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