# Small mutation rates

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### Wu B, Gokhale CS, Wang L, Traulsen A.

*How small are small mutation rates?*

*J. Math. Biol.*

### May 28;64(5):803–27, 2011

Mutations or random explorations seem to be inherent in human nature. Mutations and selection work in concert to develop the population to a possible equilibrium state. Traditionally analyzing multiplayer games with mutations is possible if we assume the mutation rate to be negligible (Fudenberg and Imhof, 2008). Consider the situation when we have three strategies A, B and C. Every time a reproductive event occurs, there is a small probability that the offspring will be of a random strategy and not necessarily like the parent. The average time between two mutations is thus $ \mu^{-1} $. We also know that the time for fixation of a single neutral mutant is $ N (N-1) $ (Antal and Scheuring, 2006). Therefore if the mutation probability, $\mu$, is much smaller than $N^{-2}$ then the time between two mutations will be much larger than the time required for either extinction or fixation of a single mutant. Thus at a time we will have to deal with only two strategies. For two strategies we can calculate exactly the fixation probability of a mutant in finite populations (Nowak, 2006a) even for multiplayer games (Gokhale and Traulsen, 2010). For small mutation rates the transition probabilities between different strategies consist of just the fixation probabilities. For example for the three strategies A, B and C the transition matrix,

consists of the fixation probability of strategy A in a population consisting predominantly of strategy B is given by $ \rho_{AB} $. Each element of the matrix denotes the probability of the row strategy to move into the column strategy. Strategy A (first row) can either stay in an all A population (first column) or move to a population of B (second column) individuals or in a population of C (third column) individuals. Since these are the only three probable events, the sum of all elements in a row is one. To find which strategy does the best at the mutation selection equilibrium we need to know which strategy has the highest frequency on an average in the stationary distribution. The stationary distribution of the system is given by the normalised right eigenvector for the largest eigenvalue of the transition matrix T.

However this analysis is only valid for small mutation rates. The question asked in this section is,

- How small do the mutation rates have to be so that the error due to the approximation is below a tolerable threshold?

This issue was can be tackled by using time scale separation analysis based on Antal and Scheuring (2006). Since then this approach has been used extensively in many papers to explore the the dynamics of strategies in the limit of strong selection and weak mutations (Imhof et al., 2005; Fudenberg and Imhof, 2006; Traulsen and Nowak, 2007; Hauert et al., 2007, 2008; Van Segbroeck et al., 2009; Sigmund et al., 2010). Our approach takes the route of the stationary distribution. If we let the system evolve for a long time it will reach an equilibrium state such that we can denote it by a distribution of the frequencies of the different strategies. For small mutation rates this distribution is approximated by the ones based on the fixation probabilities. We check for the difference between these two distributions. To check mathematically if the approximation is “good” we calculate the total variation distance between the distributions. Herein we also provide a numerically accessible bound which can be calculated for any given system of two player games and two strategies. Hence now it is possible to determine exactly how low the mutation rate should be to reduce the error below a certain threshold.