Killingback et al. (2006) Evolution in group-structured populations resolves commons tragedy

Killingback, T., Bieri, J., & Flatt, T. (2006). Evolution in group-structured populations can resolve the tragedy of the commons. Proceedings of the Royal Society B, 273, 1477-1481. 

A simple simulation model of evolution in group-structured populations shows that group structure can allow prosocial behavior in the public goods game to evolve.

Model

The basic Public Goods game
-- n agents play the PGG. Each gives investment x_i and the game multiplies all investments by k.

The model with reproduction

• n agents are initially distributed across m groups, each of equal size, with x_i drawn from low initial values.
• Individuals in each group reproduce in proportion to their fitness.
• The strategy can mutate with some small probability.
• A fraction d individuals from each group randomly disperses to another group. 

 Results

With only one group, starting with a uniform distribution of contribution values, the population evolves to zero contribution, even when k > n. 

With groups, starting with very low initial investments, the population evolves to maximum investments.

Those result only hold for a narrow range of dispersal rates.