Spatial evolutionary game theory: Hawks and Doves revisited

In the children's game of rock-scissors-paper, players each choose one of three strategies. A rock beats a pair of scissors, scissors beat a sheet of paper and paper beats a rock, so the strategies form a competitive cycle. Although cycles in competitive ability appear to be reasonably rare among terrestrial plants, they are common among marine sessile organisms and have been reported in other contexts. Here we consider a system with three species in a competitive loop and show that this simple ecology exhibits two counter-intuitive phenomena. First, the species that is least competitive is expected to have the largest population and, where there are oscillations in a finite population, to be the least likely to die out. As a consequence an apparent weakening of a species leads to an increase in its population. Second, evolution favours the most competitive individuals within a species, which leads to a decline in its population. This is analogous to the tragedy of the commons, but here, rather than leading to a collapse, the 'tragedy' acts to maintain diversity.

Heterogeneity reproductive ability promotes cooperation in spatial prisoner's dilemma game

The strategy selection process plays a crucial role in evolutionary dynamics when we study the spontaneous emergence of cooperation under the framework of evolutionary game theory. In most previous studies, it is assumed that individuals choose role models randomly from their neighbors. However, by considering the heterogeneity of individuals' influence in the real society, preferential selection is more realistic. Here, we associate an individuals' attractiveness with payoff satisfaction, which characterized the possibility of changing his strategy. We propose a preferential selection mechanism based on payoff satisfaction and memory in the spatial prisoner's dilemma games on square lattice networks and Erdős-Rényi networks and Barabási-Albert scale-free networks. People might prefer to learn from those with higher satisfaction in real life. Therefore, they are usually more preferable to choose individuals with higher satisfaction when selecting learning role model. We introduce an expected payoff to describe the level of individual satisfaction and memory length to represent the memory capacity of the individual. However, the expectation of defectors and cooperators may be different obviously in the real society. And in general, the expectation of the defector is higher than that of the cooperator. So we want to explore the distribution of expected payoffs that can improve the cooperation level. We investigate the effects of expected payoffs and memory length on cooperation. The results show that cooperation could be promoted when individuals prefer to choose more satisfied neighbors. And the size of expected payoffs and memory length also has different effects on the evolutionary game process. Besides, we investigate the robustness of the network topology, and find that the qualitative features of the results are unchanged.

In this paper, we introduce an aspiration-dependent persistence into the spatial prisoner's dilemma game aimed at promoting cooperation in a population. Based on aspiration, a player could adjust his/her strategy persistence level during the evolutionary process. In particular, he/she will hold a strategy longer if it brings him/her a satisfactory payoff, otherwise he/she will decrease the time duration of keeping the present choice. We also introduce a tunable parameter to characterize the sensitivity of strategy persistence to aspiration and investigate the effects of the sensitivity on the evolution of cooperation. The results show that aspiration-dependent strategy persistence could effectively promote cooperation. At an intermediate aspiration, diverse strategy persistence levels among population could be formed during the evolution, which ultimately results in the highest cooperation level. Then, we present intuitive explanations for the existence of optimal aspiration by scrutinizing the microscopic evolution of cooperation. Moreover, we find that, in the population where individuals are more sensitive to aspiration, cooperation could be further promoted by assembling greater cooperator clusters with the help of diverse strategy persistence distribution. Our results highlight the more realistic scenario where aspiration-dependent persistence is involved in the spatial prisoner's dilemma game.

We introduce chaotic variations, modelled by a spatially extended Lorenz system, to the payoffs of the spatial prisoner's dilemma game and study their effects on the evolution of cooperation. We show that chaotic variations of appropriate amplitude promote cooperation over a wide range of payoff parameters at which defection is the only strategy in a sterile environment. An appropriately pronounced chaotic environment can assure permanent domination of cooperation by full anonymity of players and without the aid of secondary strategies, thus designating chaotic payoff variations as a general and stand-alone mechanism for cooperation in the spatial prisoner's dilemma game.

We study the evolution of cooperation in the spatial prisoner's dilemma game where players are allowed to establish new interactions with others. By employing a simple coevolutionary rule entailing only two crucial parameters, we find that different selection criteria for the new interaction partners as well as their number vitally affect the outcome of the game. The resolution of the social dilemma is most probable if the selection favors more successful players and if their maximally attainable number is restricted. While the preferential selection of the best players promotes cooperation irrespective of game parametrization, the optimal number of new interactions depends somewhat on the temptation to defect. Our findings reveal that the “making of new friends” may be an important activity for the successful evolution of cooperation, but also that partners must be selected carefully and their number limited.

Inspired by the realistic process of taking decisions in social life, we have proposed a repeated thinking mechanism in the evolutionary spatial prisoner's dilemma game where players are denoted by the vertices and play games with their direct neighbors. Under our mechanism, a player i will randomly select a neighbor j and then deliberate for M times before strategy updating. It will remain unchanged if not all M considerations suggest it to learn the strategy of j. We mainly focus on the evolution of cooperation in the systems. Interestingly, we find that the cooperation level fC is remarkably promoted and fC has a monotonic dependence on the caution parameter M, indicating that being cautious facilitates the emergence and persistence of cooperation. We give a simple but clear explanation for this cooperation promotion via detecting the cooperator–defector transition process. Moreover, the robustness of this mechanism is also examined on different noise levels and game models.

To deeply understand the emergence of cooperation in natural, social and economical systems, we present an improved fitness evaluation mechanism with memory in spatial prisoner's dilemma game on regular lattices. In our model, the individual fitness is not only determined by the payoff in the current game round, but also by the payoffs in previous round bins. A tunable parameter, termed as the memory strength (μ), which lies between 0 and 1, is introduced into the model to regulate the ratio of payoffs of current and previous game rounds in the individual fitness calculation. When μ = 0, our model is reduced to the standard prisoner's dilemma game; while μ = 1 represents the case in which the payoff is totally determined by the initial strategies and thus it is far from the realistic ones. Extensive numerical simulations indicate that the memory effect can substantially promote the evolution of cooperation. For μ < 1, the stronger the memory effect, the higher the cooperation level, but μ = 1 leads to a pathological state of cooperation, but can partially enhance the cooperation in the very large temptation parameter. The current results are of great significance for us to account for the role of memory effect during the evolution of cooperation among selfish players.

Interaction state Q-learning promotes cooperation in the spatial prisoner's dilemma game

The impact of initial cooperation fraction on the evolutionary fate in a spatial prisoner's dilemma game

Donation of richer individual can support cooperation in spatial voluntary prisoner's dilemma game

Reciprocal rewards promote the evolution of cooperation in spatial prisoner's dilemma game

Conformity is a common phenomenon existing both in humans and in social animals, which has strong impact on collective behaviors. The effects of conformity on cooperation in a population have been widely explored in evolutionary games. Conformity-driven individuals tend to follow the strategy adopted by the majority of their neighbors. In most previous studies, the individuals take the half-number of their neighbors as the threshold for identifying the majority. However, in reality, the individuals may determine the majority based on a different threshold. For example, they might not conform unless the proportion of the majority reaches three-quarters. In contrast, it is also possible that the individuals have a lower reference than the half for majority. Here, we study the effects of the conformity threshold on cooperation in spatial prisoner's dilemma games. Our results show that there always exists an optimal conformity threshold for the population to maximize the cooperation level. Moreover, we find that the optimal conformity threshold depends on the payoff parameter of the game. Besides, the robustness of the results has been checked and we find that the simulation results are qualitatively unchanged when the different network scheme is used.

The paradox of cooperation among selfish individuals still puzzles scientific communities. Although a large amount of evidence has demonstrated that the cooperator clusters in spatial games are effective in protecting the cooperators against the invasion of defectors, we continue to lack the condition for the formation of a giant cooperator cluster that ensures the prevalence of cooperation in a system. Here, we study the dynamical organization of the cooperator clusters in spatial prisoner's dilemma game to offer the condition for the dominance of cooperation, finding that a phase transition characterized by the emergence of a large spanning cooperator cluster occurs when the initial fraction of the cooperators exceeds a certain threshold. Interestingly, the phase transition belongs to different universality classes of percolation determined by the temptation to defect b. Specifically, on square lattices, 1 < b < 4/3 leads to a phase transition pertaining to the class of regular site percolation, whereas 3/2 < b < 2 gives rise to a phase transition subject to invasion percolation with trapping. Our findings offer a deeper understanding of cooperative behavior in nature and society.

Role of memory effect in the evolution of cooperation based on spatial prisoner's dilemma game