# Strategy abundance

### 2011 Aug;283(1):180–91.

Multiplayer games are the representations of many social dilemmas. Even for multiplayer games we can use the replicator equation with a complicated payoff structure to derive the time evolution of the strategies. This includes only the effect of selection. Including mutations in a given evolutionary game is relatively easy if we assume the mutation rate to be very small. This allows us to derive important quantities such as fixation probabilities with relative mathematical ease. For high mutation rates the concept of fixation itself becomes problematic and so does fixation probability. Even with high mutation rates, if a system continues to evolve for a long time then we can calculate the average frequency of a strategy. This average frequency of a strategy in the stationary distribution (hereafter termed as abundance) for arbitrary mutation rates has been calculated previously by (Antal et al., 2009a,b,c). The procedure can even be applied in some cases when a population is structured Tarnita et al., 2009. The analysis has remained possible only for two player games.

We develop an approach for estimating the abundance for multiple players and multiple strategies.. The theory hinges on the calculation of the following term, the average change in the frequency of strategy $k$ under weak selection ($\delta \ll 1$),

Once we know this then we can add the effect of mutations ($u$) which gives us the abundance of a strategy (here strategy $k$) in the mutation-selection equilibrium (Antal et al., 2009a,b,c) as,

For the calculations we employ tools from coalescence theory ((Kingman et al., 2009), a,b,c,(Wakeley, 2008)). Small mutation rates make sense in genetical sense but for cultural traits such as fashion or plastic behaviour, high mutation rates are more realistic ((Traulsen et al.,2010),(Grujic et al., 2010)). The theory developed herein can be used for a variety of applications ranging from finding the abundance of alleles in an allelic polymorphism to the best strategy in a social setting.

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