Cooperation and Stability through Periodic Impulse

Basic games, where each individual chooses between two strategies, illustrate several issues that immediately emerge from the standard approach that applies strategic reasoning, based on rational decisions, to predict population behavior where no rationality is assumed. These include how mutual cooperation (which corresponds to the best outcome from the population perspective) can evolve when the only individually rational choice is to defect, illustrated by the Prisoner’s Dilemma (PD) game, and how individuals can randomize between two strategies when neither is individually rational, illustrated by the Battle of the Sexes (BS) game that models male-female conflict over parental investment in offspring. We examine these questions from an evolutionary perspective where the evolutionary dynamics includes an impulsive effect that models sudden changes in collective population behavior. For the PD game, we show analytically that cooperation can either coexist with defection or completely take over the population, depending on the strength of the impulse. By extending these results for the PD game, we also show that males and females each evolve to a single strategy in the BS game when the impulsive effect is strong and that weak impulses stabilize the randomized strategies of this game

Counterintuitive properties of the fixation time in network-structured populations

Evolutionary dynamics on graphs can lead to many interesting and counterintuitive findings. We study the Moran process, a discrete time birth-death process, that describes the invasion of a mutant type into a population of wild-type individuals. Remarkably, the fixation probability of a single mutant is the same on all regular networks. But non-regular networks can increase or decrease the fixation probability. While the time until fixation formally depends on the same transition probabilities as the fixation probabilities, there is no obvious relation between them. For example, an amplifier of selection, which increases the fixation probability and thus decreases the number of mutations needed until one of them is successful, can at the same time slow down the process of fixation. Based on small networks, we show analytically that (i) the time to fixation can decrease when links are removed from the network and (ii) the node providing the best starting conditions in terms of the shortest fixation time depends on the fitness of the mutant. Our results are obtained analytically on small networks, but numerical simulations show that they are qualitatively valid even in much larger populations