Spot-fare inspection in urban bus transportation systems: strategy and unpredictability under a Stackelberg game approach

This study addresses the operational implementation of a spot-fare inspection strategy on a proof-of-payment urban bus transportation system, where opportunistic passengers can evade fare payment by the most convenient path. The spot-fare inspection strategy defines the frequency at which the transit authority should control sites of the transportation network to inhibit the action of opportunistic passengers. The operational implementation is done using an unpredictable allocation schedule, where the transit authority selects an allocation schedule of $n$ sites to be controlled (one for each inspection team) each day with some probability. The challenge is to determine the set of allocation schedules and their respective probabilities of being selected whose systematic day-to-day application matches the inspection frequencies defined by the spot strategy. The interaction between transit authority and opportunistic passengers is modeled as a Leader-Follower Stackelberg game, where the decision of opportunistic passengers to evade the fare payment and the path to take depends on the passengers’ observations on the inspection frequencies set by the transit authority. We consider that the transit authority implements a vehicle selective inspection policy and an on board passengers mass inspection policy, with and without interruption of the bus schedule, representing two real approaches to fare inspection.