Bayesian Replicator Dynamics and the Inversion of Strict Dominance in Bimatrix Games

Replicator dynamics, the central model of evolutionary game theory, describe populations evolving under selection and agents learning by trial and error. They also admit a third use: continuous-time Bayesian inference, in which an agent planning against an opponent accumulates evidence for each candidate strategy. Here we show that this use divides all two-player games into two classes. With the standard expected-utility payoff, the dynamics can be expressed as Bayesian updating only in games (with any number of actions) where no player has a dominant strategy. The logarithmic payoff of Ay and Erb admits the Bayesian representation in every game. Coordination, anti-coordination, and cyclic games share qualitative dynamics under both payoff definitions, with equilibria displaced and selection thresholds shifted. In contrast, the flows of every game with a dominant strategy are reversed between the payoff definitions, with the strictly dominant strategy eliminated under logarithmic payoffs. Cooperation therefore emerges in the Prisoner's Dilemma, and games of common interest converge to mutual defection. For two-action games the classification is complete and rests on an exact Hamiltonian structure shared by both payoff definitions. The structure gives a behavioural test: cycles in the classic Matching Pennies game have amplitude-independent period (isochronous) under logarithmic payoffs, but not under expected utility. Logarithmic evidence integration thus determines which strategic outcomes dynamical selection can reach, wherever learning approximates Bayesian updating.