The spatial arrangement of individuals is thought to overcome the dilemma of cooperation: When cooperators engage in clusters they might share the benefit of cooperation while being more protected against non-cooperating individuals, which benefit from cooperation but save the cost of cooperation. This is paradigmatically shown by the spatial prisoner's dilemma model. Here, we study this model in one and two spatial dimensions, but explicitly take into account that in biological setups fitness collection and selection are separated processes occurring mostly on vastly different time scales. This separation is particularly important to understand the impact of mobility on the evolution of cooperation. We find that even small diffusive mobility strongly restricts cooperation since it enables non-cooperative individuals to invade cooperative clusters. Thus, in most biological scenarios, where the mobility of competing individuals is an irrefutable fact, the spatial prisoner's dilemma alone cannot explain stable cooperation but additional mechanisms are necessary for spatial structure to promote the evolution of cooperation. The breakdown of cooperation is analyzed in detail. We confirm the existence of a phase transition, here controlled by mobility and costs, which distinguishes between purely cooperative and non-cooperative absorbing states. While in one dimension the model is in the class of the Voter Model, it belongs to the Directed Percolation (DP) universality class in two dimensions.
Here, we study the evolution of specialization using realistic computer simulations of bacteria that secrete two public goods in a dynamic fluid. Through this first principles approach, we find physical factors such as diffusion, flow patterns, and decay rates are as influential as fitness economics in governing the evolution of community structure, to the extent that when mechanical factors are taken into account, (1) Generalist communities can resist becoming specialists, despite the invasion fitness of specialization, (2) Generalist and specialists can both resist cheaters despite the invas…
Cooperation is ubiquitous in every level of living organisms. It is known that spatial (network) structure is a viable mechanism for cooperation to evolve. Until recently, it has been difficult to predict whether cooperation can evolve at a network (population) level. To address this problem, Pinheiro et al. proposed a numerical metric, called Average Gradient of Selection (AGoS) in 2012. AGoS can characterize and forecast the evolutionary fate of cooperation at a population level. However, stochastic mutation of strategies was not considered in the analysis of AGoS. Here we analyzed the evolu…
We study the evolution of cooperation in a finite population interacting according to a simple model of like-with-like assortment. Evolution proceeds as a Moran process, and payoffs from the underlying cooperator-defector game are translated to positive fitnesses by an exponential transformation. These evolutionary dynamics can arise, for example, in a nest-structured population with rare migration. The use of the exponential transformation, rather than the usual linear one, is appropriate when interactions have multiplicative fitness effects, and allows for a tractable characterization of the…
It is well known that cooperation cannot be an evolutionary stable strategy for a non-iterative game in a well-mixed population. In contrast, structured populations favor cooperation since cooperators can benefit each other by forming local clusters. Previous studies have shown that scale-free networks strongly promote cooperation. However, little is known about the invasion mechanism of cooperation in scale-free networks. To study microscopic and macroscopic behaviors of cooperators' invasion, we conducted computational experiments of the evolution of cooperation in scale-free networks where,…
Inclusive Fitness Theory (IFT) was proposed half a century ago by W.D. Hamilton to explain the emergence and maintenance of cooperation between individuals that allows the existence of society. Contemporary evolutionary ecology identified several factors that increase inclusive fitness, in addition to kin-selection, such as assortation or homophily, and social synergies triggered by cooperation. Here we propose an Extend Inclusive Fitness Theory (EIFT) that includes in the fitness calculation all direct and indirect benefits an agent obtains by its own actions, and through interactions with ki…
Evolution depends on the fitness of organisms, the expected rate of reproducing. Directly getting offspring is the most basic form of fitness, but fitness can also be increased indirectly by helping genetically related individuals (such as kin) to increase their fitness. The combined effect is known as inclusive fitness. Here it is argued that a further elaboration of fitness has evolved, particularly in humans. It is called extensive fitness and it incorporates producing organisms that are merely similar in phenotype. The evolvability of this mechanism is illustrated by computations on a simp…
Evolution of cooperation in the prisoner's dilemma and the public goods game is studied, where initially players belong to two independent structured populations. Simultaneously with the strategy evolution, players whose current utility exceeds a threshold are rewarded by an external link to a player belonging to the other population. Yet as soon as the utility drops below the threshold, the external link is terminated. The rewarding of current evolutionary fitness thus introduces a time-varying interdependence between the two populations. We show that, regardless of the details of the evoluti…
Cooperation is a persistent behavioral pattern of entities pooling and sharing resources. Its ubiquity in nature poses a conundrum. Whenever two entities cooperate, one must willingly relinquish something of value to the other. Why is this apparent altruism favored in evolution? Classical solutions assume a net fitness gain in a cooperative transaction which, through reciprocity or relatedness, finds its way back from recipient to donor. We seek the source of this fitness gain. Our analysis rests on the insight that evolutionary processes are typically multiplicative and noisy. Fluctuations ha…
We compare the likelihood of different socially relevant features to allow the evolutionary emergence and maintenance of cooperation in a generalized variant of the iterated Prisoners Dilemma game. Results show that the average costs/benefit balance of cooperation is the primary constraint for its establishment and maintenance. Behavior increasing inclusive fitness such as assortation, homophily, kin-selection and tagging of individuals, is second in importance. Networks characteristics were the least important in favoring the establishment and maintenance of cooperation, despite being the mos…
In human societies the probability of strategy adoption from a given person may be affected by the personal features. Now we investigate how an artificially imposed restricted ability to reproduce, overruling ones fitness, affects an evolutionary process. For this purpose we employ the evolutionary prisoner's dilemma game on different complex graphs. Reproduction restrictions can have a facilitative effect on the evolution of cooperation that sets in irrespective of particularities of the interaction network. Indeed, an appropriate fraction of less fertile individuals may lead to full supremac…
Range expansion is a universal process in biological systems, and therefore plays a part in biological evolution. Using a quantitative individual-based method based on the stochastic process, we identify that enhancing the inherent self-proliferation advantage of cooperators relative to defectors is a more effective channel to promote the evolution of cooperation in range expansion than weakening the benefit acquisition of defectors from cooperators. With this self-proliferation advantage, cooperators can rapidly colonize virgin space and establish spatial segregation more readily, which acts …
In spatial evolutionary games the fitness of each individual is traditionally determined by the payoffs it obtains upon playing the game with its neighbors. Since defection yields the highest individual benefits, the outlook for cooperators is gloomy. While network reciprocity promotes collaborative efforts, chances of averting the impending social decline are slim if the temptation to defect is strong. It is therefore of interest to identify viable mechanisms that provide additional support for the evolution of cooperation. Inspired by the fact that the environment may be just as important as…
Cooperation is ubiquitous in nature, but explaining its existence remains a central interdisciplinary challenge. Cooperation is most difficult to explain in the Prisoner's Dilemma game, where cooperators always lose in direct competition with defectors despite increasing mean fitness. Here we demonstrate how spatial population expansion, a widespread natural phenomenon, promotes the evolution of cooperation. We engineer an experimental Prisoner's Dilemma game in the budding yeast Saccharomyces cerevisiae to show that, despite losing to defectors in nonexpanding conditions, cooperators increase…
Ethnocentrism is a behavioral strategy seen on every scale of social interaction. Game-theory models demonstrate that evolution selects ethnocentrism because it boosts cooperation, which increases reproductive fitness. However, some believe that interethnic unions have the potential to foster universal cooperation and overcome in-group biases in humans. Here, we use agent-based computer simulations to test this hypothesis. Cooperative intergroup mating does lend an advantage to a universal cooperation strategy when the cost/benefit ratio of cooperation is low and local population diversity is …
Families are places of affection and cooperation, but also of conflict. In his famous paper Parent-Offspring Conflict, Robert L. Trivers builds upon W. D. Hamilton's concept of inclusive fitness to argue for genetic conflict in parent-offspring relationships, and to derive numerical predictions on the intensity of the conflict. We propose a mathematical model of game theory that depicts how each member of a family allocates her resource budget to maximize her inclusive fitness; this latter is made of the sum of personal fitness plus the sum of relatives fitnesses weighted by Wright's coefficie…
The diversity in wealth and social status is present not only among humans, but throughout the animal world. We account for this observation by generating random variables that determ ine the social diversity of players engaging in the prisoner's dilemma game. Here the term social diversity is used to address extrinsic factors that determine the mapping of game pay offs to individual fitness. These factors may increase or decrease the fitness of a player depending on its location on the spatial grid. We consider different distributions of extrin sic factors that determine the social diversity …
Existing theoretical models of evolution focus on the relative fitness advantages of different mutants in a population while the dynamic behavior of the population size is mostly left unconsidered. We here present a generic stochastic model which combines the growth dynamics of the population and its internal evolution. Our model thereby accounts for the fact that both evolutionary and growth dynamics are based on individual reproduction events and hence are highly coupled and stochastic in nature. We exemplify our approach by studying the dilemma of cooperation in growing populations and show…
Societies consisting of cooperative individuals seem to require for their continuing success that defectors be policed. The precise connection between punishers and benefits, population structure, and division of labour, however, remains ill-understood. Many models assume costly "peer punishment" to enforce cooperation, but results in the economics literature suggest that this assumption may not be generally valid. In many human and animal societies, there is a division of labour between a purely supportive majority and a dedicated minority of police-like enforcers. Here we present several ext…
Cooperative behaviors are defined as the production of common goods benefitting all members of the community at the producer's cost. They could seem to be in contradiction with natural selection, as non-cooperators have an increased fitness compared to cooperators. Understanding the emergence of cooperation has necessitated the development of concepts and models (inclusive fitness, multilevel selection, ...) attributing deterministic advantages to this behavior. In contrast to these models, we show here that cooperative behaviors can emerge by taking into account the stochastic nature of evolu…
Cancer cells obtain mutations which rely on the production of diffusible growth factors to confer a fitness benefit. These mutations can be considered cooperative, and studied as public goods games within the framework of evolutionary game theory. The population structure, benefit function and update rule all influence the evolutionary success of cooperators. We model the evolution of cooperation in epithelial cells using the Voronoi tessellation model. Unlike traditional evolutionary graph theory, this allows us to implement global updating, for which birth and death events are spatially deco…
Whether or not to change strategy depends not only on the personal success of each individual, but also on the success of others. Using this as motivation, we study the evolution of cooperation in games that describe social dilemmas, where the propensity to adopt a different strategy depends both on individual fitness as well as on the strategies of neighbors. Regardless of whether the evolutionary process is governed by pairwise or group interactions, we show that plugging into the "wisdom of groups" strongly promotes cooperative behavior. The more the wider knowledge is taken into account th…
The functioning of animal as well as human societies fundamentally relies on cooperation. Yet, defection is often favorable for the selfish individual, and social dilemmas arise. Selection by individuals' fitness, usually the basic driving force of evolution, quickly eliminates cooperators. However, evolution is also governed by fluctuations that can be of greater importance than fitness differences, and can render evolution effectively neutral. Here, we investigate the effects of selection versus fluctuations in social dilemmas. By studying the mean extinction times of cooperators and defecto…
In the framework of the paradigmatic prisoner's dilemma, we investigate the evolutionary dynamics of social dilemmas in the presence of "cooperation facilitators". In our model, cooperators and defectors interact as in the classical prisoner's dilemma game, where selection favors defection. However, here the presence of a small number of cooperation facilitators enhances the fitness (reproductive potential) of cooperators, while it does not alter that of defectors. In a finite population of size N, the dynamics of the prisoner's dilemma with facilitators is characterized by the probability tha…
Evolution of cooperation with Q-learning: the impact of information perception
School of Physics and Information Technology, Shaanxi Normal University（陕西师范大学物理与信息技术学院）；School of Physics, Ningxia University（宁夏大学物理学院）；School of Xinjiang Institute of Engineering Control Engineering College, Xinjiang Institute of Engineering（新疆工程学院控制工程学院）；School of Computer Science, Soochow University（苏州大学计算机学院）
Strategy changes are an essential part of evolutionary games. Here we introduce a simple rule that, depending on the value of a single parameter $w$, influences the selection of players that are considered as potential sources of the new strategy. For positive $w$ players with high payoffs will be considered more likely, while for negative $w$ the opposite holds. Setting $w$ equal to zero returns the frequently adopted random selection of the opponent. We find that increasing the probability of adopting the strategy from the fittest player within reach, i.e. setting $w$ positive, promotes the …
The n-person Prisoner's Dilemma is a widely used model for populations where individuals interact in groups. The evolutionary stability of populations has been analysed in the literature for the case where mutations in the population may be considered as isolated events. For this case, and assuming simple trigger strategies and many iterations per game, we analyse the rate of convergence to the evolutionarily stable populations. We find that for some values of the payoff parameters of the Prisoner's Dilemma this rate is so low that the assumption, that mutations in the population are infrequen…
Situations involving cooperative behaviour are widespread among animals and humans alike. Game theory and evolutionary dynamics have provided the theoretical and computational grounds to understand what are the mechanisms that allow for such cooperation. Studies in this area usually take into consideration different behavioural strategies and investigate how they can be fixed in the population under evolving rules. However, how those strategies emerged from basic evolutionary mechanisms continues to be not fully understood. To address this issue, here we study the emergence of cooperative stra…
Through the lens of game theory, cooperation is frequently considered an unsustainable strategy: if an entire population is cooperating, each indi- vidual can increase its overall fitness by choosing not to cooperate, thereby still receiving all the benefit of its cooperating neighbors while no longer expending its own energy. Observable cooperation in naturally-occurring public goods games is consequently of great interest, as such systems offer insight into both the emergence and sustainability of cooperation. Here we consider a population that obeys a public goods game on a network of discr…
A non-local rare mutations model for quasispecies and Prisoner's dilemma: numerical assessment of qualitative behaviour
Università di Napoli “Parthenope”（那不勒斯帕尔特诺佩大学）；Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche（意大利国家研究委员会计算应用研究所）
An active participation of players in evolutionary games depends on several factors, ranging from personal stakes to the properties of the interaction network. Diverse activity patterns thus have to be taken into account when studying the evolution of cooperation in social dilemmas. Here we study the weak prisoner's dilemma game, where the activity of each player is determined in a probabilistic manner either by its degree or by its payoff. While degree-correlated activity introduces cascading failures of cooperation that are particularly severe on scale-free networks with frequently inactive …
Cooperation is ubiquitous across all levels of biological systems ranging from microbial communities to human societies. It, however, seemingly contradicts the evolutionary theory, since cooperators are exploited by free-riders and thus are disfavored by natural selection. Many studies based on evolutionary game theory have tried to solve the puzzle and figure out the reason why cooperation exists and how it emerges. Network reciprocity is one of the mechanisms to promote cooperation, where nodes refer to individuals and links refer to social relationships. The spatial arrangement of mutant in…
