Appendix 1: Covariances for repeated interaction model
The expectations we need to calculate our required covariances are:
$$ \begin{array}{rll} {\rm{E}}(P)&=&{\rm{E}}(P')={\rm{E}}(GP) \\ &=&g(1-g)+g^2 \left(1+\frac{w}{1-w}\right),\end{array} $$
(28)
$${\rm{E}}(GP')=g^2 \left(1+\frac{w}{1-w}\right),$$
(29)
and
$$ \begin{array}{rll}{\rm{E}}(\min(P,P'))&=&{\rm{E}}(G \min(P,P')) \\ &=&g^2 \left(1+\frac{w}{1-w}\right).\end{array} $$
(30)
These expectations give the following covariances:
$$ \begin{array}{rll}&&{\rm{Cov}}(G,P) \\ &&=(1-g)\left(g(1-g)+g^2\left(1+\frac{w}{1-w}\right)\right), \end{array}$$
(31)
$${\rm{Cov}}(G,P')=(1-g)g^2\left(\frac{w}{1-w}\right),$$
(32)
and
$${\rm{Cov}}(G,\min(P,P'))=(1-g)g^2\left(1+\frac{w}{1-w}\right).$$
(33)
Appendix 2: Expectations for Greenbeard model
The required expectations for the greenbeard model can be calculated as:
$${\rm{E}}(G)={\rm{E}}(G')=g, $$
(34)
$${\rm{E}}(Q)={\rm{E}}(Q')=q, $$
(35)
$${\rm{E}}(GG'Q)=g^2 {\rm{P}}(\textrm{greenbeard} | \textrm{altruist}), $$
(36)
and
$${\rm{E}}(G'Q)={\rm{E}}(GQ')={\rm{E}}(G^2Q')=qg. $$
(37)
Appendix 3: Expectations for graph model
To calculate the required expectations for the ‘death-birth’ model of altruism on graphs, we begin by calculating the expected genotype of the actors competing for reproduction, and the unconditioned expected genotype of their neighbours,
$${\rm{E}}(G)={\rm{E}}(G')=g. $$
(38)
To calculate the joint expectation E(GG′) required to determine Cov(G,G′) it would seem we should be able to directly apply Eq. 16, however this would be incorrect, as we must take account of the fact that all competitors interacted with a common neighbour in determining their payoff, the site being replaced. This focal site is an altruist with probability g, otherwise it is a defector. Thus, we modify Eq. 16 to calculate the joint expectation of G and G′ as
$$ \begin{array}{rll}{\rm{E}}(GG')&=&{\rm{E}}(G'|G=1) \\ &=&\frac{g}{k} + \frac{k-1}{k}\left(g+\frac{1-g}{k-1} \right), \end{array} $$
(39)
where the first term is the expected genotype of the site being replaced, and the second term is the conditional expectation of Eq. 16, weighted by the remaining proportion of neighbours (k − 1)/k. The variance and covariance presented in the main text follow directly from the above results.
Appendix 4: Analysis of synergism model
The expectations required for the synergism model are:
$${\rm{E}}(G)={\rm{E}}\left(G^2\right)=g, $$
(40)
and
$${\rm{E}}(GG')={\rm{E}}\left(G^2G'\right)=g^2, $$
(41)
As mentioned in the main text, it is of interest to briefly note that frequency-dependent social dilemmas such as the stag hunt are often referred to as having ‘risk dominant’ and ‘payoff dominant’ equilibria (Harsanyi and Selten 1988). These refinements of the Nash equilibrium concept arose in classical game theory. In evolutionary game theory, under the replicator dynamics for example, equilibrium selection is frequency-dependent, and a ‘risk dominant’ equilibrium can be defined as that having the greatest basin of attraction (Samuelson 1998), but the covariance approach naturally captures uncertainty and thus might provide a different and interesting perspective. For example a partial simplification from Eq. 20 to Eq. 24 could be written
$${\rm{Var}}(G)d>{\rm{E}}(1-G)c, $$
(42)
where G is the random variable corresponding to the population frequency of stag hunters, p. Thus, the risk-dominant stag hunting equilibrium can be thought of as being favoured when the variance (i.e. the uncertainty (Zidek and van Eeden 2003)) in the synergistic benefit from both individuals cooperating, exceeds the magnitude of the cost of altruism multiplied by the expected frequency of non-cooperators.