## Evolutionary games in set structured populations

May 8, 2013 3 Comments

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 individuals and sets, where each individual can belong to of the sets. Strategy and set membership are heritable (with mutation probabilities and , 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:

Where is the population-scaled set mutation rate, even when this is zero we get cooperation when . Alternatively, to simplify we can take the limit of to get:

If we allow the maximum number of sets () and take the further limit of large populations then this becomes a very simple:

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 of:

where is a structural constant given by:

Which in the limit of small population-scaled strategy mutation (), and becomes:

Since the structural constant is always greater than 1, and since we typically care about large , it is more enlightening to look at the reciprocal that in the limit of large becomes:

And simplifies the general equation for cooperation to emerge to:

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

## Mutation-bias driving the evolution of mutation rates

March 31, 2016 by Julian Xue 4 Comments

In classic game theory, we are often faced with multiple potential equilibria between which to select with no unequivocal way to choose between these alternatives. If you’ve ever heard Artem justify dynamic approaches, such as evolutionary game theory, then you’ve seen this equilibrium selection problem take center stage. Natural selection has an analogous ‘problem’ of many local fitness peaks. Is the selection between them simply an accidental historical process? Or is there a method to the madness that is independent of the the environment that defines the fitness landscape and that can produce long term evolutionary trends?

Two weeks ago, in my first post of this series, I talked about an idea Wallace Arthur (2004) calls “developmental bias”, where the variation of traits in a population can determine which fitness peak the population evolves to. The idea is that if variation is generated more frequently in a particular direction, then fitness peaks in that direction are more easily discovered. Arthur hypothesized that this mechanism can be responsible for long-term evolutionary trends.

A very similar idea was discovered and called “mutation bias” by Yampolsky & Stoltzfus (2001). The difference between mutation bias and developmental bias is that Yampolsky & Stoltzfus (2001) described the idea in the language of discrete genetics rather than trait-based phenotypic evolution. They also did not invoke developmental biology. The basic mechanism, however, was the same: if a population is confronted with multiple fitness peaks nearby, mutation bias will make particular peaks much more likely.

In this post, I will discuss the Yampolsky & Stoltzfus (2001) “mutation bias”, consider applications of it to the evolution of mutation rates by Gerrish et al. (2007), and discuss how mutation is like and unlike other biological traits.

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Filed under Commentary, Models, Reviews Tagged with evolution, mutation, supply driven evolution