On the convergence of fictitious play algorithm in repeated games via the geometrical approach

TL;DR

This paper introduces a new class of 3x3 games without IIP where fictitious play converges, using a geometrical approach and a novel projection mapping to analyze the dynamics.

Contribution

It identifies a new class of games with convergence of fictitious play using geometric methods and extends the projection technique to higher-dimensional games.

Findings

Identified 3x3 games without IIP with FPP.

Developed a new projection mapping for dynamical systems.

Proved convergence of CFP in the new class of games.

Abstract

As the earliest and one of the most fundamental learning dynamics for computing NE, fictitious play (FP) has being receiving incessant research attention and finding games where FP would converge (games with FPP) is one central question in related fields. In this paper, we identify a new class of games with FPP, i.e., $3\times3$ games without IIP, based on the geometrical approach by leveraging the location of NE and the partition of best response region. During the process, we devise a new projection mapping to reduce a high-dimensional dynamical system to a planar system. And to overcome the non-smoothness of the systems, we redefine the concepts of saddle and sink NE, which are proven to exist and help prove the convergence of CFP by separating the projected space into two parts. Furthermore, we show that our projection mapping can be extended to higher-dimensional and degenerate games.