Variable investment, the Continuous Prisoner's Dilemma, and the origin of cooperation

Abstract

Cooperation is fundamental to many biological systems. A common metaphor for studying the evolution of cooperation is the Prisoner's Dilemma, a game with two strategies: cooperate or defect. However, cooperation is rarely all or nothing, and its evolution probably involves the gradual extension of initially modest degrees of assistance. The inability of the Prisoner's Dilemma to capture this basic aspect limits its use for understanding the evolutionary origins of cooperation. Here we consider a framework for cooperation based on the concept of investment: an act which is costly, but which bene¢ts other individuals, where the cost and bene¢t depend on the level of investment made. In the resulting Continuous Prisoner's Dilemma the essential problem of cooperation remains: in the absence of any additional structure non-zero levels of investment cannot evolve. However, if investments are considered in a spatially structured context, sel¢sh individuals who make arbitrarily low investments can be invaded by higher-investing mutants. This results in the mean level of investment evolving to signi¢cant levels, where it is maintained inde¢nitely. This approach provides a natural solution to the fundamental problem of how cooperation gradually increases from a non-cooperative state.

Fundamental clusters in spatial 2×2 games

“…These requirements are a combination of suitable pairs of S,T -values together with a minimal cluster size. The study of such minimal clusters in spatial games was initiated by Killingback et al (1999). We continue this work and extend it to 2 Â 2 games in general.…”

“…Second, no smaller cluster is able to ful¢l the ¢rst condition. The role of such clusters, and 3 Â 3 clusters in particular, in determining the long-term dynamic behaviour of spatial games was ¢rst pointed out by Killingback et al (1999).…”

“…Even in this simple and completely deterministic situation, where individuals have no means to develop complex strategies, lattice dynamics turns out to be complex (see, for example, Killingback & Doebeli 1998), leading to intriguing dynamic patterns. Recent developments and some extensions are given in Nowak & Sigmund (2000).…”

“…Including spatial dimensions has proven to be a very fruitful extension to evolutionary game theory (see, for example, Nowak & May 1992;Lindgren & Nordahl 1994;Herz 1994;Killingback & Doebeli 1998;Eshel et al 1999Eshel et al , 1998Hartvigsen et al 2000). Instead of considering a well mixed population where all individuals interact with each other, the individuals are bound to lattice sites and interact only with their neighbours.…”

Does mobility decrease cooperation?

“…These spatial games, where the interactions are localized and non random, have been studied and extended in many ways (see, for example, Refs. [1,2,4,9,10,15,19,22,23,24,25,29,30,31,34,37,38,40,41,44,45,46,47,49,52]). Once the population is spatially structured, a natural question concerns the effects of mobility that, along with other important biological factors, is often neglected [28]: is it possible to evolve and sustain cooperation in a population of mobile agents, where retaliation can be avoided by moving away from the former partner?…”