The Inspection Game

A strictly competitive game with no pure equilibrium: the inspector wants to match, the evader wants to miss.

There is a comparative static here that surprises everybody and is genuinely useful. Because each side mixes in order to make the OTHER indifferent, raising the penalty for being caught does not change how often violators are caught — it changes how often inspectors need to inspect. Tougher penalties reduce inspection frequency, not violation frequency, unless the inspection rate is held fixed by something outside the game. That result is counter-intuitive, well established, and routinely ignored in policy.

How it feels

• One side gains exactly when they coincide, the other exactly when they do not.
• Any regularity in one side's behaviour is immediately exploitable.
• Coverage is partial and cannot be made complete.
• Both sides are watching each other for a pattern.

Who is playing

Inspector and evader, plus whoever sets the inspector's budget. Coverage rate is chosen outside the game, and it is the only lever that actually moves the violation rate.

Where it lands

Both sides mix; a stable rate of violation and a stable rate of detection, neither of which moves when penalties change.

What moves it

Use genuine randomisation, audit it for pattern, and understand that penalty severity and inspection frequency do different jobs. If you want the violation rate down, the frequency is the lever.

Not to be confused with

Nearly every misread in practice is a shape mistaken for its neighbour — and each of those mistakes points you at the wrong fix.

Where you have already met it