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.
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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.
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Chalub, F.A.C.C., Souza, M.O. From Fixation Probabilities to d-player Games: An Inverse Problem in Evolutionary Dynamics. Bull Math Biol 81, 4625–4642 (2019). https://doi.org/10.1007/s11538-018-00566-w
- Wright–Fisher process
- Fixation probability
- Game theory
- Inverse problems
Mathematics Subject Classification