Sampled fictitious play for multi-action stochastic dynamic programs

We introduce a class of finite-horizon dynamic optimization problems that we call multiaction stochastic dynamic programs (DPs). Their distinguishing feature is that the decision in each state is a multi-dimensional vector. These problems can in principle be solved using Bellman’s backward recursion. However, complexity of this procedure grows exponentially in the dimension of the decision vectors. This is called the curse of action-space dimensionality. To overcome this computational challenge, we propose an approximation algorithm rooted in the game theoretic paradigm of Sampled Fictitious Play (SFP). SFP solves a sequence of DPs with a one-dimensional action-space, which are exponentially smaller than the original multi-action stochastic DP. In particular, the computational effort in a fixed number of SFP iterations is linear in the dimension of the decision vectors. We show that the sequence of SFP iterates converges to a local optimum, and present a numerical case study in manufacturing where SFP is able to find solutions with objective values within 1% of the optimal objective value hundreds of times faster than the time taken by backward recursion. In this case study, SFP solutions are also better by a statistically significant margin than those found by a one-step lookahead heuristic. ∗Corresponding author. Industrial and Systems Engineering, Box 352650, The University of Washington, Seattle, WA 98195. Email: [email protected].