Heterogeneity in evolutionary games: an analysis of the risk perception

Abstract

<jats:p>In this study, we analyse the relationship between heterogeneity and cooperation. Previous investigations suggest that this relation is non-trivial, as some authors found that heterogeneity sustains cooperation, while others obtained different results. Among the possible forms of heterogeneity, we focus on the individual perception of risks and rewards related to a generic event, which can appear in a number of social and biological systems. The modelling approach is based on the framework of evolutionary game theory. To represent this kind of heterogeneity, we implement small and local perturbations on the pay-off matrix of simple two-strategy games, such as the Prisoner’s Dilemma. So, while usually the pay-off is considered to be a global and time-invariant structure, i.e. it is the same for all individuals of a population at any time, in our model its value is continuously affected by small variations, in both time and space (i.e. position on a lattice). We found that such perturbations can be beneficial or detrimental to cooperation, depending on their setting. Notably, cooperation is strongly supported when perturbations act on the main diagonal of the pay-off matrix, whereas when they act on the off-diagonal the resulting effect is more difficult to quantify. To conclude, the proposed model shows a rich spectrum of possible equilibria, whose interpretation might offer insights and enrich the description of several systems.</jats:p>

Strategy equilibrium in dilemma games with off-diagonal payoff perturbations

“…In this work, we studied payoff perturbations considering simple social dilemmas, with the main goal to clarify and quantify their effects when only the off-diagonal is interested, following the results reported in [8]. Beyond analyzing perturbative methods in evolutionary games, the proposed model has the potential to actively contribute to a vivid debate, i.e., that on the relationship between heterogeneity and cooperation.…”

“…Perturbations have also been used in evolutionary game theory for studying the emergence of cooperation [5][6][7], as recently reported in [8]. Notably, the latter shows that small and random perturbations on the payoff matrix of social dilemmas can strongly influence the strategy equilibrium of a population.…”

“…However, while their effect on the main diagonal has been clarified, i.e., they support cooperative behaviors, the effect given by off-diagonal perturbations is still unclear. Thus, here we aim to clarify this specific aspect of the perturbative method presented in [8] and to obtain a complete description of the dynamics of evolutionary games, affected by small and random perturbations on their payoff matrix. To this end, our analyses consider a variety of conditions, as three different update rules, and various games (e.g., stag hunt, snowdrift, and harmony).…”

“…As observed in [8], perturbations on the payoff matrix of dilemma games actually are responsible for a form of population heterogeneity, as they provide players with a diversified perception of risks and rewards. For this reason our investigation, while focused on clarifying the role of perturbations in social dilemmas, can provide findings of potential interest for the debate on the relationships between heterogeneity and cooperation in evolutionary games.…”

“…We deem it relevant to mention previous seminal works, such as [55][56][57][58][59], that (to the best of our knowledge) first studied such disorder in EGT. More recently, the effect of payoff noise on phase transitions has been the core of much research that has shed light on how different types of perturbations can lead to the emergence of cooperation [8,27,33,52,53,[60][61][62][63][64][65][66]. So, here, starting from the model proposed in [8], which introduces payoff perturbation in the context of the imitation update rule for all payoff entries, we perform a full analysis under different settings, and then we study the microscopical mechanism that leads to the observed macroscopic results.…”