Fictitious Play in Extensive-Form Games of Imperfect Information
Jason Castiglione, Gürdal Arslan
TL;DR
The paper investigates the long-run dynamics of fictitious play in repeated extensive-form games with imperfect information and perfect recall, where players hold erroneous beliefs about off-path moves and update decisions using empirical frequencies. It develops FP with local best replies and analyzes both the convergence of beliefs (empirical frequencies) and the realized play-paths, proving that, in a class of identical-interest EF games, empirical frequencies converge to the Nash equilibria set $\,\mathcal{E}\,$ under a non-increasing optimality-gap condition, with perfect information and almost perfect information falling into this class. It further strengthens the result by introducing inertia and fading memory, showing that FP-induced play-paths converge almost surely to an essentially pure Nash path, with empirical frequencies concentrating on that path. These results extend the Monderer-Shapley convergence from normal-form to extensive-form identical-interest games and establish the first strong convergence guarantees for FP in a substantial class of EF games, highlighting robust learning dynamics in strategic settings with imperfect information.
Abstract
We study the long-term behavior of the fictitious play process in repeated extensive-form games of imperfect information with perfect recall. Each player maintains incorrect beliefs that the moves at all information sets, except the one at which the player is about to make a move, are made according to fixed random strategies, independently across all information sets. Accordingly, each player makes his moves at any of his information sets to maximize his expected payoff assuming that, at any other information set, the moves are made according to the empirical frequencies of the past moves. We extend the well-known Monderer-Shapley result [1] on the convergence of the empirical frequencies to the set of Nash equilibria to a certain class of extensive-form games with identical interests. We then strengthen this result by the use of inertia and fading memory, and prove the convergence of the realized play-paths to an essentially pure Nash equilibrium in all extensive-form games of imperfect information with identical interests.