From fixation probabilities to d-player games: an inverse problem in evolutionary dynamics

TL;DR

This paper develops a method to infer the underlying game interactions in a population from observed fixation probabilities, using inverse modeling and d-player game approximation to reveal hidden interaction structures.

Contribution

It introduces an inverse approach to deduce the game-theoretic interactions from fixation patterns, extending the analysis with d-player game approximations for detailed insights.

Findings

Any fixation pattern can be realized by a fitness function in the Wright-Fisher model.

Such fitness functions can be approximated by d-player games with arbitrary precision.

The resulting payoff matrix reveals interaction structures not apparent from fixation data.

Abstract

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 FACC Chalub and MO Souza, J. Math. Biol. 75: 1735, 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.