Strategies for convex potential games and an application to decision-theoretic online learning

The backwards induction method due to Bellman~\cite{bellman1952theory} is a popular approach to solving problems in optimiztion, optimal control, and many other areas of applied math. In this paper we analyze the backwords induction approach, under min/max conditions. We show that if the value function is has strictly positive derivatives of order 1-4 then the optimal strategy for the adversary is Brownian motion. Using that fact we analyze different potential functions and show that the Normal-Hedge potential is optimal.