On the asymptotic behavior of a discrete time inspection game

Abstract

In many material processing and storing plants an inspector performs during some reference time interval, e.g. one year, a number of inspections because it can not be excluded that the plant operator acts illegally by violating agreed rules, e.g., diverts valuable material. The inspections guarantee that any illegal action is detected at the earliest inspection following the beginning of that illegal action. We assume that the inspector wants to choose the time points for his inspections such that the time which elapses between the beginning of the illegal action and its detection is minimized whereas the operator wants to start his illegal action such that the elapsed time is maximized. Therefore, this inspection problem is modelled as a zero‐sum game with strategies and payoffs as described. Depending on the concrete situation the start of the illegal action and the inspections can take place either at a finite number of time points or at every time point of a reference period. The first case can be modelled as a zero‐sum game with finite pure strategy sets while the latter one leads to a zero‐sum game with infinite pure strategy sets and discontinuous payoff kernel.

The aim of this contribution is to demonstrate the close relation between both games for the case of one interim inspection.

Key words:

• inspection game
• zero‐sum game
• continuous time game
• discontinuous payoff kernel