fig. 3.

In the weak selection limit, the modal strategy is fair for intermediate mutation. Shown are the most common strategies p (blue) and q (red) as functions of the mutation rate, calculated analytically in the weak selection limit (see SI Single-Population Formation for details). For low mutation 0 ≤ Nu ≤ 1, p = q = 1/3 is the most common strategy. As mutation increases, the optimum proposal increases to 1/2 and the optimum threshold decreases to 0. For intermediate mutation rates, we observe both key features of real-world ultimate game behavior: p > q > 0.

Fig. 4.

A behavioral experiment in which subjects play a one-shot anonymous UG confirms two predictions of our model. (A) Subjects that report a less-clear understanding of who in their community is more versus less successful (i.e., that developed their strategies under weaker selection) make larger offers and larger demands. (B) Subjects that report first impressions to be less reliable (i.e., that developed their strategies under higher mutation rates) make larger offers but not larger demands. Error bars indicate SEM. For visualization, subjects are divided into two groups in each panel using a median split on question responses.