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Louis Kaplow & Steven Shavell, Optimal Law Enforcement with Self-Reporting of Behavior, 102 J. Pol. Econ. 583 (1994).

Abstract: Self-reporting--the reporting by parties of their own behavior to an enforcement authority--is a commonly observed aspect of law enforcement, such as in the context of environmental and safety regulation. We add self-reporting to the model of the control of harmful externalities through probabilistic law enforcement, and we characterize the optimal scheme. Self-reporting offers two advantages over schemes without self-reporting: enforcement resources are saved because individuals who report their harmful acts need not be detected, and risk is reduced because individuals who report their behavior bear certain rather than uncertain sanctions.