Indirect Reciprocity With Optional Interactions and Private Information and Stochastic Evolution of Staying Together

Abstract

We explore indirect reciprocity with optional interactions and private information and stochastic evolution of staying together. Indirect reciprocity is cooperation based on reputation in a group of players; my behavior toward you also depends on what you have done to others. In the study of indirect reciprocity, we provide closed formulas for payoffs of cooperators and defectors and efficient algorithms to find the payoffs using game-theoretical models on cooperation. We find conditions for cooperation to be favored and conditions for coexistence of cooperators and defectors. We also provide a closed formula for fixation probability of staying together (ST). ST means that replicating units do not separate after reproduction, but remain attached to each other or in close proximity. We first study indirect reciprocity with optional interactions. We assume that cooperators can reject to play with identified defectors so that they do not have to risk their own reputations. We have three different models depending on whether the identity of defectors is uncovered in Defector-Defector interactions and how the defectors interact with uncovered defectors. We provide analytic results on replicator dynamics and analysis of the replicator equations. We also study indirect reciprocity with private information. We model indirect reciprocity such that information on interactions is private and is shared privately or selectively. This is as opposed to modeling indirect reciprocity with public information on interactions. With public information, an activity that is observed by at least one individual will be known to everyone in the population. In this model of indirect reciprocity with private information, we calculate the critical benefit-to-cost ratio such that cooperation is successful. We also explore a mathematical model of stochastic emergence of multicellularity through staying together (ST). We provide a simple closed-form expression for the fixation probability of ST in weak selection and we analyze the formula for special cases. We find a very neat result for the fixation probability of ST when it has an upper bound on the sizes of ST complexes and the population size is large.