# Imperfect Inspection Games Over Time

We consider an inspection game played on a 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. The inspection is not observed by the operator unless the inspector calls an alarm. The inspection is imperfect; it has a Type One Error which means that the inspector may call a false alarm (with probability alfa), and a Type Two Error which means that inspection may fail to detect (with probability beta) a violation which did occur. We first solve the game when alfa and beta are fixed and given. Then we consider the more general model in which the error probability alfa is chosen strategically by the inspector and may depend on the time of inspection. This yields two equilibria; one with constant alfa (and beta) and one with alfa increasing in time. The latter cannot be solved analytically. Consequently we solve a numerical example in which the inspction consist of obsderving a normally distributed signal.