Penalty and Crime with Lumpy Choices
| Date | 01 October 2004 |
| DOI | 10.1177/0951629804046148 |
| Published date | 01 October 2004 |
| Subject Matter | Articles |
PENALTY AND CRIME WITH LUMPY CHOICES
SOME FURTHER CONSIDERATIONS
Laurent Franckx
ABSTRACT
This paper clarifies an issue in the Hirshleifer and Rasmusen–Tsebelis con-
troversy on the effects of penalties on crime: what is the effect of penalties if
the transgression of law has a discrete nature and if the law enforcer cannot
act as Stackelberg leader? We differentiate between technical (compliance
costs) and institutional (penalties) parameters in the potential transgressor’s
payoff’s functions. Depending on the penalty structure, we obtain equilibria
either in pure or in mixed strategies. We confirm that the role played by penal-
ties in mixed strategy equilibria is fundamentally different from the role they
play in pure strategies. We also identify sufficient conditions for the imple-
mentability and uniqueness of given equilibria when there are restrictions on
the penalties and/or on the incentive schemes for the law enforcers. Finally,
we give a rationalization for the use of mixed strategies as a solution concept
in law enforcement games.
KEY WORDS .economic analysis of law enforcement
1. Introduction
In the first modern economic analysis of crime and punishment, Becker
(1968) suggested that optimal law enforcement would consist in combining
the highest possible penalties with the lowest possible probabilities of
apprehension.
In a series of challenging papers, Tsebelis (1990a, b, 1993) has argued that
this economic approach to crime is fundamentally flawed. According to
Journal of Theoretical Politics 16(4): 403–421 Copyright &2004 Sage Publications
DOI: 10.1177/0951629804046148 London, Thousand Oaks, CA and New Delhi
I would like to thank James Johnson, co-editor of this journal, and three anonymous referees
for their constructive and encouraging comments on this paper. I would also like to thank
Matthew Braham, Winston Harrington, Luc Lauwers, Gerd Mu
¨hlheusser, Stef Proost, Erik
Schokkaert and Frans Spinnewyn for helpful suggestions on an earlier version. I benefited
from the interaction with participants at the ERASMUS Programme in Law and Economics
Annual Conference 2001 (Hamburg), at the Sixth Spring Meeting of Young Economists (Copen-
hagen) and at the 2001 Annual Meeting of the European Public Choice Society (Paris). Finally,
Isabelle Brose provided useful editorial assistance. The usual disclaimer applies.
Tsebelis, higher penalties would not lead to lower crime but to lower crime
enforcement. In equilibrium, the level of crime in society would then be inde-
pendent from penalties.
This point has been criticized in turn (see, for instance, Weissing and
Ostrom (1991) and Hirshleifer and Rasmusen (1992)).
In this paper, we further investigate some of the points raised by Hirsh-
leifer and Rasmusen (1992) and Tsebelis’ (1993) answer. We show that a
more explicit treatment of the payoff structure of the police–criminal game
allows not only clarification of several of the issues raised in this discussion
but also several new topics (such as implementability and uniqueness of
the equilibria) to be tackled providing a justification for the use of mixed
strategies as a solution concept in the enforcement game.
Let us first summarize briefly the controversy.
Hirshleifer and Rasmusen (1992) assert that Tsebelis’ results (in their
terminology, the Payoff Irrelevance Proposition [PIP] are due to some very
specific assumptions used).
First, Tsebelis assumes that the police–criminal game is simultaneous.
Therefore, he uses the Nash Equilibrium solution concept: the criminals’
actions must be optimal, given the police’s behavior, but the police’s behavior
must also be optimal, given the criminals’ behavior. Tsebelis then obtains an
equilibrium in mixed strategies, where the probability of compliance does not
depend on the magnitude of the penalty. Indeed, in a mixed-strategy Nash
Equilibrium, criminals choose the probability of compliance to make the
police indifferent between enforcing and not enforcing the law. As long as
the penalty has no intrinsic utility for the police, it should thus not play a
role in the equilibrium strategy of the criminals.1Hirshleifer and Rasmusen
argue that the police–criminal game is more appropriately modeled as a
sequential game, where the police announce a policy and stick to it, even if
it is not an ex post optimal response to the criminals’ behavior. They argue
that PIP does not hold if the police can act as a Stackelberg leader.
A second point raised by Hirshleifer and Rasmusen (1992) is that in
Tsebelis’ approach, both criminals and police face lumpy choices. Hirshleifer
and Rasmusen argue that it is more appropriate to assume that criminals and
police actually choose their crime level and their enforcement effort along a
continuum. They then show that PIP does not hold under this alternative
approach, even using the Nash Equilibrium as a solution concept. Indeed,
with continuous action spaces, they obtain equilibria in pure strategies.
Moreover, they also consider a game with lumpy choices. They compare
the equilibrium with the equilibrium in an equivalent game with continuous
choices and obtain mixed results:
404 JOURNAL OF THEORETICAL POLITICS 16(4)
1. A similar result has been obtained by Holler (1993).
To continue reading
Request your trial