Revisiting the "fallacy of averages" in ecology: Expected gain per unit time equals expected gain divided by expected time

Two-strategy games with time constraints on regular graphs

Evolutionary game theory is a powerful method for modelling animal conflicts. The original evolutionary game models were used to explain specific biological features of interest, such as the existence of ritualised contests, and were necessarily simple models that ignored many properties of real populations, including the duration of events and spatial and related structural effects. Both of these areas have subsequently received much attention. Spatial and structural effects have been considered in evolutionary graph theory, and a significant body of literature has been built up to deal with situations where the population is not homogeneous. More recently a theory of time constraints has been developed to take account of the fact that different events can take different times, and that interaction times can explicitly depend upon selected strategies, which can, in turn, influence the distribution of different opponent types within the population. Here, for the first time, we build a model of time constraint games which explicitly considers a spatial population, by considering a population evolving on an underlying graph, using two graph dynamics, birth-death and death-birth. We consider one short time scale along which frequencies of pairs and singles change as individuals interact with their neighbours, and another, evolutionary time scale, along which frequencies of strategies change in the population. We show that for graphs with large degree, both dynamics reproduce recent results from well-mixed time constraint models, including two ESSs being common in Hawk-Dove and Prisoner's Dilemma games, but for low degree there can be marked differences. For birth-death processes the effect of the graph degree is small, whereas for death-birth dynamics there is a large effect. The general prediction for both Hawk-Dove and Prisoner's dilemma games is that as the graph degree decreases, i.e., as the number of neighbours decreases, mixed ESS do appear. In particular, for the Prisoner's dilemma game this means that cooperation is easier to establish in situations where individuals have low number of neighbours. We thus see that solutions depend non-trivially on the combination of graph degree, dynamics and game.

Defectors' intolerance of others promotes cooperation in the repeated public goods game with opting out

The theoretical and experimental research on opting out (also called conditional dissociation) in social dilemmas has concentrated on the effect this behavior has on the level of cooperation when used against defectors. The intuition behind this emphasis is based on the common property of social dilemmas that individuals are worse off the more their opponents defect. However, this article shows clearly that other opting out mechanisms are better at increasing cooperative behavior. In fact, by analyzing the stable Nash equilibria for the repeated multi-player public goods game with opting out, our results provide a strong argument that the best opting out rule is one whereby the only groups that voluntarily stay together between rounds are those that are homogeneous (i.e., those groups that are either all cooperators or all defectors), when these groups stay together for enough rounds. This outcome emerges when defectors are completely intolerant of individuals who cooperate (e.g., defectors exhibit xenophobic behavior toward cooperators) and so opt out whenever their group has a cooperator in it. The strong preference by defectors to be with like-minded individuals causes all heterogeneous groups to disband after one round.