Populations with interactions of multiple players driven by heterogeneous updating rules: Aspiration and Imitation as an examples

Updating by imitation, where individuals imitate successful strategies used by others, has been widely used to study the evolution of cooperation. However recent behavioral experiences such Lasius Niger (Gruter et al. [1]) and ninespined sticklebacks (van Bergen et al. [2])) question how this updating rule can be applied because their results are not compatible with the predictions given by updating by imitation. An alternative updating rule which can be used to explain self-evaluation in many species is the aspiration-led update. When an individual is selected to revise its decision, it compares its payoff to a satisfaction index with its current strategy. If it is satisfied with its strategy, then it will be more likely to keep its strategy. Otherwise, it will be more likely to imitate others. Using imitation or aspiration, studies always assume a homogeneous population where all individuals use the same updating rule. However, recently some studies on human interactions conclude that an updating rule depends on the personality of the individual. Then, a heterogeneous population, where individuals used different updating rules, becomes a necessity to understand human behavior given that updating rules can change dramatically the evolutionary outcomes. In this project, we will investigate the heterogeneity effect in a well-mixed population by using the concept of fixation probability. We will study a conditions for selection to favor strategy A (fixation probability of A is greater than its value in the neutral model) and to favor A against B (fixation probability of A is greater than the fixation probability of B). In a structured population we will investigate the evolutionary outcome by comparing the frequency of A under selection in the stead state to its frequency under the neutral model. For a large population size and by using a Kingmann coalescent process, we are attending a simple condition expressed in terms of the payoff matrix and some measure identities. Studies using aspiration dynamic assume that individuals compare its payoffs to a some constant which is unrealistic because this satisfaction index reflects individual experiences. Therefore, we will generalize the aspiration dynamic where the satisfaction index depends on individual. Under strong selection, we will investigate the spread cooperation in some additive public good games as the Prisoners Dilemma, Snow Drift and Stag Hunt.