A double-edged sword: Benefits and pitfalls of heterogeneous punishment in evolutionary inspection games

Heterogeneous contributions can jeopardize cooperation in the public goods game

When studying social dilemma games, a crucial question arises regarding the impact of general heterogeneity on cooperation, which has been shown to have positive effects in numerous studies. Here, we demonstrate that heterogeneity in the contribution value for the focal public goods game can jeopardize cooperation. We show that there is an optimal contribution value in the homogeneous case that most benefits cooperation depending on the lattice. In a heterogeneous scenario, where strategy and contribution coevolve, cooperators making contributions higher than the optimal value end up harming those who contribute less. This effect is notably detrimental to cooperation in the square lattice with von Neumann neighborhood, while it can have no impact in other lattices. Furthermore, in parameter regions where a higher-contributing cooperator cannot normally survive alone, the exploitation of lower-value contribution cooperators allows their survival, resembling a parasitic behavior. To obtain these results, we examined the effect of various distributions for the contribution values in the initial condition and we conducted Monte Carlo simulations.

The probabilistic pool punishment proportional to the difference of payoff outperforms previous pool and peer punishment

The public goods game is a multiplayer version of the prisoner’s dilemma game. In the public goods game, punishment on defectors is necessary to encourage cooperation. There are two types of punishment: peer punishment and pool punishment. Comparing pool punishment with peer punishment, pool punishment is disadvantageous in comparison with peer punishment because pool punishment incurs fixed costs especially if second-order free riders (those who invest in public goods but do not punish defectors) are not punished. In order to eliminate such a flaw of pool punishment, this study proposes the probabilistic pool punishment proportional to the difference of payoff. In the proposed pool punishment, each punisher pays the cost to the punishment pool with the probability proportional to the difference of payoff between his/her payoff and the average payoff of his/her opponents. Comparing the proposed pool punishment with previous pool and peer punishment, in pool punishment of previous studies, cooperators who do not punish defectors become dominant instead of pool punishers with fixed costs. However, in the proposed pool punishment, more punishers and less cooperators coexist, and such state is more robust against the invasion of defectors due to mutation than those of previous pool and peer punishment. The average payoff is also comparable to peer punishment of previous studies.

Symbiotic behaviour in the public goods game with altruistic punishment

Finding ways to overcome the temptation to exploit one another is still a challenge in behavioural sciences. In the framework of evolutionary game theory, punishing strategies are frequently used to promote cooperation in competitive environments. Here, we introduce altruistic punishers in the spatial public goods game. This strategy acts as a cooperator in the absence of defectors, otherwise it will punish all defectors in their vicinity while bearing a cost to do so. We observe three distinct behaviours in our model: i) in the absence of punishers, cooperators (who don't punish defectors) are driven to extinction by defectors for most parameter values; ii) clusters of punishers thrive by sharing the punishment costs when these are low; iii) for higher punishment costs, punishers, when alone, are subject to exploitation but in the presence of cooperators can form a symbiotic spatial structure that benefits both. This last observation is our main finding since neither cooperation nor punishment alone can survive the defector strategy in this parameter region and the specificity of the symbiotic spatial configuration shows that lattice topology plays a central role in sustaining cooperation. Results were obtained by means of Monte Carlo simulations on a square lattice and subsequently confirmed by a pairwise comparison of different strategies' payoffs in diverse group compositions, leading to a phase diagram of the possible states.

Environmental Governance Cooperative Behavior among Enterprises with Reputation Effect Based on Complex Networks Evolutionary Game Model

This paper first portrays the equilibrium payoff of enterprise’s cooperation of environmental governance based on the Cournot model. Secondly, the evolutionary game model in complex networks is adopted to depict the evolution of environmental governance cooperative behavior among enterprises. Further, the evolutionary process of environmental governance cooperative behavior of enterprises is simulated considering the supervision behavior of government and the reputation evaluation behavior of environmental social organization. The results show that the cooperation level of enterprise group under self-organization condition will reach a low level; the supervision of government can enhance the cooperation level of enterprise group with high betrayal tempatation while it has limited effect on enterprise group with low betrayal tempatation. The reputation evaluation behavior of environmental social organization can realize reputation effect to improve the the cooperation level of enterprise group with high betrayal tempatation. The enhance of reputation sensitivity can optimize equilibrium distribution of reputation and it can strengthen the reputation effect on cooperation level. Based on the analysis above, the suggestions to effectively improve cooperation level are given.

Wealth-relative effects in cooperation games

This paper investigates cooperation games in which poor agents do not benefit from cooperation with wealthy agents. They instead benefit from considering wealth relative to decision payoffs of fitness or wealth. Of concern is the effect of cooperation on participants, their rational self-interest and choices, and not the evolution of cooperation directly. The accumulation of fitness or wealth has been shown in the literature to lead to different optimal strategies for wealthy and poor players in Chicken games. The effect could have important explanatory power if it were more broadly applicable. First we empirically compare two published results, one involving the temptation parameter vs. degree of cooperation in Prisoner's Dilemma, and the other a surprising result from a public goods game with participants from different cultures, networks and wealth in which a fixed rather than relative payoff scheme was used. Using the temptation data to calibrate the public goods behavior suggests wealth factors can provide an explanation for the results. Second we show using simulation that adding a survival threshold to a wealth or fitness accumulating Iterated Prisoner's Dilemma produces a wealth relative effect. We clarify previous results to show the poor must avoid survival risk, regardless of whether this is associated with cooperation or defection. We do this by introducing the Farmer's Game, a simulation of Iterated Prisoner's Dilemma with wealth accumulation and a survival threshold. This is used to evaluate the Tit-for-Tat strategy and four variants. Equilibrium payoffs keep the game scaled to social relevance, with a fraction of all payoffs externalized as a turn cost parameter. Findings include poor performance of Tit-for-Tat near the survival threshold, superior performance of low risk strategies for both poor and wealthy players, dependence of survival of the poor near the threshold on Tit-for-Tat forgiveness, unexpected optimization of forgiveness without encountering a social dilemma, improved performance of a diverse mix of strategies, and a more abrupt threshold of social catastrophe for the better performing mix. Lastly we compare cooperating and non-cooperating societies using the simulation and discover disturbing connections between cooperation and familiar non-egalitarian wealth distribution patterns.

Evolutionary dynamics of organised crime and terrorist networks

Crime is pervasive into modern societies, although with different levels of diffusion across regions. Its dynamics are dependent on various socio-economic factors that make the overall picture particularly complex. While several theories have been proposed to account for the establishment of criminal behaviour, from a modelling perspective organised crime and terrorist networks received much less attention. In particular, the dynamics of recruitment into such organisations deserve specific considerations, as recruitment is the mechanism that makes crime and terror proliferate. We propose a framework able to model such processes in both organised crime and terrorist networks from an evolutionary game theoretical perspective. By means of a stylised model, we are able to study a variety of different circumstances and factors influencing the growth or decline of criminal organisations and terrorist networks, and observe the convoluted interplay between agents that decide to get associated to illicit groups, criminals that prefer to act on their own, and the rest of the civil society.

Benefits of asynchronous exclusion for the evolution of cooperation in stochastic evolutionary optional public goods games

Mechanisms and conditions for the spontaneous emergence of cooperation in multi-player social dilemma games remain an open question. This paper focuses on stochastic evolutionary optional public goods games with different exclusion strategies. We introduce four strategy types in the population, namely, cooperation, defection, loner and exclusion. Synchronous and asynchronous exclusion forms have been compared in finite-sized, well-mixed and structured populations. In addition, we verify that the asynchronous exclusion mechanism is indeed better than the synchronous exclusion mechanism in all cases. The benefits of the asynchronous exclusion are measured by comparing the probability that the system chooses the cooperative states in the two situations. In the well-mixed population cases, only when the investment amplification factor is small and the probability of exclusion success is high will the asynchronous exclusion mechanism have a relatively large advantage in promoting cooperation. However, in the structured population cases, the range of the investment amplification factor, in which the asynchronous exclusion mechanism has relatively large advantages in promoting cooperation, is somewhat different and is mainly in the middle of the interval under our parameters. Our study further corroborated that when non-participation and exclusion strategies exist, a structured population does not necessarily promote cooperation compared with a well-mixed population for some parameter combinations. Thus, we acquire a good understanding of the emergence of cooperation under different exclusion mechanisms.

Environment-based preference selection promotes cooperation in spatial prisoner's dilemma game

The impact of environment on individuals is particularly critical. In evolutionary games, adopting the strategy of the neighbor who performs better is nontrivial for the survival and maintenance of cooperation, in that such an action may help the agents to obtain higher benefit and more obvious evolutionary advantages. Inspired by this idea, we investigate the effect of the environment-based preference selection on the evolution of cooperation in spatial prisoner’s dilemma. A simple rule, incorporating individual preference selection via an adjustable parameter α to explore how the selection of the potential strategy sources influences individual behavior traits, is considered. Because social interaction may not be the only way of generating payoffs, we assume that the individual’s income is also affected by the environment. Besides, taking into account individual differences, we introduce the heterogeneity of the environment. Through numerous computing simulations, we find that environment-based preference selection, which accelerates the microscopic organization of cooperator clusters to resist the aggression of defectors, can truly promote cooperation within a large range of parameters. Our study indicates that the combination of heterogeneity and preference selection may be key for the sustainability of cooperation in structured populations.

Evolutionary dynamics in the public goods games with switching between punishment and exclusion

Pro-social punishment and exclusion are common means to elevate the level of cooperation among unrelated individuals. Indeed, it is worth pointing out that the combined use of these two strategies is quite common across human societies. However, it is still not known how a combined strategy where punishment and exclusion are switched can promote cooperation from the theoretical perspective. In this paper, we thus propose two different switching strategies, namely, peer switching that is based on peer punishment and peer exclusion, and pool switching that is based on pool punishment and pool exclusion. Individuals adopting the switching strategy will punish defectors when their numbers are below a threshold and exclude them otherwise. We study how the two switching strategies influence the evolutionary dynamics in the public goods game. We show that an intermediate value of the threshold leads to a stable coexistence of cooperators, defectors, and players adopting the switching strategy in a well-mixed population, and this regardless of whether the pool-based or the peer-based switching strategy is introduced. Moreover, we show that the pure exclusion strategy alone is able to evoke a limit cycle attractor in the evolutionary dynamics, such that cooperation can coexist with other strategies.

Inequality as information: Wealth homophily facilitates the evolution of cooperation

Free-riding produces inequality in the prisoners’ dilemma: cooperators suffer costs that defectors avoid, thus putting them at a material disadvantage to their anti-social peers. This inequality, accordingly, conveys information about a social partner’s choices in past game play and raises the possibility that agents can use the aggregation of past payoffs—i.e. wealth—to identify a social partner who uses their same strategy. Building on these insights, we study a computational model in which agents can employ a strategy—when playing multiple one-shot prisoners’ dilemma games per generation—in which they view other agents’ summed payoffs from previous games, choose to enter a PD game with the agent whose summed payoffs most-closely approximate their own, and then always cooperate. Here we show that this strategy of wealth homophily—labelled COEQUALS (“CO-operate with EQUALS”)—can both invade an incumbent population of defectors and resist invasion. The strategy succeeds because wealth homophily leads agents to direct cooperation disproportionately toward others of their own type—a phenomenon known as “positive assortment”. These findings illuminate empirical evidence indicating that viewable inequality degrades cooperation and they show how a standard feature of evolutionary game models—viz. the aggregation of payoffs during a generation—can double as an information mechanism that facilitates positive assortment.

Strategy intervention for the evolution of fairness

The 'irrational' preference for fairness has attracted increasing attention. Although previous studies have focused on the effects of spitefulness on the evolution of fairness, they did not consider non-monotonic rejections shown in behavioral experiments. In this paper, we introduce a non-monotonic rejection in an evolutionary model of the Ultimatum Game. We propose strategy intervention to study the evolution of fairness in general structured populations. By sequentially adding five strategies into the competition between a fair strategy and a selfish strategy, we arrive at the following conclusions. First, the evolution of fairness is inhibited by altruism, but it is promoted by spitefulness. Second, the non-monotonic rejection helps fairness overcome selfishness. Particularly for group-structured populations, we analytically investigate how fairness, selfishness, altruism, and spitefulness are affected by population size, mutation, and migration in the competition among seven strategies. Our results may provide important insights into understanding the evolutionary origin of fairness.

Stochastic evolutionary voluntary public goods game with punishment in a Quasi-birth-and-death process

Traditional replication dynamic model and the corresponding concept of evolutionary stable strategy (ESS) only takes into account whether the system can return to the equilibrium after being subjected to a small disturbance. In the real world, due to continuous noise, the ESS of the system may not be stochastically stable. In this paper, a model of voluntary public goods game with punishment is studied in a stochastic situation. Unlike the existing model, we describe the evolutionary process of strategies in the population as a generalized quasi-birth-and-death process. And we investigate the stochastic stable equilibrium (SSE) instead. By numerical experiments, we get all possible SSEs of the system for any combination of parameters, and investigate the influence of parameters on the probabilities of the system to select different equilibriums. It is found that in the stochastic situation, the introduction of the punishment and non-participation strategies can change the evolutionary dynamics of the system and equilibrium of the game. There is a large range of parameters that the system selects the cooperative states as its SSE with a high probability. This result provides us an insight and control method for the evolution of cooperation in the public goods game in stochastic situations.

Individual mobility promotes punishment in evolutionary public goods games

In explaining the pressing issue in biology and social sciences how cooperation emerges in a population of self-interested individuals, researchers recently pay intensive attentions to the role altruistic punishment plays. However, as higher-order cooperators, survival of punishers is puzzling due to their extra cost in regulating norm violators. Previous works have highlighted the importance of individual mobility in promoting cooperation. Yet its effect on punishers remains to be explored. In this work we incorporate this feature into modeling the behavior of punishers, who are endowed with a choice between leaving current place or staying and punishing defectors. Results indicate that optimal mobility level of punishers is closely related to the cost of punishing. For considerably large cost, there exists medium tendency of migration which favors the survival of punishers. This holds for both the direct competition between punishers and defectors and the case where cooperators are involved, and can also be observed when various types of punishers with different mobility tendencies fight against defectors simultaneously. For cheap punishment, mobility does not provide with punishers more advantage even when they are initially rare. We hope our work provide more insight into understanding the role individual mobility plays in promoting public cooperation.

Modelling the role of correctional services on gangs: insights through a mathematical model

Research has shown that gang membership increases the chances of offending, antisocial behaviour and drug use. Gang membership should be acknowledged as part of crime prevention and policy designs, and when developing interventions and preventative programmes. Correctional services are designed to rehabilitate convicted offenders. We formulate a deterministic mathematical model using nonlinear ordinary differential equations to investigate the role of correctional services on the dynamics of gangs. The recruitment into gang membership is assumed to happen through an imitation process. An epidemic threshold value, [Formula: see text], termed the gang reproduction number, is proposed and defined herein in the gangs' context. The model is shown to exhibit the phenomenon of backward bifurcation. This means that gangs may persist in the population even if [Formula: see text] is less than one. Sensitivity analysis of [Formula: see text] was performed to determine the relative importance of different parameters in gang initiation. The critical efficacy ε* is evaluated and the implications of having functional correctional services are discussed.

A remarkable effect of the combination of probabilistic peer-punishment and coevolutionary mechanism on the evolution of cooperation

In the previous studies, the author has proposed the probabilistic peer-punishment based on the difference of payoff, and presented that the proposed peer-punishment utilizes its mechanism for preventing antisocial punishment like retaliation of a defector on a cooperator, effectively enhances the evolution of cooperation, and greatly increases the average payoff of all players in various parameters regarding static three types of topology of connections. Here, this study introduces both activities of breaking and creating connections of every player based on his/her preference to the model of the proposed peer-punishment. Every player will keep connections with his/her preferable players, whereas he/she will frequently break connections with his/her dissatisfied other players. Therefore, the new model of this study is the combination of probabilistic peer-punishment and coevolutionary mechanism that not only strategy of players but also connections between players evolve. This study discovers new knowledge that such combination induces high-level evolution of cooperation and great increase of the average payoff of all players in the condition where cooperation is hard to evolve.

Competitions between prosocial exclusions and punishments in finite populations

Prosocial punishment has been proved to be a powerful mean to promote cooperation. Recent studies have found that social exclusion, which indeed can be regarded as a kind of punishment, can also support cooperation. However, if prosocial punishment and exclusion are both present, it is still unclear which strategy is more advantageous to curb free-riders. Here we first study the direct competition between different types of punishment and exclusion. We find that pool (peer) exclusion can always outperform pool (peer) punishment both in the optional and in the compulsory public goods game, no matter whether second-order sanctioning is considered or not. Furthermore, peer exclusion does better than pool exclusion both in the optional and in the compulsory game, but the situation is reversed in the presence of second-order exclusion. Finally, we extend the competition among all possible sanctioning strategies and find that peer exclusion can outperform all other strategies in the absence of second-order exclusion and punishment, while pool exclusion prevails when second-order sanctioning is possible. Our results demonstrate that exclusion is a more powerful strategy than punishment for the resolution of social dilemmas.

Evolution of cooperation by the introduction of the probabilistic peer-punishment based on the difference of payoff

There are two types of costly punishment, i.e. peer-punishment and pool-punishment. While peer-punishment applies direct face to face punishment, pool-punishment is based on multi-point, collective interaction among group members. Regarding those two types of costly punishment, peer-punishment is especially considered to have the flaws that it lowers the average payoff of all players as well as pool-punishment does, and facilitates antisocial behaviour like retaliation of a defector on a cooperator. Here, this study proposes the new peer-punishment that punishment to an opponent player works at high probability when an opponent one is uncooperative, and the difference of payoff between a player and an opponent one becomes large in order to prevent such antisocial behaviour. It is natural to think that players of high payoff do not expect to punish others of lower payoff because they do not have any complaints regarding their economic wealth. The author shows that the introduction of the proposed peer-punishment increases both the number of cooperative players and the average payoff of all players in various types of topology of connections between players.