  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.
