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The concept of fitness is central to evolution, but it quantifies only theexpected number of offspring an individual will produce. The actual number ofoffspring is also subject to noise, arising from environmental or demographicstochasticity. In nature, individuals who are more fecund tend to have greatervariance in their offspring number -- sometimes far greater than the Poissonvariance assumed in classical models of population genetics. Here, we develop amodel for the evolution of two types reproducing in a population ofnon-constant size. The frequency-dependent fitness of each type is determinedby pairwise interactions in a prisoner's dilemma game, but the offspring numberis subject to an exogenously controlled variance that may depend upon the mean.Whereas defectors are preferred by natural selection in classical well-mixedpopulations, since they always have greater fitness than cooperators, we showthat large offspring variance can reverse the direction of evolution and favorcooperation. Reproductive over-dispersion produces qualitatively new dynamicsfor other types of social interactions, as well, which cannot arise inpopulations with a fixed size or Poisson offspring variance.

The arrow of evolution when the offspring variance is large