Mixed strategy under generalized public goods games

Evolutionary public goods game on the birandom geometric graph

To investigate the evolution of cooperation in spatial public goods games, this paper establishes a birandom geometric graph, in which two types of nodes, representing players and public goods respectively, are placed at random locations in the unit square. Each public good has a limit influence range and the individuals that fall into the same range engage in a public good game. In contrast to the classical network models consisting of only one type of nodes, the birandom geometric graph provides a natural way to describe the scenarios where individuals and public resources are independent of each other. Numerical simulations reveal that cooperation can be significantly promoted when the group size and the average number of groups that each player participates in are relatively small, which is at odds with the results on the square lattice, but is consistent with a body of empirical evidence reported by Ostrom and Olson et al. Analysis of the evolutionary process suggests that the facilitation of cooperation is due primarily to the formation of the cooperative clusters, which can effectively resist the invasion of the defectors.

Evolution of cooperation with interactive identity and diversity

Interactive identity and interactive diversity are generally regarded as two typical interaction patterns in living systems. The former describes that in each generation every individual behaves identically to all of its opponents, and the latter allows each individual to behave diversely to its distinct opponents. Most traditional research on the evolution of cooperation, however, has been confined to populations with a uniform interaction pattern. Here we study the cooperation conundrum in a diverse population comprising players with interactive identity and with interactive diversity. We find that in homogeneous networks a small fraction of players taking interactive diversity are enough to stabilize cooperation for a wide range of payoff values even in a noisy environment. When assigned to heterogeneous networks, players in high-degree nodes taking interactive diversity significantly strengthen systems' resilience against the shifty environment and enlarge the survival region of cooperation. However, they fail to establish a homogeneous strategy 'cloud' in the neighborhood and thus can not coordinate players in low-degree nodes to reach a socially optimal cooperation level. The most favorable outcome emerges when players in high-degree nodes take interactive identity and meanwhile others adopt interactive diversity. Our findings reveal the significance of the two typical interaction patterns and could be a good heuristic in coordinating them to achieve the social optimum in cooperation.

Intermediate-Range Migration Furnishes a Narrow Margin of Efficiency in the Two-Strategy Competition

It is well-known that the effects of spatial selection on the two-strategy competition can be quantified by the structural coefficient σ under weak selection. We here calculate the accurate value of σ in group-structured populations of any finite size. In previous similar models, the large population size has been explicitly required for obtaining σ, and here we analyze quantitatively how large the population should be. Unlike previous models which have only involved the influences of the longest and the shortest migration rang on σ, we consider all migration ranges together. The new phenomena are that an intermediate range maximizes σ for medium migration probabilities which are of the tiny minority and the maximum value is slightly larger than those for other ranges. Furthermore, we find the ways that migration or mutation changes σ can vary significantly through determining analytically how the high-frequency steady states (distributions of either strategy over all groups) impact the expression of σ obtained before. Our findings can be directly used to resolve the dilemma of cooperation and provide a more intuitive understanding of spatial selection.

Collective action problem in heterogeneous groups

I review the theoretical and experimental literature on the collective action problem in groups whose members differ in various characteristics affecting individual costs, benefits and preferences in collective actions. I focus on evolutionary models that predict how individual efforts and fitnesses, group efforts and the amount of produced collective goods depend on the group's size and heterogeneity, as well as on the benefit and cost functions and parameters. I consider collective actions that aim to overcome the challenges from nature or win competition with neighbouring groups of co-specifics. I show that the largest contributors towards production of collective goods will typically be group members with the highest stake in it or for whom the effort is least costly, or those who have the largest capability or initial endowment. Under some conditions, such group members end up with smaller net pay-offs than the rest of the group. That is, they effectively behave as altruists. With weak nonlinearity in benefit and cost functions, the group effort typically decreases with group size and increases with within-group heterogeneity. With strong nonlinearity in benefit and cost functions, these patterns are reversed. I discuss the implications of theoretical results for animal behaviour, human origins and psychology.

Cooperation in group-structured populations with two layers of interactions

Recently there has been a growing interest in studying multiplex networks where individuals are structured in multiple network layers. Previous agent-based simulations of games on multiplex networks reveal rich dynamics arising from interdependency of interactions along each network layer, yet there is little known about analytical conditions for cooperation to evolve thereof. Here we aim to tackle this issue by calculating the evolutionary dynamics of cooperation in group-structured populations with two layers of interactions. In our model, an individual is engaged in two layers of group interactions simultaneously and uses unrelated strategies across layers. Evolutionary competition of individuals is determined by the total payoffs accrued from two layers of interactions. We also consider migration which allows individuals to move to a new group within each layer. An approach combining the coalescence theory with the theory of random walks is established to overcome the analytical difficulty upon local migration. We obtain the exact results for all “isotropic” migration patterns, particularly for migration tuned with varying ranges. When the two layers use one game, the optimal migration ranges are proved identical across layers and become smaller as the migration probability grows.

Socioecological regime shifts in the setting of complex social interactions

The coupling between social and ecological system has become more ubiquitous and predominant in the current era. The strong interaction between these systems can bring about regime shifts which in the extreme can lead to the collapse of social cooperation and the extinction of ecological resources. In this paper, we study the occurrence of such regime shifts in the context of a coupled social-ecological system where social cooperation is established by means of sanction that punishes local selfish act and promotes norms that prescribe nonexcessive resource extraction. In particular, we investigate the role of social networks on social-ecological regimes shift and the corresponding hysteresis effects caused by the local ostracism mechanism under different social and ecological parameters. Our results show that a lowering of network degree reduces the hysteresis effect and also alters the tipping point, which is duly verified by our numerical results and analytical estimation. Interestingly, the hysteresis effect is found to be stronger in scale-free network in comparison with random network even when both networks have the same average degree. These results provide deeper insights into the resilience of these systems, and can have important implications on the management of coupled social-ecological systems with complex social interactions.

Cheating is evolutionarily assimilated with cooperation in the continuous snowdrift game

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We fully analyze continuous snowdrift games with quadratic payoff functions in diversified populations.

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It is well known that classical snowdrift games maintain the coexistence of cooperators and cheaters.

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We clarify that the continuous snowdrift games often lead to assimilation of cooperators and cheaters.

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Allowing the gradual evolution of cooperative behavior can facilitate social inequity aversion in joint ventures.

It is well known that in contrast to the Prisoner’s Dilemma, the snowdrift game can lead to a stable coexistence of cooperators and cheaters. Recent theoretical evidence on the snowdrift game suggests that gradual evolution for individuals choosing to contribute in continuous degrees can result in the social diversification to a 100% contribution and 0% contribution through so-called evolutionary branching. Until now, however, game-theoretical studies have shed little light on the evolutionary dynamics and consequences of the loss of diversity in strategy. Here, we analyze continuous snowdrift games with quadratic payoff functions in dimorphic populations. Subsequently, conditions are clarified under which gradual evolution can lead a population consisting of those with 100% contribution and those with 0% contribution to merge into one species with an intermediate contribution level. The key finding is that the continuous snowdrift game is more likely to lead to assimilation of different cooperation levels rather than maintenance of diversity. Importantly, this implies that allowing the gradual evolution of cooperative behavior can facilitate social inequity aversion in joint ventures that otherwise could cause conflicts that are based on commonly accepted notions of fairness.

A tale of two contribution mechanisms for nonlinear public goods

Amounts of empirical evidence, ranging from microbial cooperation to collective hunting, suggests public goods produced often nonlinearly depend on the total amount of contribution. The implication of such nonlinear public goods for the evolution of cooperation is not well understood. There is also little attention paid to the divisibility nature of individual contribution amount, divisible vs. non-divisible ones. The corresponding strategy space in the former is described by a continuous investment while in the latter by a continuous probability to contribute all or nothing. Here, we use adaptive dynamics in finite populations to quantify and compare the roles nonlinearity of public-goods production plays in cooperation between these two contribution mechanisms. Although under both contribution mechanisms the population can converge into a coexistence equilibrium with an intermediate cooperation level, the branching phenomenon only occurs in the divisible contribution mechanism. The results shed insight into understanding observed individual difference in cooperative behavior.