## Evolutionary games in set structured populations

We have previously discussed the importance of population structure in evolutionary game theory, and looked at the Ohtsuki-Nowak transform for analytic studies of games on one of the simplest structures — random regular graphs. However, there is another extremely simple structure to consider: a family of inviscid sets. We can think of each agent as belonging to one or more sets and interacting with everybody that shares a set with them. If there is only one set then we are back to the case on a completely inviscid population. If we associate a set with each edge in a graph and restrict them to have constant size then we have standard evolutionary graph theory. However, it is more natural to allow sets to grow larger if their members have high fitness.

Tarnita et al. (2009) consider a population of $N$ individuals and $M$ sets, where each individual can belong to $K$ of the sets. Strategy and set membership are heritable (with mutation probabilities $u$ and $v$, respectively), and interactions are only with agents that share a set (if two agents share more than one set then they interact more than once). However, reproduction is inviscid: a random individual is selected to die and everybody competes to replace them with a child. This set-dependent interaction, makes the model equivalent to the earliest models of ethnocentrism, but the model is not equivalent to more modern approaches to ethnocentrism. Since sets cannot reproduce and migration (through mutation) between sets is purely random, the model also cannot capture group selection. However, cooperation for the Prisoners’ dilemma still emerges in this model, if we have: $\frac{b}{c} \geq \frac{K}{M - K}(\nu + 2) + \frac{M}{M - K}\frac{\nu^2 + 3\nu + 3}{\nu(\nu + 2)}$

Where $\nu = 2Nv$ is the population-scaled set mutation rate, even when this is zero we get cooperation when $\frac{b}{c} \geq \frac{3}{2} + \frac{7K}{2(M - K)}$. Alternatively, to simplify we can take the limit of $M >> K = 1$ to get: $\frac{b}{c} \geq 1 + \frac{1}{\nu}(1 + \frac{1}{\nu + 2}) + \frac{\nu + 2}{M}$

If we allow the maximum number of sets ( $M = N$) and take the further limit of large populations $N >> 2$ then this becomes a very simple: $\frac{b}{c} \geq 1 + 2v$

Finally, in the supplementary materials, the authors derive a very nice relationship for evolution of cooperation on arbitrary cooperate-defect games given by a payoff matrix $\begin{pmatrix}1 & U \\ V & 0\end{pmatrix}$ of: $\sigma \geq V - U$

where $\sigma$ is a structural constant given by: $\sigma = \frac{1 + \nu + \mu}{3 + \nu + \mu)}\frac{K(\nu^2 + 2\nu + \nu\mu) + M(3 + 2\nu + \mu}{K(\nu^2 + 2\nu + \nu\mu) + M(1 + \mu)}$

Which in the limit of small population-scaled strategy mutation ( $\mu = 0$), and $M >> K$ becomes: $\sigma = 1 + \nu + \frac{\nu(\nu + 1)}{\nu + 3}$

Since the structural constant is always greater than 1, and since we typically care about large $\nu$, it is more enlightening to look at the reciprocal that in the limit of large $\nu >> 3$ becomes: $\frac{1}{\sigma - 1} = \frac{1}{2\nu}$

And simplifies the general equation for cooperation to emerge to: $1 + 2\nu \geq V - U$

Which can be seen as a relaxation of the classic game theory concept of risk-dominance. Tarnita, C., Antal, T., Ohtsuki, H., & Nowak, M. (2009). Evolutionary dynamics in set structured populations Proceedings of the National Academy of Sciences, 106 (21), 8601-8604 DOI: 10.1073/pnas.0903019106