Boundary-only stability of stochastic FTRL limit sets

Establish whether, for finite games and standard regularizers, the only stochastically asymptotically stable limit sets of the stochastic follow-the-regularized-leader (S-FTRL) dynamics are entirely contained in the boundary of the game’s strategy space; that is, determine if every stochastically asymptotically stable set for S-FTRL must lie on the boundary.

Background

Throughout the paper the authors analyze stochastic follow-the-regularized-leader (S-FTRL) dynamics and show that randomness drives play toward the boundary while classifying stable attractors via closedness: a span of pure strategies is stochastically asymptotically stable if and only if it is closed. They also show that in harmonic games no proper face is stochastically asymptotically stable.

Given the pervasive tendency of trajectories under uncertainty to drift toward extremal (boundary) regions, the authors propose a stronger principle: that any stochastically asymptotically stable limit set should be entirely on the boundary. This would sharpen their main stability characterization by eliminating the possibility of interior stable limit sets altogether.