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 chooses a best response to the empirical distribution of opponents' past actions by maximizing \emph{expected} payoff. 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 study Nonlinear Fictitious Play (NL-FP), in which each pure action is evaluated by a general, possibly nonlinear functional of the payoff distribution induced by the opponent's empirical play. We first identify the single-crossing property (SCP) that preserves the classical geometry for FP in $2 \times 2$ games. Furthermore, we conduct a novel geometric analysis of the convergence behavior of NL-FP in $2 \times 2$ games, even when the SCP fails. Beyond $2 \times 2$ games, we show that convergence of NL-FP in $2 \times n$ ($n\ge3$) games may fail for continuous-time CVaR-FP in a $2 \times 3$ game. In addition, we identify a finite-switching condition that restores convergence of continuous-time NL-FP in general \(2\times n\) games, which provides new insights into the global landscape of the dynamics even for classical FP in such games.