Individualised aspiration dynamics: Calculation by proofs

Cooperation is key for the evolution of biological systems ranging from bacteria communities to human societies. Evolutionary processes can dramatically alter the cooperation level. Evolutionary processes are typically of two classes: comparison based and self-evaluation based. The fate of cooperation is extremely sensitive to the details of comparison based processes. For self-evaluation processes, however, it is still unclear whether the sensitivity remains. We concentrate on a class of self-evaluation processes based on aspiration, where all the individuals adjust behaviors based on their own aspirations. We prove that the evolutionary outcome with heterogeneous aspirations is the same as that of the homogeneous one for regular networks under weak selection limit. Simulation results further suggest that it is also valid for general networks across various distributions of personalised aspirations. Our result clearly indicates that self-evaluation processes are robust in contrast with comparison based rules. In addition, our result greatly simplifies the calculation of the aspiration dynamics, which is computationally expensive.