Evolutionary dynamics of collective action when individual fitness derives from group decisions taken in the past

Many actions take some time to have an impact – their effects only appear at some point in the future. Such time lags turn out to be ubiquitous among living organisms. Here we study the impact of time lags in the evolutionary dynamics of cooperative collective action. We consider a population in which individuals interact via a N-Person Stag Hunt dilemma and must opt to cooperate or defect. In the absence of any delay, the replicator dynamics reveals the existence of regimes in which two internal fixed points appear simultaneously. We show that the presence of time delay in the fitness of individuals leads to a delayed replicator equation exhibiting new evolutionary profiles, each profile being separated by critical values of the delay that we determine explicitly. When we break the symmetry in the time lags, we show that, generally, defectors take more advantage from delay than cooperators. Finally, when we take into consideration, approximately, effects associated with the finite population size, we find that counter-intuitive evolutionary outcomes may occur, resulting from the interplay between delay and the basins of attraction in the neighborhood of the internal fixed-points, and which may lead to full cooperation in conditions under which the outcome would be Full Defection in infinite populations.

► Evolutionary fixed points are not changed by delays between action and change.
► Delays may have profound effects in the dynamics of collective action.
► Delays may render cooperation viable when otherwise it would never be possible.
► Defectors profit more from delays than cooperators.
► History functions may play an important role in the evolution of collective action.