  Monte Carlo Tree Search (MCTS), most famously used in game-play artificial
intelligence (e.g., the game of Go), is a well-known strategy for constructing
approximate solutions to sequential decision problems. Its primary innovation
is the use of a heuristic, known as a default policy, to obtain Monte Carlo
estimates of downstream values for states in a decision tree. This information
is used to iteratively expand the tree towards regions of states and actions
that an optimal policy might visit. However, to guarantee convergence to the
optimal action, MCTS requires the entire tree to be expanded asymptotically. In
this paper, we propose a new technique called Primal-Dual MCTS that utilizes
sampled information relaxation upper bounds on potential actions, creating the
possibility of "ignoring" parts of the tree that stem from highly suboptimal
choices. This allows us to prove that despite converging to a partial decision
tree in the limit, the recommended action from Primal-Dual MCTS is optimal. The
new approach shows significant promise when used to optimize the behavior of a
single driver navigating a graph while operating on a ride-sharing platform.
Numerical experiments on a real dataset of 7,000 trips in New Jersey suggest
that Primal-Dual MCTS improves upon standard MCTS by producing deeper decision
trees and exhibits a reduced sensitivity to the size of the action space.
