Prediction with Five Experts and Geometric Stopping: A Probabilistic Construction and Analytic Verification
TL;DR
It is shown that the alternating COMB control attains the limiting Hamiltonian maximum exactly when the two highest scores coincide and the third- and fourth-highest scores coincide, and that the alternating COMB control attains the limiting Hamiltonian maximum exactly when the two highest scores coincide.
Abstract
We present a solution of the limiting five-expert prediction problem with geometric stopping, based on stochastic calculus and partial differential equations. If $\delta$ is the stopping probability, the minimax expected regret from tied initial scores is $45\pi^2/(512\sqrt{2\delta})+o(\delta^{-1/2})$ as $\delta\downarrow0$. The adversarial rank control selecting the best and third-best experts maximizes the limiting Hamiltonian at every state. Following the four-expert construction of Bayraktar, Ekren, and Zhang (2020), we represent the value correction as a discounted boundary-local-time expectation for a degenerate obliquely reflected Brownian motion. The hyperbolic systems governing its boundary traces are derived and solved explicitly. To verify the nonlinear Hamilton--Jacobi--Bellman equation, we combine equality directions, hyperbolic comparison principles, and averaging across controls. These arguments reduce the 48 regional control inequalities to lower-dimensional boundary problems, with the remaining signs established by explicit monotonicity arguments. We prove global $C^2$ regularity across both regional interfaces and changes in rank ordering. We also show that the alternating COMB control attains the limiting Hamiltonian maximum exactly when the two highest scores coincide and the third- and fourth-highest scores coincide.