From Fixation Probabilities to d-player Games: An Inverse Problem in Evolutionary Dynamics

  • Fabio A. C. C. Chalub
  • Max O. SouzaEmail author
Special Issue: Modelling Biological Evolution: Developing Novel Approaches


The probability that the frequency of a particular trait will eventually become unity, the so-called fixation probability, is a central issue in the study of population evolution. Its computation, once we are given a stochastic finite population model without mutations and a (possibly frequency dependent) fitness function, is straightforward and it can be done in several ways. Nevertheless, despite the fact that the fixation probability is an important macroscopic property of the population, its precise knowledge does not give any clear information about the interaction patterns among individuals in the population. Here we address the inverse problem: from a given fixation pattern and population size, we want to infer what is the game being played by the population. This is done by first exploiting the framework developed in Chalub and Souza (J Math Biol 75:1735–1774, 2017), which yields a fitness function that realises this fixation pattern in the Wright–Fisher model. This fitness function always exists, but it is not necessarily unique. Subsequently, we show that any such fitness function can be approximated, with arbitrary precision, using d-player game theory, provided d is large enough. The pay-off matrix that emerges naturally from the approximating game will provide useful information about the individual interaction structure that is not itself apparent in the fixation pattern. We present extensive numerical support for our conclusions.


Wright–Fisher process Fixation probability Game theory Inverse problems 

Mathematics Subject Classification

91A06 91A22 91A80 92D15 



FACCC was partially supported by FCT/Portugal Strategic Projects UID/MAT/00297/2013 and UID/MAT/00297/2019 (Centro de Matemática e Aplicações, Universidade Nova de Lisboa) and by a “Investigador FCT” grant. MOS was partially supported by CNPq under Grants # 486395/2013-8 and # 309079/2015-2, and by CAPES—Finance Code 001. We also thank the useful comments by two anonymous reviewers, which helped to improve the manuscript.


  1. Alesina A, Galuzzi M (2000) Vincent’s theorem from a modern point of view. Rend Circ Mat Palermo, Ser II Suppl 64:179–191MathSciNetzbMATHGoogle Scholar
  2. Allgower E L, Georg K (2012) Numerical continuation methods: an introduction, vol 13. Springer, BerlinzbMATHGoogle Scholar
  3. Bürger R (2000) The mathematical theory of selection, recombination and mutation. Wiley, Chichester ISBN 0-471-98653-4/hbkzbMATHGoogle Scholar
  4. Chalub FACC, Souza MO (2009) From discrete to continuous evolution models: a unifying approach to drift-diffusion and replicator dynamics. Theor Pop Biol 76(4):268–277zbMATHCrossRefGoogle Scholar
  5. Chalub FACC, Souza MO (2014) The frequency-dependent Wright–Fisher model: diffusive and non-diffusive approximations. J Math Biol 68(5):1089–1133MathSciNetzbMATHCrossRefGoogle Scholar
  6. Chalub FACC, Souza MO (2016) Fixation in large populations: a continuous view of a discrete problem. J Math Biol 72(1–2):283–330MathSciNetzbMATHCrossRefGoogle Scholar
  7. Chalub FACC, Souza MO (2017) On the stochastic evolution of finite populations. J Math Biol 75(6):1735–1774MathSciNetzbMATHCrossRefGoogle Scholar
  8. Chalub FA, Souza MO (2018) Fitness potentials and qualitative properties of the Wright–Fisher dynamics. J Theor Biol 457:57–65 (ISSN 0022-5193)MathSciNetCrossRefGoogle Scholar
  9. Czuppon P, Gokhale CS (2018) Disentangling eco-evolutionary effects on trait fixation. Theor Pop Biol 124:93–107zbMATHCrossRefGoogle Scholar
  10. Czuppon P, Traulsen A (2018) Fixation probabilities in populations under demographic fluctuations. J Math Biol 77(4):1233–1277MathSciNetzbMATHCrossRefGoogle Scholar
  11. Ethier SN, Kurtz TG (1986) Markov processes: characterization and convergence. Wiley series in probability and mathematical statistics: probability and mathematical statistics. Wiley, New York (ISBN 0-471-08186-8)zbMATHCrossRefGoogle Scholar
  12. Ewens WJ (2004) Mathematical population genetics. I: theoretical introduction, 2nd edn. Interdisciplinary mathematics 27. Springer, New YorkzbMATHCrossRefGoogle Scholar
  13. Fisher RA (1922) On the dominance ratio. Proc R Soc Edinb 42:321–341. CrossRefGoogle Scholar
  14. Gokhale CS, Traulsen A (2010) Evolutionary games in the multiverse. Proc Natl Acad Sci USA 107(12):5500–5504MathSciNetCrossRefGoogle Scholar
  15. Gokhale CS, Traulsen A (2014) Evolutionary multiplayer games. Dyn Games Appl 4(4):468–488MathSciNetzbMATHCrossRefGoogle Scholar
  16. Gzyl H, Palacios JL (2003) On the approximation properties of bernstein polynomials via probabilistic tools. Boletín de la Asociación Matemática Venezolana 10(1):5–13MathSciNetzbMATHGoogle Scholar
  17. Hartl DL, Clark AG (2007) Principles of population genetics. Sinauer, MassachussetsGoogle Scholar
  18. Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  19. Huang W, Hauert C, Traulsen A (2015) Stochastic game dynamics under demographic fluctuations. Proc Natl Acad Sci 112(29):9064–9069CrossRefGoogle Scholar
  20. Imhof LA, Nowak MA (2006) Evolutionary game dynamics in a Wright–Fisher process. J Math Biol 52(5):667–681MathSciNetzbMATHCrossRefGoogle Scholar
  21. Kurokawa S, Ihara Y (2009) Emergence of cooperation in public goods games. Proc R Soc B Biol Sci 276(1660):1379–1384CrossRefGoogle Scholar
  22. Lane JM, Riesenfeld RF (1983) A geometric proof for the variation diminishing property of b-spline approximation. J Approx Theory 37(1):1–4MathSciNetzbMATHCrossRefGoogle Scholar
  23. Lessard S (2011) On the robustness of the extension of the one-third law of evolution to the multi-player game. Dyn Games Appl 1(3):408–418MathSciNetzbMATHCrossRefGoogle Scholar
  24. Nowak MA (2006) Evolutionary dynamics: exploring the equations of life. The Belknap Press of Harvard University Press, CambridgezbMATHGoogle Scholar
  25. Pacheco JM, Santos FC, Souza MO, Skyrms B (2009) Evolutionary dynamics of collective action in n-person stag hunt dilemmas. Proc R Soc Lond B Biol Sci 276(1655):315–321CrossRefGoogle Scholar
  26. Peña J, Lehmann L, Nöldeke G (2014) Gains from switching and evolutionary stability in multi-player matrix games. J Theor Biol 346:23–33CrossRefGoogle Scholar
  27. Phillips GM (2003) Interpolation and approximation by polynomials. CMS books in mathematics. Springer, New YorkCrossRefGoogle Scholar
  28. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in C: the art of scientific computing. Cambridge University Press, CambridgezbMATHGoogle Scholar
  29. Rouillier F, Zimmermann P (2004) Efficient isolation of polynomial’s real roots. J Comput Appl Math 162(1):33–50MathSciNetzbMATHCrossRefGoogle Scholar
  30. Smith JM (1982) Evolution and the theory of games. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  31. Smith JM, Price GR (1973) The logic of animal conflict. Nature 246(5427):15–18zbMATHCrossRefGoogle Scholar
  32. Souza MO, Pacheco JM, Santos FC (2009) Evolution of cooperation under n-person snowdrift games. J Theor Biol 260(4):581–588MathSciNetzbMATHCrossRefGoogle Scholar
  33. Taylor PD, Jonker LB (1978) Evolutionary stable strategies and game dynamics. Math Biosci 40(1–2):145–156MathSciNetzbMATHCrossRefGoogle Scholar
  34. Taylor HM, Karlin S (1998) An introduction to stochastic modeling, 3rd edn. Academic Press Inc., San Diego, CA ISBN 0-12-684887-4zbMATHGoogle Scholar
  35. Wright S (1937) The distribution of gene frequencies in populations. Proc Natl Acad Sci USA 23:307–320zbMATHCrossRefGoogle Scholar
  36. Wright S (1938) The distribution of gene frequencies under irreversible mutations. Proc Natl Acad Sci USA 24:253–259zbMATHCrossRefGoogle Scholar
  37. Wu B, Traulsen A, Gokhale CS (2013) Dynamic properties of evolutionary multi-player games in finite populations. Games 4(2):182–199MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.Departamento de Matemática and Centro de Matemática e Apliçoes, Faculdade de Ciências e TecnologiaUniversidade Nova de LisboaCaparicaPortugal
  2. 2.Instituto de Matemática e Estatística, Universidade Federal FluminenseNiteróiBrasil

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