# Extrapolating weak selection in evolutionary games

## Abstract

This work is inspired by a 2013 paper from Arne Traulsen’s lab at the Max Plank Institute for Evolutionary Biology (Wu et al. in PLoS Comput Biol 9:e1003381, 2013). They studied evolutionary games when the mutation rate is so small that each mutation goes to fixation before the next one occurs. It has been shown that for \(2 \times 2\) games the ranking of the strategies does not change as strength of selection is increased (Wu et al. in Phys Rev 82:046106, 2010). The point of the 2013 paper is that when there are three or more strategies the ordering can change as selection is increased. Wu et al. (2013) did numerical computations for a fixed population size *N*. Here, we will instead let the strength of selection \(\beta = c/N\) where *c* is fixed and let \(N\rightarrow \infty \) to obtain formulas for the invadability probabilities \(\phi _{ij}\) that determine the rankings. These formulas, which are integrals on [0, 1], are intractable calculus problems, but can be easily evaluated numerically. Here, we use them to derive simple formulas for the ranking order when *c* is small or *c* is large.

## Mathematics Subject Classification

91A22 92D25## Notes

### Acknowledgements

We would like to thanks Ben Allen and Philipp Altrock for comments on the paper and for bringing some important references to our attention. Arne Traulsen and Bin Wu made many useful suggestions and helped us refocus the paper on our positive contributions.

## References

- Antal T, Scheuring I (2006) Fixation of strategies for an evolutionary game. Bull Math Biol 68:1923–1944MathSciNetCrossRefzbMATHGoogle Scholar
- Durrett R (2014) Spatial evolutionary games with small selection coefficients. Electron J Probab 19:121MathSciNetzbMATHGoogle Scholar
- Freidlin MI, Wentzell AD (1984) Random perturbations of dynamical systems. Springer, New YorkCrossRefzbMATHGoogle Scholar
- Fudenberg D, Imhof LA (2006) Imitation processes with small mutations. J Econ Theory 131:251–262MathSciNetCrossRefzbMATHGoogle Scholar
- Jeong HC, Oh SY, Allen B, Nowak MA (2014) Optional games on cycles and complete graphs. J Theor Biol 356:98–112MathSciNetCrossRefGoogle Scholar
- Karlin S, McGregor J (1957) The classification of birth and death processes. Trans Am Math Soc 86:366–400MathSciNetCrossRefzbMATHGoogle Scholar
- Nanda M, Durrett R (2017) Spatial evolutionary games with weak selection. Proc Natl Acad Sci USA 114:6046–6051MathSciNetCrossRefzbMATHGoogle Scholar
- Sample C, Allen B (2017) The limits of weak selection and large population size in evolutionary game theory. J Math Biol 75:1285–1317MathSciNetCrossRefzbMATHGoogle Scholar
- Taylor C, Fudenberg D, Saskai A, Nowak MA (2004) Evolutionary game dynamics in finite populations. Bull Math Biol 66:1621–1644MathSciNetCrossRefzbMATHGoogle Scholar
- Wu B, Altrock PM, Wang L, Traulsen A (2010) Universality of weak selection. Phys Rev E 82:046106CrossRefGoogle Scholar
- Wu B, Gokhale CS, Wang L, Traulsen A (2012) How small are small mutation rates? J Math Biol 64:803–827MathSciNetCrossRefzbMATHGoogle Scholar
- Wu B, Garcia J, Hauert C, Traulsen A (2013) Extrapolating weak selection in evolutionary games. PLOS Comput Biol 9:e1003381CrossRefGoogle Scholar