When enforcement is imperfect, so the probability that any given violation will be punished is less than 100%, it is often said that the ideal penalty (insofar as deterrence is concerned) equals the harm caused by the violation multiplied by one over the probability of punishment. In most contexts where enforcement is imperfect, however, the probability of punishment will decline with any improvement in a defendant's behavior. If so, the deterrent effect will differ depending on whether the multiplier is calculated (1) case by case, to reflect each defendant's actual probability of punishment, or (2) on an average basis, to reflect the average probability of punishment facing all defendants. The deterrent effect will also be different if the law uses (3) a constant fine, based on the average probability of punishment and the average harm.
This paper analyzes the deterrent effects of all three regimes (focusing on the latter two, which are much more common in real legal systems). Under the latter two regimes, optimal deterrence generally is not achieved by following the conventional wisdom, and setting the expected punishment equal to the expected harm divided by the probability of punishment. Under the second regime, optimal penalties will be (weakly) lower than this benchmark; under the third regime, optimal penalties could be either higher or lower. The paper also discusses the administrative differences between the regimes, such as the informational demands they place on the legal system, or the effect of each regime on risk-aversion and on litigation costs.