Journal of Mathematical Biology

, Volume 78, Issue 1–2, pp 135–154 | Cite as

Extrapolating weak selection in evolutionary games

  • Zhuoqun Wang
  • Rick DurrettEmail author


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 



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.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Nanjing UniversityNanjingChina
  2. 2.Duke UniversityDurhamUSA

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