Optimal law enforcement with multiple criminalorganizations
Ken Yahagi ∗September, 8, 2016

Abstract
This paper develops a model with regard to the optimal law enforcement on organized crime with multiple criminal organizations (the Maﬁas) which regulate
the criminal market by extortion. It enables us to consider strategic interaction relationships among the criminal organizations and the government. This
paper shows that welfare eﬀects of introducing multiple criminal organizations
depends on how they face a competition in this criminal market. If each criminal organization faces a ﬁerce competition, it is hard for criminal organizations
to make proﬁts and play the role as a regulator. On the other hand, if there
exists no competition among criminal organizations, introducing multiple criminal organizations contributes to the social welfare improvement. This indicatesthat the classical view stressing the desirable eﬀects of monopolistic criminalorganization is not always supported. Furthermore, by considering the costly
conﬂicts among criminal organizations, this paper also explores whether harsh
penalties on the criminal market induce them to engage in conﬂicts and showsthat it can be diﬃcult to sustain the desirable criminal markets regulated bymultiple criminal organizations.
keywords: Conﬂict, Oligopoly, Organized CrimeJEL Classiﬁcation: D74, K4, L13,

∗ Graduate School of Economics, Waseda University, 1-6-1, Nishiwaseda, Shinjuku-ku,
Tokyo, 169-8050, Japan. Email: yhgkn@moegi.waseda.jp

1

1

Introduction

The literature on the economics of crime has focused on the rational individual’s
decision problem as regards whether to engage in criminal activities. Such ananalysis was originally started with Becker’s (1968) seminal work. Based on
the Becker’s approach, a lot of researchers have tried to examine what is the
optimal deterrence policy the government should take and its eﬀect on social
welfare and eﬃciency. See Garoupa (1997) and Polinsky and Shavell (2000) foran overview.
Even if organized crime is also an important issue for economics, the economic analysis of organized crime is still scarce as pointed out by Fiorentini
and Peltzman (1995). However, there exist also some important studies aboutorganized crime. Almost previous papers have stressed the welfare comparisons
between monopolistic criminal markets (with a criminal organization) and competitive criminal markets (without criminal organization) because a criminalorganization is thought of a monopolistic ﬁrm, i.e., Schelling (1971), Buchanan(1973), Gambetta and Reuter (1995) and Garoupa (2000)1 . According to sucha monopolistic view, the monopolistic criminal market is desirable than competitive criminal market because social bads, like criminal activities, supplied
by a criminal organization become smaller.
However, such a monopolistic view on a criminal organization is true ? In
reality, we often observe miserable conﬂicts among criminal organizations as inJapan, Italy and other countries. Considering such an actual aﬀair, focusing
on only monopolistic markets provides limited meaningful implications. Also,
Fiorentini (1995) argues that there exists no convincing reason to support themonopolistic view of illegal markets. Therefore, by extending to multiplicity ofcriminal organizations, this paper tries to give an important and new insight into
deterrence policies against them. Furthermore, we also aim to examine whether
the multiplicity of criminal organization is beneﬁcial for the social welfare andeﬃciency.
Motivated by these observations, apart from the monopolistic view, we examine the oligopolistic criminal market controlled by multiple criminal organizations and its eﬀect on optimal law enforcement the government takes. We
extend the optimal law enforcement with a monopolistic criminal organization
originally proposed in Garoupa (2000) to incorporating the multiplicity of criminal organizations. Following Garoupa, the role of criminal organizations is toregulate the criminal market by extortion. This means that potential oﬀenders
must buy a license from them to commit an illegal act2 .
Since the main diﬀerence between a monopoly and an oligopoly is intro1 Dick (1995) does not stress the monopolistic aspect but alternatively take transaction
costs approach. Grossman (1995) considers a criminal organization as a competitor of thestate and an alternative provider of public goods rather than considering a monopolistic ﬁrm.
2 There exist some other papers extending Garoupa (2000) to diﬀerent directions. One ofthem is Chang et al. (2005) incorporating the possibility of coexisting of individual criminals
and organized crime. The role of criminal organization is the same, but potential oﬀenders canchoose whether to be a member of criminal organization. Another paper is Garoupa (2007)
which focuses on internal organization aspects between the principal and agents.

2

ducing a strategic interaction, we consider diﬀerent two types of competition.
The ﬁrst competition is to attract potential oﬀenders in entering the criminal
markets. If each criminal organization can diﬀerentiate their criminal markets,competition among them may occur. On the other hand, if each criminal organization does not need to diﬀerentiate, they face no competition. Hence, we
model these two diﬀerent competition structures as regards to the ﬁrst typecompetition.
The second competition is to acquire the monopolization by engaging in
conﬂicts with violence. As observed in realty, conﬂicts for market opportunities
among criminal organizations are constantly continuing3 . Such an incentive forwaging in wars is derived by the beneﬁts from obtaining the monopoly proﬁts.
Thus, if criminal organizations prefer the monopoly proﬁts to the oligopoly
proﬁts, conﬂicts will be emerged.
By incorporating these two feature, this paper gives new insights whichhave not been pointed out before. First, this paper shows that the classicalview stressing the superiority of a monopolistic criminal organization does notalways hold. Whether to hold such a classical view depends on how the strategic
relationship among criminal organizations is. If criminal organizations face completely ﬁerce competition in collecting potential oﬀenders, oligopolistic criminal
markets can not be sustained because criminal organizations can not obtainproﬁts and, as a result, the superiority of introducing criminal organizations as
stressed in a monopoly case is not achieved. Hence, each criminal organization
has an incentive to wage wars and, as a result, monopolizes criminal markets.
This result is very intuitive. This indicates that the government must accountfor the extra cost caused in conﬂicts, so the optimal law enforcement may become harsher and the expenditure on enforcing law also increases. This result
contrasts to the one proposed in Garoupa (2000) which stresses a criminal organization contributes to less harsh deterrence policy. On the other hand, if
criminal organizations face no severe competition like above, oligopolistic criminal markets will contribute to improvement of the social welfare eﬃciency. Even
if this result reinforces the superiority of introducing criminal organizations, but
this does not support the desirability of the monopolistic criminal market. However, even if an oligopolistic criminal market is desirable for the society, criminalorganizations with strong military power prefer conﬂicts to peace. This mightcause social welfare loss as discussed in the above case and duopolistic market
cannot be sustained.
Second, this paper models the violence and its eﬀect in an explicit way.
For this framework, we can obtain the analytical strength as regards to theinteraction between the incentive to wage wars and the enforcement of law. It
is unclear whether the harsh penalty by the government causes more conﬂicts.
This paper gives some explanations to this complicated problem and showsthat harsher penalties do not always bring about conﬂicts. Our framework has
common with Castillo (2015), but the motivation is diﬀerent because Castillo
3 Hill (2004) examines that how inter Japanese criminal organizations disputes arise andexhibits some resolutions and diﬃculties. Also, Catanzaro (1994) also argues the conﬂicts
among Italian criminal organizations.

3focuses on how the peaceful equilibrium can be achieved in a repeated gamesetting4 . Furthermore, the role of criminal organizations is diﬀerent from oursettings as a regulator of criminal markets.
There exist some other papers focusing on the economic eﬀects of oligopolistic criminal organizations in bads markets. Fiorentini (1995) models the illegal
goods market, like drugs, and considers the oligopolistic competition in priceand quantities. Fiorentini shows that whether a monopolization contributes tothe reduction of social bads depends on the type of government’s strategy allowing the biased deterrence policy against some organizations. Masour et al.
(2006) examines the model in which a group formation of criminal organizations is endogenous without the use of violence. Kugler et al. (2005) considers
the possibility of corruption between the law enforcer and multiple criminalorganizations.
This paper organizes as follows. Section 2 describes the basic model of the
optimal law enforcement. In section 2.1, we introduce the benchmark result incase of without criminal organization which was originally proposed in Garoupa(2000). Section 3 introduces the model developed by Garoupa and consider
our extensions. Focusing on the diﬀerent form of strategic relations, we derive
the optimal strategy each economic actor should take in section 3.1 and 3.2.
In section 4, we add the possibility of engaging in conﬂicts and its eﬀects in a
duopoly market case. Finally, section 5 concludes.

2

A model without criminal organization

In this section, we introduce the basic model on the optimal enforcement oflaw which is identical to Garoupa (2000). We introduce risk-neutral potential
oﬀenders choosing whether to commit an illegal act that beneﬁts the oﬀender byb, which vary across potential oﬀenders, and harms the rest of society by h > 1.
The government does not know any oﬀenders’ b but knows its distribution: b is
uniformly distributed over [0, 1]. The assumption h > 1 means that committing
an illegal act is not socially beneﬁcial.
The government chooses a sanction f and a detection and conviction probability p. The expenditure on detection and conviction is C(p), which is increasing function of p. For simplicity, we assume C(p) = cp where c > 0 is acost parameter. Also, we assume 0 ≤ f ≤ F where F is the maximum feasible
sanction, which can be interpreted as the maximum wealth of individuals. The
objective function of the government is to maximize the social welfare: the sumof oﬀenders’ beneﬁts minus the harm caused by them and enforcement cost.
We followed the standard assumption that oﬀenders’ beneﬁts are taken into
consideration for the social welfare.

4 Castillo (2015) considers the possibility of conﬂicts among drug traﬃcking cartels. Castillo
models violence in an explicit way and examines the incentives of cartels to wage a war or
stay in peace. Castillo shows that there can be an oligopolistic market in equilibrium andsuch an oligopolistic and peaceful equilibrium also contributes to the reduction of violence.

4

2.1

A competitive criminal market

This subsection introduces the main result proposed by Garoupa (2000). As an
important bench mark, we consider a no criminal organization situation. We
call it a competitive criminal market because, as we mentioned in introduction,
there exists no criminal organization which can regulate to enter this criminal
market. Potential oﬀenders commit an illegal act if and only if b ≥ pf . Thus,
the social welfare is
 1
W =
pf

(b − h)db − cp(1)

The government maximizes the social welfare W with respect to p and f . Let
us use the subscript C to denote the results obtained in a competitive criminal
market.
Proposition 1 (Garoupa (2000)). In a competitive criminal market, theoptimal ﬁne is the maximum ﬁne (f = F ). The optimal detection probability
is pC F = h − c/F .

3

A model with multiple criminal organizations

In this section, we introduce criminal organizations (the Maﬁas). The role ofthe Maﬁa is to regulate the criminal market by extortion. This means that
potential oﬀenders must buy a license from the the Maﬁa to commit an illegal
act. Garoupa (2000) treats criminal organizations as a vertical structure: theMaﬁa extracts some rents from potential oﬀenders with extortion 5 . This paper
aims to extend the Garoupa’s model to an oligopolistic competition among
multiple criminal organizations by incorporating the strategic aspects. Formally,
there exist the n Maﬁas and each potential oﬀender must pay ei to one of theMaﬁa i ∈ {1, 2, ..., n} to commit an illegal act. The Maﬁa i tries to maximize
proﬁts with respect to ei .
What is the strategic relationship among the Maﬁas? We introduce diﬀerent
two types of competition: (a) competition for attracting potential oﬀenders and(b) competition for the monopolization. The type (a) competition indicates
that, for example, if Maﬁa 1 demands an excessive license fee and Maﬁa 1 demands no license fee, potential oﬀenders would want to enter the latter criminal
market. In this case, as in the price competition like the Bertrand competition,
demanding a high price will cause the low proﬁt for the Maﬁa. However, is there
always such a ﬁerce competitions among the Maﬁas? If potential oﬀenders can
not move across each district, severe competition will not arise. Also, collecting
information about the maﬁa may be diﬃcult for potential oﬀenders and there
5 This paper will consider only costless extortion situations although Garoupa also considers
extortion is costly.

5may be no competition among the Maﬁas. Therefore, as for (a) competition,
we consider further diﬀerent two types of strategic relations among the Maﬁas:(a.1) no competition and (a.2) Bertrand competition.
As for type (b) competition, if the Maﬁas ﬁnd the proﬁt in a monopolistic
criminal market is larger than in an oligopolistic criminal market, the Maﬁamay have an incentive to wage a war even if a war takes a cost. In this case,the government must take care of the social welfare loss caused in conﬂicts.
Thus, the strategy the government should take will be changed. At ﬁrst, weconsider only (a.1) and (a.2) competition in section 3.1 and 3.2. The type (b)competition will be considered in section 4.

3.1

A criminal market with multiple criminal organizations: no competition case

If there exists no competition among the Maﬁas, the potential oﬀenders would
enter the criminal markets controlled by Maﬁa i with some probability qi . For
simplicity, we assume that qi = 1/n, ∀i. Thus, the expectedn beneﬁt for potential
oﬀenders from committing an illegal act is b − pf − i ei /n.
n Thus, potential
oﬀenders enter the criminal market if and only if b ≥ pf + i ei /n. Also, theproﬁts of the Maﬁa i are
 1
πi =

pf +

n
i

ei
ei /n ndb.

(2)

Following Garoupa (2000), we consider two types of game: (1) Nash-Cournot
game and (2) Stackelberg game. In (1), the government and the Maﬁas decide
their strategies simultaneously. In (2), only the government moves ﬁrst and all
the Maﬁas move later6 .
Nash-Cournot game. The social welfare function is
 1
W =
pf +

n
i

ei /n

(b − h)db − cp.

(3)

The government maximizes the social welfare function with respect to p andf subject to 0 ≤ f ≤ F . According to the standard maximization problem with
Lagrangean method, the optimal ﬁne is also maximum ﬁne as in Proposition
1. Hence, we obtain the government’s reaction function against the Maﬁas’
strategies:
pF = h −nei /n − c/F.

(4)i

Also, according to the standard maximization problem, the Maﬁa i’s reaction
function against the government strategy are:6 Garoupa (2000) provides the reasons to take these two approaches.

6

ei =

n(1 − pf ) −nj=i ej.
(5)2Moreover, because all the Maﬁa have the same objective function, we assume
the symmetric equilibrium, so ei = ej , ∀ i and j. In the following analysis, we
assume the interior solutions.
Proposition 2. In no competition (NC) case with multiple criminal organizations and a Nash-Cournot game situation, the optimal ﬁne is the maximum ﬁne(f = F ). As given the number of the Maﬁas, the optimal detection probability
is pN C F = (n + 1)h − n − (n + 1)c/F . pN C F decreases as n increases.
Proof. As already mentioned above, we use the Lagrangean method. Basically,
we follow the way of proof as in Garoupa (2000). Deﬁne the Lagrangean as
L = W + λ(F − f ) where λ is the Lagrangean multiplier . The ﬁrst orderconditions aren
Lf = p(h − pf −ei /n) − λ = 0 and(6)i

Lp = f (h − pf −nei /n) − c = 0.

(7)

i
∗
Suppose that
nthe optimal ﬁne, f ∗ is not maximal. From (6), we must have∗ ∗h − p f − i ei /n = 0 where p is optimal detection probability. However,this is impossible according to (7). Hence, the optimal sanction f ∗ must be
maximal, f ∗ = F , and λ∗ > 0. Also, dpN C F/dn = h − 1 − c/F < 0 because we
assume 0 < pC F = h − c/F < 1. Q.E.D.

Note that in case of n = 0, pN C F = h − c/F = pC and in case of n = 1,
p F = 2h − 1 − 2c/F . In particular, the latter results correspond to themonopolistic criminal market proposed in Garoupa (2000). From a theoreticalview point, this result can be interpreted as the generalization of the optimallaw enforcement with organized crime originally proposed by Garoupa. This
result indicates that as the number of the Maﬁas increases, the government
would reduce the expenditure of the enforcement of law. Also, the equilibrium
criminal rate and the equilibrium proﬁts for the Maﬁas do not depend on n.
What is the eﬀects of multiple criminal organizations on social welfare ? The
answer is summarized as follows:NC

Corollary. In no competition case with multiple criminal organizations and aNC
Nash-Cournot game situation, the equilibrium social welfare is W C < Wn=1<
NC
NC
Wn=2 < · · · < Wn=n .
This means that social welfare will improve as the number of Maﬁas increases. The equilibrium extortion by the Maﬁas eN C = n(1 − h + c/F ) in7

creases with n. This means that each Maﬁa tries more extortion because ofrivalry among the Maﬁas and, as a result, it is diﬃcult for oﬀenders to enter themarket. Thus, criminal market controlled by oligopolistic maﬁas can contributeto the social welfare eﬃciency compared to the monopolistic market. Therefore,
the classical view about organized crime claiming the desirability of monopolistic bads markets is not supported as long as potential oﬀenders can not choosemarkets freely.
Stackelberg game. In the Stackelberg game situation, the government moves
ﬁrst and the Maﬁas moves later.
 Thus, the ﬁrst order conditions for the Maﬁan(1−pf )−

n

ej

j=i
i s extortion are ei =.As in Nash-Cournot game, we assume the2symmetric equilibrium. Thus, the social welfare function is 1
W =n/(n+1)+pf /(n+1)(b − h)db − cp.

(8)Therefore, according to the standard optimization problem with Lagrangean
method, we have the results below.
Proposition 3. In no competition case with multiple criminal organizations
and a Stackelberg game situation, the optimal ﬁne is the maximum ﬁne (f =F ). As given the number of the Maﬁas, the optimal detection probability is
pN C F = (n + 1)h − n − (n + 1)2 c/F . pN C F decreases as n increases.
Proof.. By the same argument as in proof of Proposition 2, deﬁne the Lagrangean as L = W + λ(F − f ) where λ is the Lagrangean multiplier . The ﬁrst
order conditions are
Lf = p(h − pf − n/(n + 1) + pf /(n + 1)) − λ = 0 and(9)

Lp = f (h − pf − n/(n + 1) + pf /(n + 1)) − c = 0.

(10)

Suppose that the optimal ﬁne, f ∗ is not maximal. From (9), we must haveh − p∗ f ∗ − n/(n + 1) + pf /(n + 1) = 0 where p∗ is optimal detection probability.
However, this is impossible according to (10). Hence, the optimal sanction f ∗must be maximal, f ∗ = F , and λ∗ > 0. pN C F decreases as n increases as in
the proof in Proposition 2. Q.E.D.
As in Proposition 2, as the number of Maﬁas increases, the expected punishment pN C F decreases. The main diﬀerence between Nash and Stackelberg
game is that extortion will be high in Stackelberg game. The equilibrium crime
rate decreases and the proﬁts increase as the number of the Maﬁas increases.
Thus, it is not clear whether the social welfare improves in this case.

8

3.2

A criminal market with multiple criminal organizations: Bertrand competition case

If a competition among the Maﬁas is like a Bertrand price competition, thepotential oﬀenders would enter the criminal market which gives the highest
expected beneﬁts. Therefore, potential oﬀenders enter the criminal market controlled by the Maﬁa i ∈ {1, 2, ..., n} if and only if b − pf − ei > b − pf − ej
for j = i. In other words, criminal market i is preferred to any other criminal
markets if and only if ej > ei . This condition is very intuitive. In the same way,
we can deﬁne the proﬁts for the Maﬁa i as
⎧
⎪
⎨
πi =

⎪
⎩

1
e db
pf +ei i1
1
M pf +ei ei db

0

if ej > ei ∀j = i
if ej = ei with |j| = M − 1, and ek > ei f or other k (11)
if ej < ei ∀j = i.

In this Bertrand competition case, the Nash Equilibrium is e∗i = 0, ∀ i and∗
πi = 0, ∀ i in Cournot and Stakelberg game. As a result, potential oﬀenders join
the criminal market i if and only if b ≥ pf . This results means that the proﬁt
for the Maﬁas becomes 0 and there exists no regulation on criminal markets.
Therefore, we have the same results as in Proposition1.

Proposition 4. In Bertrand competition (BC) case with multiple criminalorganizations, the optimal ﬁne is the maximum ﬁne (f = F ). The optimal
detection probability is pBC F = h − c/F . This is the same as in a competitive
criminal market.
This proposition says that as long as the Maﬁas face a ﬁerce competition
like the Bertrand price competition, introducing the Maﬁas will not contributeto improve the social welfare compared to a competitive criminal market case.
Therefore, the social welfare improving eﬀects made by the Maﬁa as in Garoupa(2000) is not applied in oligopolistic criminal markets. Furthermore, the classical view about organized crime claiming the superiority of monopolistic bads
markets is also supported as long as potential oﬀenders can choose markets
easily.
Remark some extensions. Let us consider that each Maﬁa can provide theoﬀenders the advantage in committing a crime. That is, the probability ofapprehension and conviction varies across the criminal market controlled by theMaﬁa i. We assume that pi = p(p, θi ) is a function of p and the parameterθi diﬀerentiating an apprehending probability among the Maﬁas and is strictly
smaller than the apprehending probability in a competitive criminal market,
pi < p. This is because the Maﬁa can provide some information to oﬀenders7 .
Thus, if we introduce the heterogeneity of apprehending probability, the optimal
strategy the government take will be altered.
7 Garoupa (2007) examines this problem more deeply.

9Motivated by the above extension, we derive another results. Potential offenders enter the criminal market i ∈ {1, 2, ..., n} if and only if b − pi f − ei >maxj=i b − pj f − ej .Without loss of generality, we assume p1 < p2 < ... < pn
as given the value of p. Entering criminal markets controlled by the Maﬁa 1
is beneﬁcial for potential oﬀenders as long as every Maﬁas demand the same
license fee. In this case, the Maﬁa 1 will make use an advantage to gain the
proﬁts, so the zero proﬁts equilibrium does not occur as in Proposition 4. In
this case, the equilibrium extortion is e∗1 = (p2 − p1 )f − , and e∗j = 0 ∀j = 1
where  is small enough 8 . As a result, the Maﬁa 1 can obtain the monopolization in this criminal market. As for the government strategy, the government
1
maximize the new social welfare W = p2 f + (b − h)db − cp.. This means thatthe government’s strategy becomes less eﬀective, so the government may makeless investment in detection. As a result, the equilibrium crime rate increases
and it is not clear whether the social welfare improves.

4

An extension: the possibility of conﬂicts

In this section, we will extend the basic model introduced in section 3 to incorporating the possibility of conﬂicts among the Maﬁas. One of the main diﬀerence
between the legal goods market and illegal goods market is the use of violence.
In illegal goods market, the Maﬁas often resort to violence to exclude otherMaﬁas from this markets and monopolize the rent extracting.
What should be the optimal law enforcement for the government in a criminal market with multiple criminal organizations in case of the possibility the
costly conﬂict? What is the relation between the deterrence policy and the incentive for the Maﬁas to wage a war? To answer these questions, we integrate
the simple conﬂict theory, i.e., Garﬁnkel and Skaperdas (2007) and Konrad(2009), into our basic model.
For simplicity, we assume that there exist two Maﬁas, Maﬁa 1 and 2. Thus,
our setting is a duopolistic criminal market case. The game proceeds as follows. At ﬁrst, the government announces the detection probability, p, and thesanction, f . After observing the government strategy, the Maﬁa 1 and 2 decide
whether to wage a war. If there exists no war, the duopolistic proﬁts are realized. If there exists a war, the winner obtains the monopolistc proﬁts and theloser gets no proﬁts. Thus, this setting is an extension of a Stackelberg gamesituation.
2
According to our results, monopolistic proﬁts will be π M ≡ ( 1−pf2 ) . The
duopolistic proﬁts in Bertrand competition case (Duopolistic Bertrand Competition) is π DBC ≡ 0 and in no comeptition case (Duopolistic No Competition)
8 If e > 0 for some j = 1 , potential oﬀender’s expected beneﬁts in criminal market 1 andjj = 1 are b − p1 f − (p2 − p1 )f +  and b − pj f − ej . Hence, Maﬁa 1 can collects all oﬀenders
if and only if  + ej + (pj − p2 )f > 0. This condition is always satisﬁed and Maﬁa j’s proﬁt iszero. On the other hand, if e∗j = 0 for all j = 1, in order for Maﬁa 1 to obtain proﬁts, Maﬁa
1 sets e1 such that b − p1 f − e1 > maxj=i b − pj f . Hence, e1 mast satisfy (pj − p1 )f > e1 forall j = 1. Hence, we must have e∗1 = (p2 − p1 )f − .

102is π DN C ≡ ( 1−pf3 ) . Based on the basic conﬂict theory proposed by Tullock(1980), we deﬁne the expected proﬁts from engaging in conﬂict. In case of conﬂicts, Maﬁa 1 and 2 can invest military resources g1 and g2 to win conﬂicts.
The winning probability for each Maﬁa depends on ratio of the amount of invested resources. Let pi be the winning probability of the Maﬁa i, thus wehave p1 = βg1 /(βg1 + g2 ) and p2 = g2 /(βg1 + g2 ) where β ≥ 1. The value β
represents the relative ability of Maﬁa 1 in terms of eﬀectivities of investments
and we assume that the value is exogenously given. This setting means that if
every Maﬁas invest the same resources, the winning probability of Maﬁa 1 isgreater than Maﬁa 2. Therefore, the expected proﬁts for Maﬁa 1 and 2 are

π1W =

βg1
g2
π M − g1 and π2W =π M − g2 .
βg1 + g2
βg1 + g2

(12)

The subscript W represents the situation in a conﬂict and war. Thus, fromthe standard optimization problem, we have the ﬁrst order conditions for each
Maﬁa:
dπ1W
βg2
=
π M − 1 = 0 and
g1(βg1 + g2 )2

(13)

dπ2W
βg1
=
π M − 1 = 0.
dg2
(βg1 + g2 )2(14)

Therefore, we have Lemme 1.
Lemma 1. The equilibrium invested resources for the Maﬁa 1 and 2 are g1∗ =g2∗ = β/(1 + β)2 π M . The equilibrium expected proﬁts for the Maﬁa 1 and 2 areβ1π1W = ( 1+β)2 π M and π2W = ( 1+β)2 π M .
Hence, in case of Bertrand competition, it is always true that π1W > π1DBC =0 according to the discussion in section 3.2. Thus, the Maﬁa has an incentive to
avoid a Bertrand competition and try to monopolize a criminal market. Also,in case of no competition, if β > (≤)2, π1W > (≤)π1DN C . This indicates thatas long as the Maﬁa has an advantage in conﬂict, the Maﬁa 1 would like towage a war. As a result, the social welfare function becomes diﬀerent from theprevious one. Let us denote r > 1 as the harm caused in the process conﬂicts9 .
At ﬁrst, we examine the no competition case and the social welfare function issummarized below.

W DN C =1(b − h)db − (g1∗ + g2∗ )r − cp
1/2+pf /2
1(b − h)db − cp if β ≤ 2.
2/3+pf /3

if β > 2(15)

9 In case of r = 1, the society suﬀers from only dissipation, i.e., Nitzan (1991). However,in case of r > 1, the society suﬀers from not only dissipation but also harms related with
conﬂicts.

11In case of β > 2, the new social welfare function corresponds to the monopolistic
criminal market and contains the social welfare loss caused by conﬂicts. On theother hand, in case of β ≤ 2, the new social welfare function corresponds to the
duopolistic criminal market and no conﬂict situation. Therefore, we obtain the
optimal government strategy.
Proposition 5. In no competition case under the possibility of conﬂict, theoptimal ﬁne is the maximum ﬁne (f = F ). The optimal detection probability
is pW F = {2h − 1 − 4c/F + (4βr)/(1 + β)2 }/{1 + (4βr)/(1 + β)2 } if β > 2 andpW F = 3h − 2 − 9c/F if β ≤ 2.
Proof. By the same argument as in proof of Proposition 2, deﬁne the Lagrangean as L = W + λ(F − f ) where λ is the Lagrangean multiplier . In caseof β > 2, the ﬁrst order conditions are
Lf = p(h/2 − 1/4 − βr/(1 + β)2 − pf /4 − pf βr/(1 + β)2 ) − λ = 0 and (16)
Lp = f (h/2 − 1/4 − βr/(1 + β)2 − pf /4 − pf βr/(1 + β)2 ) − c = 0.

(17)

∗

Suppose that the optimal ﬁne, f is not maximal. From (16), we must have(h/2 − 1/4 − βr/(1 + β)2 − p∗ f ∗ /4 − pf βr/(1 + β)2 ) = 0 where p∗ is optimal
detection probability. However, this is impossible according to (17). Hence, theoptimal sanction f ∗ must be maximal, f ∗ = F , and λ∗ > 0. In case of β ≤ 2,
this proof follows as in Proposition 3. Q.E.D.
Also, we can derive the optimal strategy in Bertrand competition with duopoly
Maﬁa case.
Corollary. In Bertrand competition case under the possibility of conﬂict, theoptimal ﬁne is the maximum ﬁne (f = F ). The optimal detection probability
is pW F = {2h − 1 − 4c/F + (4βr)/(1 + β)2 }/{1 + (4βr)/(1 + β)2 }.
This result indicates that if we consider the possibility of conﬂicts, the expected punishment in a competitive and monopolistic criminal market, pC F
and pM F , may be smaller than pW F as long as r is large. This means thatthe introducing the Maﬁas does not always contribute to the reduction of theexpenditure on law enforcement, which contrasts with Garoupa (2000). Even
if extortion by the maﬁa is not costly, multiplicity of the Maﬁas will induce
harsher penalties. Even if the duopoly criminal market is desirable, it cannot
be achieved because of diﬃculties to avoid conﬂicts.
Let us remark about the value β and its eﬀect. What is the equilibrium if
each Maﬁa can choose its eﬀectiveness in conﬂicts? Let βH > 2 and βL = 1 be
the two values the Maﬁa can choose. The Maﬁa has to pay s in obtaining βHand nothing in βL . Each Maﬁa can choose either βH or βL before deciding toengage in conﬂicts. According to the payoﬀ matrix in Figure 1, choosing βH is
optimal for the Maﬁa 1 and 2 if the cost parameter s is low. This means that if
12

Maﬁa 1/ Maﬁa 2βH
βL = 1βH
π DN C − s, π DN C − s
βH
1
( 1+β
)2 π M ,( 1+β)2 π M − sHHβL = 1,
βH
1
2 M
( 1+β
)
π
− s, ( 1+β)2 π M
H
H
DN C
DN C
π
,π

Figure 1: The payoﬀ matrix for the Maﬁa 1 and 2

the Maﬁa can obtain weapons for conﬂicts with low cost, the optimal strategy
for the Maﬁas is no conﬂict and the equilibrium payoﬀ for them becomes smaller.
This outcome is similar to the Prisoner’s dilemma. This implicates that if thegovernment does not regulate on the weapons the Maﬁas use, the low Maﬁa’s
proﬁt can be achieved and there exists no conﬂict. Hence, in order to achieve low
proﬁts of the Maﬁas, less regulations on the weapons can be optimal. However,in reality, because the government tries to regulate the weapon, so the cost
parameter s is high. As a result, no conﬂict equilibrium can be achieved but
the equilibrium proﬁts for the Maﬁa become high.
In the last part of this section, we consider the heterogeneity of a detection
and conviction probability. We assume that a detection and conviction probability in duopolistic market can be diﬀerent from monopolistic market. If there exists such a heterogeneity, the incentive to engage in conﬂicts can be diﬀered. Let
pM = pM (p) and pD = pD (p) be the detection probability function in monopolistic and duopolistic criminal market and pM can be smaller or bigger than pD .
M
ββTherefore, the expected proﬁts become π1W = ( 1+β)2 π M = ( 1+β)2 ( 1−p2 f )2
M

D

1
1
and π2W = ( 1+β)2 π M = ( 1+β)2 ( 1−p2 f )2 . Also, π DN C ≡ ( 1−p3 f )2 in no competition case. Focusing on no competition case, we examine the incentive forMaﬁa 1 in four cases: (1) βH and pD < pM ,(2) βH and pD > pM ,(3) βL andpD < pM and (4) βL and pD > pM .
In case of (1) and (2), because the Maﬁa 1 has a relative advantage in conﬂicts(βH ), the Maﬁa 1 may have an incentive for the war. In case of (1) and (3),a detection and conviction probability in monopolistic market is more eﬀective
than one in duopolistic market, pM > pD . This condition holds, for example, if
the government has to allocate resources for each market controlled by the Maﬁa
1 and 2. On the contrary as in (2) and (4), a detection and conviction probability
in monopolistic market is less eﬀective than one in duopolistic market, pM < pD .
This condition holds, for example, if a detection probability depends on thenumber of oﬀenders. As the number of oﬀenders increases, it is more diﬃcult
to detect the oﬀenders.
At ﬁrst, we examine (2) and (3). In (2), the Maﬁa 1 always prefers conﬂicts
for any government strategies p 10 . Therefore, the social welfare function is1
W DN C = 1/2+pf /2 (b − h)db − (g1∗ + g2∗ )r − cp. Also in (3), the Maﬁa 1 and2 always prefer no conﬂict for any government strategies. Therefore, the socialβ)2 (1−pM f )2 is always larger than π DN C ≡ ( 13 )2 (1−pD f )22(1+β)
DM
and p > p .

10 This is because π M = (in case of βH

131

welfare function is W DN C = 2/3+pf /3 (b − h)db − cp.
As for (1), since the value of β is high, the Maﬁa has a relative advantage in
conﬂicts and monopolistic proﬁt is also high in case of low detection probability. However, since pD < pM , in case of high detection probability p, duopilistic
proﬁts will be high. Thus, there exists the threshold value of a detection probability, p∗ , such that if p < p∗ , the Maﬁa 1 prefers war and if p > p∗ , Maﬁa 1
prefers peace. Hence, the new social welfare function is diﬀerent from the above
case11 . As for (4), the incentive for causing war is completely diﬀerent from (1).
The low detection probability will make the Maﬁa 1 cause war and the highdetection probability may bring about no conﬂict equilibrium.
By incorporating the diﬀerence of detection probability between a monopoly
and a duopoly market, the incentive to wage wars will be varied. We can givethe theoretical explanations whether the harsh penalties induce more wars.

5

Concluding remarks

This paper develops a model about the optimal law enforcement on organizedcrime by extending the model originally proposed by Garoupa (2000) to multiple
criminal organizations, i.e., the Maﬁas. The role of the Maﬁas is to regulate thecriminal market by extortion. Based on this framework, we can examine theinteraction among criminal organizations and the government and see its eﬀect
which has not been dealt with in the previous literature. This paper asks thequestion: Will the multiplicity of the Maﬁas contribute to the improvement ofsocial welfare?
This paper considers two types of structures of competition among criminalorganizations. The ﬁrst one is criminal organizations face a ﬁerce competition
for collecting potential oﬀenders in criminal markets to make proﬁts. The second one is that there exists no such a competition. If criminal organizations
suﬀer from the severe competitions, introducing criminal organizations can nothave social welfare improving eﬀects. On the other hand, if there is no competition among them, an oligopolistic criminal markets will contribute to thesocial welfare improvement. Hence, this paper shows that whether a competition among criminal organizations contributes to social welfare improvement
depends on the structure of strategic relations. These results make some differences from the previous literature stressing the desired eﬀects of criminalorganizations as in Buchanan (1973), Schelling (1971) and Garoupa (2000). In
other words, a monopolistic criminal market controlled by one Maﬁa does notalways preferable.
In addition to a competition for collecting oﬀenders, this paper also considers
the possibility of conﬂicts. By considering the social welfare loss caused in
conﬂicts, the government has to change its optimal strategy. Also, we canobtain new insight as regards to the incentive for engaging in conﬂicts because
11 The new social welfare function will be W

p < p∗ and W =1
2/3+pD f /3=

1

1/2+pM f /2∗(b − h)db − cp if p > p .

14(b − h)db − (g1∗ + g2∗ )r − cp if

it has been unclear whether harsher penalties cause wars. We show that aslong as the social welfare loss caused by conﬂicts is large, the government may
have an incentive to take the severe punishment. This is almost consistent withactual policies against organized crime the government takes. As a result, it is
not always true that the Maﬁas contribute to the less severe punishment andthe reduction of its expenditure as stressed in the previous literature. As forthe incentive for engaging in conﬂicts, we show that harsher penalties do notalways give the Maﬁas an incentive to wage wars.
This paper’s model has some insuﬃciencies. First, this paper considers
only two diﬀerent extreme competition situations. However, actual competition among criminal organizations lies between these two extreme case. Hence,
more general model should be developed. Second, we should consider the possibility of collusions among the Maﬁas. In case that the government sets harsh
penalties, waging wars is not the only option for the Maﬁas, on the contrary,
they may try cooperative activities against the government.

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