Aspiration dynamics generate robust predictions in heterogeneous populations

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.

Cooperation among unequal players with aspiration-driven learning

Direct reciprocity promotes the evolution of cooperation when players are sufficiently equal, such that they have similar influence on each other. In the light of ubiquitous inequality, this raises the question of how reciprocity evolves among unequal players. Existing studies on inequality mainly focus on payoff-driven learning rules, which rely on the knowledge of others' strategies. However, inferring one's strategy is a difficult task even if the whole interaction history is known. Here, we consider aspiration-driven learning rules, where players seek strategies that satisfy their aspirations based on their own information. Under aspiration-driven learning rules, we explore the evolutionary dynamics among players with inequality in endowments and productivity. We model the interactions among unequal players with asymmetric games and characterize the condition where cooperation is feasible. Remarkably, we find that aspiration-driven learning rules lead to a higher level of cooperation than payoff-driven ones over a wide range of inequality. Moreover, our results show that high aspiration levels are conducive to the evolution of cooperation when more productive players are equipped with higher endowments. Our work highlights the advantages of aspiration-driven learning for promoting cooperation among unequal players and suggests that aspiration-based decision-making may be more beneficial for the collective.

Cooperation transitions in social games induced by aspiration-driven players

Cooperation and defection are social traits whose evolutionary origin is still unresolved. Recent behavioral experiments with humans suggested that strategy changes are driven mainly by the individuals' expectations and not by imitation. This work theoretically analyzes and numerically explores an aspiration-driven strategy updating in a well-mixed population playing games. The payoffs of the game matrix and the aspiration are condensed into just two parameters that allow a comprehensive description of the dynamics. We find continuous and abrupt transitions in the cooperation density with excellent agreement between theory and the Gillespie simulations. Under strong selection, the system can display several levels of steady cooperation or get trapped into absorbing states. These states are still relevant for experiments even when irrational choices are made due to their prolonged relaxation times. Finally, we show that for the particular case of the prisoner dilemma, where defection is the dominant strategy under imitation mechanisms, the self-evaluation update instead favors cooperation nonlinearly with the level of aspiration. Thus, our work provides insights into the distinct role between imitation and self-evaluation with no learning dynamics.

Evolutionary dynamics on sequential temporal networks

Population structure is a well-known catalyst for the evolution of cooperation and has traditionally been considered to be static in the course of evolution. Conversely, real-world populations, such as microbiome communities and online social networks, frequently show a progression from tiny, active groups to huge, stable communities, which is insufficient to be captured by constant structures. Here, we propose sequential temporal networks to characterize growing networked populations, and we extend the theory of evolutionary games to these temporal networks with arbitrary structures and growth rules. We derive analytical rules under which a sequential temporal network has a higher fixation probability for cooperation than its static counterpart. Under neutral drift, the rule is simply a function of the increment of nodes and edges in each time step. But if the selection is weak, the rule is related to coalescence times on networks. In this case, we propose a mean-field approximation to calculate fixation probabilities and critical benefit-to-cost ratios with lower calculation complexity. Numerical simulations in empirical datasets also prove the cooperation-promoting effect of population growth. Our research stresses the significance of population growth in the real world and provides a high-accuracy approximation approach for analyzing the evolution in real-life systems.

Optimization of institutional incentives for cooperation in structured populations

The application of incentives, such as reward and punishment, is a frequently applied way for promoting cooperation among interacting individuals in structured populations. However, how to properly use the incentives is still a challenging problem for incentive-providing institutions. In particular, since the implementation of incentive is costly, to explore the optimal incentive protocol, which ensures the desired collective goal at a minimal cost, is worthy of study. In this work, we consider the positive and negative incentives for a structured population of individuals whose conflicting interactions are characterized by a Prisoner's Dilemma game. We establish an index function for quantifying the cumulative cost during the process of incentive implementation, and theoretically derive the optimal positive and negative incentive protocols for cooperation on regular networks. We find that both types of optimal incentive protocols are identical and time-invariant. Moreover, we compare the optimal rewarding and punishing schemes concerning implementation cost and provide a rigorous basis for the usage of incentives in the game-theoretical framework. We further perform computer simulations to support our theoretical results and explore their robustness for different types of population structures, including regular, random, small-world and scale-free networks.

Evolutionary dynamics under partner preferences

The fact that people often have preference rankings for their partners is a distinctive aspect of human behavior. Little is known, however, about how this talent as a powerful force shapes human behavioral traits, including those which should not have been favored by selection, such as cooperation in social dilemma situations. Here we propose a dynamic model in which network-structured individuals can switch their interaction partners within neighborhoods based on their preferences. For the partner switching, we propose two interruption regimes: dictatorial regime and negotiating regime. In the dictatorial regime, focal individuals are able to suspend interactions out of preferences unilaterally. In the negotiating regime, either focal individuals or the associated partners agree to suspend, then these interactions can be successfully suspended. We investigate the evolution of cooperation under both preference-driven partner switching regimes in the context of both the weakened variant of the donation game and the standard one. Specifically, we theoretically approximate the critical conditions for cooperation to be favored by weak selection in the weakened donation game where cooperators bear a unit cost to provide a benefit for each active neighbor and simulate the evolutionary dynamics of cooperation in the standard donation game to test the robustness of the analytical results. Under dictatorial regime, selection of cooperation becomes harder when individuals have preferences for either cooperator or defector partners, implying that the expulsion of defectors by cooperators is overwhelmed by the chasing of defectors towards cooperators. Under negotiating regime, both preferences for cooperator and defector partners can significantly favor the evolution of cooperation, yet underlying mechanisms differ greatly. For preferences over cooperator partners, cooperator-cooperator interaction relationships are reinforced and the associated mutual reciprocity can resist and assimilate defectors. For preferences over defector partners, defector-defector interaction relationships are anchored, weakening defectors' exploitation over cooperators. Cooperators are thus offered much time space to interact among cospecies and spread. Our work may help better understand the critical role of preference-based adaptive partner switching in promoting the evolution of cooperation.

Evolutionary dynamics of zero-determinant strategies in repeated multiplayer games

Several studies have confirmed the existence of zero-determinant (ZD) strategies in repeated social dilemmas since Press and Dyson's ingenious discovery of ZD strategies in iterated prisoner's dilemmas. However, less research studies evolutionary performance of multiplayer ZD strategies, especially from a theoretical perspective. Here, we use a state-clustering method to theoretically analyze evolutionary dynamics of two representative ZD strategies: generous ZD strategies and extortionate ZD strategies. We consider two new settings for multiplayer ZD strategies: competitions with all ZD strategies and competitions with all memory-one strategies, apart from the competitions between these strategies and some classical ones. Moreover, we investigate the influence of the level of generosity and extortion on evolutionary dynamics of generous and extortionate ZD strategies, which was commonly ignored in previous studies. Theoretical results show that players with limited generosity are at an advantageous place and extortioners extorting more severely hold their ground more readily. Our results may provide new insights into better understanding evolutionary dynamics of ZD strategies in repeated multiplayer games.

Impact of the baseline payoff on evolutionary outcomes

Do individuals enjoying a higher baseline payoff behave similarly in competitive scenarios compared to their counterparts? The classical replicator equation does not answer such a question since it is invariant to the background or baseline payoff of individuals. In reality, however, if one's baseline payoff is higher than the possible payoffs of an interaction (or game), the individual may respond generously or indifferently if s(he) is satisfied with the prevailing benchmark payoff. This work intends to explore such a phenomenon within the realm of pairwise interactions-taking the prisoner's dilemma as a metaphor-in well-mixed finite and infinite populations. In this framework, a player uses the payoff (comprising baseline and game payoffs) -expectation difference to estimate a degree of eagerness and, with that degree of eagerness, revises his or her strategy with a certain probability. We adopt two approaches to explore such a context, naming them as the Fermi and imitation processes, in which the former uses a pairwise Femi function and the latter considers the relative fitness to estimate probabilities for strategy revision. In a finite population, we examine the effect of intensities to payoff-expectation and strategic payoff differences (denoted by k_{1} and k_{2}, respectively) as well as the level of contentment (ω) on the fixation probability and fixation time (for a single defector). We observe that the fixation probability surges with the increase of intensity parameters. Nevertheless, the maximum fixation probability may require a substantially larger time to fixate, especially when the expectation is lower than the baseline payoff. This means that cooperators can persist for a longer period of time. A higher expectation or greed, however, considerably reduces the fixation time. Interestingly, our numerical simulation reveals that both approaches are equivalent under weak k_{2}(≪1) in the Fermi process. We further derive mean-field equations for both approaches in the context of an infinite population, where we observe two possible evolutionary consequences: either full-scale defection or the persistence of the initial frequency of cooperators. The latter scenario indicates players' uninterested or neutral behavior in relation to the interaction due to their satisfaction on the baseline payoff.

Imitation and aspiration dynamics bring different evolutionary outcomes in feedback-evolving games

Feedback-evolving games characterize the interplay between the evolution of strategies and environments. Rich dynamics have been derived for such games under the premise of the replicator equation, which unveils persistent oscillations between cooperation and defection. Besides replicator dynamics, here we have employed aspiration dynamics, in which individuals, instead of comparing payoffs with opposite strategies, assess their payoffs by self-evaluation to update strategies. We start with a brief review of feedback-evolving games with replicator dynamics and then comprehensively discuss such games with aspiration dynamics. Interestingly, the tenacious cycles, as perceived in replicator dynamics, cannot be observed in aspiration dynamics. Our analysis reveals that a parameter θ —which depicts the strength of cooperation in enhancing the environment—plays a pivotal role in comprehending the dynamics. In particular, with the symmetric aspiration level, if replete and depleted states, respectively, experience Prisoner's Dilemma and Trivial games, the rich environment is achievable only when θ  > 1. The case θ  < 1 never allows us to reach the replete state, even with a higher cooperation level. Furthermore, if cooperators aspire less than defectors, then the enhanced state can be achieved with a relatively lower θ value compared with the opposite scenario because too much expectation from cooperation can be less beneficial.