Active linking in evolutionary games

Population dynamics and games of variable size

This work introduces the concept of Variable Size Game Theory (VSGT), in which the number of players in a game is a strategic decision made by the players themselves. We start by discussing the main examples in game theory: dominance, coexistence, and coordination. We show that the same set of pay-offs can result in coordination-like or coexistence-like games depending on the strategic decision of each player type. We also solve an inverse problem to find a d-player game that reproduces the same fixation pattern of the VSGT. In the sequel, we consider a game involving prosocial and antisocial players, i.e., individuals who tend to play with large groups and small groups, respectively. In this game, a certain task should be performed, that will benefit one of the participants at the expense of the other players. We show that individuals able to gather large groups to perform the task may prevail, even if this task is costly, providing a possible scenario for the evolution of eusociality. The next example shows that different strategies regarding game size may lead to spontaneous separation of different types, a possible scenario for speciation without physical separation (sympatric speciation). In the last example, we generalize to three types of populations from the previous analysis and study compartmental epidemic models: in particular, we recast the SIRS model into the VSGT framework: Susceptibles play 2-player games, while Infectious and Removed play a 1-player game. The SIRS epidemic model is then obtained as the replicator equation of the VSGT. We finish with possible applications of VSGT to be addressed in the future.

Deterministic theory of evolutionary games on temporal networks

Recent empirical studies have revealed that social interactions among agents in realistic networks merely exist intermittently and occur in a particular sequential order. However, it remains unexplored how to theoretically describe evolutionary dynamics of multiple strategies on temporal networks. Herein, we develop a deterministic theory for studying evolutionary dynamics of any [Formula: see text] pairwise games in structured populations where individuals are connected and organized by temporally activated edges. In the limit of weak selection, we derive replicator-like equations with a transformed payoff matrix characterizing how the mean frequency of each strategy varies over time, and then obtain critical conditions for any strategy to be evolutionarily stable on temporal networks. Interestingly, the re-scaled payoff matrix is a linear combination of the original payoff matrix with an additional one describing local competitions between any pair of different strategies, whose weights are solely determined by network topology and selection intensity. As a particular example, we apply the deterministic theory to analysing the impacts of temporal networks in the mini-ultimatum game, and find that temporally networked population structures result in the emergence of fairness. Our work offers theoretical insights into the subtle effects of network temporality on evolutionary game dynamics.

Influence analysis of network evolution on Parrondo effect

Parrondo's paradox is a scheme used to describe an interesting paradoxical situation that a losing Game A and a losing Game B played randomly or periodically will produce a winning result. Here, a dynamic process of network evolution of Link A + Game B is proposed to yield the Parrondo effect. Game B with two asymmetric branches depends on the relative comparison between the capital of the network node and the average capital of all its neighbors. Simulation results demonstrate that network structure evolution make the losing Game B produce a paradoxical effect of winning and would be advantageous to the development of the "ratcheting" mechanism in Game B. Furthermore, the underlying paradoxical mechanism is analyzed to illustrate the parameter space where the "strong" Parrondo effect occurs. Then influence of two types of network connection is analyzed, indicating that the "agitating" effect of the network is basically the same when a node connects to a neighbor's neighbor or randomly chooses a node other than its neighbors. Further, a higher frequency of network evolution yields a larger parameter region where the "strong" Parrondo effect emerges.

Wild jackdaws can selectively adjust their social associations while preserving valuable long-term relationships

Influential theories of the evolution of cognition and cooperation posit that tracking information about others allows individuals to adjust their social associations strategically, re-shaping social networks to favour connections between compatible partners. Crucially, to our knowledge, this has yet to be tested experimentally in natural populations, where the need to maintain long-term, fitness-enhancing relationships may limit social plasticity. Using a social-network-manipulation experiment, we show that wild jackdaws (Corvus monedula) learned to favour social associations with compatible group members (individuals that provided greater returns from social foraging interactions), but resultant change in network structure was constrained by the preservation of valuable pre-existing relationships. Our findings provide insights into the cognitive basis of social plasticity and the interplay between individual decision-making and social-network structure.

Inhibition and activation of interactions in networked weak prisoner's dilemma

In the framework of the coevolution dynamics of the weak prisoner's dilemma, inspired by prior empirical research, we present a coevolutionary model with local network dynamics in a static network framework. Viewing the edges of the network as social interactions between individuals, when individuals play the weak prisoner's dilemma game, they accumulate both payoffs and social interaction willingness based on a payoff matrix of the social interaction willingness we constructed. The edges are then inhibiting or activating based on the social interaction willingness of the two individuals, and individuals only interact with others through activated edges, resulting in local network dynamics in a static network framework. Individuals who receive more cooperation will be more likely to activate the edges around them, meaning they will participate in more social interactions. Conversely, individuals who receive more defects will do the opposite. Specifically, we investigate the evolutionary dynamics of cooperation under different levels of sensitivity to social interaction willingness and the temptation to defect. Through the simulation, we find that sparse cooperator clusters can expand greatly when social interaction sensitivity and temptation to defect are low. In contrast, dense cooperator clusters form rapidly in a high social interaction sensitivity, which protects the cooperation from high temptation.

Assortment and Reciprocity Mechanisms for Promotion of Cooperation in a Model of Multilevel Selection

In the study of the evolution of cooperation, many mechanisms have been proposed to help overcome the self-interested cheating that is individually optimal in the Prisoners' Dilemma game. These mechanisms include assortative or networked social interactions, other-regarding preferences considering the payoffs of others, reciprocity rules to establish cooperation as a social norm, and multilevel selection involving simultaneous competition between individuals favoring cheaters and competition between groups favoring cooperators. In this paper, we build on recent work studying PDE replicator equations for multilevel selection to understand how within-group mechanisms of assortment, other-regarding preferences, and both direct and indirect reciprocity can help to facilitate cooperation in concert with evolutionary competition between groups. We consider a group-structured population in which interactions between individuals consist of Prisoners' Dilemma games and study the dynamics of multilevel competition determined by the payoffs individuals receive when interacting according to these within-group mechanisms. We find that the presence of each of these mechanisms acts synergistically with multilevel selection for the promotion of cooperation, decreasing the strength of between-group competition required to sustain long-time cooperation and increasing the collective payoff achieved by the population. However, we find that only other-regarding preferences allow for the achievement of socially optimal collective payoffs for Prisoners' Dilemma games in which average payoff is maximized by an intermediate mix of cooperators and defectors. For the other three mechanisms, the multilevel dynamics remain susceptible to a shadow of lower-level selection, as the collective outcome fails to exceed the payoff of the all-cooperator group.

On the evolutionary language game in structured and adaptive populations

We propose an evolutionary model for the emergence of shared linguistic convention in a population of agents whose social structure is modelled by complex networks. Through agent-based simulations, we show a process of convergence towards a common language, and explore how the topology of the underlying networks affects its dynamics. We find that small-world effects act to speed up convergence, but observe no effect of topology on the communicative efficiency of common languages. We further explore differences in agent learning, discriminating between scenarios in which new agents learn from their parents (vertical transmission) versus scenarios in which they learn from their neighbors (oblique transmission), finding that vertical transmission results in faster convergence and generally higher communicability. Optimal languages can be formed when parental learning is dominant, but a small amount of neighbor learning is included. As a last point, we illustrate an exclusion effect leading to core-periphery networks in an adaptive networks setting when agents attempt to reconnect towards better communicators in the population.

Evolution of trustfulness in the case where resources for cooperation are sometimes absent

It is worth investigating the existence of cooperation, which is costly for the actor but beneficial to the recipient (precisely because it is costly for the former). If players, when they approach defectors, stop their relationship with them, cooperation can pay off and favorably emerge in the course of evolutionary dynamics. The present study examines the situation in which animals, even when they want to cooperate, sometimes lack the necessary resources, and are thereby prevented from cooperating with others. In addition, it is also considered that the underlying information about the presence or absence of these resources can be conveyed to the opponent player. Here, the opponent who defects-has no resources for cooperation-may be a cooperator or a defector. Therefore, it is not clear which behavior is more likely to evolve, if it is keeping the interaction with such an opponent (i.e., being trustful) or stopping the interaction with such an opponent (i.e., being not trustful). By using evolutionary game theory, it is revealed that those who want to keep the interaction with those without the resources to cooperate are favored by natural selection. This study sheds new light on the role of keeping and stopping interaction in the evolution of cooperation under variable availability of resources.

Network rewiring promotes cooperation in an aspirational learning model

We analyze a cooperative decision-making model that is based on individual aspiration levels using the framework of a public goods game in static and dynamic networks. Sensitivity to differences in payoff and dynamic aspiration levels modulates individual satisfaction and affects subsequent behavior. The collective outcome of such strategy changes depends on the efficiency with which aspiration levels are updated. Below a threshold learning efficiency, cooperators dominate despite short-term fluctuations in strategy fractions. Categorizing players based on their satisfaction level and the resulting strategy reveal periodic cycling between the different categories. We explain the distinct dynamics in the two phases in terms of differences in the dominant cyclic transitions between different categories of cooperators and defectors. Allowing even a small fraction of nodes to restructure their connections can promote cooperation across almost the entire range of values of learning efficiency. Our work reinforces the usefulness of an internal criterion for strategy updates, together with network restructuring, in ensuring the dominance of altruistic strategies over long time-scales.

Best Reply Player Against Mixed Evolutionarily Stable Strategy User

We consider matrix games with two phenotypes (players): one following a mixed evolutionarily stable strategy and another one that always plays a best reply against the action played by its opponent in the previous round (best reply player, BR). We focus on iterated games and well-mixed games with repetition (that is, the mean number of repetitions is positive, but not infinite). In both interaction schemes, there are conditions on the payoff matrix guaranteeing that the best reply player can replace the mixed ESS player. This is possible because best reply players in pairs, individually following their own selfish strategies, develop cycles where the bigger payoff can compensate their disadvantage compared with the ESS players. Well-mixed interaction is one of the basic assumptions of classical evolutionary matrix game theory. However, if the players repeat the game with certain probability, then they can react to their opponents’ behavior. Our main result is that the classical mixed ESS loses its general stability in the well-mixed population games with repetition in the sense that it can happen to be overrun by the BR player.

Trust-based partner switching among partitioned regions promotes cooperation in public goods game

In this paper, the coevolution mechanism of trust-based partner switching among partitioned regions on an adaptive network is studied. We investigate a low-information approach to building trust and cooperation in public goods games. Unlike reputation, trust scores are only given to players by those with whom they have a relationship in the game, depending on the game they play together. A player's trust score for a certain neighbor is given and known by that player only. Players can adjust their connections to neighbors with low trust scores by switching their partners to other players. When switching partners, players divide other nodes in the network into three regions: immediate neighbors as the known region, indirectly connected second-order neighbors as the intermediate region, and other nodes as the unknown region. Such choices and compartmentalization often occur in global and regional economies. Our results show that preference for switching to partners in the intermediate region is not conducive to spreading cooperation, while random selection has the disadvantage of protecting the cooperator. However, selecting new partners in the remaining two regions based on the average trust score of the known region performs well in both protecting partners and finding potential cooperators. Meanwhile, by analyzing the parameters, we find that the influence of vigilance increasing against unsatisfactory behavior on evolution direction depends on the level of cooperation reward.

Avoiding the bullies: The resilience of cooperation among unequals

Can egalitarian norms or conventions survive the presence of dominant individuals who are ensured of victory in conflicts? We investigate the interaction of power asymmetry and partner choice in games of conflict over a contested resource. Previous models of cooperation do not include both power inequality and partner choice. Furthermore, models that do include power inequalities assume a static game where a bully's advantage does not change. They have therefore not attempted to model complex and realistic properties of social interaction. Here, we introduce three models to study the emergence and resilience of cooperation among unequals when interaction is random, when individuals can choose their partners, and where power asymmetries dynamically depend on accumulated payoffs. We find that the ability to avoid bullies with higher competitive ability afforded by partner choice mostly restores cooperative conventions and that the competitive hierarchy never forms. Partner choice counteracts the hyper dominance of bullies who are isolated in the network and eliminates the need for others to coordinate in a coalition. When competitive ability dynamically depends on cumulative payoffs, complex cycles of coupled network-strategy-rank changes emerge. Effective collaborators gain popularity (and thus power), adopt aggressive behavior, get isolated, and ultimately lose power. Neither the network nor behavior converge to a stable equilibrium. Despite the instability of power dynamics, the cooperative convention in the population remains stable overall and long-term inequality is completely eliminated. The interaction between partner choice and dynamic power asymmetry is crucial for these results: without partner choice, bullies cannot be isolated, and without dynamic power asymmetry, bullies do not lose their power even when isolated. We analytically identify a single critical point that marks a phase transition in all three iterations of our models. This critical point is where the first individual breaks from the convention and cycles start to emerge.

A game-theoretical dynamic imitation model on networks

A game-theoretical model is constructed to capture the effect of imitation on the evolution of cooperation. This imitation describes the case where successful individuals are more likely to be imitated by newcomers who will employ their strategies and social networks. Two classical repeated strategies 'always defect (ALLD)' and 'tit-for-tat (TFT)' are adopted. Mathematical analyses are mainly conducted by the method of coalescence theory. Under the assumption of a large population size and weak selection, the results show that the evolution of cooperation is promoted in this dynamic network. As we observed that the critical benefit-to-cost ratio is smaller compared to that in well-mixed populations. The critical benefit-to-cost ratio approaches a specific value which depends on three parameters, the repeated rounds of the game, the effective strategy mutation rate, and the effective link mutation rate. Specifically, for a very high value of the effective link mutation rate, the critical benefit-to-cost ratio approaches 1. Remarkably, for a low value of the effective link mutation rate, by letting the effective strategy mutation is nearly equal to zero, the critical benefit-to-cost ratio approaches [Formula: see text] for the resulting highly connected networks, which allows TFT to be evolutionary stable. It illustrates that dominance of TFTs is associated with more connected networks. This research can enrich the theory of the coevolution of game strategy and network structure with dynamic imitation.

How strategy environment and wealth shape altruistic behaviour: cooperation rules affecting wealth distribution in dynamic networks

Societies rely on individual contributions to sustain public goods that benefit the entire community. Several mechanisms, that specify how individuals change their decisions based on past experiences, have been proposed to explain how altruists are not outcompeted by selfish counterparts. A key aspect of such strategy updates involves a comparison of an individual's latest payoff with that of a random neighbour. In reality, both the economic and social milieu often shapes cooperative behaviour. We propose a new decision heuristic, where the propensity of an individual to cooperate depends on the local strategy environment in which she is embedded as well as her wealth relative to that of her neighbours. Our decision-making model allows cooperation to be sustained and also explains the results of recent experiments on social dilemmas in dynamic networks. Final cooperation levels depend only on the extent to which the strategy environment influences altruistic behaviour but are largely unaffected by network restructuring. However, the extent of wealth inequality in the community is affected by a subtle interplay between the environmental influence on a person's decision to contribute and the likelihood of reshaping social ties, with wealth-inequality levels rising with increasing likelihood of network restructuring in some situations.

Two-strategy games with time constraints on regular graphs

Evolutionary game theory is a powerful method for modelling animal conflicts. The original evolutionary game models were used to explain specific biological features of interest, such as the existence of ritualised contests, and were necessarily simple models that ignored many properties of real populations, including the duration of events and spatial and related structural effects. Both of these areas have subsequently received much attention. Spatial and structural effects have been considered in evolutionary graph theory, and a significant body of literature has been built up to deal with situations where the population is not homogeneous. More recently a theory of time constraints has been developed to take account of the fact that different events can take different times, and that interaction times can explicitly depend upon selected strategies, which can, in turn, influence the distribution of different opponent types within the population. Here, for the first time, we build a model of time constraint games which explicitly considers a spatial population, by considering a population evolving on an underlying graph, using two graph dynamics, birth-death and death-birth. We consider one short time scale along which frequencies of pairs and singles change as individuals interact with their neighbours, and another, evolutionary time scale, along which frequencies of strategies change in the population. We show that for graphs with large degree, both dynamics reproduce recent results from well-mixed time constraint models, including two ESSs being common in Hawk-Dove and Prisoner's Dilemma games, but for low degree there can be marked differences. For birth-death processes the effect of the graph degree is small, whereas for death-birth dynamics there is a large effect. The general prediction for both Hawk-Dove and Prisoner's dilemma games is that as the graph degree decreases, i.e., as the number of neighbours decreases, mixed ESS do appear. In particular, for the Prisoner's dilemma game this means that cooperation is easier to establish in situations where individuals have low number of neighbours. We thus see that solutions depend non-trivially on the combination of graph degree, dynamics and game.

Evolution of cooperation in an epithelium

Cooperation is prevalent in nature, not only in the context of social interactions within the animal kingdom but also on the cellular level. In cancer, for example, tumour cells can cooperate by producing growth factors. The evolution of cooperation has traditionally been studied for well-mixed populations under the framework of evolutionary game theory, and more recently for structured populations using evolutionary graph theory (EGT). The population structures arising due to cellular arrangement in tissues, however, are dynamic and thus cannot be accurately represented by either of these frameworks. In this work, we compare the conditions for cooperative success in an epithelium modelled using EGT, to those in a mechanical model of an epithelium—the Voronoi tessellation (VT) model. Crucially, in this latter model, cells are able to move, and birth and death are not spatially coupled. We calculate fixation probabilities in the VT model through simulation and an approximate analytic technique and show that this leads to stronger promotion of cooperation in comparison with the EGT model.

Co-Evolution of Complex Network Public Goods Game under the Edges Rules

The reconnection of broken edges is an effective way to avoid drawback for the commons in past studies. Inspired by this, we proposed a public goods game model under the edges rules, where we evaluate the weight of edges by their nodes' payoff. The results proved that the game obtains a larger range of cooperation with a small gain factor by this proposed model by consulting Monte Carlo simulations (MCS) and real experiments. Furthermore, as the following the course of game and discussing the reason of cooperation, in the research, we found that the distribution entropy of the excess average degree is able to embody and predict the presence of cooperation.

The effect of network topology on optimal exploration strategies and the evolution of cooperation in a mobile population

We model a mobile population interacting over an underlying spatial structure using a Markov movement model. Interactions take the form of public goods games, and can feature an arbitrary group size. Individuals choose strategically to remain at their current location or to move to a neighbouring location, depending upon their exploration strategy and the current composition of their group. This builds upon previous work where the underlying structure was a complete graph (i.e. there was effectively no structure). Here, we consider alternative network structures and a wider variety of, mainly larger, populations. Previously, we had found when cooperation could evolve, depending upon the values of a range of population parameters. In our current work, we see that the complete graph considered before promotes stability, with populations of cooperators or defectors being relatively hard to replace. By contrast, the star graph promotes instability, and often neither type of population can resist replacement. We discuss potential reasons for this in terms of network topology.

Computation and Simulation of Evolutionary Game Dynamics in Finite Populations

The study of evolutionary dynamics increasingly relies on computational methods, as more and more cases outside the range of analytical tractability are explored. The computational methods for simulation and numerical approximation of the relevant quantities are diverging without being compared for accuracy and performance. We thoroughly investigate these algorithms in order to propose a reliable standard. For expositional clarity we focus on symmetric 2 × 2 games leading to one-dimensional processes, noting that extensions can be straightforward and lessons will often carry over to more complex cases. We provide time-complexity analysis and systematically compare three families of methods to compute fixation probabilities, fixation times and long-term stationary distributions for the popular Moran process. We provide efficient implementations that substantially improve wall times over naive or immediate implementations. Implications are also discussed for the Wright-Fisher process, as well as structured populations and multiple types.

A mathematical formalism for natural selection with arbitrary spatial and genetic structure

We define a general class of models representing natural selection between two alleles. The population size and spatial structure are arbitrary, but fixed. Genetics can be haploid, diploid, or otherwise; reproduction can be asexual or sexual. Biological events (e.g. births, deaths, mating, dispersal) depend in arbitrary fashion on the current population state. Our formalism is based on the idea of genetic sites. Each genetic site resides at a particular locus and houses a single allele. Each individual contains a number of sites equal to its ploidy (one for haploids, two for diploids, etc.). Selection occurs via replacement events, in which alleles in some sites are replaced by copies of others. Replacement events depend stochastically on the population state, leading to a Markov chain representation of natural selection. Within this formalism, we define reproductive value, fitness, neutral drift, and fixation probability, and prove relationships among them. We identify four criteria for evaluating which allele is selected and show that these become equivalent in the limit of low mutation. We then formalize the method of weak selection. The power of our formalism is illustrated with applications to evolutionary games on graphs and to selection in a haplodiploid population.

Coevolution of Environmental Perception and Cooperative Behavior in Evacuation Crowd

For the evacuation crowd of social agents, environment plays a big effect on the behavior and decision of the agents. When facing the uncertain environment, the behavior and decision of agents depend heavily on the perception of environment. Therefore, the cooperation between agents and their perception of environment may coexist during evacuation. Here we establish a mechanism to analyze the coevolution between the cooperation of agents and the perception of environment. In detail, we use a regular square lattice with periodic boundaries, where two payoff matrices are used to describe two kinds of games between neighbors in the safe and dangerous environments. For individual agent, its perception can be adjusted by interacting with neighboring agents. When the environment is generally considered dangerous, the fraction of cooperative agents keeps at a high level, even if the value of b is very large. When all the agents think that the environment is safe, the fraction of cooperation will decrease as the value of b increases.

Coevolution of nonlinear group interactions and strategies in well-mixed and structured populations

In microbial populations and human societies, the rule of nonlinear group interactions strongly affects the intraspecific evolutionary dynamics, which leads to the variation of the strategy composition eventually. The consequence of such variation may retroact to the rule of the interactions. This correlation indicates that the rule of nonlinear group interactions may coevolve with individuals' strategies. Here, we develop a model to investigate such coevolution in both well-mixed and structured populations. In our model, positive and negative correlations between the rule and the frequency of cooperators are considered, with local and global information. When the correlation refers to the global information, we show that in well-mixed populations, the coevolutionary outcomes cover the scenarios of defector dominance, coexistence, and bi-stability. Whenever the population structure is considered, its impact on the coevolutionary dynamics depends on the type of the correlation: with a negative (positive) correlation, population structure promotes (inhibits) the evolution of cooperation. Furthermore, when the correlation is based on the more accessible local information, we reveal that a negative correlation pushes cooperators into a harsh situation whereas a positive one lowers the barriers for cooperators to occupy the population. All our analytical results are validated by numerical simulations. Our results shed light on the power of the coevolution of nonlinear group interactions and evolutionary dynamics on generating various evolutionary outcomes, implying that the coevolutionary framework may be more appropriate than the traditional cases for understanding the evolution of cooperation in both structureless and structured populations.

Evolutionary prisoner's dilemma games coevolving on adaptive networks

We study a model for switching strategies in the Prisoner's Dilemma game on adaptive networks of player pairings that coevolve as players attempt to maximize their return. We use a node-based strategy model wherein each player follows one strategy at a time (cooperate or defect) across all of its neighbors, changing that strategy and possibly changing partners in response to local changes in the network of player pairing and in the strategies used by connected partners. We compare and contrast numerical simulations with existing pair approximation differential equations for describing this system, as well as more accurate equations developed here using the framework of approximate master equations. We explore the parameter space of the model, demonstrating the relatively high accuracy of the approximate master equations for describing the system observations made from simulations. We study two variations of this partner-switching model to investigate the system evolution, predict stationary states, and compare the total utilities and other qualitative differences between these two model variants.

Reinforcement learning account of network reciprocity

Evolutionary game theory predicts that cooperation in social dilemma games is promoted when agents are connected as a network. However, when networks are fixed over time, humans do not necessarily show enhanced mutual cooperation. Here we show that reinforcement learning (specifically, the so-called Bush-Mosteller model) approximately explains the experimentally observed network reciprocity and the lack thereof in a parameter region spanned by the benefit-to-cost ratio and the node’s degree. Thus, we significantly extend previously obtained numerical results.

Public goods games on adaptive coevolutionary networks

Productive societies feature high levels of cooperation and strong connections between individuals. Public Goods Games (PGGs) are frequently used to study the development of social connections and cooperative behavior in model societies. In such games, contributions to the public good are made only by cooperators, while all players, including defectors, reap public goods benefits, which are shares of the contributions amplified by a synergy factor. Classic results of game theory show that mutual defection, as opposed to cooperation, is the Nash Equilibrium of PGGs in well-mixed populations, where each player interacts with all others. In this paper, we explore the coevolutionary dynamics of a low information public goods game on a complex network in which players adapt to their environment in order to increase individual payoffs relative to past payoffs parameterized by greediness. Players adapt by changing their strategies, either to cooperate or to defect, and by altering their social connections. We find that even if players do not know other players' strategies and connectivity, cooperation can arise and persist despite large short-term fluctuations.

Multilevel Evolutionary Algorithm that Optimizes the Structure of Scale-Free Networks for the Promotion of Cooperation in the Prisoner's Dilemma game

Understanding the emergence of cooperation has long been a challenge across disciplines. Even if network reciprocity reflected the importance of population structure in promoting cooperation, it remains an open question how population structures can be optimized, thereby enhancing cooperation. In this paper, we attempt to apply the evolutionary algorithm (EA) to solve this highly complex problem. However, as it is hard to evaluate the fitness (cooperation level) of population structures, simply employing the canonical evolutionary algorithm (EA) may fail in optimization. Thus, we propose a new EA variant named mlEA-C_PD-SFN to promote the cooperation level of scale-free networks (SFNs) in the Prisoner’s Dilemma Game (PDG). Meanwhile, to verify the preceding conclusions may not be applied to this problem, we also provide the optimization results of the comparative experiment (EA_cluster), which optimizes the clustering coefficient of structures. Even if preceding research concluded that highly clustered scale-free networks enhance cooperation, we find EA_cluster does not perform desirably, while mlEA-C_PD-SFN performs efficiently in different optimization environments. We hope that mlEA-C_PD-SFN may help promote the structure of species in nature and that more general properties that enhance cooperation can be learned from the output structures.

Evolution of cooperative imitators in social networks

Many evolutionary game models for network reciprocity are based on an imitation dynamics, yet how semirational imitators prevail has seldom been explained. Here we use a model to investigate the coevolutionary dynamics of cooperation and partnership adjustment in a polygenic population of semirational imitators and rational payoff maximizers. A rational individual chooses a strategy best responding to its neighbors when updating strategy and switches to a new partner who can bring it the maximal payoff from all candidates when adjusting the partnership. In contrast, a semirational individual imitates its neighbor's strategy directly and adjusts its partnership based upon a simple reputation rule. Individual-based simulations show that cooperation cannot evolve in a population of all best responders even if they can switch their partners to somebody who can reward them best in game playing. However, when imitators exist, a stable community that consists of cooperative imitators emerges. Further, we show that a birth-death selection mechanism can eliminate all best responders, cultivating a social regime of all cooperative imitators. Compared with parallel simulations that assume fixed networks, cooperative imitators are evolutionarily favored, provided they are able to adjust their partners.

The ecology and evolution of social behavior in microbes

Cooperation has been studied extensively across the tree of life, from eusociality in insects to social behavior in humans, but it is only recently that a social dimension has been recognized and extensively explored for microbes. Research into microbial cooperation has accelerated dramatically and microbes have become a favorite system because of their fast evolution, their convenience as lab study systems and the opportunity for molecular investigations. However, the study of microbes also poses significant challenges, such as a lack of knowledge and an inaccessibility of the ecological context (used here to include both the abiotic and the biotic environment) under which the trait deemed cooperative has evolved and is maintained. I review the experimental and theoretical evidence in support of the limitations of the study of social behavior in microbes in the absence of an ecological context. I discuss both the need and the opportunities for experimental investigations that can inform a theoretical framework able to reframe the general questions of social behavior in a clear ecological context and to account for eco-evolutionary feedback.

Disease Spreading in Time-Evolving Networked Communities

Human communities are organized in complex webs of contacts that may be represented by a graph or network. In this graph, vertices identify individuals and edges establish the existence of some type of relations between them. In real communities, the possible edges may be active or not for variable periods of time. These so-called temporal networks typically result from an endogenous social dynamics, usually coupled to the process under study taking place in the community. For instance, disease spreading may be affected by local information that makes individuals aware of the health status of their social contacts, allowing them to reconsider maintaining or not their social contacts. Here we investigate the impact of such a dynamical network structure on disease dynamics, where infection occurs along the edges of the network. To this end, we define an endogenous network dynamics coupled with disease spreading. We show that the effective infectiousness of a disease taking place along the edges of this temporal network depends on the population size, the number of infected individuals in the population and the capacity of healthy individuals to sever contacts with the infected, ultimately dictated by availability of information regarding each individual’s health status. Importantly, we also show how dynamical networks strongly decrease the average time required to eradicate a disease.

Models of social evolution: can we do better to predict 'who helps whom to achieve what'?

Models of social evolution and the evolution of helping have been classified in numerous ways. Two categorical differences have, however, escaped attention in the field. Models tend not to justify why they use a particular assumption structure about who helps whom: a large number of authors model peer-to-peer cooperation of essentially identical individuals, probably for reasons of mathematical convenience; others are inspired by particular cooperatively breeding species, and tend to assume unidirectional help where subordinates help a dominant breed more efficiently. Choices regarding what the help achieves (i.e. which life-history trait of the helped individual is improved) are similarly made without much comment: fecundity benefits are much more commonly modelled than survival enhancements, despite evidence that these may interact when the helped individual can perform life-history reallocations (load-lightening and related phenomena). We review our current theoretical understanding of effects revealed when explicitly asking 'who helps whom to achieve what', from models of mutual aid in partnerships to the very few models that explicitly contrast the strength of selection to help enhance another individual's fecundity or survival. As a result of idiosyncratic modelling choices in contemporary literature, including the varying degree to which demographic consequences are made explicit, there is surprisingly little agreement on what types of help are predicted to evolve most easily. We outline promising future directions to fill this gap.

Stochastic win-stay-lose-shift strategy with dynamic aspirations in evolutionary social dilemmas

In times of plenty expectations rise, just as in times of crisis they fall. This can be mathematically described as a win-stay-lose-shift strategy with dynamic aspiration levels, where individuals aspire to be as wealthy as their average neighbor. Here we investigate this model in the realm of evolutionary social dilemmas on the square lattice and scale-free networks. By using the master equation and Monte Carlo simulations, we find that cooperators coexist with defectors in the whole phase diagram, even at high temptations to defect. We study the microscopic mechanism that is responsible for the striking persistence of cooperative behavior and find that cooperation spreads through second-order neighbors, rather than by means of network reciprocity that dominates in imitation-based models. For the square lattice the master equation can be solved analytically in the large temperature limit of the Fermi function, while for other cases the resulting differential equations must be solved numerically. Either way, we find good qualitative agreement with the Monte Carlo simulation results. Our analysis also reveals that the evolutionary outcomes are to a large degree independent of the network topology, including the number of neighbors that are considered for payoff determination on lattices, which further corroborates the local character of the microscopic dynamics. Unlike large-scale spatial patterns that typically emerge due to network reciprocity, here local checkerboard-like patterns remain virtually unaffected by differences in the macroscopic properties of the interaction network.

Evolutionary mixed games in structured populations: Cooperation and the benefits of heterogeneity

Evolutionary games on networks traditionally involve the same game at each interaction. Here we depart from this assumption by considering mixed games, where the game played at each interaction is drawn uniformly at random from a set of two different games. While in well-mixed populations the random mixture of the two games is always equivalent to the average single game, in structured populations this is not always the case. We show that the outcome is, in fact, strongly dependent on the distance of separation of the two games in the parameter space. Effectively, this distance introduces payoff heterogeneity, and the average game is returned only if the heterogeneity is small. For higher levels of heterogeneity the distance to the average game grows, which often involves the promotion of cooperation. The presented results support preceding research that highlights the favorable role of heterogeneity regardless of its origin, and they also emphasize the importance of the population structure in amplifying facilitators of cooperation.

Social niche construction and evolutionary transitions in individuality

Social evolution theory conventionally takes an externalist explanatory stance, treating observed cooperation as explanandum and the positive assortment of cooperative behaviour as explanans. We ask how the circumstances bringing about this positive assortment arose in the first place. Rather than merely push the explanatory problem back a step, we move from an externalist to an interactionist explanatory stance, in the spirit of Lewontin and the Niche Construction theorists. We develop a theory of 'social niche construction' in which we consider biological entities to be both the subject and object of their own social evolution. Some important cases of the evolution of cooperation have the side-effect of causing changes in the hierarchical level at which the evolutionary process acts. This is because the traits (e.g. life-history bottlenecks) that act to align the fitness interests of particles (e.g. cells) in a collective can also act to diminish the extent to which those particles are bearers of heritable fitness variance, while augmenting the extent to which collectives of such particles (e.g. multicellular organisms) are bearers of heritable fitness variance. In this way, we can explain upward transitions in the hierarchical level at which the Darwinian machine operates in terms of particle-level selection, even though the outcome of the process is a collective-level selection regime. Our theory avoids the logical and metaphysical paradoxes faced by other attempts to explain evolutionary transitions.

Benefits of tolerance in public goods games

Leaving the joint enterprise when defection is unveiled is always a viable option to avoid being exploited. Although loner strategy helps the population not to be trapped into the tragedy of the commons state, it could offer only a modest income for nonparticipants. In this paper we demonstrate that showing some tolerance toward defectors could not only save cooperation in harsh environments but in fact results in a surprisingly high average payoff for group members in public goods games. Phase diagrams and the underlying spatial patterns reveal the high complexity of evolving states where cyclic dominant strategies or two-strategy alliances can characterize the final state of evolution. We identify microscopic mechanisms which are responsible for the superiority of global solutions containing tolerant players. This phenomenon is robust and can be observed both in well-mixed and in structured populations highlighting the importance of tolerance in our everyday life.

Population Fluctuation Promotes Cooperation in Networks

We consider the problem of explaining the emergence and evolution of cooperation in dynamic network-structured populations. Building on seminal work by Poncela et al., which shows how cooperation (in one-shot prisoner's dilemma) is supported in growing populations by an evolutionary preferential attachment (EPA) model, we investigate the effect of fluctuations in the population size. We find that a fluctuating model - based on repeated population growth and truncation - is more robust than Poncela et al.'s in that cooperation flourishes for a wider variety of initial conditions. In terms of both the temptation to defect, and the types of strategies present in the founder network, the fluctuating population is found to lead more securely to cooperation. Further, we find that this model will also support the emergence of cooperation from pre-existing non-cooperative random networks. This model, like Poncela et al.'s, does not require agents to have memory, recognition of other agents, or other cognitive abilities, and so may suggest a more general explanation of the emergence of cooperation in early evolutionary transitions, than mechanisms such as kin selection, direct and indirect reciprocity.

Costs for switching partners reduce network dynamics but not cooperative behaviour

Social networks represent the structuring of interactions between group members. Above all, many interactions are profoundly cooperative in humans and other animals. In accordance with this natural observation, theoretical work demonstrates that certain network structures favour the evolution of cooperation. Yet, recent experimental evidence suggests that static networks do not enhance cooperative behaviour in humans. By contrast, dynamic networks do foster cooperation. However, costs associated with dynamism such as time or resource investments in finding and establishing new partnerships have been neglected so far. Here, we show that human participants are much less likely to break links when costs arise for building new links. Especially, when costs were high, the network was nearly static. Surprisingly, cooperation levels in Prisoner's Dilemma games were not affected by reduced dynamism in social networks. We conclude that the mere potential to quit collaborations is sufficient in humans to reach high levels of cooperative behaviour. Effects of self-structuring processes or assortment on the network played a minor role: participants simply adjusted their cooperative behaviour in response to the threats of losing a partner or of being expelled.

Neighbourhood reaction in the evolution of cooperation

Combining evolutionary games with adaptive networks, an entangled model between strategy evolution and structure adaptation is researched in this paper. We consider a large population of cooperators C and defectors D placed in the networks, playing the repeated prisoner׳s dilemma (PD) games. Because of the conflicts between social welfare and personal rationality, both strategy and structure are allowed to change. In this paper, the dynamics of strategy originates form the partner imitation based on social learning and the dynamics of structure is driven by the active linking and neighbourhood reaction. Notably, the neighbourhood reaction is investigated considering the changes of interfaces between cooperators and defectors, where some neighbours may get away from the interface once the focal agent changes to different strategy. A rich landscape is demonstrated by changing various embedding parameters, which sheds light upon that reacting promptly to the shifted neighbour will promote the prevalence of cooperation. Our model encapsulates the dynamics of strategy, reaction and structure into the evolutionary games, which manifests some intriguing principles in the competition between two groups in natural populations, artificial systems and even human societies.

The duality of spatial death-birth and birth-death processes and limitations of the isothermal theorem

Evolutionary models on graphs, as an extension of the Moran process, have two major implementations: birth-death (BD) models (or the invasion process) and death-birth (DB) models (or voter models). The isothermal theorem states that the fixation probability of mutants in a large group of graph structures (known as isothermal graphs, which include regular graphs) coincides with that for the mixed population. This result has been proved by Lieberman et al. (2005 Nature 433, 312-316. (doi:10.1038/nature03204)) in the case of BD processes, where mutants differ from the wild-types by their birth rate (and not by their death rate). In this paper, we discuss to what extent the isothermal theorem can be formulated for DB processes, proving that it only holds for mutants that differ from the wild-type by their death rate (and not by their birth rate). For more general BD and DB processes with arbitrary birth and death rates of mutants, we show that the fixation probabilities of mutants are different from those obtained in the mass-action populations. We focus on spatial lattices and show that the difference between BD and DB processes on one- and two-dimensional lattices is non-small even for large population sizes. We support these results with a generating function approach that can be generalized to arbitrary graph structures. Finally, we discuss several biological applications of the results.

The effects of reputational and social knowledge on cooperation

The emergence and sustenance of cooperative behavior is fundamental for a society to thrive. Recent experimental studies have shown that cooperation increases in dynamic networks in which subjects can choose their partners. However, these studies did not vary reputational knowledge, or what subjects know about other's past actions, which has long been recognized as an important factor in supporting cooperation. They also did not give subjects access to global social knowledge, or information on who is connected to whom in the group. As a result, it remained unknown how reputational and social knowledge foster cooperative behavior in dynamic networks both independently and by complementing each other. In an experimental setting, we show that global reputational knowledge is crucial to sustaining a high level of cooperation and welfare. Cooperation is associated with the emergence of dense and clustered networks with highly cooperative hubs. Global social knowledge has no effect on the aggregate level of cooperation. A community analysis shows that the addition of global social knowledge to global reputational knowledge affects the distribution of cooperative activity: cooperators form a separate community that achieves a higher cooperation level than the community of defectors. Members of the community of cooperators achieve a higher payoff from interactions within the community than members of the less cooperative community.