Converse between CCE convergence rate and intrinsic time

Prove or disprove whether, under the self-tuned intrinsic-time regret-matching schedule, an improved coarse-correlated-equilibrium rate $d(\sigma_T,\mathrm{CCE})=o(T^{-1/2})$ necessarily implies $\max_k V_T^{(k)}=o(T)$.

Background

The paper proves the forward implication that sublinear realized intrinsic time yields an improved CCE distance rate. However, the ledger contains a potentially offsetting retempering drift, and at the headline constant its residual coefficient vanishes. The authors therefore cannot derive the converse from their envelope and explicitly state that the converse is undecided.

References

So in the setting of Theorem~\ref{thm:k-player-ledger} under the self-tuned schedule, whether $d(\sigma_T,)=o(T{-1/2})$ forces $\max_k V_T{(k)}=o(T)$ is undecided, and deciding it needs a lower bound the envelope cannot supply at that constant.

— The concentration game: Bayesian updating, regret, and information  (2608.18061 - Balsubramani, 18 Aug 2026) in Appendix, Section \ref{app:cce-converse}; Section 6.2, General-sum play and equilibrium consequences

An exact growth-rate law for $V_T{(k)}$, and a matching lower bound on the non-stabilizing polymatrix class, remain open.

— The concentration game: Bayesian updating, regret, and information  (2608.18061 - Balsubramani, 18 Aug 2026) in Section 8.1, Exact values and rates