Decelerated invasion and waning-moon patterns in public goods games with delayed distribution

Abstract

We study the evolution of cooperation in the spatial public goods game, focusing on the effects that are brought about by the delayed distribution of goods that accumulate in groups due to the continuous investments of cooperators. We find that intermediate delays enhance network reciprocity because of a decelerated invasion of defectors, who are unable to reap the same high short-term benefits as they do in the absence of delayed distribution. Long delays, however, introduce a risk because the large accumulated wealth might fall into the wrong hands. Indeed, as soon as the curvature of a cooperative cluster turns negative, the engulfed defectors can collect the heritage of many generations of cooperators and by doing so start a waning-moon pattern that nullifies the benefits of decelerated invasion. Accidental meeting points of growing cooperative clusters may also act as triggers for the waning-moon effect, thus linking the success of cooperators with their propensity to fail in a rather bizarre way. Our results highlight that "investing in the future" is a good idea only if that future is sufficiently near and not likely to be burdened by inflation. Large segments of our economy rely on continuous investments into a common pool that, at least so we are told, will or are very likely to pay off sometime in the future. Retirement plans and investments into the public health care and education system fall under this category, while somewhat shorter-term examples might include investing some fraction of personal wealth into a startup business or becoming a shareholder of a company with bright prospects. Either way, more often than not, the payoff from an investment is likely not going to be immediate, but rather it will set in with a delay.The public goods game [1] captures the essential social dilemma that is linked with investments into a common pool. Those that contribute are cooperators, while those that do not are defectors. All the contributions within a group are multiplied to take into account synergetic effects of cooperation and the resulting amount is divided equally among all group members irrespective of their strategies. If nobody invests, the group fails to harvest the benefits of a collective investment and the society may evolve towards the "tragedy of the commons" The game in its traditional form, however, does not take into account delays that might occur before the investments can actually be capitalized upon. Taking such delays into account essentially means altering the time scales of evolutionary dynamics [17][18][19]. Previous research showed that the diversity in reproduction time scales [20] [25][26][27], have also been noted as beneficial for resolving social dilemmas. The application of a delayed distribution of accumulating goods, however, cannot be considered as a simple separation of time scales between learning (strategy adoption) and interaction (payoff accumulation) processes because cooperators, as we will see, lose their payoff permanently between two consecutive distributions, ...

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