Finite-time 4-expert prediction problem

We explicitly solve the nonlinear PDE that is the continuous limit of dynamic programing equation of the expert prediction problem in finite horizon setting with N = 4 experts. The expert prediction problem is formulated as a zero sum game between a player and an adversary. By showing that the solution is (Formula presented.) we are able to show that the comb strategies, as conjectured by Gravin et al., form an asymptotic Nash equilibrium. We also prove the “Finite vs Geometric regret” conjecture proposed in arXiv:1409.3040 for N = 4, and show that this conjecture in fact follows from the conjecture that the comb strategies are optimal for all N.