Evolutionary consequences of behavioral diversity

The effect of environmental information on evolution of cooperation in stochastic games

Many human interactions feature the characteristics of social dilemmas where individual actions have consequences for the group and the environment. The feedback between behavior and environment can be studied with the framework of stochastic games. In stochastic games, the state of the environment can change, depending on the choices made by group members. Past work suggests that such feedback can reinforce cooperative behaviors. In particular, cooperation can evolve in stochastic games even if it is infeasible in each separate repeated game. In stochastic games, participants have an interest in conditioning their strategies on the state of the environment. Yet in many applications, precise information about the state could be scarce. Here, we study how the availability of information (or lack thereof) shapes evolution of cooperation. Already for simple examples of two state games we find surprising effects. In some cases, cooperation is only possible if there is precise information about the state of the environment. In other cases, cooperation is most abundant when there is no information about the state of the environment. We systematically analyze all stochastic games of a given complexity class, to determine when receiving information about the environment is better, neutral, or worse for evolution of cooperation.

Mutation enhances cooperation in direct reciprocity

Direct reciprocity is a powerful mechanism for the evolution of cooperation based on repeated interactions between the same individuals. But high levels of cooperation evolve only if the benefit-to-cost ratio exceeds a certain threshold that depends on memory length. For the best-explored case of one-round memory, that threshold is two. Here, we report that intermediate mutation rates lead to high levels of cooperation, even if the benefit-to-cost ratio is only marginally above one, and even if individuals only use a minimum of past information. This surprising observation is caused by two effects. First, mutation generates diversity which undermines the evolutionary stability of defectors. Second, mutation leads to diverse communities of cooperators that are more resilient than homogeneous ones. This finding is relevant because many real-world opportunities for cooperation have small benefit-to-cost ratios, which are between one and two, and we describe how direct reciprocity can attain cooperation in such settings. Our result can be interpreted as showing that diversity, rather than uniformity, promotes evolution of cooperation.

Group reciprocity and the evolution of stereotyping

Stereotypes are generalized beliefs about groups of people, which are used to make decisions and judgements about them. Although such heuristics can be useful when decisions must be made quickly, or when information is lacking, they can also serve as the basis for prejudice and discrimination. In this paper, we study the evolution of stereotypes through group reciprocity. We characterize the warmth of a stereotype as the willingness to cooperate with an individual based solely on the identity of the group they belong to. We show that when stereotype groups are large, such group reciprocity is less likely to evolve, and stereotypes tend to be negative. We also show that, even when stereotypes are broadly positive, individuals are often overly pessimistic about the willingness of those they stereotype to cooperate. We then show that the tendency for stereotyping itself to evolve is driven by the costs of cognition, so that more people are stereotyped with greater coarseness as costs increase. Finally we show that extrinsic 'shocks', in which the benefits of cooperation are suddenly reduced, can cause stereotype warmth and judgement bias to turn sharply negative, consistent with the view that economic and other crises are drivers of out-group animosity.

Evolution of direct reciprocity in group-structured populations

People tend to have their social interactions with members of their own community. Such group-structured interactions can have a profound impact on the behaviors that evolve. Group structure affects the way people cooperate, and how they reciprocate each other's cooperative actions. Past work has shown that population structure and reciprocity can both promote the evolution of cooperation. Yet the impact of these mechanisms has been typically studied in isolation. In this work, we study how the two mechanisms interact. Using a game-theoretic model, we explore how people engage in reciprocal cooperation in group-structured populations, compared to well-mixed populations of equal size. In this model, the population is subdivided into groups. Individuals engage in pairwise interactions within groups while they also have chances to imitate strategies outside the groups. To derive analytical results, we focus on two scenarios. In the first scenario, we assume a complete separation of time scales. Mutations are rare compared to between-group comparisons, which themselves are rare compared to within-group comparisons. In the second scenario, there is a partial separation of time scales, where mutations and between-group comparisons occur at a comparable rate. In both scenarios, we find that the effect of population structure depends on the benefit of cooperation. When this benefit is small, group-structured populations are more cooperative. But when the benefit is large, well-mixed populations result in more cooperation. Overall, our results reveal how group structure can sometimes enhance and sometimes suppress the evolution of cooperation.

Adapting paths against zero-determinant strategies in repeated prisoner's dilemma games

Long-term cooperation, competition, or exploitation among individuals can be modeled through repeated games. In repeated games, Press and Dyson discovered zero-determinant (ZD) strategies that enforce a special relationship between two players. This special relationship implies that a ZD player can unilaterally impose a linear payoff relationship to the opponent regardless of the opponent's strategies. A ZD player also has a property that can lead the opponent to an unconditional cooperation if the opponent tries to improve its payoff. This property has been mathematically confirmed by Chen and Zinger. Humans often underestimate a payoff obtained in the future. However, such discounting was not considered in their analysis. Here, we mathematically explored whether a ZD player can lead the opponent to an unconditional cooperation even if a discount factor is incorporated. Specifically, we represented the expected payoff with a discount factor as the form of determinants and calculated whether the values obtained by partially differentiating each factor in the strategy vector become positive. As a result, we proved that the strategy vector ends up as an unconditional cooperation even when starting from any initial strategy. This result was confirmed through numerical calculations. We extended the applicability of ZD strategies to real world problems.

Criticality and Griffiths phases in random games with quenched disorder

The perceived risk and reward for a given situation can vary depending on resource availability, accumulated wealth, and other extrinsic factors such as individual backgrounds. Based on this general aspect of everyday life, here we use evolutionary game theory to model a scenario with randomly perturbed payoffs in a prisoner's dilemma game. The perception diversity is modeled by adding a zero-average random noise in the payoff entries and a Monte Carlo simulation is used to obtain the population dynamics. This payoff heterogeneity can promote and maintain cooperation in a competitive scenario where only defectors would survive otherwise. In this work, we give a step further, understanding the role of heterogeneity by investigating the effects of quenched disorder in the critical properties of random games. We observe that payoff fluctuations induce a very slow dynamic, making the cooperation decay behave as power laws with varying exponents, instead of the usual exponential decay after the critical point, showing the emergence of a Griffiths phase. We also find a symmetric Griffiths phase near the defector's extinction point when fluctuations are present, indicating that Griffiths phases may be frequent in evolutionary game dynamics and play a role in the coexistence of different strategies.

A Dual Role for Behavior in Evolution and Shaping Organismal Selective Environments

The hypothesis that evolved behaviors play a determining role in facilitating and impeding the evolution of other traits has been discussed for more than 100 years with little consensus beyond an agreement that the ideas are theoretically plausible in accord with the Modern Synthesis. Many recent reviews of the genomic, epigenetic, and developmental mechanisms underpinning major behavioral transitions show how facultative expression of novel behaviors can lead to the evolution of obligate behaviors and structures that enhance behavioral function. Phylogenetic and genomic studies indicate that behavioral traits are generally evolutionarily more labile than other traits and that they help shape selective environments on the latter traits. Adaptive decision-making to encounter resources and avoid stress sources requires specific sensory inputs, which behaviorally shape selective environments by determining those features of the external world that are biologically relevant. These recent findings support the hypothesis of a dual role for behavior in evolution and are consistent with current evolutionary theory.

Conditions for the existence of zero-determinant strategies under observation errors in repeated games

Repeated games are useful models to analyze long term interactions of living species and complex social phenomena. Zero-determinant (ZD) strategies in repeated games discovered by Press and Dyson in 2012 enforce a linear payoff relationship between a focal player and the opponent. This linear relationship can be set arbitrarily by a ZD player. Hence, a subclass of ZD strategies can fix the opponent's expected payoff and another subclass of the strategies can exceed the opponent for the expected payoff. Since this discovery, theories for ZD strategies are extended to cope with various natural situations. It is especially important to consider the theory of ZD strategies for repeated games with a discount factor and observation errors because it allows the theory to be applicable in the real world. Recent studies revealed their existence of ZD strategies even in repeated games with both factors. However, the conditions for the existence has not been sufficiently analyzed. Here, we mathematically analyzed the conditions in repeated games with both factors. First, we derived the thresholds of a discount factor and observation errors which ensure the existence of Equalizer and positively correlated ZD (pcZD) strategies, which are well-known subclasses of ZD strategies. We found that ZD strategies exist only when a discount factor remains high as the error rates increase. Next, we derived the conditions for the expected payoff of the opponent enforced by Equalizer as well as the conditions for the slope and base line payoff of linear lines enforced by pcZD. As a result, we found that, as error rates increase or a discount factor decreases, the conditions for the linear line that Equalizer or pcZD can enforce become strict.

Symbiotic behaviour in the public goods game with altruistic punishment

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) are driven to extinction by defectors for most parameter values; ii) clusters of punishers thrive by sharing the punishment costs when these are low; iii) for higher punishment costs, punishers, when alone, are subject to exploitation but in the presence of cooperators can form a symbiotic spatial structure that benefits both. This last observation is our main finding since neither cooperation nor punishment alone can survive the defector strategy in this parameter region and the specificity of the symbiotic spatial configuration shows that lattice topology plays a central role in sustaining cooperation. Results were obtained by means of Monte Carlo simulations on a square lattice and subsequently confirmed by a pairwise comparison of different strategies' payoffs in diverse group compositions, leading to a phase diagram of the possible states.

Zero-determinant strategies under observation errors in repeated games

Zero-determinant (ZD) strategies are a novel class of strategies in the repeated prisoner's dilemma (RPD) game discovered by Press and Dyson. This strategy set enforces a linear payoff relationship between a focal player and the opponent regardless of the opponent's strategy. In the RPD game, games with discounting and observation errors represent an important generalization, because they are better able to capture real life interactions which are often noisy. However, they have not been considered in the original discovery of ZD strategies. In some preceding studies, each of them has been considered independently. Here, we analytically study the strategies that enforce linear payoff relationships in the RPD game considering both a discount factor and observation errors. As a result, we first reveal that the payoffs of two players can be represented by the form of determinants as shown by Press and Dyson even with the two factors. Then, we search for all possible strategies that enforce linear payoff relationships and find that both ZD strategies and unconditional strategies are the only strategy sets to satisfy the condition. We also show that neither Extortion nor Generous strategies, which are subsets of ZD strategies, exist when there are errors. Finally, we numerically derive the threshold values above which the subsets of ZD strategies exist. These results contribute to a deep understanding of ZD strategies in society.

Strategy equilibrium in dilemma games with off-diagonal payoff perturbations

We analyze the strategy equilibrium of dilemma games considering a payoff matrix affected by small and random perturbations on the off-diagonal. Notably, a recent work [Proc. R. Soc. A 476, 20200116 (2020)1364-502110.1098/rspa.2020.0116] reported that while cooperation is sustained by perturbations acting on the main diagonal, a less clear scenario emerges when perturbations act on the off-diagonal. Thus, the second case represents the core of this investigation, aimed at completing the description of the effects that payoff perturbations have on the dynamics of evolutionary games. Our results, achieved by analyzing the proposed model under a variety of configurations as different update rules, suggest that off-diagonal perturbations actually constitute a nontrivial form of noise. In particular, the most interesting effects are detected near the phase transition, as perturbations tend to move the strategy distribution towards nonordered states of equilibrium, supporting cooperation when defection is pervading the population, and supporting defection in the opposite case. To conclude, we identified a form of noise that, under controlled conditions, could be used to enhance cooperation and greatly delay its extinction.

The Evolution of Cooperation in One-Dimensional Mobile Populations with Deterministic Dispersal

I investigate how different dispersal patterns affect the evolution of cooperation in a spatially-structured population. I consider a finite fixed-size population of cooperators and free-riders residing on a one-dimensional lattice with periodic boundaries. Individuals interact via a multiplayer game, which is a version of a public goods game, and the population evolves via a Moran process. Individuals try to improve their interactions by evaluating the current state of the environment and moving to locations with better payoffs. I ran stochastic simulations of the evolution of this Markov process and found that if individuals disperse deterministically to locations with the best payoffs, then cooperation can still be maintained even in the worst-case scenarios, albeit at reduced levels compared to the better-case scenarios. This contrasts with an earlier investigation of probabilistic dispersal patterns, which resulted in the breakdown of cooperation in sparse populations with small interaction neighborhoods, a high mobility rate, and a large dispersal range.

Dynamics of Cooperation in Minority Games in Alliance Networks

Alliance networks are the underlying structures of social systems in business, management, and society. The sustainability and dynamics of a social system rely on the structural evolutions of the topologies. Understanding the evolution sheds light on the dynamics and sustainability of a social system. Minority game models have been successfully applied across social science, economy, management, and engineering. They provide simple yet applicable modeling to articulate the evolutionary cooperation dynamics of competitive players in binary decision situations. By extending the minority games played in alliance networks, the cooperation in structured systems of different network topologies is analyzed. In this model, local and global score strategies are considered with and without cooperation rewiring options. The cooperation level, the score, and the topological properties are investigated. The research uses a numerical simulation approach on random networks, scale-free networks, and small-world networks. The results suggest that the network rewiring strategy leads to higher systemic performance with a higher score and a higher level of stability in decision-making. Competitive decision-making can lead to a higher level of cooperation from a poor initial start. However, stubbornness in decision-making can lead to a poor situation when cooperation is discouraged. Players with local or global information adopt local and global score strategies. The results show that local strategies might lead to imbalance, while a global strategy might achieve a relatively stable outcome. This work contributes to bridge minority games in structured networks to study the cooperation between formation and evolution, and calls for future minority game modeling on social networks.

Indirect reciprocity with private, noisy, and incomplete information

Indirect reciprocity is a mechanism for cooperation based on shared moral systems and individual reputations. It assumes that members of a community routinely observe and assess each other and that they use this information to decide who is good or bad, and who deserves cooperation. When information is transmitted publicly, such that all community members agree on each other's reputation, previous research has highlighted eight crucial moral systems. These "leading-eight" strategies can maintain cooperation and resist invasion by defectors. However, in real populations individuals often hold their own private views of others. Once two individuals disagree about their opinion of some third party, they may also see its subsequent actions in a different light. Their opinions may further diverge over time. Herein, we explore indirect reciprocity when information transmission is private and noisy. We find that in the presence of perception errors, most leading-eight strategies cease to be stable. Even if a leading-eight strategy evolves, cooperation rates may drop considerably when errors are common. Our research highlights the role of reliable information and synchronized reputations to maintain stable moral systems.

Social norm complexity and past reputations in the evolution of cooperation

Indirect reciprocity is the most elaborate and cognitively demanding of all known cooperation mechanisms, and is the most specifically human because it involves reputation and status. By helping someone, individuals may increase their reputation, which may change the predisposition of others to help them in future. The revision of an individual's reputation depends on the social norms that establish what characterizes a good or bad action and thus provide a basis for morality. Norms based on indirect reciprocity are often sufficiently complex that an individual's ability to follow subjective rules becomes important, even in models that disregard the past reputations of individuals, and reduce reputations to either 'good' or 'bad' and actions to binary decisions. Here we include past reputations in such a model and identify the key pattern in the associated norms that promotes cooperation. Of the norms that comply with this pattern, the one that leads to maximal cooperation (greater than 90 per cent) with minimum complexity does not discriminate on the basis of past reputation; the relative performance of this norm is particularly evident when we consider a 'complexity cost' in the decision process. This combination of high cooperation and low complexity suggests that simple moral principles can elicit cooperation even in complex environments.

Single equalizer strategy with no information transfer for conflict escalation

In an iterated two-person game, for instance prisoner's dilemma or the snowdrift game, there exist strategies that force the payoffs of the opponents to be equal. These equalizer strategies form a subset of the more general zero-determinant strategies that unilaterally set the payoff of an opponent. A challenge in the attempts to understand the role of these strategies in the evolution of animal behavior is the lack of iterations in the fights for mating opportunities or territory control. We show that an arbitrary two-parameter strategy may possess a corresponding equalizer strategy which produces the same result: statistics of the fight outcomes in the contests with competitors are the same for each of these two strategies. Therefore, analyzing only the equalizer strategy space may be sufficient to predict animal behavior if nature, indeed, reduces (marginalizes) complex strategies to equalizer strategy space. The work's main finding is that there is a unique equalizer strategy that predicts fight outcomes without symmetric cooperation responses. The lack of symmetric cooperation responses is a common trait in conflict escalation contests that generally require a clear winner. In addition, this unique strategy does not assess information of the opponent's state. The method bypasses the standard analysis of evolutionary stability. The results fit well the observations of combat between male bowl and doily spiders and support an empirical assumption of the war of attrition model that the species use only information regarding their own state during conflict escalation.

Partners and rivals in direct reciprocity

Reciprocity is a major factor in human social life and accounts for a large part of cooperation in our communities. Direct reciprocity arises when repeated interactions occur between the same individuals. The framework of iterated games formalizes this phenomenon. Despite being introduced more than five decades ago, the concept keeps offering beautiful surprises. Recent theoretical research driven by new mathematical tools has proposed a remarkable dichotomy among the crucial strategies: successful individuals either act as partners or as rivals. Rivals strive for unilateral advantages by applying selfish or extortionate strategies. Partners aim to share the payoff for mutual cooperation, but are ready to fight back when being exploited. Which of these behaviours evolves depends on the environment. Whereas small population sizes and a limited number of rounds favour rivalry, partner strategies are selected when populations are large and relationships stable. Only partners allow for evolution of cooperation, while the rivals' attempt to put themselves first leads to defection.

Evolution of cooperation with interactive identity and diversity

Interactive identity and interactive diversity are generally regarded as two typical interaction patterns in living systems. The former describes that in each generation every individual behaves identically to all of its opponents, and the latter allows each individual to behave diversely to its distinct opponents. Most traditional research on the evolution of cooperation, however, has been confined to populations with a uniform interaction pattern. Here we study the cooperation conundrum in a diverse population comprising players with interactive identity and with interactive diversity. We find that in homogeneous networks a small fraction of players taking interactive diversity are enough to stabilize cooperation for a wide range of payoff values even in a noisy environment. When assigned to heterogeneous networks, players in high-degree nodes taking interactive diversity significantly strengthen systems' resilience against the shifty environment and enlarge the survival region of cooperation. However, they fail to establish a homogeneous strategy 'cloud' in the neighborhood and thus can not coordinate players in low-degree nodes to reach a socially optimal cooperation level. The most favorable outcome emerges when players in high-degree nodes take interactive identity and meanwhile others adopt interactive diversity. Our findings reveal the significance of the two typical interaction patterns and could be a good heuristic in coordinating them to achieve the social optimum in cooperation.

Evolutionary dynamics of group formation

Group formation is a quite ubiquitous phenomenon across different animal species, whose individuals cluster together forming communities of diverse size. Previous investigations suggest that, in general, this phenomenon might have similar underlying reasons across the interested species, despite genetic and behavioral differences. For instance improving the individual safety (e.g. from predators), and increasing the probability to get food resources. Remarkably, the group size might strongly vary from species to species, e.g. shoals of fishes and herds of lions, and sometimes even within the same species, e.g. tribes and families in human societies. Here we build on previous theories stating that the dynamics of group formation may have evolutionary roots, and we explore this fascinating hypothesis from a purely theoretical perspective, with a model using the framework of Evolutionary Game Theory. In our model we hypothesize that homogeneity constitutes a fundamental ingredient in these dynamics. Accordingly, we study a population that tries to form homogeneous groups, i.e. composed of similar agents. The formation of a group can be interpreted as a strategy. Notably, agents can form a group (receiving a ‘group payoff’), or can act individually (receiving an ‘individual payoff’). The phase diagram of the modeled population shows a sharp transition between the ‘group phase’ and the ‘individual phase’, characterized by a critical ‘individual payoff’. Our results then support the hypothesis that the phenomenon of group formation has evolutionary roots.