  We consider a sequential inspection game where an inspector uses a limited
number of inspections over a larger number of time periods to detect a
violation (an illegal act) of an inspectee. Compared with earlier models, we
allow varying rewards to the inspectee for successful violations. As one
possible example, the most valuable reward may be the completion of a sequence
of thefts of nuclear material needed to build a nuclear bomb. The inspectee can
observe the inspector, but the inspector can only determine if a violation
happens during a stage where he inspects, which terminates the game; otherwise
the game continues. Under reasonable assumptions for the payoffs, the
inspector's strategy is independent of the number of successful violations.
This allows to apply a recursive description of the game, even though this
normally assumes fully informed players after each stage. The resulting
recursive equation in three variables for the equilibrium payoff of the game,
which generalizes several other known equations of this kind, is solved
explicitly in terms of sums of binomial coefficients. We also extend this
approach to non-zero-sum games and, similar to Maschler (1966), "inspector
leadership" where the inspector commits to (the same) randomized inspection
schedule, but the inspectee acts legally (rather than mixes as in the
simultaneous game) as long as inspections remain.
