Hamilton’s rule in non-additive games

Recently a number of authors have questioned both the validity and utility of inclusive fitness. One particular claim is that Hamilton’s rule applies only to additive games. Additive games represent a vanishingly small subset of all games and do not capture a number of interesting qualitative behaviours which are present in non-additive games. Thus, if these criticisms were correct, inclusive fitness would be a severely limited theoretical tool. We show these criticisms are not valid by demonstrating that any symmetric game can be transformed into an additive payoff matrix in such a way that the action of selection remains unchanged. The result comes with a caveat, however, which is that terms in the payoff matrix must themselves be frequency dependent. Despite this, we demonstrate the utility of inclusive fitness by means of applying Hamilton’s rule to two such non-additive games. The central claim of inclusive fitness is that relatedness is the key to cooperation, we show that this remains true even for non-additive games.