Efficiency and resilience of cooperation in asymmetric social dilemmas

Abstract

Direct reciprocity is a powerful mechanism for cooperation in social dilemmas. The very logic of reciprocity, however, seems to require that individuals are symmetric, and that everyone has the same means to influence each others' payoffs. Yet in many applications, individuals are asymmetric. Herein, we study the effect of asymmetry in linear public good games. Individuals may differ in their endowments (their ability to contribute to a public good) and in their productivities (how effective their contributions are). Given the individuals' productivities, we ask which allocation of endowments is optimal for cooperation. To this end, we consider two notions of optimality. The first notion focuses on the resilience of cooperation. The respective endowment distribution ensures that full cooperation is feasible even under the most adverse conditions. The second notion focuses on efficiency. The corresponding endowment distribution maximizes group welfare. Using analytical methods, we fully characterize these two endowment distributions. This analysis reveals that both optimality notions favor some endowment inequality: More productive players ought to get higher endowments. Yet the two notions disagree on how unequal endowments are supposed to be. A focus on resilience results in less inequality. With additional simulations, we show that the optimal endowment allocation needs to account for both the resilience and the efficiency of cooperation.

Keywords: direct reciprocity; evolution of cooperation; inequality; public good games; social dilemmas.

Fig. 1.

Schematic representation of the model. Players engage in a repeated linear asymmetric public good game. In every round, each player $i$ receives an endowment $e_i$. (A) Players choose how much to contribute toward the production of the public good, $c_i$, from their available endowment, $e_i$. (B) All individual contributions are multiplied by individual productivity factors, $r_i$. (C) The size of the public good is defined as the sum of all effective contributions. (D) After its production, the public good is divided equally among all players. (E) Individual payoffs are equal to the $n$th share of the public good plus the remaining share of the endowment that players did not contribute toward the public good. (F) Without loss of generality, we assume $\sum _{i=1}^n{e}_i=1$. In the case of a three-player game, we can represent the endowment distributions in a simplex, where each point corresponds to a vector $\mathbf{e}=(e_1,e_2,e_3)$. (G) We aim to identify the optimal endowment distribution with respect to different objectives.

Fig. 2.

Resilience-maximizing endowment distribution. (A) To demonstrate the degree of inequality in the resilience-maximizing endowment distribution, we construct three examples of games, all of which have the same level of heterogeneity in productivities. In this first example, three players with comparatively low productivities interact. (B) When the ratio $r_i/n$ is low, then the resilience-maximizing endowment distribution is close to $(1/n,\cdots ,1/n)$. (C) With productivities close to $1$, cooperation is challenging: ${\delta }_{ \text{min}}=0.620$. (D–F) Cooperation becomes more attractive and much easier to sustain. ${\delta }_{ \text{min}}=0.005$. The resulting degree of inequality of ${\mathbf{e}}^{\ast }$ increases. (G–I) With high productivities, but more players, the endowment distribution is again close to $(1/n,\cdots ,1/n)$. The required continuation probability is now ${\delta }_{ \text{min}}=0.238$, which is lower than in the previous example, but higher than in the example with low productivities. (I) We demonstrate the general principle by systematically varying group size while keeping productivities fixed (up to multiplicity). We plot the degree of inequality of the resulting resilience-maximizing endowment distribution measured by the ratio between the highest and the lowest endowments in the allocation. As the ratio $r_i/n$ decreases, the resilience-maximizing endowment allocation approaches $(1/n,\dots ,1/n)$.

Fig. 3.

Trade-off between efficiency and resilience of cooperation. We demonstrate the difference between the resilience- and efficiency-maximizing endowment distributions in a three-player example. (A) We chose the productivity vector to be $r=(2.7,1.5,1.1)$. (B) The resulting resilience-maximizing endowment distribution yields the total social welfare of $\mathrm{\Phi }=2.025$. (C) We find that the efficiency-maximizing endowment at $\delta =0.3$ is more unequal and yields a total group payoff of $\mathrm{\Phi }=2.174$. (D and E) We formulate a multi-objective optimization problem where both the resilience and efficiency are varied. The Pareto optimal values are shown by the pink line. (F) We run simulations to test which of the endowment distributions performs best when players adopt straz tegies based on a stochastic learning process. We find that in general, the highest cooperation levels are achieved along the Pareto frontier. Indeed, the total maximum group payoff of $1.729$ is achieved at $e_{ \text{maxW}}=(0.65,0.35,0)$.

Fig. 4.

Evolutionary simulations of the group welfare. In order to gain a deeper insight into the behavior of the dynamics, we provide results of extensive simulations of a two-player game for a wide range of parameters of the dynamics, that is, the error rate and the intensity of selection. (A–C) We first study the evolution of cooperation with equal productivities. Here, the dashed vertical lines bound the region where cooperation is sustainable at $\delta = 1$ according to the analytical model. We choose three error rate values for comparison: very rare errors, very frequent errors, and the error rate ${\epsilon }^{\ast }=0.019$ that yields the highest welfare at $\mathbf{e}={\mathbf{e}}^{\ast }$ (indicated by the solid vertical line). As can be seen, some (rare) errors can help the evolution of cooperation by ensuring stability of cooperative strategies such as WSLS (57). Here, there is no unique ${\mathbf{e}}^{†}$, since all endowment distributions where full cooperation is sustainable (shaded in blue), yield identical welfare. Near the boundaries of this interval, we observe very low cooperation rates, while the highest group welfare is observed at the resilience-maximizing endowment in the center. (D–F) Next, we consider a two-player game with unequal productivities given as $r_1=1.3$ and $r_2=1.9$. We employ the same logic for the choice of the parameters and obtain ${\epsilon }^{\ast }=0.043$. As can be seen in panel (E), the highest group welfare is no longer attained at ${\mathbf{e}}^{\ast }$ but at a point in between ${\mathbf{e}}^{†}$ and ${\mathbf{e}}^{\ast }$. For ${\epsilon }^{\ast }$, we denote that point as $e_{ \text{maxW}}$. It is equal to $(0.21,0.79)$. As with equal productivities, we find that higher selection strength increases welfare. Here, it also shifts the endowment $e_{ \text{maxW}}$ closer to $e^{†}$.