Box 1. Optimal choosiness
From the following article:
The coevolution of choosiness and cooperation
John M. McNamara, Zoltan Barta, Lutz Fromhage & Alasdair I. Houston
Nature 451, 189-192(10 January 2008)
doi:10.1038/nature06455
In a single round of the game, an individual that expends effort x receives payoff W(x, x') when its partner's effort is x'. Here we consider the special case of the continuous prisoner's dilemma for which
where C(x) is a strictly increasing function of x, and b is a positive constant. For this payoff the Nash effort in a single round is the minimum effort x* = 0.
Consider a population in which the efforts of members of the pool of single individuals are distributed with probability density function f(x) and mean 
. Then (see Supplementary Information) an individual that is never dismissed by others should dismiss its partner if and only if its partner's effort is below y*, where
where 
is a decreasing function of the mortality M, and S is the search cost. Individuals that are dismissed by some co-players have a lower optimal threshold (Supplementary Information). From equation (2) it is easy to show that y* increases as S and M decrease (Supplementary Information).
It can be shown (Supplementary Information) that if all population members behave optimally then all expend the minimum effort x* = 0. Thus complete non-cooperation is evolutionarily stable unless effects such as mutation maintain a significant amount of non-adaptive variation in the population. Equation (2) provides an intuitive explanation of this result. If we assume a monomorphic population with all individuals expending effort 
, then equation (2) reduces to 
. If all individuals adopt this optimal threshold and 
, it pays to reduce effort below 
because this can be done without incurring the risk of being dismissed.
So how much variation is necessary for cooperation to evolve? We expect mean effort to evolve so that it is greater than most acceptance thresholds (so ensuring the cost of dismissal is not paid). Thus cooperation should evolve when y* exceeds the mean effort in the population. Approximating this mean by 
, equation (2) predicts cooperation to evolve when the variation, as measured by 
(the mean absolute deviation of effort from the population mean), is sufficiently high:
Equivalently, the mortality rate must be sufficiently low:
Because 
(the mean among single individuals) is typically less than the overall population mean, these are minimal criteria.
