Evolutionary game dynamics of combining the imitation and aspiration-driven update rules

Two-stage strategy update rule based on learning cost in weak prisoner's dilemma

When players are dissatisfied with their actual payoffs, they will change the actuality by learning strategy of neighbors. The more effort players put in, the more likely they are to succeed in learning. Inspired by this, this paper proposes a two-stage strategy update rule based on learning cost. The players first decide whether to learn strategy according to the updating willingness. If the players imitate the strategy of neighbors, they need to pay the learning cost. Results show that for the well-mixed population, if the updating willingness is homogeneous and remains unchanged, reducing the updating willingness or increasing the learning cost can extend the life cycle of cooperators. If the updating willingness is heterogeneous and dynamically adjusted based on the difference between the actual payoff and the expected payoff, increasing aspiration value and learning cost promotes cooperation. For the structured population, if the updating willingness is homogeneous and remains unchanged, the moderate learning cost is beneficial for cooperators to resist the temptation of defection, and reducing updating willingness makes the system maintain cooperation within a larger parameter range. If the updating willingness is heterogeneous and dynamically adjusted, the larger learning cost and the appropriate aspiration value promote cooperation. This study highlights the complex dynamics of cooperation in paid strategy learning, contributing to the theory of cooperation in the evolutionary game.

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.

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.

Evolution of cooperation in social dilemmas under the coexistence of aspiration and imitation mechanisms

Imitation and aspiration update rules are frequently observed in human and animal populations. While the imitation process entails payoff comparisons with surroundings, the aspiration process refers to self-evaluation. This work explores the evolution of cooperation in dyadic games under the coexistence of these two dynamics in an infinitely large well-mixed population. Two situations have been explored: (i) individuals adopt either an imitation or aspiration update rule with a certain probability, and (ii) the entire population is divided into two groups where one group only uses imitative rules and the other obeys aspiration updating alone. Both premises have been modeled by taking an infinite approximation of the finite population. In particular, the second mixing principle follows an additive property: the outcome of the whole population is the weighted average of outcomes from imitators and aspiration-driven individuals. Our work progressively investigates several variants of aspiration dynamics under strong selection, encompassing symmetric, asymmetric, and adaptive aspirations, which then coalesce with imitative dynamics. We also demonstrate which of the update rules performs better, under different social dilemmas, by allowing the evolution of the preference of update rules besides strategies. Aspiration dynamics always outperform imitation dynamics in the prisoner's dilemma, however, in the chicken and stag-hunt games the predominance of either update rule depends on the level of aspirations as well as on the extent of greed and fear present in the system. Finally, we examine the coevolution of strategies, aspirations, and update rules which leads to a binary state of obeying either imitation or aspiration dynamics. In such a circumstance, when aspiration dynamics prevail over imitation dynamics, cooperators and defectors coexist to an equal extent.

Vaccinating behaviour guided by imitation and aspiration

Vaccinating decisions can be influenced by imitation as well as self-evaluation or aspiration. This work analyses vaccinating behaviours by coupling both imitation and aspiration update rules, adopting an existing set-up of the mean-field vaccination game. We incorporate the imitation mechanism with several variants of the aspiration protocol, encompassing constant and adaptive aspirations. Equations of the combined dynamics have been derived by grouping the population according to (i) vaccinating strategies and (ii) healthy and infected status within each strategy. If aspiration levels are fixed but differentiated by vaccinating strategies, then vaccinators aspiring less than non-vaccinators are found to ameliorate the vaccination coverage, thereby yielding a less infectious state. The adaptive aspirations maintain a positive correlation with the vaccine efficacy while keeping the opposite relation with vaccination cost. When vaccinating strategies, aspirations and update rules are allowed to evolve synchronously, then either the imitation or aspiration process takes over the entire population. If aspiration rules prevail, then vaccinees and non-vaccinees coexist equally (according to (i)) or vaccine uptake follows a non-monotonic trend with the efficacy (according to (ii)). The imitative rule performs better when vaccination is less expensive or cheap, while aspiration updating safeguards the tenacity of vaccinees despite vaccination being expensive.

Evolutionary dynamics of the delayed replicator-mutator equation: Limit cycle and cooperation

Game theory deals with strategic interactions among players and evolutionary game dynamics tracks the fate of the players' populations under selection. In this paper, we consider the replicator equation for two-player-two-strategy games involving cooperators and defectors. We modify the equation to include the effect of mutation and also delay that corresponds either to the delayed information about the population state or in realizing the effect of interaction among players. By focusing on the four exhaustive classes of symmetrical games-the Stag Hunt game, the Snowdrift game, the Prisoners' Dilemma game, and the Harmony game-we analytically and numerically analyze the delayed replicator-mutator equation to find the explicit condition for the Hopf bifurcation bringing forth stable limit cycle. The existence of the asymptotically stable limit cycle imply the coexistence of the cooperators and the defectors; and in the games, where defection is a stable Nash strategy, a stable limit cycle does provide a mechanism for evolution of cooperation. We find that while mutation alone can never lead to oscillatory cooperation state in two-player-two-strategy games, the delay can change the scenario. On the other hand, there are situations when delay alone cannot lead to the Hopf bifurcation in the absence of mutation in the selection dynamics.