Exact value of the per-round-capped concentration game

Determine the exact minimax value of the per-round-capped concentration game for horizons $T\ge 2$, including the two-expert case and general numbers of experts, beyond the known one-round and cumulative-budget results.

Background

The paper obtains exact values for one round in the two-expert case and for arbitrary numbers of experts when the comparator relative-entropy budget saturates the simplex. It also establishes a horizon-free closed form under a cumulative information budget at constant scale from a finite horizon onward. In contrast, the per-round-capped game couples the rounds through the evolving learner distribution and has no known closed form beyond one round; the authors explicitly identify this as an open problem.

References

What remains open at $T\ge2$ is the per-round-capped game, whose reachable set still grows with the horizon through $V=T\beta$ (Section~\ref{sec:discussion}).

— The concentration game: Bayesian updating, regret, and information  (2608.18061 - Balsubramani, 18 Aug 2026) in Section 5.1, From per-round caps to a cumulative budget; Section 8.1, Exact values and rates

For $\Gamma<\Gamma_{\max}$ the optimal play moves off the uniform toward the prior --- already visible at $K=2$, where it sits strictly between the two --- and no comparable closed form is known beyond $K=2$ (Section~\ref{sec:discussion}).

— The concentration game: Bayesian updating, regret, and information  (2608.18061 - Balsubramani, 18 Aug 2026) in Appendix, Section \ref{app:one-round-exact}, Proposition \ref{prop:saturated-exact}; Section 8.1, Exact values and rates

But the loss couples the rounds through the path-dependent $p_t$, so the optimal schedule is again this control problem and a matching lower bound for the surrogate is unknown.

— The concentration game: Bayesian updating, regret, and information  (2608.18061 - Balsubramani, 18 Aug 2026) in Section 5.2, Exact variable-temperature decomposition; Section 8.1, Exact values and rates