Final Report Summary - GAMES (Game theory and cooperation from economics to evolutionary biology) The objectives of the project were to study the origin and stability of cooperation, still one of the major unsolved questions in evolutionary biology and in the social sciences. The study of cooperation among multiple individuals is essentially the study of public goods games (PGGs), which has a long tradition in economics. It has long been argued that rational, self-interested behaviour leads to the inefficient provision of public goods or the overexploitation of common resources (the 'tragedy of the commons') because of the incentive to free-ride on the contribution of other group members. Evolutionary game theory has also, more recently, focused on PGGs in order to understand, for example, the production of diffusible molecules in microbes. Variations of the simplest PGG, the n-person prisoner's dilemma (NPD), are well known. In the NPD the benefit of the public good is a linear function of the number of contributors. While linearity makes the NPD analytically tractable, it is also an unrealistic assumption. No examples of linear public goods in nature have been reported, whereas non-linear (sigmoid) benefits have been described for cooperative behaviour in animals and for the production of diffusible molecules in microbes. This is likely to be a common feature of public goods in biology, as the effect of enzyme production is generally a sigmoid, saturating function of its concentration. General non-linear public goods have been so far beyond analytical tractability. During the course of the project we used economic game theory to understand the evolution of cooperation in multi-player social dilemmas. We have completed the analysis of the comparative statics of threshold public goods games (the volunteer's dilemma) showing that the contribution to public goods decreases with group size. We have then gone further to analyse more general non-linear games using a new approach based on Bernstein polynomials, a rather neglected piece of mathematics, that allows to analyse all N-player 2-strategies PGGs, including non-linear games that are impossible to analyse using the standard approach of evolutionary game theory. Moreover, this approach establishes an equivalence between two-strategy games and more realistic games with continuous strategies. Analysing the Bernstein coefficient of the gradient of selection of the replicator dynamics, we can now characterize analytically the dynamics of any non-linear game, and compute the equilibria for complex social dilemmas that have been so far beyond the reach of evolutionary game theory. This has implications for social dilemmas such as the contribution to measures against global warming. We can show that, counterintuitively, the amount of cooperation in public goods games is maximised at intermediate levels of threshold uncertainty. We find that the optimal uncertainty level is a function of group size, the cost of cooperation and a constant. This suggests a practical way for improving efficiency in the solution of social dilemmas by increasing uncertainty rather than using incentives for contributors or sanctions against free-riders.
