Imperfect inspection games over time

We consider an inspection game played on a continuous finite time interval. The inspector wishes to detect a violation as soon as possible after it has been made by the operator. The loss to the inspector is assumed to be linear in the duration of the time elapsed between the violation and its detection. This paper is mostly an extension of Diamond's models for a single inspection, which includes the uncertainty aspect, by relaxing the assumption that the inspection is perfect. Here the inspection is imperfect; it has a Type One Error which means that the inspector may call a false alarm (with probability α), and a Type Two Error which means that the inspection may fail to detect (with probability β) a violation which did occur. In addition we will assume that the inspection is silent, i.e., the operator is unaware of the inspection when it takes place, unless the inspector calls a false alarm.