Direct reciprocity is a powerful mechanism for the evolution of cooperation on the basis of repeated interactions1,2,3,4. It requires that interacting individuals are sufficiently equal, such that everyone faces similar consequences when they cooperate or defect. Yet inequality is ubiquitous among humans5,6 and is generally considered to undermine cooperation and welfare7,8,9,10. Most previous models of reciprocity do not include inequality11,12,13,14,15. These models assume that individuals are the same in all relevant aspects. Here we introduce a general framework to study direct reciprocity among unequal individuals. Our model allows for multiple sources of inequality. Subjects can differ in their endowments, their productivities and in how much they benefit from public goods. We find that extreme inequality prevents cooperation. But if subjects differ in productivity, some endowment inequality can be necessary for cooperation to prevail. Our mathematical predictions are supported by a behavioural experiment in which we vary the endowments and productivities of the subjects. We observe that overall welfare is maximized when the two sources of heterogeneity are aligned, such that more productive individuals receive higher endowments. By contrast, when endowments and productivities are misaligned, cooperation quickly breaks down. Our findings have implications for policy-makers concerned with equity, efficiency and the provisioning of public goods.
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All evolutionary simulations and numerical calculations have been performed with MATLAB R2014A. We provide the respective scripts in the Supplementary Information. These scripts can be used to compute the payoffs of the players, to simulate the introspection dynamics and to numerically compute the expected dynamics.
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This work was supported by the European Research Council Start Grant 279307: Graph Games (to K.C.), Austrian Science Fund (FWF) NFN Grant S11407-N23 Rigorous Systems Engineering/Systematic Methods in Systems Engineering (to K.C.), Office of Naval Research Grant N00014-16-1-2914 (to M.A.N.), Defense Advanced Research Projects Agency Grant W911NF-18-2-0265 (to M.A.N.), and the John Templeton Foundation Grant 55832 (to M.A.N.). C.H. acknowledges support from the ISTFELLOW program.
The authors declare no competing interests.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Peer review information Nature thanks Joe Yuichiro Wakano and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Extended data figures and tables
In Figs. 2, 3, we show how often players cooperate on average. a–e, Here we depict evolutionary trajectories over time, for the five treatments considered in our experiment. We assume that players can choose among the 16 pure memory-1 strategies. Top, five single runs of the introspection dynamics. Bottom, expected trajectories of the introspection dynamics, which can be derived explicitly (Supplementary Information section 4.3). These expected trajectories represent the cooperation rate over time as we average over many realizations of the process. We observe substantial cooperation in three of the five cases: in the treatments with full equality (a), productivity inequality (c) and aligned inequality (d).
Extended Data Fig. 2 Under endowment inequality and misaligned inequality, players fail to coordinate on WSLS.
Here, we consider the long-run dynamics of the games considered in Extended Data Fig. 1. For each pair (p1, p2) of pure memory-1 strategies, we can compute how often the respective strategy pair is played according to the invariant distribution of the evolutionary process. a, c, d, Under full equality, productivity inequality or aligned inequality, players typically coordinate on a WSLS equilibrium, as indicated by the coloured square in the centre of the dotted lines. b, e, Under endowment inequality or misaligned inequality, players fail to coordinate on a unique equilibrium. Instead, most of the evolving strategies prescribe to defect against the opponent. We note that in those treatments in which players have different endowments, the low-endowment player faces a reduced strength of selection (because the endowment of this player is reduced from 0.5 to 0.25). As a consequence, the marginal distribution of the low-endowment player in b, e is more uniform than the marginal distribution of the high-endowment player.
Extended Data Fig. 3 An equilibrium analysis explains why cooperation emerges in only three of the five treatments.
Using the same two-player setup as in Extended Data Figs. 1, 2, we explored how much players contribute on average when we simultaneously vary the endowment (x axis) as well as their productivity r1 of player 1. For each parameter combination, we record the total contributions of the player and how often they use WSLS according to the invariant distribution of the evolutionary process (indicated in shades of grey). We compare these evolutionary results with the region for which WSLS is an equilibrium (indicated by dashed lines) and with the region for which Grim is an equilibrium (dotted lines); see Supplementary Information for details. The coloured symbols indicate which parameter combinations have been used for the experimental treatments. a–c, For equal productivities, the full equality treatment (1) is in the region in which cooperation can evolve, whereas the unequal endowment treatment (2) is not. d–f, For unequal productivities, only the misaligned inequality treatment (5) is outside the region in which cooperation can evolve.
a–c, To explore the robustness of our theoretical predictions, we varied the expected number of rounds played between two players (a), the selection strength (b) and the rate at which players commit an implementation error (c). Although the quantitative results depend on these parameters, the qualitative ordering of the five treatments is the same across all considered scenarios. Except for the parameters explicitly varied on the x axis, all parameters are the same as in Extended Data Figs. 1, 2.
Extended Data Fig. 5 Cooperation in an asymmetric game in which players derive different payoffs from the public good.
Instead of considering players who differ in their productivity, here we consider an asymmetric two-player public goods game in which players differ in the share of the public goods that they get (the exact model is specified in the Supplementary Information). We vary two parameters, player 1’s share of the initial endowment, and player 1’s share of the public good. For each parameter combination, we record the average contributions of the players over the course of the evolutionary process (indicated in the grey colour). For games in which players get different shares of the public good, we note that the game is a social dilemma only if neither player’s share is too large (otherwise that player would always have an incentive to cooperate, no matter what the co-player does). However, if both players get an intermediate share of the public good, full cooperation can again evolve when WSLS is an equilibrium.
a, Here we represent finite-state automata for a game between two players in which players can either contribute their full endowment (C) or nothing (D). A finite-state automaton consists of three components: a set of states (represented by the large circles), the action played in each state (represented by the colour of the circle and the letters ‘C’ and ‘D’) and a transition rule (represented by arrows; the associated letter shows for which of the co-player’s actions the respective arrow is taken). Finite-state automata are able to implement all memory-1 strategies. In addition, they can encode strategies that depend on arbitrarily long sequences of past actions. b, To model evolution among finite-state automata, we use a previously published mutation scheme15,64. When a mutation occurs, the direction of a random arrow is changed, the action in a randomly chosen state is changed, a random state is removed or a state is added. c, Using this more general strategy space, we repeated the simulations in Fig. 3. Although overall cooperation rates are slightly lower, all qualitative results remain unchanged.
For each of the five experimental treatments, we compare the average contributions and the average payoff of the two players. Grey bars indicate the theoretical prediction based on evolutionary simulations. Coloured bars depict the outcome of the experiment. Error bars represent the respective 95% confidence intervals. Asterisks indicate statistical differences based on two-tailed Wilcoxon signed-rank tests. The number of groups per treatment is 42, 42, 40, 39, 40 for treatments 1–5, respectively. a, b, Under full equality, the two players contribute a similar share of their endowment and they obtain approximately equal payoffs. Under endowment inequality, the cooperation rates of both players are reduced, with the contributions of the high-endowment player (player 1) being significantly lower than the contributions of player 2. c, d, For productivity inequality and aligned inequality, we find no differences in the relative contributions of the players. For misaligned inequality, the relative contributions of the better-endowed but less-productive player 2 are considerably reduced. For both aligned and misaligned inequality, the two players earn significantly different payoffs. Nevertheless, the player with the lower payoff in the aligned inequality treatment derives a similar payoff as the two player types under productivity inequality. For details, see Supplementary Information.
a–e, For each of the five treatments, we show the average contributions of the players over the course of the experiment. In all treatments the contributions are relatively stable over time, except for a significant negative trend in the treatment with endowment inequality (b) (see Supplementary Information for details).
a–e, To analyse the joint contribution decisions of the two players, we plot here how often player 1 has contributed y1 tokens while player 2 has contributed y2 tokens, for each pair (y1, y2). a, c, d, Under full equality (a), productivity inequality (c) and aligned inequality (d), most individual decisions are mutually cooperative. b, e, By contrast, under endowment inequality (b) and misaligned inequality (e), contributions are more scattered. e, Moreover, in the treatment with misaligned inequality, we observe that a substantial fraction of high-endowment players only matches the absolute contributions of the other player. For example, in 12.4% of the rounds, the low-endowment player contributes all 25 tokens at their disposal, and the high-endowment player contributes the same absolute amount of tokens (corresponding to 1/3 of this player’s endowment).
a, b, To explore whether subjects apply reciprocal strategies, we show the fraction of rounds in which subjects match or exceed their co-player’s relative contribution from the previous round. That is, if player 1 has contributed x% of their endowment in round t, we record whether or not player 2 contributes at least x% of their endowment in round t + 1. Note that reciprocal strategies do not automatically yield high cooperation rates, because mutually defecting players are also reciprocal. Error bars represent the respective confidence intervals. Statistically significant differences were analysed using a two-tailed Wilcoxon signed-rank test. ***P < 0.001. Sample sizes are 42, 42, 40, 39, 40 for the treatments 1–5, respectively. Generally, we find high levels of reciprocity; only in the treatment with misaligned inequality does the high-endowment low-productivity player 2 exhibit a strongly reduced reciprocity rate. See Supplementary Information for details.
This file contains a Supplementary Discussion, Supplementary Tables 1 and 2, and Supplementary References. Supplementary Table 1 summarizes the statistical results for our treatments with equal productivities. Supplementary Table 2 summarizes the statistical results for our treatments with unequal productivities.