Traditionally, resource limitation in evolutionary game theory is assumed just to impose a constant population size. Here we show that resource limitations may generate dynamical payoffs able to alter an original prisoner's dilemma, and to allow for the stable coexistence between unconditional cooperators and defectors in well-mixed populations. This is a consequence of a self-organizing process that turns the interaction payoff matrix into evolutionary neutral, and represents a resource-based control mechanism preventing the spread of defectors. To our knowledge, this is the first example of coexistence in well-mixed populations with a game structure different from a snowdrift game.
Do people discriminate between men and women when they have the option to punish defectors or reward cooperators? Here we report on four pre-registered experiments, that shed some light on this question. Study 1 (N=544) shows that people do not discriminate between genders when they have the option to punish (reward) defectors (cooperators) in a one-shot prisoner's dilemma with third-party punishment/reward. Study 2 (N=253) extends Study 1 to a different method of punishing/rewarding: participants are asked to rate the behaviour of a defector/cooperator on a scale of 1 to 5 stars. In this case…
In the study of the evolution of cooperation, resource limitations are usually assumed just to provide a finite population size. Recently, however, agent-based models have pointed out that resource limitation may modify the original structure of the interactions and allow for the survival of unconditional cooperators in well-mixed populations. Here, we present analytical simplified versions of two types of agent-based models recently published: one in which the limiting resource constrains the ability of reproduction of individuals but not their survival, and a second one where the limiting re…
Recent experimental evidence [Grujić et al., PLoS ONE 5, e13749 (2010)] on the spatial Prisoner's Dilemma suggests that players choosing to cooperate or not on the basis of their previous action and the actions of their neighbors coexist with steady defectors and cooperators. We here study the coexistence of these three strategies in the multiplayer iterated Prisoner's Dilemma by means of the replicator dynamics. We consider groups with n = 2, 3, 4 and 5 players and compute the payoffs to every type of player as the limit of a Markov chain where the transition probabilities between actions are…
Consider a cooperation game on a spatial network of habitat patches, where players can relocate between patches if they judge the local conditions to be unfavorable. In time, the relocation events may lead to a homogeneous state where all patches harbor the same relative densities of cooperators and defectors or they may lead to self-organized patterns, where some patches become safe havens that maintain an elevated cooperator density. Here we analyze the transition between these states mathematically. We show that safe havens form once a certain threshold in connectivity is crossed. This thre…
Despite the wealth of empirical and theoretical studies, the origin and maintenance of cooperation is still an evolutionary riddle. In this context, ecological life-history traits which affect the efficiency of selection may play a role, though these are often ignored. We consider here species such as bacteria, fungi, invertebrates and plants which exhibit resting stages in the form of a quiescent state or a seedbank. When quiescent, individuals are inactive and reproduce upon activation, while under seed bank parents produce offspring remaining dormant for different amount of time. We assume …
Many mathematical frameworks of evolutionary game dynamics assume that the total population size is constant and that selection affects only the relative frequency of strategies. Here, we consider evolutionary game dynamics in an extended Wright-Fisher process with variable population size. In such a scenario, it is possible that the entire population becomes extinct. Survival of the population may depend on which strategy prevails in the game dynamics. Studying cooperative dilemmas, it is a natural feature of such a model that cooperators enable survival, while defectors drive extinction. Alt…
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…
Economic experiments have shown that punishment can increase public goods game contributions over time. However, the effectiveness of punishment is challenged by second-order freeriding and antisocial punishment. The latter implies that non-cooperators punish cooperators, while the former implies unwillingness to shoulder the cost of punishment. Here we extend the theory of cooperation in the spatial public goods game by considering four competing strategies, which are traditional cooperators and defectors, as well as cooperators who punish defectors and defectors who punish cooperators. We sh…
Social, biological and economic networks grow and decline with occasional fragmentation and re-formation, often explained in terms of external perturbations. We show that these phenomena can be a direct consequence of simple imitation and internal conflicts between 'cooperators' and 'defectors'. We employ a game-theoretic model of dynamic network formation where successful individuals are more likely to be imitated by newcomers who adopt their strategies and copy their social network. We find that, despite using the same mechanism, cooperators promote well-connected highly prosperous networks …
Whether or not cooperation is favored in evolutionary games on graphs depends on the population structure and spatial properties of the interaction network. Population structures can be expressed as configurations. Such configurations extend scenarios with a single cooperator among defectors to any number of cooperators and any arrangement of cooperators and defectors. Thus, as a single cooperator can be interpreted as a lone mutant, the discussion about fixation properties based on configurations also applies to multiple mutants. For interaction networks modeled as regular graphs and for weak…
Stability of evolutionary dynamics of non-repeated Prisoner's Dilemma game with non-uniform interaction rates [1], via benefit and cost dilemma is studied . Moreover, the stability condition (b+c/b-c)2 < r1r3 is derived in case of coexistence between cooperators and defectors .If r1,r3 -> infinity cooperation is the dominant strategy and defectors can no longer exploit cooperators.
In structured populations the spatial arrangement of cooperators and defectors on the interaction graph together with the structure of the graph itself determines the game dynamics and particularly whether or not fixation of cooperation (or defection) is favored. For a single cooperator (and a single defector) and a network described by a regular graph the question of fixation can be addressed by a single parameter, the structure coefficient. As this quantity is generic for any regular graph, we may call it the generic structure coefficient. For two and more cooperators (or several defectors) …
Dense cooperative networks are an essential element of social capital for a prosperous society. These networks enable individuals to overcome collective action dilemmas by enhancing trust. In many biological and social settings, network structures evolve endogenously as agents exit relationships and build new ones. However, the process by which evolutionary dynamics lead to self-organization of dense cooperative networks has not been explored. Our large group prisoner's dilemma experiments with exit and partner choice options show that core-periphery segregation of cooperators and defectors dr…
According to the standard protocol of spatial public goods game, a cooperator player invests not only into his own game but also into the games organized by neighboring partners. In this work, we relax this assumption by allowing cooperators to decide which neighboring group to prefer instead of supporting them uniformly. In particular, we assume that they select their most successful neighbor and focus external investments exclusively into the related group. We show that this very simple alteration of the dynamical rule results in a surprisingly positive evolutionary outcome -- cooperators pr…
We study the evolution of cooperation in spatial public goods games with four competing strategies: cooperators, defectors, punishing cooperators, and punishing defectors. To explore the robustness of the cooperation-promoting effect of costly punishment, besides the usual strategy adoption dynamics we also apply strategy mutations. As expected, frequent mutations create kind of well-mixed conditions, which support the spreading of defectors. However, when the mutation rate is small, the final stationary state does not significantly differ from the state of the mutation-free model, independent…
We consider a stochastic model for evolution of group-structured populations in which interactions between group members correspond to the Prisoner's Dilemma or the Hawk-Dove game. Selection operates at two organization levels: individuals compete with peer group members based on individual payoff, while groups also compete with other groups based on average payoff of group members. In the Prisoner's Dilemma, this creates a tension between the two levels of selection, as defectors are favored at the individual level, whereas groups with at least some cooperators outperform groups of defectors …
Cooperation is beneficial for the species as a whole, but, at the level of an individual, defection pays off. Natural selection is then expected to favor defectors and eliminate cooperation. This prediction is in stark contrast with the abundance of cooperation at all levels of biological systems: from cells cooperating to form a biofilm or an organism to ecosystems and human societies. Several explanations have been proposed to resolve this paradox, including direct reciprocity, kin, and group selection. However, our work builds upon an observation that selection on cooperators might depend b…
We study the conditions for persistent cooperation in an off-lattice model of mobile agents playing the Prisoner's Dilemma game with pure, unconditional strategies. Each agent has an exclusion radius rP, which accounts for the population viscosity, and an interaction radius rint, which defines the instantaneous contact network for the game dynamics. We show that, differently from the rP=0 case, the model with finite-sized agents presents a coexistence phase with both cooperators and defectors, besides the two absorbing phases, in which either cooperators or defectors dominate. We provide, in a…
Deliberate deceptiveness intended to gain an advantage is commonplace in human and animal societies. In a social dilemma, an individual may only pretend to be a cooperator to elicit cooperation from others, while in reality he is a defector. With this as motivation, we study a simple variant of the evolutionary prisoner's dilemma game entailing deceitful defectors and conditional cooperators that lifts the veil on the impact of such two-faced behavior. Defectors are able to hide their true intentions at a personal cost, while conditional cooperators are probabilistically successful at identify…
Ecology and evolution are inherently linked, and studying a mathematical model that considers both holds promise of insightful discoveries related to the dynamics of cooperation. In the present article, we use the prisoner's dilemma (PD) game as a basis for long-term apprehension of the essential social dilemma related to cooperation among unrelated individuals. We upgrade the contemporary PD game with an inclusion of evolution-induced act of punishment as a third competing strategy in addition to the traditional cooperators and defectors. In a population structure, the abundance of ecological…
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 …
Limitation of resources has been recently introduced as a mechanism for the survival and coexistence of cooperators with defectors in well-mixed populations. Here we examine the same model on a scale-free network. A prisoner's dilemma game on a scale-free network has shown coexistence of cooperators and defectors for the entire range of parameters. Our results show that by introducing the network to the limited resources model, the cooperator-dominated region in the parameter space expands comparing to the results of well-mixed population and the coexistence region becomes narrower. The effect…
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) ar…
Whether or not cooperation is favored over defection in evolutionary games can be assigned by structure coefficients for any arrangement of cooperators and defectors on any network modeled as a regular graph. We study how these structure coefficients relate to a scaling of dilemma strength in social dilemma games. It is shown that some graphs permit certain arrangements of cooperators and defectors to possess particularly large structure coefficients. Moreover, these large coefficients imply particularly large sections of a bounded parameter plane spanned by scaling gamble-intending and risk-a…
We study the evolutionary dynamics of the prisoner's dilemma game in which cooperators and defectors interact with another actor type called exiters. Rather than being exploited by defectors, exiters exit the game in favour of a small payoff. We find that this simple extension of the game allows cooperation to flourish in well-mixed populations when iterations or reputation are added. In networked populations, however, the exit option is less conducive to cooperation. Instead, it enables the coexistence of cooperators, defectors, and exiters through cyclic dominance. Other outcomes are also po…
The fact that individuals will most likely behave differently in different situations begets the introduction of conditional strategies. Inspired by this, we study the evolution of cooperation in the spatial public goods game, where besides unconditional cooperators and defectors, also different types of conditional cooperators compete for space. Conditional cooperators will contribute to the public good only if other players within the group are likely to cooperate as well, but will withhold their contribution otherwise. Depending on the number of other cooperators that are required to elicit…
When the interactions between cooperators (C) and defectors (D) can be partially avoided within a population, there may be an overall enhancement of cooperation. One example of such screening mechanism occurs in the presence of risk-averse agents (loners, L) that are neutral towards others, i.e., both L and its opponent, whatever its strategy, receive the same payoff. Their presence in the Prisoner's Dilemma (PD) game sustains the coexistence of cooperators and defectors far beyond the level attained in their absence. Another screening mechanism is a heterogeneous landscape obtained, for examp…
Synchronization, cooperation, and chaos are ubiquitous phenomena in nature. In a population composed of many distinct groups of individuals playing the prisoner's dilemma game, there exists a migration dilemma: No cooperator would migrate to a group playing the prisoner's dilemma game lest it should be exploited by a defector; but unless the migration takes place, there is no chance of the entire population's cooperator-fraction to increase. Employing a randomly rewired coupled map lattice of chaotic replicator maps, modelling replication-selection evolutionary game dynamics, we demonstrate th…
Utilizing common resources is always a dilemma for community members. While cooperator players restrain themselves and consider the proper state of resources, defectors demand more than their supposed share for a higher payoff. To avoid the tragedy of the common state, punishing the latter group seems to be an adequate reaction. This conclusion, however, is less straightforward when we acknowledge the fact that resources are finite and even a renewable resource has limited growing capacity. To clarify the possible consequences, we consider a coevolutionary model where beside the payoff-driven …
Population structure and spatial heterogeneity are integral components of evolutionary dynamics, in general, and of evolution of cooperation, in particular. Structure can promote the emergence of cooperation in some populations and suppress it in others. Here, we provide results for weak selection to favor cooperation on regular graphs for any configuration, meaning any arrangement of cooperators and defectors. Our results extend previous work on fixation probabilities of single, randomly placed mutants. We find that for any configuration cooperation is never favored for birth-death (BD) updat…
Inspired by the fact that people have diverse propensities to punish wrongdoers, we study a spatial public goods game with defectors and different types of punishing cooperators. During the game, cooperators punish defectors with class-specific probabilities and subsequently share the associated costs of sanctioning. We show that in the presence of different punishing cooperators the highest level of public cooperation is always attainable through a selection mechanism. Interestingly, the selection not necessarily favors the evolution of punishers who would be able to prevail on their own agai…
Diffusion-induced instabilities promote cooperation in eco-evolutionary networks
Bagchi School of Public Health, Ahmedabad University（艾哈迈达巴德大学巴吉公共卫生学院）；Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata（印度科学教育与研究学院加尔各答分校物理科学系）；Department of Mathematics and Namur Institute for Complex Systems - naXys, University of Namur（南纳尔大学数学与南纳尔复杂系统研究所-naXys）；Faculty of Natural Sciences and Mathematics, University of Maribor（马里博尔大学自然科学与数学学院）；Community Healthcare Center Dr. Adolf Drolc Maribor（马里博尔阿道夫·德罗尔克博士社区健康中心）；Department of Physics, Kyung Hee University（庆熙大学物理系）；University College, Korea University（韩国大学综合研究生院）；Physics and Applied Mathematics Unit, Indian Statistical Institute（印度统计物理学与应用数学单元）
Situations where individuals have to contribute to joint efforts or share scarce resources are ubiquitous. Yet, without proper mechanisms to ensure cooperation, the evolutionary pressure to maximize individual success tends to create a tragedy of the commons (such as over-fishing or the destruction of our environment). This contribution addresses a number of related puzzles of human behavior with an evolutionary game theoretical approach as it has been successfully used to explain the behavior of other biological species many times, from bacteria to vertebrates. Our agent-based model distingui…
Cooperation is the backbone of modern human societies, making it a priority to understand how successful cooperation-sustaining mechanisms operate. Cyclic dominance, a non-transitive setup comprising at least three strategies wherein the first strategy overrules the second which overrules the third which, in turn, overrules the first strategy, is known to maintain bio-diversity, drive competition between bacterial strains, and preserve cooperation in social dilemmas. Here, we present a novel route to cyclic dominance in voluntary social dilemmas by adding to the traditional mix of cooperators,…
We present a novel system of equations to describe the evolution of self-organized structured societies (biological or human) composed of several trait groups. The suggested approach is based on the combination of ideas employed in the theory of biological populations, system theory, and utility theory. The evolution equations are defined as utility rate equations, whose parameters are characterized by the utility of each group with respect to the society as a whole and by the mutual utilities of groups with respect to each other. We analyze in detail the cases of two groups (cooperators and d…
Cooperators that refuse to participate in sanctioning defectors create the second-order free-rider problem. Such cooperators will not be punished because they contribute to the public good, but they also eschew the costs associated with punishing defectors. Altruistic punishers - those that cooperate and punish - are at a disadvantage, and it is puzzling how such behaviour has evolved. We show that sharing the responsibility to sanction defectors rather than relying on certain individuals to do so permanently can solve the problem of costly punishment. Inspired by the fact that humans have str…
Cooperative behaviour has been extensively studied as a choice between cooperation and defection. However, the possibility to not participate is also frequently available. This type of problem can be studied through the optional public goods game. The introduction of the "Loner" strategy, allows players to withdraw from the game, which leads to a cooperator-defector-loner cycle. While prosocial punishment can help increase cooperation, anti-social punishment -- where defectors punish cooperators -- causes its downfall in both experimental and theoretical studies. In this paper, we introduce so…
Cooperation is a difficult proposition in the face of Darwinian selection. Those that defect have an evolutionary advantage over cooperators who should therefore die out. However, spatial structure enables cooperators to survive through the formation of homogeneous clusters, which is the hallmark of network reciprocity. Here we go beyond this traditional setup and study the spatiotemporal dynamics of cooperation in a population of populations. We use the prisoner's dilemma game as the mathematical model and show that considering several populations simultaneously give rise to fascinating spati…
ETH Zürich（苏黎世联邦理工学院）；Complexity Science Hub Vienna（维也纳复杂性科学中心）；University of Maribor（马里博尔大学）；China Medical University Hospital, China Medical University（中国医药大学附设医院，中国医药大学）；Alma Mater Europaea（欧洲母校大学）
In a non-sustainable, "over-populated" world, what might the use of nanotechnology-based targeted, autonomous weapons mean for the future of humanity? In order to gain some insights, we make a simplified game-theoretical thought experiment. We consider a population where agents play the public goods game, and where in parallel an epidemic unfolds. Agents that are infected defectors are killed with a certain probability and replaced by susceptible cooperators. We show that such "nanowars", even if aiming to promote good behavior and planetary health, fail not only to promote cooperation, but th…
We study the evolution of cooperation in the spatial public goods game, focusing on the effects that are brought about by the delayed distribution of goods that accumulate in groups due to the continuous investments of cooperators. We find that intermediate delays enhance network reciprocity because of a decelerated invasion of defectors, who are unable to reap the same high short-term benefits as they do in the absence of delayed distribution. Long delays, however, introduce a risk because the large accumulated wealth might fall into the wrong hands. Indeed, as soon as the curvature of a coop…
I study two mechanisms based on punishment to promote cooperation in the two-population snowdrift game. The first mechanism follows the traditional approach in the literature and is based on the inclusion of a third additional strategy in the payoff matrix of the stage-game. The second mechanism consists of letting cooperators to punish defectors with a given exogenous frequency. While both mechanisms share the same result regarding the minimum required level of punishment in order to eliminate defectors in both populations, stability in the mechanism following the second approach is more robu…
The emergence and maintenance of cooperative behavior is a fascinating topic in evolutionary biology and social science. The public goods game (PGG) is a paradigm for exploring cooperative behavior. In PGG, the total resulting payoff is divided equally among all participants. This feature still leads to the dominance of defection without substantially magnifying the public good by a multiplying factor. Much effort has been made to explain the evolution of cooperative strategies, including a recent model in which only a portion of the total benefit is shared by all the players through introduci…
Cooperation among individuals has been key to sustaining societies. However, natural selection favors defection over cooperation. Cooperation can be favored when the mobility of individuals allows cooperators to form a cluster (or group). Mobility patterns of animals sometimes follow a Lévy flight. A Lévy flight is a kind of random walk but it is composed of many small movements with a few big movements. The role of Lévy flights for cooperation has been studied by Antonioni and Tomassini. They showed that Lévy flights promoted cooperation combined with conditional movements triggered by neighb…
Extortion strategies can dominate any opponent in an iterated prisoner's dilemma game. But if players are able to adopt the strategies performing better, extortion becomes widespread and evolutionary unstable. It may sometimes act as a catalyst for the evolution of cooperation, and it can also emerge in interactions between two populations, yet it is not the evolutionary stable outcome. Here we revisit these results in the realm of spatial games. We find that pairwise imitation and birth-death dynamics return known evolutionary outcomes. Myopic best response strategy updating, on the other han…
We study the problem of the emergence of cooperation in the spatial Prisoner's Dilemma. The pioneering work by Nowak and May showed that large initial populations of cooperators can survive and sustain cooperation in a square lattice with imitate-the-best evolutionary dynamics. We revisit this problem in a cost-benefit formulation suitable for a number of biological applications. We show that if a fixed-amount reward is established for cooperators to share, a single cooperator can invade a population of defectors and form structures that are resilient to re-invasion even if the reward mechanis…
In the traditional setup of public goods game all players are involved in every available groups and the mutual benefit is shared among competing cooperator and defector strategies. But in real life situations the group formation of players could be more sophisticated because not all players are attractive enough for others to participate in a joint venture. What if only those players can initiate a group formation and establish a game who are successful enough to the neighbors? To elaborate this idea we employ a modified protocol and demonstrate that a carefully chosen threshold to establish …
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…
Populations composed of a collection of subpopulations (demes) with random migration between them are quite common occurrences. The emergence and sustenance of cooperation in such a population is a highly researched topic in the evolutionary game theory. If the individuals in every deme are considered to be either cooperators or defectors, the migration dilemma can be envisaged: The cooperators would not want to migrate to a defector-rich deme as they fear of facing exploitation; but without migration, cooperation can not be established throughout the network of demes. With a view to studying …
The basic social dilemma is frequently captured by a public goods game where participants decide simultaneously whether to support a common pool or not and after the enhanced contributions are distributed uniformly among all competitors. What if the result of common efforts is {\it not} distributed immediately, but it is reinvested and added to the pool for a next round? This extension may not only result in an enhanced benefit for group members but also opens new strategies for involved players because they may act in distinct rounds differently. In this work we focus on the simplest case whe…
In recent years, there has been growing interest in studying evolutionary games with environmental feedback. Previous studies exclusively focus on two-player games. However, extension to multi-player game is needed to study problems such as microbial cooperation and crowdsourcing collaborations. Here, we study coevolutionary public goods games where strategies coevolve with the multiplication factors of group cooperation. Asymmetry can arise in such environmental feedback, where games organized by focal cooperators may have a different efficiency than the ones by defectors. Our analysis shows …
We study the evolution of cooperation in the collective-risk social dilemma game, where the risk is determined by a collective target that must be reached with individual contributions. All players initially receive endowments from the available amount of common resources. While cooperators contribute part of their endowment to the collective target, defectors do not. If the target is not reached, the endowments of all players are lost. In our model, we introduce a feedback between the amount of common resources and the contributions of cooperators. We show that cooperation can be sustained on…
We explore the minimal conditions for sustainable cooperation on a spatially distributed population of memoryless, unconditional strategies (cooperators and defectors) in presence of unbiased, non contingent mobility in the context of the Prisoner's Dilemma game. We find that cooperative behavior is not only possible but may even be enhanced by such an ``always-move'' rule, when compared with the strongly viscous (``never-move'') case. In addition, mobility also increases the capability of cooperation to emerge and invade a population of defectors, what may have a fundamental role in the probl…
Game theory deals with strategic interactions among players and evolutionary game dynamics tracks the fate of the players' populations under selection. In this paper, we consider the replicator equation for two-player-two-strategy games involving cooperators and defectors. We modify the equation to include the effect of mutation and also delay that corresponds either to the delayed information about the population state or in realizing the effect of interaction among players. By focusing on the four exhaustive classes of symmetrical games -- the Stag Hunt game, the Snowdrift game, the Prisoner…
We study the evolution of cooperation as a birth-death process in spatially extended populations. The benefit from the altruistic behavior of a cooperator is implemented by decreasing the death rate of its direct neighbors. The cost of cooperation is the increase of a cooperator's death rate proportional to the number of its neighbors. When cooperation has higher cost than benefit, cooperators disappear. Then the dynamics reduces to the contact process with defectors as the single particle type. Increasing the benefit-cost ratio above 1, the extinction transition of the contact process splits …
