Evolution of cooperation on dynamical graphs

There are two key characteristic of animal and human societies: (1) degree heterogeneity, meaning that not all individual have the same number of associates; and (2) the interaction topology is not static, i.e. either individuals interact with different set of individuals at different times of their life, or at least they have different associations than their parents. Earlier works have shown that population structure is one of the mechanisms promoting cooperation. However, most studies had assumed that the interaction network can be described by a regular graph (homogeneous degree distribution). Recently there are an increasing number of studies employing degree heterogeneous graphs to model interaction topology. But mostly the interaction topology was assumed to be static. Here we investigate the fixation probability of the cooperator strategy in the prisoner's dilemma, when interaction network is a random regular graph, a random graph or a scale-free graph and the interaction network is allowed to change.     

A game theoretical model of deforestation in human-environment relationships

We studied a two-person game regarding deforestation in human-environment relationships. Each landowner manages a single land parcel where the state of land-use is forested, agricultural, or abandoned. The landowner has two strategies available: forest conservation and deforestation. The choice of deforestation provides a high return to the landowner, but it degrades the forest ecosystem services produced on a neighboring land parcel managed by a different landowner. Given spatial interactions between the two landowners, each landowner decides which strategy to choose by comparing the expected discounted utility of each strategy. Expected discounted utility is determined by taking into account the current and future utilities to be received, according to the state transition on the two land parcels. The state transition is described by a Markov chain that incorporates a landowner's choice about whether to deforest and the dynamics of agricultural abandonment and forest regeneration. By considering a stationary distribution of the Markov chain for land-use transitions, we derive explicit conditions for Nash equilibrium. We found that a slow regeneration of forests favors mutual cooperation (forest conservation). As the forest regenerates faster, mutual cooperation transforms to double Nash equilibria (mutual cooperation and mutual defection), and finally mutual defection (deforestation) leads to a unique Nash equilibrium. Two different types of social dilemma emerge in our deforestation game. The stag-hunt dilemma is most likely to occur under an unsustainable resource supply, where forest regenerates extremely slowly but agricultural abandonment happens quite rapidly. In contrast, the prisoner's dilemma is likely under a persistent or circulating supply of resources, where forest regenerates rapidly and agricultural abandonment occurs slowly or rapidly. These results show how humans and the environment mutually shape the dilemma structure in forest management, implying that solutions to dilemmas depend on environmental properties.     

Evolution of cooperation with shared costs and benefits

The quest to determine how cooperation evolves can be based on evolutionary game theory, in spite of the fact that evolutionarily stable strategies (ESS) for most non-zero-sum games are not cooperative. We analyse the evolution of cooperation for a family of evolutionary games involving shared costs and benefits with a continuum of strategies from non-cooperation to total cooperation. This cost-benefit game allows the cooperator to share in the benefit of a cooperative act, and the recipient to be burdened with a share of the cooperator's cost. The cost-benefit game encompasses the Prisoner's Dilemma, Snowdrift game and Partial Altruism. The models produce ESS solutions of total cooperation, partial cooperation, non-cooperation and coexistence between cooperation and non-cooperation. Cooperation emerges from an interplay between the nonlinearities in the cost and benefit functions. If benefits increase at a decelerating rate and costs increase at an accelerating rate with the degree of cooperation, then the ESS has an intermediate level of cooperation. The game also exhibits non-ESS points such as unstable minima, convergent-stable minima and unstable maxima. The emergence of cooperative behaviour in this game represents enlightened self-interest, whereas non-cooperative solutions illustrate the Tragedy of the Commons. Games having either a stable maximum or a stable minimum have the property that small changes in the incentive structure (model parameter values) or culture (starting frequencies of strategies) result in correspondingly small changes in the degree of cooperation. Conversely, with unstable maxima or unstable minima, small changes in the incentive structure or culture can result in a switch from non-cooperation to total cooperation (and vice versa). These solutions identify when human or animal societies have the potential for cooperation and whether cooperation is robust or fragile.     

The importance of mechanisms for the evolution of cooperation

Studies aimed at explaining the evolution of phenotypic traits have often solely focused on fitness considerations, ignoring underlying mechanisms. In recent years, there has been an increasing call for integrating mechanistic perspectives in evolutionary considerations, but it is not clear whether and how mechanisms affect the course and outcome of evolution. To study this, we compare four mechanistic implementations of two well-studied models for the evolution of cooperation, the Iterated Prisoner''s Dilemma (IPD) game and the Iterated Snowdrift (ISD) game. Behavioural strategies are either implemented by a 1 : 1 genotype–phenotype mapping or by a simple neural network. Moreover, we consider two different scenarios for the effect of mutations. The same set of strategies is feasible in all four implementations, but the probability that a given strategy arises owing to mutation is largely dependent on the behavioural and genetic architecture. Our individual-based simulations show that this has major implications for the evolutionary outcome. In the ISD, different evolutionarily stable strategies are predominant in the four implementations, while in the IPD each implementation creates a characteristic dynamical pattern. As a consequence, the evolved average level of cooperation is also strongly dependent on the underlying mechanism. We argue that our findings are of general relevance for the evolution of social behaviour, pleading for the integration of a mechanistic perspective in models of social evolution.     

活跃连接下Wright-Fisher过程的演化动态

在活跃连接下讨论了2×2对称博弈中频率相依Wright-Fisher过程的演化动力学,得到弱选择下策略的固定概率.利用固定概率讨论囚徒困境与雪堆博弈的演化动态,给出促使合作行为的条件.最后对所得结果进行数值模拟,发现活跃连接可以有效地促进合作行为.     

Stochastic evolutionary dynamics of direct reciprocity

Evolutionary game theory is the study of frequency-dependent selection. The success of an individual depends on the frequencies of strategies that are used in the population. We propose a new model for studying evolutionary dynamics in games with a continuous strategy space. The population size is finite. All members of the population use the same strategy. A mutant strategy is chosen from some distribution over the strategy space. The fixation probability of the mutant strategy in the resident population is calculated. The new mutant takes over the population with this probability. In this case, the mutant becomes the new resident. Otherwise, the existing resident remains. Then, another mutant is generated. These dynamics lead to a stationary distribution over the entire strategy space. Our new approach generalizes classical adaptive dynamics in three ways: (i) the population size is finite; (ii) mutants can be drawn non-locally and (iii) the dynamics are stochastic. We explore reactive strategies in the repeated Prisoner''s Dilemma. We perform ‘knock-out experiments’ to study how various strategies affect the evolution of cooperation. We find that ‘tit-for-tat’ is a weak catalyst for the emergence of cooperation, while ‘always cooperate’ is a strong catalyst for the emergence of defection. Our analysis leads to a new understanding of the optimal level of forgiveness that is needed for the evolution of cooperation under direct 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.     

The face of another: anonymity and facial symmetry influence cooperation in social dilemmas

In the present paper, we study how the morphological features related to developmental physiology of other participants influence the decision to cooperate in a social dilemma. To that end, we let a large sample of men play a prisoner's dilemma game, both anonymously and against a series of counterparts whose photographs were shown. We focus on three characteristics already linked to cooperative behavior and with described effects on the adult facial shape: facial fluctuating asymmetry, a frequently employed but debated proxy for developmental instability; the degree of facial dimorphism, related to levels of testosterone during puberty; and the second to fourth digit ratio, related to relative prenatal testosterone exposure. We find significantly higher cooperation rates in anonymous interactions than in the non-anonymous round. We also find that individuals are more likely to cooperate with more asymmetric counterparts, and that more asymmetric participants were less likely to believe that their counterpart would cooperate. Variables related to exposure to testosterone during development do not display any effect. We conclude by discussing how out-of-the-game rewards can explain our results.     

Effects of increasing the number of players and memory size in the iterated Prisoner's Dilemma: a numerical approach

The Prisoner''s Dilemma has become a paradigm for the evolution of altruistic behaviour. Here we present results of numerical simulations of the infinitely iterated stochastic simultaneous Prisoner''s Dilemma considering players with longer memory, encounters of more than two players as well as different pay-off values. This provides us with a better foundation to compare theoretical results to experimental data. We show that the success of the strategy Pavlov, regardless of its simplicity, is far more general by having an outstanding role in the iterated N-player N-memory Prisoner''s Dilemma. Besides, we study influences of increased memory sizes in the iterated two-player Prisoner''s Dilemma, and present comparisons to results of experiments with first-year students.     

A quantum probability explanation for violations of ‘rational’ decision theory

Two experimental tasks in psychology, the two-stage gambling game and the Prisoner''s Dilemma game, show that people violate the sure thing principle of decision theory. These paradoxical findings have resisted explanation by classical decision theory for over a decade. A quantum probability model, based on a Hilbert space representation and Schrödinger''s equation, provides a simple and elegant explanation for this behaviour. The quantum model is compared with an equivalent Markov model and it is shown that the latter is unable to account for violations of the sure thing principle. Accordingly, it is argued that quantum probability provides a better framework for modelling human decision-making.     

The excuse principle can maintain cooperation through forgivable defection in the Prisoner's Dilemma game

Reciprocal altruism describes a situation in which an organism acts in a manner that temporarily reduces its fitness while increasing another organism''s fitness, but there is an ultimate fitness benefit based on an expectation that the other organism will act in a similar manner at a later time. It creates the obvious dilemma in which there is always a short-term benefit to cheating, therefore cooperating individuals must avoid being exploited by non-cooperating cheaters. This is achieved by following various decision rules, usually variants of the tit-for-tat (TFT) strategy. The strength of TFT, however, is also its weakness—mistakes in implementation or interpretation of moves, or the inability to cooperate, lead to a permanent breakdown in cooperation. We show that pied flycatchers (Ficedula hypoleuca) use a TFT with an embedded ‘excuse principle’ to forgive the neighbours that were perceived as unable to cooperate during mobbing of predators. The excuse principle dramatically increases the stability of TFT-like behavioural strategies within the Prisoner''s Dilemma game.     

Individual variation and the resolution of conflict over parental care in penduline tits

Eurasian penduline tits (Remiz pendulinus) have an unusually diverse breeding system consisting of frequent male and female polygamy, and uniparental care by the male or the female. Intriguingly, 30 to 40 per cent of all nests are deserted by both parents. To understand the evolution of this diverse breeding system and frequent clutch desertion, we use 6 years of field data to derive fitness expectations for males and females depending on whether or not they care for their offspring. The resulting payoff matrix corresponds to an asymmetric Snowdrift Game with two alternative evolutionarily stable strategies (ESSs): female-only and male-only care. This, however, does not explain the polymorphism in care strategies and frequent biparental desertion, because theory predicts that one of the two ESSs should have spread to fixation. Using a bootstrapping approach, we demonstrate that taking account of individual variation in payoffs explains the patterns of care better than a model based on the average population payoff matrix. In particular, a model incorporating differences in male attractiveness closely predicts the observed frequencies of male and female desertion. Our work highlights the need for a new generation of individual-based evolutionary game-theoretic models.     

Female cleaner fish cooperate more with unfamiliar males

Joint group membership is of major importance for cooperation in humans, and close ties or familiarity with a partner are also thought to promote cooperation in other animals. Here, we present the opposite pattern: female cleaner fish, Labroides dimidiatus, behave more cooperatively (by feeding more against their preference) when paired with an unfamiliar male rather than with their social partner. We propose that cooperation based on asymmetric punishment causes this reversed pattern. Males are larger than and dominant to female partners and are more aggressive to unfamiliar than to familiar female partners. In response, females behave more cooperatively with unfamiliar male partners. Our data suggest that in asymmetric interactions, weaker players might behave more cooperatively with out-group members than with in-group members to avoid harsher punishment.     

A behavioral analysis of altruism

Altruistic acts have been defined, in economic terms, as “…costly acts that confer economic benefits on other individuals” (Fehr and Fischbacher, 2003). In multi-player, one-shot prisoner's dilemma games, a significant number of players behave altruistically; their behavior benefits each of the other players but is costly to them. We consider three potential explanations for such altruism. The first explanation, following a suggestion by the philosopher Derek Parfit, assumes that players devise a strategy to avoid being free-loaders—and that in the present case this strategy dictates cooperation. The second explanation says that cooperators reject the one-shot aspect of the game and behave so as to maximize reward over a series of choices extending beyond the present situation (even though reward is not maximized in the present case). This explanation assumes that people may learn to extend the boundaries of their selves socially (beyond their own skin) as well as temporally (beyond the present moment). We propose a learning mechanism for such behavior analogous to the biological, evolutionary mechanism of group selection. The third explanation assumes that people's altruism is based on a straightforward balancing of undiscounted costs to themselves against discounted benefits to others (social discounting). The three proposed explanations of altruism complement each other.     

Evolving learning rules and emergence of cooperation in spatial prisoner's dilemma

In the evolutionary Prisoner's dilemma (PD) game, agents play with each other and update their strategies in every generation according to some microscopic dynamical rule. In its spatial version, agents do not play with every other but, instead, interact only with their neighbours, thus mimicking the existing of a social or contact network that defines who interacts with whom. In this work, we explore evolutionary, spatial PD systems consisting of two types of agents, each with a certain update (reproduction, learning) rule. We investigate two different scenarios: in the first case, update rules remain fixed for the entire evolution of the system; in the second case, agents update both strategy and update rule in every generation. We show that in a well-mixed population the evolutionary outcome is always full defection. We subsequently focus on two-strategy competition with nearest-neighbour interactions on the contact network and synchronised update of strategies. Our results show that, for an important range of the parameters of the game, the final state of the system is largely different from that arising from the usual setup of a single, fixed dynamical rule. Furthermore, the results are also very different if update rules are fixed or evolve with the strategies. In these respect, we have studied representative update rules, finding that some of them may become extinct while others prevail. We describe the new and rich variety of final outcomes that arise from this co-evolutionary dynamics. We include examples of other neighbourhoods and asynchronous updating that confirm the robustness of our conclusions. Our results pave the way to an evolutionary rationale for modelling social interactions through game theory with a preferred set of update rules.