Fictitious Play When Agents Are Not Expected-Utility Maximizers: A Geometric Analysis in $2$-Action Games

Fictitious play (FP) is one of the earliest and most fundamental dynamics in learning in games, where each agent independently best responds to the \emph{expected} payoff with respect to the empirical distribution of opponents' past actions. However, learning agents may not always be expected-utility maximizers: they may use \emph{nonlinear} evaluation functionals to account for the uncertainty induced by the opponents, e.g., via risk measures. In this work, we introduce $\rho$-fictitious play (\rhoFP), in which each pure action is evaluated by a general functional $\rho$ of the payoff distribution induced by the opponent's empirical play. We first identify the single-crossing property (SCP) on $\rho$ that preserves the classical geometry for FP in 2-action games. Furthermore, we conduct a novel geometric analysis of the convergence behavior of \rhoFP in 2 × 2 games, even when the SCP fails. In addition, we show that convergence of $\rho$-FP in 2 × $n$ ($n\ge3$) games does not always hold, by establishing nonconvergence of the continuous-time CVaR-FP in a 2 × 3 game.