A sequential Stackelberg game for dynamic inspection problems

Abstract

We introduce an inspection game where one inspector has the role of monitoring a group of inspectees. The inspector has the resources to visit only a few of them. Visits are performed sequentially with no repetitions. The inspectees report and share the sequence of inspections as they occur, but otherwise, they do not cooperate. We formulate two Stackelberg models, a static game where the inspector commits to play a sequence of visits announced at the start of the game, and a dynamic game where visits will depend on who was visited previously. In the static game, we characterize the (randomized) inspection paths in an equilibrium using linear programs. In the dynamic game, we determine the inspection paths in an equilibrium using backward induction. Our paper focuses on the mathematical structure of the equilibria of this sequential inspection game, where the inspector can perform exactly two visits. In the static game, the inspection paths are solutions to a transportation problem. We use this equivalence to determine an explicit solution to the game and to show that set of inspection path probabilities in an equilibrium, projected onto its first and second visit marginals, is convex. We discuss how the static and dynamic games relate to each other and how to use these models in practical settings.