Relaxation of convexity in the state variable for fictitious play

Determine whether the convexity assumption in the state variable can be relaxed for the fictitious-play learning rule, including whether this relaxation is possible when the idiosyncratic volatility is strictly positive, by using more subtle arguments.

Background

The convergence analysis of the proposed fictitious-play procedure assumes that the terminal cost is convex in the state variable and that the running cost is convex, strongly in the control variable, in the finite-dimensional state and control arguments. The author notes that variants with small convexity defects or suitable lambda-convexity can be treated, but a general removal of convexity in the state variable was not obtained. The unresolved issue concerns whether this restriction is intrinsic to the control-averaging learning rule or can be overcome by alternative analytical methods, even when the idiosyncratic noise is nondegenerate.

References

It is unknown to us whether this is a feature specific to the learning rule eq: fictitious play or if the convexity in $x$ can be relaxed by using more subtle arguments.

— Learning controls in mean field games: the fictitious play and related gradient descent  (2610.06292 - Meynard, 5 Oct 2026) in Remark following Hypothesis 2 (Hypothesis \ref{hyp: convexity in x}), Section 1, Technical setting and notations

It is however unknown to us whether simple displacement monotonicity is sufficient to obtain convergence (as opposed to flat monotonicity+ convexity in the finite dimensional variables), and we leave such extension to the interested reader.

— Learning controls in mean field games: the fictitious play and related gradient descent  (2610.06292 - Meynard, 5 Oct 2026) in Section 2, Strong convergence in the convex setting, paragraph following Theorem \ref{thm: cv monotone setting}