On the Multistage Shortest Path Problem Under Distributional Uncertainty

In this paper, we consider an ambiguity-averse multistage network game between a user and an attacker. The arc costs are assumed to be random variables that satisfy prescribed first-order moment constraints for some subsets of arcs and individual probability constraints for some particular arcs. The user aims at minimizing its cumulative expected loss by traversing between two fixed nodes in the network, while the attacker’s objective is to maximize the user’s expected loss by selecting a distribution of arc costs from the family of admissible distributions. In contrast to most of the related studies, both the user and the attacker can dynamically adjust their decisions at particular nodes of the user’s path. By observing the user’s decisions, the attacker may reveal some additional distributional information associated with the arcs emanated from the current user’s position. It is shown that the resulting multistage distributionally robust shortest path problem (DRSPP) admits a linear mixed-integer programming reformulation (MIP). In particular, we distinguish between acyclic and general graphs by introducing different forms of non-anticipativity constraints. Finally, we perform a numerical study, where the quality of adaptive decisions and computational tractability of the proposed MIP reformulation are explored with respect to several classes of synthetic network instances.