Article
Property rights’ emergence
in illicit drug markets
Rationality and Society
2021, Vol. 33(1) 52–105
Ó The Author(s) 2020
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DOI: 10.1177/1043463120968288
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Jefferson DP Bertolai
and Luiz GDS Scorzafave
University of São Paulo (FEARP/USP), Brazil
Abstract
Governance rules are efficient mechanisms in the sense that they increase people’s
welfare. They emerge even when the state is unable or refuses to create and
enforce them. We study a situation in which this demand for governance manifests
itself through the emergence of property rights in illicit drug markets: a privatelyprovided governance. Specifically, we propose a model for property rights emergence in illicit drug markets as predicted by the theory on governance provided by
prison gangs. It is studied a situation in which an agreement among criminals,
resembling property rights enforceability on its allocative effect, can emerge in illicit
drug markets. Our Mechanism Design approach shows that a change inside the
prison system, from a competitive environment to the hegemony of a group of
criminals, implies the equilibrium in illicit markets to shift from warfare to peace:
the hegemonic group is shown to desire to promote the collusive agreement when
it is able to do so. This contrasts with the equilibrium under no hegemony, in which
the possibility to conquer consumers/territories drives violence up to a positive
level. The novel empirical perspective implied by the model is explored using data
from Brazil, a context for which the theory of governance provided by prison gangs
has been pointed as a key explanation.
Keywords
Drug-dealing game, peaceful equilibrium, property rights, violence, PCC
Corresponding author:
Jefferson DP Bertolai, School of Economics, Business and Accounting at Ribeirão Preto –
University of São Paulo, Av. Bandeirantes 3900, Ribeirão Preto, São Paulo 14040-905, Brazil.
Email: jbertolai@fearp.usp.br
Bertolai and Scorzafave
53
Introduction
In Alchian (2008), property right is defined as a socially enforced right to
select uses of an economic good. Also, this right to choose among alternative uses allows the exchange for similar rights over other goods. Most
important for our purposes, society enforces the right to sell the good by
detecting, convicting and punishing those (other than the right’s owner)
trying to restrict or to perform such exchange.
In that sense, one can see illicit markets as those for which society has
assigned the property right over the corresponding good to itself and has
decided to not sell it.1 Licit markets, on the other hand, can be seen as those
for which society has assigned property rights to dealers and enforces them,
for example, by forbidding the use of violence among them as a competition instrument.
The extent to which property rights are enforceable depends on society’s
ability to detect, convict and punish those (other than the right’s owner) trying to restrict exchanges in licit markets and/or to perform exchanges in
illicit markets. If society’s ability is high enough, no trading takes place in
illicit markets and no violence is performed in licit markets in order to
restrict property rights. On the other hand, if society’s ability is not high
enough, trading occurs in illicit markets, and violence takes place in both
licit and illicit markets as a competition instrument.
The most obvious feature determining society’s ability to enforce property rights is trading profitability. For highly profitable markets, both
restricting other’s trading and promoting his own trade are worth taking
the risk to be punished by society. More subtle features are detection and
conviction capacity. For example, in societies where the respect to a relevant amount of human rights2 is required when convicting and punishing
someone, both police work in detecting and evidencing crimes is a sophisticated activity and the state’s power to punish criminals is bounded. In
such societies, if police work does not produce sufficient evidence of
guilty while preserving criminals’ human rights, judicial system is not able
to convict them. Even though conviction is achieved, society cannot make
punishment arbitrarily high without violating criminals’ human rights. As
a result, criminals assess bounded expected punishment.
Such reasoning about profitability and boundness can be seen as fundamental for the standard economic model to the decision individuals make
between legal and illegal activities.3 According to this approach, potential
criminals decide going illegal if expected profitability from the illegal
activity is greater than the bounded expected punishment they are exposed
to. Applied to our discussion about licit and illicit markets, this model
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Rationality and Society 33(1)
predicts that property rights should not be fully enforceable in markets for
which profitability is greater than expected punishment. We make this
point for the illicit-markets case in subsection ‘‘A drug dealing game’’ by
presenting a game intended to model an illicit-drug retail market and showing that both violence and trading take place if expected punishment is low
and profitability is high.
A subtle situation to be considered in Alchian (2008)’s definition is the
case property right emerges even without society’s enforceability through
state law and judicial system. This would require an agent, other than the
society-sponsored state, to be able and to desire to enforce such rights.
Following Skarbek (2011, 2012, 2014, 2016)’s theory on governance provided by prison gangs, and inspired on Lessing (2010)’s consolidation-propagation-projection conceptual framework, we study conditions under
which this agent can be a group of criminals that becomes hegemonic
inside prison system. Hegemony is modeled as a change inside the prison
system, from a competitive environment to monopoly over prison-life matters, which endows the hegemonic group with the ability to distort inmates’
welfare. Consistent with Skarbek (2011, 2012, 2014, 2016)’s theory and
Lessing (2010)’s conceptual framework, this ability establishes a channel
of influence over out-of-prison criminals that can be used to enforce property rights in illicit drug markets. In section ‘‘Property rights in illicit markets’’, we show that trading without violence among dealers (‘property
rights’) is enforced in the illicit market model presented in subsection ‘‘A
drug dealing game’’ as a rational-choice phenomenon if the hegemonic
prison gang has sufficiently high power to distort expected welfare inside
the prison system.
Closely related literature
Our model makes use of standard tools found on the Economics of Conflict
literature4 to study the market for illicit drugs. The conflict among dealers
is modeled as a game in which each player seeks to obtain some prize (drug
profits) by investing some effort or resources (violence). For this matter,
we build on Burrus (1999)’s model on competition among drug dealers.
In studying how participants in the illicit markets can collude to sustain
a peaceful equilibrium inside the Economics of Conflict’s framework, we
are closely related to Castillo (2013) and Castillo and Kronick (2020).
Cooperative behavior among drug dealers are justified in these works by
resorting to the combination of high level of patience and infinite horizon.
Under this framework, authors study how policies can eliminate collusion
Bertolai and Scorzafave
55
by shortening criminals’ horizon on illegal activities. Our results on cooperative behavior, on the other hand, hold on a two-stage model and are sustained by the prison gangs’ influence studied by Skarbek (2011) and
Lessing (2010) and extensively documented by Lessing (2014, 2017b,
2017c) and Lessing and Willis (2019). Our simpler model allows us to
study the cooperative behavior using the mechanism design approach.
That way, we are not only able to establish that the peaceful arrangement
is a subgame-perfect Nash equilibrium, as in Castillo (2013) and Castillo
and Kronick (2020), but also that it is the best equilibrium for the game
designer (in our case, the prison gang).
As in Burrus (1999), the prize from the conflict in our model is the control over retail drug markets while Castillo (2013) and Castillo and Kronick
(2020) model disputes over the control of smuggling routes. The first
approach fits well drug-consumer countries like USA and Brazil (our motivating case described in subsection ‘‘Empirical motivation’’), while the latter seeks to model drug-producer countries like Colombia.5
At a more basic theoretical level, our interpretation of the collusive
arrangement as the emergence of property rights without society enforcement is related to Skaperdas and Syropoulos (1995)’s analysis on how
property rights emerges from anarchy.6 Such a connection could be used
to see prison gang as a primitive form of state in the tradition of Olson’s
stationary bandit model (McGuire and Olson, 1996; Olson, 1993). This is
actually the framework upon which Skarbek (2011)’s theory is build.
Lessing (2017b)’s analysis also recognizes the relevance Olson’s framework for the subject in question, but it goes further in highlighting that
‘‘prison gangs ability to project power onto the streets depends not only on
state absence, but also crucially on state actions’’ (Lessing, 2017b: 3).
Because Skarbek (2011) and Lessing (2017b) are the main theoretical
references for our work, we get back to them in section ‘‘The model’’ in
order to explore their frameworks in a more detailed way.
It is important to highlight that Skarbek’s most recent work (Skarbek,
2012, 2014, 2016) has focused on prison gang governance provision inside
the prison system, while Skarbek (2011) is much more fundamental to our
work, to the extent it is focused on its provision to outside-prison activities.
While the former is in some ways similar to Leeson (2007a)’s work on pirate
anarchy, the latter builds on the stationary bandit model proposed by Olson
(1993) and McGuire and Olson (1996). Nevertheless, Skarbek (2011, 2012,
2014, 2016)’s theory on governance provided by prison gangs, as a whole,
can be seen as a theory of private provision of governance.
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Rationality and Society 33(1)
Similarly to the contributions of Skarbek (2011, 2012, 2014, 2016) and
Lessing (2010, 2017b), our work contributes to a large literature on private
provision of governance.7 Contrary to the Hobbesian view8 on the necessity of an external agent with coercive power to enforce contracts (cooperation), this literature sees the private sector as a relevant source of
governance rules’ creation and enforcement (Anderson and McChesney,
2018; Leeson, 2007b, 2012b, 2014a; Ostrom, 1990; Ostrom et al., 1992).
As discussed by Skarbek (2012),
‘‘Privately produced law and order can emerge in remarkably diverse environments (Benson, 1990, 1998; Stringham, 2007). These institutions worked reasonably well in the context of mining camps (Stewart, 2009; Umbeck, 1977a,
1977b, 1981), the wild west (Anderson and Hill, 2004; Clay, 1997), international commerce (Benson, 1989), Medieval Japanese monasteries (Adolphson
and Ramseyer, 2009), international Hawala networks (Schaeffer, 2008),
Ancient Greece (D’Amico, 2010), and in early trade (Leeson, 2005, 2006,
2007c, 2008, 2009)’’ (Skarbek, 2012: 97).
More recent work has enriched this list by studying land property rights in
rural Afghanistan (Murtazashvili and Murtazashvili, 2016), online piracy
(Harris, 2018), kidnap insurance (Shortland, 2017, 2018, 2019), piracy protection (Shortland and Varese, 2014, 2016), and superstition (Leeson,
2012a, 2014a, 2014b, 2014c; Leeson and Coyne, 2012).
From this point of view, the emergence of property rights in illicit drug
markets can be seen as another instance in which those agents most directly
involved on an economic activity manage to self-organize to solve collective action problems and secure cooperation.9 And, it is an interesting
instance: governance is privately provided for an economic activity the
state has forbidden and strives to restrain. As in the Harris (2018)’s work
on online piracy, an illegal community ‘‘self-organize and develop institutional mechanisms to ensure cooperation under conditions that are far from
ideal’’ (Harris, 2018: 920).10
This distinguishing feature can be used to see our work as similar to
Shortland and Varese (2014, 2016)’s work on piracy protection to the extent
that their work builds on the so called Protection Theory, usually applied to
study for the emergence of the modern state (Lane, 1958; Nozick, 1974;
Olson, 1993, 2000; Tilly and Besteman, 1985), to address governance issues
of criminal organizations. Comparing their findings for Somali piracy to
our findings for illicit drug markets in Brazil shall be shown useful.
Methodologically, our study on private provision of governance employs
the rational choice approach in devising a formal model that makes explicit
Bertolai and Scorzafave
57
how an hegemonic prison gang can be seen as an efficient provider of property rights in outside-prison illicit drug markets, in the sense that such provision maximizes its revenues. Also, the model provides a clear empirical
perspective for crime rates outside prison: correlation between homicides
and drug dealing vanishes after prison gang takes control of inside prison
matters. We, then, employ econometric tools for providing new and statistical evidence on the adherence of Brazilian case to the Skarbek (2011, 2012,
2014, 2016)’s theory on governance provided by prison gangs and Lessing
(2010)’s consolidation-propagation-projection conceptual framework.
More generally, the combination of a formal rational-choice model with
econometric tools provide a complementary approach to the analytical strategies usually employed in the large and interdisciplinary literature on private provision of governance discussed above.
Empirical motivation11
The sharp change in crime rates in São Paulo (the richest state in Brazil,
hereafter denominated SP) in the recent past is our motivating case.
Figures A2 and A3 in Appendix C illustrates the big picture. Time-series
for virtually all crimes in the figures have reversed trend at the beginning
of the 2000’s, after persistent growth in the 1990’s. The most striking cases
are violent crimes like murder, attempted murder, robbery followed by
death, and vehicle robbery. Less violent crimes like theft, vehicle theft,
and robbery (excluded vehicle robbery) have stopped increasing and
remained stable in the 2000’s. Finally, drug dealing records started growing at a higher rate from 2001 on.12
Figure 1 suggests that SP’s experience on murder rates is atypical for
the Brazilian context. It presents annual data on homicide rates for SP and
for the five regions in the country, from 1996 to 2011. In this period,
homicide rate in SP fell on average 6.28% per year, while the average rate
among the remaining states grew on average 2.4% per year. Therefore, the
causes for such decline seems to rest on specific features that differentiate
SP from other Brazilian states. Consistent with the theoretical framework
to be presented in subsection ‘‘A drug dealing game’’ and section
‘‘Property rights in illicit markets’’,13 we argue that two factors make SP’s
experience unique in Brazil:
(i)
imprisonment policy in SP, which has traditionally been stricter
than that of other states, has become even more aggressive in the
1990’s and;
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Rationality and Society 33(1)
Figure 1. Homicide rate – São Paulo and Brazilian regions (SP excluded).
Source: Waiselfisz (2006) and Waiselfisz (2013).
(ii)
a single group of criminals has consolidated control over prison
life and has propagated it throughout the prison system.
The first fact is illustrated in Figure 2a, which displays the evolution for
the incarceration rate14 in the 1990’s for SP and for the remaining
Brazilian states. It can be seen how particular SP’s imprisonment policy
was. In 1994, incarceration rate was 2.7 times higher in SP. Moreover, it
grew on average 3.49% per year in SP from 1994 to 2002, while it grew
only 2.76% per year for the other states.
Factor (ii) has been studied by sociological and political science literatures and widely publicized by the press in Brazil. Dias (2011), for example, presents a detailed description on the events that resulted in the
consolidation and propagation of a group of criminals over the prison system in SP. Figure 2b, reproduced from Dias (2011), clearly illustrates how
events inside the prison system follow closely trend reversion in crime
rates (as presented in Figures A2 and A3 in Appendix C). It shows that
murders in prison as a result of prisoners’ conflicts15 has sharply increased
in the 1990’s, but, starting in 2001, it fell at a rate even faster than it has
increased. Dias (2011)’s detailed presentation shows that most of such
deaths resulted from conflicts among prison gangs during prisoners rebellions. Moreover, it is documented that by 2001 a single prison gang has
Bertolai and Scorzafave
59
Figure 2. Some specific features to São Paulo’s experience. (a) Prison population
rate – SP and Brazil regions. (b) Murders in prison (SP, 1990–2010).
Source: (a) INFOPEN, SAP/SP and IBGE. (b) Dias (2011).
emerged from these conflicts as the dominating organization inside the
prison system. As a result, most of battles among prisoners ceased, as well
as the associated murders.16
In addition to factors (i) and (ii) above, by building on Skarbek (2011)’s
theory and Lessing (2010) conceptual framework, our model brings consistency for two apparently unrelated evidence on the pattern of criminal
activities, which standard economic models based on Becker (1968) are
not fully equipped to explain. First, out-of-prison criminals are required to
make monthly payments to the hegemonic prison gang, in what is known
in Brazil as ‘mensalidades’.17 Second, prison gang’s rules (‘laws’) are
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enforced fast and effectively in what is known in Brazil as ‘‘Tribunal do
Crime’’, something similar to a court where the hegemonic prison gang
establishes punishments to those not complying with their rules.18
In subsection ‘‘The empirical exploration’’, data from SP’s experience
is used to study a novel empirical perspective that emerges from our model:
correlation between violent crimes (murder) and drug dealing changes from
a positive level before the channel of influence (from inside to outside
prison system) is established to no correlation after that. Specifically, the
proposed exercise consists in estimating the following relation between
murder rate (h) and drug dealing rate (d):
h = f0 + d0 d + (f1 + d1 d)c + YX + n
ð1Þ
where the channel of influence (c) from inside prison was taken into
account, X is a vector of covariates and (f0 , f1 , d0 , d1 ) and Y are vectors
of parameters. According to our model, correlation between h and d is positive before the channel has been established (d0 . 0 since c = 0), and it is
nonexistent after the channel takes place (d0 + d1 = 0 since c = 1). As presented in Table 1, our findings strongly support the first prediction and
provide some evidence on the second one, in the sense it is not rejected at
5% significance level.
The model
Before presenting our model, it is worth to describe how the channel of
influence from inside prison over outside criminals is modeled by Skarbek
(2011), hereafter denominated SK, and by Lessing (2017b), hereafter
denominated LS.
As discussed by SK, Mexican Mafia wield substantial control over
inmates in the Los Angeles county jail system. Because drug dealers on
the street anticipate future incarceration, this endows Mexican Mafia with
the ability to extort them. Figure 3 reproduces the decision tree SK uses to
represent the problem drug dealers face under the extortion mechanism.
Three features are proposed as the most relevant for drug dealer’ decision
on whether or not to pay taxes to prison gang: incarceration probability,
prison gang’s power over inmates, and drug dealer’s relationships with
current inmates.
Paying taxes is the optimal behavior for outside-prison drug dealers
only when prison gang has strong enough control over inmates’ welfare.
Welfare distortion inside prison system is sufficient to induce tax payments
Bertolai and Scorzafave
61
when incarceration probability is high or when there is someone in jail
drug dealer cares about.
Building on Olson’s stationary bandit model (McGuire and Olson,
1996; Olson, 1993, 2000), SK sees Mexican Mafia as an ‘‘incarcerated
bandit’’ in the sense that its ability to extract resources from drug dealers
creates an incentive to provide governance institutions that mitigate market
failures. In particular, the incarcerated bandit would enforce property rights
in order to increase drug dealing expected profitability and, therefore, drug
dealers’ capacity to pay taxes.
Complementary to this framework, LS’s analysis highlights the relevance of state actions for the prison gangs’ ability to project power onto
the streets. In order to study this new perspective, LS proposes a model in
which a sequential game is played between the prison gang (G) and a
street-gang leader (S) as illustrated in Figure 4, reproduced from Lessing
(2017b). In addition to a taxation level t, already present in SK’s model, G
requires from S a behavior outside prison M. After S chooses its behavior,
nature N determines if S is arrested and goes to jail or if it remains free.
By refusing to comply with behavior M, S is arrested under probability
p. The probability of being arrested under behavior M, denoted p
~ , is
assumed greater than or equal to p. Punishment inside prison, denoted
j . 0, is lessened by a factor 1=a 2 (0, 1 if the arrested criminal has complied with M outside prison. Behavior M can also increase expected criminal profit y by a factor b ø 1. Additive utility punishments are imposed on
S when it refuses to comply with M. It amounts to g ø 0 inside prison and
to d ø 0 outside prison.
LS interpret the the maximum taxation t consistent with S complying
with (M, t) as an indirect measure of S power. It is studied how this measure depends on state actions/policies according to the punishment level j
and the degree of targeting p
~ =p they entail. LS’s analysis shows that policies with sufficiently negative impact on j and sufficiently positive impact
on p
~ =p weakens prison gang power.
A drug dealing game
We propose a model of drug dealing based on both dealers’ competition
presented in Burrus (1999) and prison gangs’ influence over out-of-prison
criminals studied in SK and LS. Two features in this market are key for
our analysis: (i) dealing drugs is an illegal activity and, therefore, no
dealer is able to legally enforce property rights over his selling post
through the judicial system. Instead, each of them protects their claimed
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Rationality and Society 33(1)
Incarceration
Likely
Strong
Gang
Weak
Gang
Pay
Taxes
Don’t Pay
Taxes
Incarceration
Unlikely
Friends
Incarcerated
No Friends
Incarcerated
Strong
Gang
Weak
Gang
Pay
Taxes
Don’t Pay
Taxes
Don’t Pay
Taxes
Figure 3. Decision tree for the Skarbek (2011)’s model.
properties and eventually violently attack their rivals. Following Burrus
(1999), we refer to these battles for selling posts as turf wars; and (ii) punishment dealers get inside the prison system when arrested can potentially
be distorted by previously arrested criminals. That establishes a (probabilistic) link between inside prison events and illegal drug markets’ functioning in that current prisoners could make treatment inside the prison system
contingent on out-of-prison behavior. Following Lessing (2014), we refer
to these group of current prisoners as the prison gang or PG.
The environment. There is a homogeneous drug, whose trading is illegal
and for which individual demand function is Q(p) = X Zp, where X . 0
and Z . 0. There is a continuum of consumers (of measure 1) and there
are two dealers in the drug market, dealer 1 and dealer 2. It will be shown
convenient to refer to dealer j 2 f1, 2g as Dj . Competition instrument is
the violence to conquer consumers/territory from the other dealer. There is
a police force to arrest those not complying with drug prohibition and/or
committing violence acts.
The timing. There are two stages, as shown in Figure 5. First and second
stages are denominated, respectively, turf war stage and trading stage.
Police force acts after the trading stage in a two-fold mission. It arrests
criminals for violence acts according to probability n 2 ½0, 1, which we
Bertolai and Scorzafave
63
Figure 4. Decision tree in Lessing (2017b)’s model.
assume to be independent among criminals, and it arrests dealer j 2 f1, 2g
for selling drugs according to the probability r 2 ½0, 1.19
The turf war stage. In the first stage, each dealer chooses how much violence to implement in order to defend himself and to conquer monopoly
power over consumers and territories. The fight is summarized by a vector
of violence amounts v = (v1 , v2 ) 2 R2+ , where vi stands for the violence
amount chosen by Di . Actions (violence) in this stage are simultaneously
chosen and determine the distribution of monopoly power over consumers/
territory: Dj gets monopoly power over a proportion tj (v) of the aggregate
demand for drugs.
With probability n dealer i 2 f1, 2g is arrested and suffers punishment
hi for each unit of violence acts he has performed. This is the present value
of costs associated to being in the jail. It includes the traditional opportunity cost in form of wages he could earn during his time in prison, but also
legal expenses during his trial and family expenses to visit him at the
prison. Also, it captures monetary value of expected welfare losses of
being inside prison. For example, individuals inside prison might face
higher probability of getting sick or being victim of aggression, robbery,
extortion or murder. We assume the cost hi is composed of a punishment
common to all prisoners, h, and a specific one, bi , by supposing that
hi = bi h for bi . 0.20
The trading stage. Suppose turf war has taken place under violence profile v. Dealer i 2 f1, 2g trade drugs under monopoly in his territory: Di
decides how much drug to sell, qi ø 0, in order to fulfill its market demand
ti (v)Q(p),21 so that qi = ti (v)Q(p). We assume that there is no cost in producing and supplying drugs and, therefore, profits equal revenues.22
With probability r, dealer i 2 f1, 2g is arrested and suffers punishment
di = ai d for each unit of drug sold, where ai . 0. This is again the present
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Rationality and Society 33(1)
Turf war stage
Trading stage
Arrests for violence
Arrests for trading
Figure 5. Timing of the two-stage game.
value of costs associated to being in the jail and is composed by common
and specific factors, d and ai , respectively.
Expected payoffs and the contest success function. After turf war v = (v1 , v2 )
has taken place and the distribution of consumers/territories (t1 (v), t2 (v)) has
being established, the expected payoff the monopolist dealer gets after selling
qi units of drug and facing punishment factors a and b is
Ui (v, qi , a, b) [ P(qi =ti (v))qi
f(1 n)rdi qi + (1 r)nhi vi + rn(hi vi + di qi )g
= P(qi =ti (v))qi rdaqi nhbvi
ð2Þ
where P(q) denotes the individual (inverse) demand function
Q1 (q) = A Bq, in which A = X =Z . 0 and B = 1=Z . 0.
The relation between market shares (t1 , t2 ) and violence efforts (v1 , v2 )
is based in Contest Success Function literature that followed Tullock
(1980)’s rent-seeking model.23 In the simplest formulation, the one we
assume here, relative success is stated as a function of the ratio of the
respective resource commitments – in our case, violence effort:
t i vi
= ,
tk vk
if vk . 0:
ð3Þ
If vk = 0, we have tk = 0 if vi . 0 and tk = ti if vi = 0. Using (3) and the fact
t1 + t2 = 1, we have the same Contest Success Function24 studied in
Hirshleifer (1989) and Burrus (1999):
8 v
j
>
, if (v1 , v2 ) 6¼ (0, 0)
<
v
+
tj (v) = 11 v2
>
: ,
otherwise:
2
Actually, our drug dealing model is heavily based in Burrus (1999)’s
model. The only modification we make in his setup is to allow punishment
inside prison system to be specific to each dealer.
Bertolai and Scorzafave
65
Comparing environments. Compared to SK’s model, only the left-handside branch of Figure 3 is considered in our environment. From the perspective of LS’s model, some features of drug dealing activity is made
more explicit in our environment: LS’s Nature player N is replaced here by
the police force and LS’s generic behavior M is particularized to violence
acts vi and drug dealing qi . This allows us to study LS’s parameter for profitability, y . 0, as an equilibrium object which emerges from both the drug
dealing in the trading stage and the market share conquered in the turf-war
stage. One can alternatively see our model as extending LS’s framework if
the profitability parameter y is interpreted in LS’s model as drug dealing
profits for a given market share. Under this perspective, our model extend
LS’s framework by explicit considering how market shares emerges in
equilibrium as a result of competition on turf wars. The extension to multiple dealers case is a natural modeling strategy for our purpose in studying
property rights emergence in illicit drug dealing markets.
Also key in our analysis, LS’s regular punishment inside prison (j . 0)
is studied here as a function of equilibrium outside-prison behavior of the
arrested criminal: dependence is assumed linear on both drug dealing, di qi ,
and violence acts, hi vi . In addition, LS’s punishment distortions inside
prison (a and g), is assumed here as proportional to punishment: the factor
is ai for drug dealing punishment and bi for violence act punishment.
As in the frameworks proposed by SK and LS, we study in section
‘‘Property rights in illicit markets’’ the incentives in our model for compliance with prison gang’s taxation t. For simplicity, though, we assume that
arrestment probabilities r and n are invariant to cooperating with the prison
gang: in LS’s notation, in our environment p
~ = p.
The subgame-perfect equilibrium. In order to compute the subgame-perfect
equilibrium for this game, we must impose that dealer’s strategies constitute a Nash equilibrium in every subgame. We do this computing player’s
strategies in a backward induction fashion, computing stage outcomes from
the last stage to the first one.
The trading equilibrium. Equilibrium outcome in the trading stage is
straightforward, since dealers always operate under monopoly. For
ra = P(0)=da and ai = a, Di ’s optimal quantity and its implied optimal
expected profit are given by
qa (ti ) [ arg max piv (x, a) = ti
xø0
ad
maxf
ra r, 0g
2B
ð4Þ
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Rationality and Society 33(1)
pa (ti ) [ piv (qa (ti ), a) = ti B
qa (ti ) 2
,
ti
ð5Þ
as established by Lemma 13 in Appendix A.
This shows that drug dealing is profitable (qa (ti ) . 0) if society’s ability
to enforce property rights (drug dealing prohibition) is not high enough
(r\P(0)=da). Also, it is made explicit how such ability depends on the
ability to detect and convict those dealing drugs (r) and the magnitude of
the expected punishment (da).
The turf war equilibrium. Taking as given the equilibrium payoffs in the
last stage, pa (t) in (5), Di ’s expected payoff in (2) equals
pa (ti (v)) nhbi vi . More specifically, it is given by
ti (v1 , v2 )B
qa ½ti (v1 , v2 ) 2
vi ðad maxf
ra r, 0gÞ2
nhbi vi :
nhbi vi =
ti (v1 , v2 )
v1 + v2
4B
ð6Þ
Then, for each conjecture vk about his opponent choice on violence, Dj
chooses vj ø 0 in order to maximize (6).25
No violence is optimal when drug dealing is not profitable (r ø ra ): violence is not attractive in equilibrium since there is no valuable property right
to be protected/attacked. For the case drug dealing is profitable for both dealers, on the other hand, there is no subgame perfect equilibrium in which no
violence is performed, as a corollary of lemma 14 in Appendix A.
Specifically, if expected punishment inside the prison system is not high
enough to make drug dealing unprofitable, then no violence (vi = 0) is not a
best response for the no-violence conjecture (vk = 0). The reasoning behind
this result is that, absent property rights, the only punishment Di gets choosing vi = e . 0, with e ! 0, is hbe under probability n, whereas the expected
reward pa (ti (e, 0)) = pa (1) is strictly positive and invariant to e. On the other
hand, when dealing drugs is not illicit and property rights holds, dealers can
resort to the judicial system to require a punishment H to their aggressor. If
such a punishment is high enough and invariant to the aggression level, so
that H . pa (1), then no violence becomes a best response to no violence.
It follows from this reasoning that, if a turf war equilibrium exists when
drug dealing is profitable, then it entails positive violence for some dealer.
In order to study existence of such equilibria, consider the conjecture
vk . 0, so that the objective function (6) is strictly concave in vi . Then,
first-order optimality condition and constraint vi ø 0 imply the following
Di ’s best response function:
Bertolai and Scorzafave
8
< 0rffiffiffiffiffiffiffiffiffiffiffiffiffi
pai (1)pffiffiffiffiffi
vi (vk ) =
vk vk
:
nhbi
67
if vk . pai (1)=nhbi
otherwise
ð7Þ
:
Observe that no violence is optimal to Di if Dk ’s violence (vk ) is sufficiently high. The cutoff violence beyond which violence is not worth
(pai (1)=nhbi ) is lower when Di ’s monopoly profits are lower and/or
expected punishment for violence acts is high. This shows that dealer k is
always able to conquer all consumers by increasing vk beyond such cutoff.
But, we know from Lemma 14 that this cannot constitute an equilibrium,
since Dk has no optimal response to no-violence conjecture (vi = 0) when
drug dealing is profitable. The following lemma describes the turf war
equilibrium implied by (7) and establishes that it entails positive level of
violence.
rai g, so that pai (1) . 0 for all
Lemma 1. Suppose that r\ mini f
i 2 f1, 2g. Then, turf war equilibrium, for given equilibrium payoffs in the
trading stage, is
pa1 (1)
pa (1)
v = (v1 , v2 ) =
u(a, b), 2 u(a, b)
ð8Þ
nhb1
nhb2
b b p (1)p (1)
a2
1 2 a1
where u(a, b) =
2 2 ½0, 1. As a consequence, equilibrium
ðb2 pa1 (1) + b1 pa2 (1)Þ
payoffs are
Ui (a, b) [ Ui ðv , qai (ti (v )), a, bÞ = pai (1)
bk pai (1)
b1 pa2 (1) + b2 pa1 (1)
2
ð9Þ
Equilibrium violence vj is obtained by computing vj = vj ½vk (vj ), where
vj (vk ) is defined in (7), and equilibrium payoffs Uj (a, b) can be computed
using straightforward algebra and (6). When (a1 , b1 ) = (a2 , b2 ) = (a, b) for
some a, b . 0, dealers receive the same treatment inside the prison system.
In this case, also studied in Burrus (1999), their equilibrium payoffs equal
pa (1)=4 and are strictly lower than the payoff both would get under no violence, which equals pa (1)=2. If dealers could coordinate to not attack each
other, they could both improve their payoffs.26
Remark 2. Lemmas 1 and 14 have shown that when society’s ability to
enforce property rights is not high enough (r\ mini rai ), violence and
trading will take place in illicit drug markets.
68
Rationality and Society 33(1)
At this point of the analysis, it is worth noting that the equilibrium presented in Lemma 1 implies a clear pattern of correlation between crime
rates. In effect, violence rates should be higher in places and periods at
which drug dealing rates are higher. If criminal data are collected in an
economy where the game described above is played in each period, then a
pattern of positive correlation should be seen between criminal data for violence and those for drug dealing. According to the model, violence should
be greater in places where there is drug dealing and, similarly, violence
should be greater in periods in which drug dealers are operative. Thus, correlation should be positive, both geographically and intertemporally.
Property rights in illicit markets
The model in subsection ‘‘A drug dealing game’’ is now extended to allow
the punishment dealers get inside prison system when arrested, (a, b), to be
distorted by previously arrested criminals. We compare two situations, which
are intended to represent the two equilibria inside prison system suggested by
Figure 2b. The first one assumes a competitive environment among prison
gangs and refers both to SP’s situation before 2000 and to weak gang in Figure
3. In this case, no prison gang is able to distort punishments (ai = bi = 1 for all
i), since prisoners are equally exposed to attacks and protection from competing identical gangs. In the second situation, both the period after 2000 and the
strong gang case in Figure 3, a hegemonic prison gang has emerged in the
sense that it has become able to distort punishments in between l 1 and
L ø 1.
Throughout this section, we assume that social choice process (whatever it is) has resulted in an ability to enforce property rights such that
r\
r1 . That way, trading and violence occur in equilibrium in the first situation, as implied by subsection ‘‘The subgame-perfect equilibrium’’
when a1 = a2 = 1. In what follows in this section, we study the second
situation.
The hegemonic prison gang’s problem
Suppose the hegemonic prison gang (PG) wants to maximize its revenues
and is be able to perfectly monitor outside-prison actions. We suppose PG
has no access to any production function so that the only way it potentially
can get revenues is by extorting/taxing outside-prison drug dealers. The
instruments for convincing them to comply with such taxation are its
perfect monitoring ability and distortions in (ai , bi ) based on Di ’s
Bertolai and Scorzafave
69
outside-prison behavior. PG’s power to distort inside-prison payoff is not
unlimited: ai 2 ½l, L and bi 2 ½l, L.27
Inspired on the motivating case described in subsection ‘‘Empirical
motivation’’, we assume PG demands payments from dealers before turf
war stage and communicates how dealers should behave outside prison
with respect to violence amounts (v1 , v2 ) and drug dealing (q1 , q2 ). In addition, PG credibly promises to distort punishment inside prison (a, b)
according to (a, b). The following definition makes clear the objects to be
chosen by PG when building its extortion mechanism.
Definition 3. An extortion mechanism is a vector of functions
e = (t, v, q, a, b) such that t 2 R2 , v : R2 ! R2 , q : R4 ! R2 , and
(a, b) : R6 ! R2 3R2 .
Component functions in the extortion mechanism e = (t, v, q, a, b) has
the following interpretation: t = (t1 , t 2 ) specifies a payment ti from each
dealer i 2 f1, 2g; vt = (v1t , v2t ) specifies a vector of violence amounts for
each payment vector t 2 R2 ; qit (v) specifies, for each possible (t , v), the
quantity to be sold in Di ’s market; bit (v, q) describes for each possible
(t, v, q), the punishment factor Di gets inside the prison system when
arrested for violence acts; and ait (v, q) specifies, for each possible (t, v, q),
the specific punishment Di gets inside the prison system when arrested for
dealing drugs.
Observe that (a, b) is allowed to depend on how dealers has behaved
outside prison, since a mechanism specifies what dealers are supposed to
do, and what expected punishment they are exposed to, for each possible
contingency could emerge from the drug dealing activity and from the violence activity, (t, v, q). Most important, once operating in the market, they
are subject to the promised punishments a and b under probabilities r and
n, respectively.
When choosing the extortion mechanism, PG is restricted to mechanisms that are feasible.
Definition 4. The extortion mechanism e = (t, v, q, a, b) is feasible if
t i ø 0, vit ø 0, qit (v) ø 0 and
l ait (v, q) L, l bit (v, q) L
for each i 2 f1, 2g, each t 2 R2+ , each v 2 R2+ and each q 2 R2+ .
Property t i ø 0 requires payments from dealers to PG to be nonnegative, that is, PG is not able to make payments to outside dealers. Condition
70
Rationality and Society 33(1)
vit ø 0 establishes that violence cannot be negative, which could be interpreted as dealers not being able to caress each other in a way that is relevant for their expected payoffs. Condition qit (v) ø 0 requires trading to be
nonnegative and properties l ait (v, q) L and l bit (v, q) L say
that PG’s power to distort prisoners’ payoffs is limited to ½l, L.
Having defined what a feasible mechanism is, we are now able to define
expected payoffs for PG and (D1 , D2 ). Let e = (t, v, q, a, b) be given. PG is
assumed to derive utility from the revenues extorted from dealers and,
therefore, its payoff function is
W (e) = W (t, v, q, a, b) = t 1 + t 2 :
ð10Þ
If Di expects the other dealer to completely comply with the extortion
mechanism e, then its expected payoff when paying t0 ø 0 and complying
with the remaining recommendations in the mechanism is
Uie (t 0 ) [ Ui ½v^t , qi^t ðv^t Þ, ai^t ½v^t , q^t ðv^t Þ, bi^t ½v^t , q^t ðv^t Þ,
where ^t = (t0 , tk ) and the function Uj ( ) is defined in (2).
In addition to feasibility, PG’s choice is required to be compatible with
dealers’ participation and dealer’s incentives, as presented in the following
definition.
Definition 5. The extortion mechanism e = (t, v, q, a, b) is said to be compatible with Di ’s participation if it is feasible and
Uie t j t j ø 0:
(PC)
Also, it is said to be incentive compatible for Di if it is feasible and (t, v, q)
is optimal for Di in each possible contingency, that is, for each t 2 R2+ and
v 2 R2+ ,
qit (v) solves max piv (x, ait ½v, (x, qkt (v)))
ð11Þ
xø0
vit
solves
max Ui ½^vt (x), ^
qit (x), (a, b)t (^vt (x), ^
q1t (x), ^q2t (x))
xø0
ti
solves
max Uie (t) t
tø0
ð12Þ
ð13Þ
where ^vt (x) [ (x, vkt ) denotes the pair of violence level observed after payment profile t when Di chooses violence amount x and Dk comply with e.
The amount ^
qit (x) [ qit (^vt (x)) is the trading level after payments t and violence levels ^vt (x). Finally, the mechanism is implementable if it is compatible with participation and with incentives for all dealers.
Bertolai and Scorzafave
71
We are now able to define what is an optimal extortion mechanism for
the prison gang.
Definition 6. The mechanism e = (t, v, q, a, b) is optimal for PG if it
solves
maxfW (e) : (PC), (11), (12), and (13)g
e
(PGp)
Remark 7. Problem (PGp) is always feasible, since PG is always able to
implement the equilibrium outcome described in subsection ‘‘The subgame-perfect equilibrium’’. It is enough to recommend punishments a and
b invariant to history, the violence amounts presented in lemma 1, and no
payments from drug dealers.
As is traditional under the mechanism design approach, in defining
what a mechanism is, we have allowed PG’s choice to lie in a quite general space. This is intended to provide robustness to the no-violence result
presented in subsection ‘‘Property rights’ optimality and enforceability’’.
Having said that, we show in subsection ‘‘Property rights’ optimality and
enforceability’’ that the optimal mechanism e = (t, v, q, a, b) is quite
simple: it promises the minimum punishment in the equilibrium path
(i.e. ait ½vt , qt (vt ) = bit ½vt , qt (vt ) = l for all i) and the worse treatment
inside prison after any deviation from recommendations (i.e.
q) = bit (v, q) = L for all (t, v, q) 6¼ ½t, vt , qt (vt ) and all i). Also, tradait (v, ing is always monopoly quantities qai (ti ) defined in (4).
Property rights’ optimality and enforceability
We now provide a quite simple necessary and sufficient condition for the
no-violence agreement among criminals (property rights enforceability) to
be the optimal extortion mechanism for PG, that is, to solve (PGp). As a
corollary of lemmas 15 and 16 in Appendix A, remark 8 presents properties the optimal extortion mechanism should satisfy, respectively, out-ofequilibrium path and in-the-equilibrium path.
Remark 8. Incentive compatibility condition (13) can be rewritten without
loss of generality as
Uie (t i ) t i ø max Uie (x) x ,
8i 2 f1, 2g:
ð14Þ
xø0
Also, under the optimal extortion mechanism
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Rationality and Society 33(1)
(i) PG chooses the harshest possible punishment after deviations from
the extortion mechanism.
(ii) PG minimizes punishment to cooperating dealers and maximizes
their profits.
As a consequence of Remark 8(ii), incentive compatibility condition
(11)
slacks
in
the
equilibrium
path
since
it
equals
pivt ðql (ti (vt )), lÞ = pl (ti (vt )) ø pL (ti (vt )) and function pa (ti (vt )) is
decreasing in a. Also, the left-hand side of both (PC) and (14) becomes
Uie (ti ) t i = pl (ti (vt )) nhvit tj : These allow us to prove the following lemma:
Lemma 9. Constraint (14) binds at the optimal solution e and, therefore,
PG’s objective function equals
2
X
½pl (ti (vt )) nhvit sup Uie (xi ) xi : 0 xi 6¼ t i ,
ð15Þ
i=1
feasibility constraint ti ø 0 becomes
pl (ti (vt )) nhvit ø sup Uie (xi ) xi : 0 xi 6¼ t i :
ð16Þ
where supff (x) : x 2 Ag is the supremum of f (A) when f is a function.
Proof. See Appendix A.
Lemma 9 implies that PG’s problem reduces to the choice of (vit , qit (vt ))
in R2+ and (ait ½vt , qit (vt ), bit ½vt , qit (vt )) in ½l, L2 for each
(i, t) 2 f1, 2g3R2+ in order to maximize (15) subject to feasibility con00
0
straint (16) and incentive constraints (ICq ) and (ICv ), presented in
Appendix A. Also, because (14) binds at the optimal solution, the properties established in lemmas 15 and 16 are not only without loss of generality, but also desired for PG. Most important, property (15) in lemma 9
shows that (if implementable) PG will optimally choose no violence in the
equilibrium path (v1t = v2t = 0). The following proposition is our main
result and establishes a (quite simple) sufficient and necessary condition
for implementability of such no-violence agreement.
Proposition 10. (Property rights’ optimality) Suppose pa (ti ) . 0 for all
i. The optimal extortion mechanism for PG features no violence in the
equilibrium path (i.e. vt = (0, 0)) if and only if
Bertolai and Scorzafave
73
ð17Þ
pl (1) ø 2pL (1):
Proof. See Appendix A.
Condition (17) is equivalent to no violence (vit = 0) being an optimal
response for Di when Dk is not using violence as a competition instrument
(vkt = 0) and payments were executed as demanded by PG (t = (t 1 , t2 )).
By complying with the no-violence agreement, Di gets half of consumers/
territories from which profit level pl (1)=2 is earned. By deviating from
the agreement, Di gets monopoly power over the whole market which provides pL (1) in profits. Because treatment inside the prison system after
complying with the agreement is better than the respective treatment after
deviating (i.e. l L) there could be room for half of the drug market
being more attractive than the whole market.
Our second result makes more transparent the conditions under which
(17) holds and, therefore, property rights emerge after a prison gang
becomes hegemonic inside the prison system. The purpose is to highlight
how important the specific features of SP’s experience (discussed in subsection ‘‘Empirical motivation’’) are for property rights becoming enforceable even without the state’s enforceability. The intuition for the necessity
result (which already appears in Skarbek (2011) and Lessing (2014)) is
that traitor dealer could protect himself from PG punishment when r = 0
because he would then never be arrested, and that cooperation with PG
would have no effect on imprisonment costs if l = L.
Corollary 11. (Collusion sustainability) The optimal extortion mechanism for PG features no violence in the equilibrium path if (r, l, L) is such
that rL r\
rl , that is,
rl\
P(0)
rL
d
(POe)
and only if police work has some effectiveness (r . 0) and PG has some
power inside the prison system (L . l).
Proof. When r\
rl , we have pl (ti ) . 0 for all i. Thus, from proposition
10, we know that no-violence in the equilibrium path is optimal if and
only if pl (1) ø 2pL (1), that is,
G(r, l, L) [ pl (1) 2pL (1)
= max½P(x)x rdlx 2 max½P(y)y rdLy ø 0:
xø0
yø0
74
Rationality and Society 33(1)
For the necessity result, observe that G(0, l, L) = maxx P(x)x\0 for
all l 1 L, and G(r, 1, 1) = p1 (1)\0 for all r 2 ½0, r). For the sufficiency result, remember that piv (q, a) is assumed strictly concave in q and
observe that its derivative at q = 0 equals P(0) rda. Then, using (21) we
have that P(0) . rda implies qa (1) . 0, since in this case
∂piv (0, a)=∂q . 0. Also, P(0) rda implies qa (1) = 0, since then
∂p(q, a)=∂q\0 for all q . 0. Now, observe that under (POe) and r\
rl ,
we have rdl\P(0) rdL and, therefore, ql (1) . 0 = qL (1). As a conclusion, pl (1) . 0 = 2pL (1).
The sufficiency result holds because by increasing detention probability
r and/or by increasing PG’s power to distort punishments L l so that
(POe) holds, the benefit in deviating from collusion is eliminated
(pL (1) ! 0), while some benefit from cooperation is still available
(pl (1) . 0).28 The necessary conditions (r . 0 and L . l) are actually
sufficient for no-violence to be the only solution to PG’s problem when
demand for drugs are linear, as in our case.
Remark 12. The optimal extortion mechanism for PG when demand function for the illegal drug is linear features no violence in the equilibrium
path if and only if r(L l) . 0.
Some insights and the empirical exploration
The model presented in subsection ‘‘A drug dealing game’’ and section
‘‘Property rights in illicit markets’’ provides a simple framework where the
theoretical perspective proposed in section ‘‘Introduction’’ and based on
Skarbek (2011) and Lessing (2010) can be discussed. In effect, the model
is a natural extension of standard economic models based on Becker (1968)
to the case treatment inside prison system can be made contingent on outof-prison behavior in a wider sense than it is traditionally done. In particular, previously arrested criminals are assumed to be able to distort punishment inside prisons based on what they observe (actually, on what they are
informed about) outside prison.
Following the mechanism design tradition, a natural questioning on the
results established in subsections ‘‘The subgame-perfect equilibrium’’ and
‘‘Property rights’ optimality and enforceability’’ concerns to the reason
property rights are not enforceable by other agents in the model. In effect,
it is reasonable to expect that people outside prison is in a better condition
to monitor what happens outside prison than people inside the prison system. Why society-sponsored state is not able to cease violence in drug dealing markets while PG is able to do so? The reason, we think, rests on the
Bertolai and Scorzafave
75
constraints faced by the state that are not necessarily faced by PG. Even
though the state is endowed with a better monitoring technology, PG is
sometimes expected to have a richer punishment technology.
As an important example of this reasoning, in societies where the
respect to a relevant amount of human rights is required when convicting
and punishing someone, the state’s power to punish criminals is bounded
and the police work in detecting and evidencing crimes is a sophisticated
activity. If police work does not produce sufficient evidence of guilty
while preserving criminals’ human rights, judicial system in such societies
is not able to convict them. This is specially critical in developing countries, like Brazil. Even though conviction is achieved, such societies cannot
make punishment arbitrarily high without violating criminals’ human
rights. Because PG is not necessarily constrained by human rights constraints, it both requires much less evidence to convict those not complying
with its orders and has a wider range of punishment instruments.29
In the model developed in subsection ‘‘A drug dealing game’’ and section ‘‘Property rights in illicit markets’’, the state and PG can be seen as
endowed with perfect monitoring technologies on the outside prison behavior of arrested criminals. The constraints faced by the state, however, limit
its capacity to arrest and convict criminals (i.e. r\1 and n\1) and to punish them inside the prison system (i.e. di = ai d\‘ and hi = bi h\‘).
On the other hand, fewer constraints on PG’s action allows it implement
faster and wider punishment. In effect, not only the treatment inside the
prison system is distorted to induce compliance to the agreement proposed
by PG, but also unilateral deviations outside prison are punished even
before arrestments have taken place. Specifically, payment t 0 to PG not
equal to t i is punished through the violence level recommended to the
other dealer: vk(t0 , tk ) . This is the counterpart in our model of LS’s punishment parameter for defectors, d. Strictly speaking, however, this outside
punishment to unilateral deviations is ultimately induced in our model by
the distorted treatment inside prisons, since the other dealer must be convinced to execute it.
The key factor studied using our model, a change inside the prison system from a competitive environment to the hegemony of a group of criminals, has been extensively documented and studied in the Brazilian
context.30 The results in subsection ‘‘A drug dealing game’’ and section
‘‘Property rights in illicit markets’’ shows that such concentration of power
inside the prison system implies the equilibrium in illicit markets outside
prison to shift from warfare to peace among market dealers, since this shift
is optimal to PG. Specifically, the hegemonic group of criminals is shown
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Rationality and Society 33(1)
to desire a collusive agreement among dealers under which no violence is
performed. This contrasts with the equilibrium under no hegemony inside
prison system, in which the possibility to conquer consumers/territories
from competitors was shown to drive violence up to a positive level.
Mapping our model and results to the Shortland and Varese (2014,
2016)’s framework and results provides another useful perspective on how
our work contributes to the large literature on private provision of governance discussed in section ‘‘Introduction’’. In their work, authorities and
local clans in Somalia, seen as stationary bandits offer shelter and protection to maritime Somali pirates. They also show that the protective choice
is not a trivial one, in the sense that it is possible for the ‘stationary bandits’
‘‘to switch from protecting criminal activities directed against outsiders, to
the protection of peaceful and productive economic endeavors. When taxation from trade is more profitable than taxation from crime, local elites stop
supporting widespread criminality.’’ (Shortland and Varese, 2014, p. 760).
The so called ‘stationary bandit’ in our framework, the PG, provides
protective services to drug dealers, and it does so in a nontrivial fashion: it
convinces the drug dealers to refrain from trying to conquer the opponents’
market share by making use of the (state-sponsored) police work on arresting criminals. Also, this distinguishing nature of PG’s enforcement tool
makes it much more complex for PG taxing legal economic activities than
the illicit ones. Qualitatively, this is the same asymmetric feature that convinces authorities and local clans in Somalia opting to offer shelter and
protection to pirates in areas remote from trade routes.
In addition to deepening Lessing (2017b)’s formalization of Skarbek
(2011)’s theory and Lessing (2010)’s conceptual framework (on the functioning of illicit drug markets and its relation to the equilibrium inside the
prison system) in order to study property rights emergence, the model
developed in subsection ‘‘A drug dealing game’’ and section ‘‘Property
rights in illicit markets’’ provides a new empirical perspective on crime
rates dynamics. As already pointed out, the equilibrium behavior under no
hegemony inside prisons, as presented in Lemma 1, implies positive spatial and temporal correlation between violence and drug dealing outside
prison system. The results on section ‘‘Property rights in illicit markets’’,
on the other hand, implies that such correlations should disappear in data
after a group of prisoners takes relevant control over inmates’ welfare.
Such empirical new perspective is explored in subsection ‘‘The empirical
exploration’’ as a further step to collect evidence on the relevance of the
SK and Lessing (2010) theoretical frameworks.
Bertolai and Scorzafave
77
The empirical exploration
As a way to explore the relation between violence and drug dealing in data
for SP, we collect annual data on homicides and drug traffic rates in all
645 SP’s municipalities. The aggregate relationship between these two
rates are presented in Figure 6, which shows the scatterplot of drug traffic
and homicides in our sample over time.31 The time evolution already suggested by Figures A2 and A3 shapes the scatterplot in Figure 6: while there
is a important decrease in homicide rate starting in year 2000, the drug
traffic has an opposite trend.
This evolution has already been discussed by other authors using different theoretical frameworks. More interesting from our new empirical perspective, a stylized fact about homicides and drug dealing is suggested by
this figure, as follows. As a rough measure of association between these
two
crime
rates,
consider
what
has
happened
with
Dht =Ddt [ (ht ht1 )=(dt dt1 ), where ht denotes homicide rate in year
t and dt denotes drug traffic rate in year t. Such measure is positive and
seems relatively stable up to year 2000, becomes negative after that and
approaches zero in the end of the period. This temporal and aggregate pattern could be interpreted in light of our study as a transition between equilibria during which positive correlation fades out as PG’s hegemony
disseminate across the prison system.32
In order to explore a more detailed measure of association between drug
traffic and homicide, we compute the evolution of the linear correlation
between them over municipalities and after taking into account factors
common to all places in each year. Specifically, we estimate the following
relationship using the Ordinary Least Squares (OLS) technique:
hit = a0 + b0 dit +
10
X
(as + bs dis )yst + eit ,
ð18Þ
s=1
where hit denotes homicide rate observed in municipality i in year
(1997 + t), dit denotes drug traffic rate observed in municipality i in year
(1997 + t). The variable yst equals 1 in year (1997 + t) if s = t and equals
zero otherwise. The variable eit is assumed to have zero mean across i and
across t. Equation (18) implies that expected homicide rate in 1997 equals
a0 if drug dealing is absent. Similarly, expected homicide rate in year
(1997 + t) equals (a0 + at ) if drug dealing is absent. The correlation
between homicide and drug traffic in 1997 is b0 , while the same measure
for year (1997 + t) equals (b0 + bt ). The estimated values for (b0 + bt ) are
reported in Figure 7a.
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Rationality and Society 33(1)
Figure 6. Drug traffic rate and homicides rate for SP.
Figure 7a shows that the pattern of association between ht and dt suggested by Figure 6 is still present after taking into account factors common
to all municipalities in each year. Estimated correlation suggests a positive
and increasing relationship before 1999 and a decreasing trend after that.
Also important, the negative association suggested by our rough measure
of relationship between homicides and drug dealing (Dht =Ddt ) has disappeared in Figure 7.
Although the period 2000–2001 is identified in the literature as a critical point on the dissemination of PG over the prison system, Dias (2011)
shows (and the construction of our measure of PG power supposes) that
such dissemination was heterogeneous in space and time. Such dispersion
pattern in data makes it possible to improve our measure of association
between ht and dt by taking into account that in a given year PG’s influence were not present in all drug dealing markets. Specifically, we use the
OLS technique to estimate the following extension of equation (18):
hit = g0 + h0 dit +
6
X
½(g l + hl dit )cit ( l) + (g l + hl dit )cit (l) + eit
ð19Þ
l=1
where hit is still homicide rate in year (1997 + t) at region i, but now the
definition of region has changed from municipality from a set of neighbors
municipalities.33 Variable dit denotes drug dealing in year (1997 + t) at
region i and has changed in the same way. The variable eit is assumed to
have zero mean across i and across t. The new variable cit (l) indicates if in
year (1997 + t) region i have been under PG’s control for l periods (years).
Bertolai and Scorzafave
79
Figure 7. Association between drug dealing and homicides. (a) Association
evolution. (b) Lagged association.
Accordingly, if (1997 + ti ) denotes the year PG took control over drug dealing markets in region i, then cit (l) = 1 if l = t ti and cit (l) = 0 otherwise.
Equation (19) implies that expected homicide rate in the year PG took
control of a region equals g 0 if drug dealing is absent. Accordingly,
expected homicide rate l 2 f6, 5, , 0, , 5, 6g periods from the
time PG’s control took place equals (g0 + g l ) if drug dealing is absent. The
correlation between homicide and drug traffic in the year PG took control
of a region is h0 , while the same measure l 2 f6, 5, , 0, , 5, 6g
periods from the time PG’s control took place equals (h0 + hl ). The estimated values for (h0 + hl ) are reported in Figure 7b as a function of l and
show that before PG control (l\0), the correlation between homicides and
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traffic was positive and 2 years after PG control in the region (l ø 2), there
is no correlation at all.
Figure 7b reveals that, by taking into account the spatial and the temporal dimensions of PG’s dissemination over the prison system, the positive correlation between violence and drug dealing identified in Figure 7a
is particular to regions without the influence from PG.34Also, the average
correlation before PG’s control in Figure 7b is higher than that measure
before 2000–2001 period, in Figure 7a. In the same fashion, the average
correlation after PG’s control in Figure 7b is much closer to zero than that
measure after 2000–2001 period, in Figure 7a.
As a final exercise in exploring correlation between violence and drug
dealing in data, we study in more detail the pattern suggested in Figure 7b of
average correlation after and before PG’s control. Accordingly, we impose
hl = h for l\0 and hl = h+ for l ø 0 in relationship (19) and reintroduce
factors common to all regions in a given year (variable yst ). Also, we take into
account factors common to all periods in a given region (known as fixed effect
for region) and some other variables usually deemed important to explain
homicide rate (as described in Appendix B). Specifically, we use econometric
techniques to estimate the following relationship
hit = (a0 + f0 + m0 ) + d0 dit + (f1 + d1 dit )cit +
10
X
s=1
as yst + mi +
K
X
uk xkit + uit
k=1
ð20Þ
where mi denotes the fixed effect for region i and xkit is the value at region i
in year (1997 + t) of the kth variable in our list of variables usually deemed
important to explain homicide rate. Error term uit is assumed to have zero
mean across i and across t. The variable cit = maxl . 0 cit (l) 2 f0, 1g indicates if region i was under PG’s influence in year (1997 + t).
Equation (20) implies that expected homicide rate at the base region
(j = 0) in year 1997 equals h0 [ (a0 + f0 + m0 ) if drug dealing and PG’s
control were both absent and xk were zero for all k.35 The correlation
between homicide and drug traffic is d0 if PG’s control is absent, while the
same measure under PG’s control equals (d0 + d1 ).
Existing empirical work on the influence of PG on violence has paid
attention to c’s effect on h without telling apart its impact through drug
dealing (d:c).36 The interest in such cases lies in estimating how relevant
PG’s influence is to explain the sharp decline in homicide rate presented in
Figure 1. Much more important than (econometrically) correcting an eventually biased estimation of f1 , our empirical exploration calls attention for
Bertolai and Scorzafave
81
a neglected perspective on data: positive correlation between violence and drug
dealing when property rights are not enforceable (d0 . 0 when c = 0) and no
correlation at all when property rights emerges as result of PG’s control
(d0 + d1 = 0 when c = 0). Although the effect of PG’s influence on homicides
(f1 + d1 d) is a relevant point to be explored, it is of secondary importance for
the empirical exploration of the model developed in this paper.
A summary on the estimate of parameters in relationship (20) is presented
in Table 1. As discussed in Appendix B, two alternative measures of PG’s
control (c) are considered. They are identified in Table 1 as ‘‘PG’s control
A’’ and ‘‘PG’s control C’’. Also discussed in detail in Appendix B, a relevant
econometric concern when dealing with relationships like (20) refers to the
fact that homicide rate and drug traffic are simultaneously determined. This
raises questioning about OLS estimator being biased. Such potential problem
is taken into account by using the Instrumental Variable (IV) technique.
Table 1 presents estimates for both techniques, OLS and IV.
In a broad sense, results in Table 1 support the findings of the previous
empirical exercises presented in Figures 6 and 7. Correlation between violence and drug dealing is positive before PG’s influence (d0 . 0) and
become much lower after it (d0 + d1 ’0). Although the hypothesis (d0 0)
cannot be rejected at a satisfactory level of confidence under OLS estimates, it is strongly rejected under IV estimation. Also, the hypothesis
(d0 + d1 = 0) is rejected at a level of confidence of 90% only in the IV estimation under measure A for PG’s control (p-value in this case is 0.074).
But it is not rejected in all cases at the level of 95% (since p-value in all
case are greater than 0.05).37
Final remarks
We have provided a model for illicit drug markets in which property rights
can emerge as a result of changes inside the prison system, as predicted by
Skarbek (2011)’s theory on governance provided by prison gangs. Instead
of resorting to infinite horizon and high patience, as in Castillo (2013) and
Castillo and Kronick (2020), we extend the two stage game in Burrus
(1999) to show that the channel of influence studied by Skarbek (2011)
and Lessing (2010, 2014), from inside to outside the prison system, makes
reasonable to expect turf wars to cease after a prison gang becomes hegemonic inside prison system. Such expectation is shown reasonable by
using the mechanism design approach. Specifically, the hegemonic group
of prisoners is shown to desire to promote the collusive agreement among
dealers under which no violence is performed when it is able to do so.
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Rationality and Society 33(1)
Table 1. Econometric estimations of relationship (20).
Homicide rate (h)
d0
PG’s control A
PG’s control C
OLS
IV
OLS
IV
+ 2:013
(0:828)
1:216
(0:278)
36:58
(8:12)
+ 0:126
(0:577)
0:029
(0:015)
+ 0:098
(0:583)
Yes
0.074
4873
0:015
(0:034)
+ 0:038
(0:036)
4:643
(1:790)
0:782
(0:171)
0:024
(0:005)
+ 0:493
(0:169)
Yes
0.190
4873
+ 2:831
(0:954)
1:536
(0:430)
+ 48:96
(12:18)
+ 0:345
(0:673)
0:039
(0:016)
+ 0:370
(0:543)
Yes
0.299
4873
+ 0:016
(0:043)
d1
0:001
Drug* PG (d:c)
(0:048)
0:117
PG (c)
f1
(1:950)
u1
0:814
Gini on
(0:179)
wages (x1 )
u2
0:026
Population
(0:007)
density (x2 )
u3
+ 0:649
Urbanization
(0:165)
rate (x3 )
Yes
Year dummies (yst ) as
p-value (joint test) d0 + d1 = 0 0.434
Number of
N
4873
observations
Drug (d)
*** Significant at 1%; ** Significant at 5%; * Significant at 10%.
It is worth emphasizing that features in our model to illicit drug markets
does not include some commonly expected reasons for the peaceful
arrangement to be attractive to drug dealers. Players do not spend productive resources in the conflict since there is no resource cost for violence
acts. Violence effort affects only the distribution of territories/consumers,
implying no losses from dead personnel or resource destruction. Also,
prison gang power to distort punishments is bounded and both demand for
drugs and police work are invariant to violence levels. This reinforces our
argument for reasonableness in predicting property rights to take place
after the inside-prison shift, since they do take place in our model even
without the features just mentioned.
We have also provided a new empirical perspective to Skarbek (2011)’s
theory and Lessing (2010) conceptual framework by exploring data for our
motivating case, the fall on violent crimes rate in São Paulo in the 2000’s.
The exploratory exercise support our model’s prediction: correlation
between drug dealing and murders is positive before PG and becomes nonexistent after PG. This must be recognized as a remarkable result, considering that the empirical exercise was guided by the theory, not the other
Bertolai and Scorzafave
83
way around. Inspired on the Skarbek (2011)’s theory and Lessing (2010)
conceptual framework, the formal model was built to make explicit how
an hegemonic prison gang can be seen as an efficient provider of property
rights in outside-prison illicit drug markets, in the sense that such provision
maximizes its revenues. Equilibrium analysis, then, clearly indicated what
in the data we should look for: temporal and spatial correlations between
homicides and drug traffic. This novel empirical perspective on the
Brazilian case would not emerge as the most natural and promising way to
analyze the data without theory’s guidance. Also, it is not trivially
expected, ex-ante, that data will confirm theory’s prediction.
More than having conclusive econometric evidence on the adherence of
our model to the observed changes in the pattern on crime rates in São
Paulo, the strength of our reasoning comes from bringing consistence for
apparently unrelated pieces of information, such as falling rates of violent
crimes both inside and outside the prison system, increasing drug dealing
in general, the existence of parallel justice system (Tribunal do Crime), and
monthly payments from outside to inside-prison criminals (Mensalidades).
More generally, this consistence makes clear the São Paulo case is another
instance (an interesting one) in which the private provision of governance
manifests itself. And, it is an interesting instance: governance is privately
provided for an economic activity the state has forbidden and strives to
restrain. The consistence fits so well to this framework, that it is difficult to
deny its relevance to this case. From this general point of view, our work
contributes to the literature on private provision of governance discussed in
section ‘‘Introduction’’.
While our model seems reasonable to describe São Paulo’s prison gang
in the 2000’s, Lessing and Willis (2019) show that this specific prison gang
has evolved in the 2010’s to promote its legitimacy among criminals by
means of mild sanctions to violators on its rules and by funding collective
benefits for member’s families. Although our model captures some benefits to inmates’ families through better treatment l, an interesting extension
of our model to accommodate Lessing and Willis (2019)’s findings would
be changing prison gang’s objective, from revenue maximization to something that depends positively on both risk sharing among gang members
and inmates welfare.
Finally, for the discussion about public policies against violence, our
framework weakens the reasoning for drug legalization, since the argument
for drug legalization as a way to fight violent crimes (Miron and Zwiebel,
1995) would have no effect if property rights have already been enforced
in illicit drug markets. Although this observation might appear surprising
84
Rationality and Society 33(1)
to some readers at first, this reasoning is another instance in which a central argument from the literature on economics of anarchy (Leeson, 2007b,
2014a) manifests itself. Namely, the attractiveness of state-provided governance must be evaluated by comparing it to the privately provided governance, not by comparing it to no governance at all.
Acknowledgements
Previous versions of this paper circulated under the title ‘‘Property rights in illicit
drug markets’’. We are grateful to anonymous referees for helpful comments and
suggestions, and to seminar participants at FEARP-USP, INSPER, EESP-FGV and
at the 2019 LACEA-LAMES Meeting. We are also thankful to the valuable support
from research assistants: Bruna Alves, Denis S. Moreira, Giovanni Di Pietra, Lı́gia
C. Godoy, Lucas Kava, Marcos Paulo C. Costa, and Vinicius J. Vilela. All remaining errors are our own.
Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.
ORCID iD
Jefferson DP Bertolai
https://orcid.org/0000-0002-2535-920X
Notes
1. For example, Demsetz (1967) describes state ownership as a situation in
which ‘‘the state may exclude anyone from the use of a right. ’’.
2. See Assembly (1948).
3. See Becker (1968).
4. See Garfinkel and Skaperdas (2007) for a comprehensive review on the economic theory of conflicts.
5. For other theoretical works focused in drug-consumer nations, see Reuter and
Kleiman (1986) and Lee (1993). For drug-producer nations, see Grossman and
Mejı́a (2008) and Poret (2002).
6. See also Hafer (2006) and Anderson and McChesney (2018).
7. A rich overview of the literature on public choice and the economics of anarchy can be found in Powell and Stringham (2009). Also, Leeson (2014a) provides a collection of interesting studies on the economics of anarchy.
8. See Hobbes (1960).
9. We thank an anonymous referee for calling our attention to this point of view.
Bertolai and Scorzafave
85
10. Although also studying private provision of enforcement to property rights,
Murtazashvili and Murtazashvili (2016)’s work on land property in rural
Afghanistan analyses private provision in a context of a state interested in
enforcing such rights but poorly equipped (organized) to do so. Our study, on
the other hand, suggests the emergence of private provision of governance on
a context the state has refused to grant property rights to individuals, but has
been unable to completely restrict drug dealers operations.
11. Although our model were motivated by São Paulo’s case, it applies not only
to the Brazilian case. Every system designed to fight criminal activities should
recognize that the way criminals are controlled inside the prison system is relevant for crime rates dynamics outside the prison system. By presenting case
studies from Brazil, California, El Salvador, Peru, South Africa and Texas,
Lessing (2010, 2014)’s analysis suggests that the connection between inside
and outside prison pointed by Skarbek (2011) is not specific to SP context.
12. It grew on average 4% per year from the third quarter 1995 to the end of 2000
and 12% from the beginning of 2001 to the first quarter of 2014.
13. And, as a consequence, also consistent with Skarbek (2011)’s theory and
Lessing’s research agenda (Lessing, 2010, 2014, 2017b, 2017c; Lessing and
Willis, 2019).
14. Specifically, the ratio of prison population to total population multiplied by
105.
15. Figure 2 does not present inmates’ deaths from other reasons. In particular, it
does not include all deaths from Carandiru Massacre in 1992, an event discussed in Lessing (2010, 2017a), for example.
16. For detailed analyses on this matter in political science literature, see
Lessing’s research agenda (Lessing, 2010, 2014, 2017a, 2017b, 2017c;
Lessing and Willis, 2019).
17. See, for example, Dias (2011), page 230, and Lessing and Willis (2019), page
591.
18. See, for example, Dias (2011), Feltran (2010), Lessing (2010), page 172, and
Lessing and Willis (2019), page 587.
19. For richer formulation in which the probability to be arrested dealing drugs
depends on the quantity sold, see Poret and Téjédo (2006).
20. One could also assume this is a cost for the dealer’s organization as a reducedform modeling of contract relation between organization and its employees.
See Polo (1996) for a rich formulation to this agency problem.
21. Because individual demand for drugs is given by Q(p) and Di has monopoly
power over ti (v) consumers, then its market demand is ti (v)Q(p).
22. Because our motivating case is a drug-consumer country, Military combats,
election campaigns, industrial struggles (strikes and lockouts), legal conflicts
(lawsuits), and even rivalries among siblings or between spouses within the
family all fall under this heading’’ we abstract from wholesale drug market.
23. As pointed out by Hirshleifer (1989), however, such formulation ‘‘ . applies
far beyond the rent-seeking context. (Hirshleifer, 1989: 101).
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Rationality and Society 33(1)
24. Axiomatization for more general contest success function has been provided
by Skaperdas (1996) and Clark and Riis (1998). For an overview on how such
functions has been used in Economics, see Garfinkel and Skaperdas (2007).
25. Observe that under the standard contest success function we assume here, the
decisions on violence amounts are strategic complements.
26. Although the conflict is a zero sum game per se, effort (violence) in this
model is probabilistically and proportionally punished through police work.
This explain why both dealers are worse off in the equilibrium presented in
Lemma 1 than under the no-violence situation.
27. Accordingly, (L l) can be view as an indicator of how powerful PG is in
distorting (a, b). PG’s power is nonexistent when l = L and gets unlimited
when (L l) ! ‘.
28. Of course, if probability of detention is high enough such that drug dealing is
not worth even under the lower punishment l (i.e. rl ø P(0)=d), then no violence is also an equilibrium. This trivial no-violence equilibrium is obviously
of no interest at this point, however.
29. On the other hand, Lessing and Willis (2019) shows that even PG can sometimes make efforts to provide much more evidence of guilty as a mechanism
to legitimate its power over outside criminals.
30. See, for example, Dias (2011), Feltran (2010) and Lessing’s research agenda
(Lessing, 2010, 2014, 2017b, 2017c; Lessing and Willis, 2019). Also, a brief
description on historical details is presented in Appendix B, for convenience.
More generally, Skarbek (2012, 2014, 2016) suggests that inside-prison governance evolves from a decentralized system of rules (based on norms, e.g.
the prisoner’s reputation) to a more centralized one (based on organizations,
the prison gangs) as a result of several features that can be observed in the SP
prison system in the 1990’s, such as large and overcrowded prison population
and lack of official supply of governance.
31. A description on data and econometric details is presented in Appendix B.
32. The PG dissemination over the state became pretty relevant around years
2000 and 2001, but it was heterogeneous in space and time. This pattern is
briefly reviewed in Appendix B (it is described in detail by Dias (2011)) and
is captured by our measure of PG’s dissemination (described in Appendix B).
Of course, our model is not able to (designed for) explain the transition
between the two equilibria. It must be seen as comparing two different steadystates.
33. In redefining our measure of region, the number of regions was reduced from
645 to 439. See Appendix B for more details.
34. The correlation is positive in the first 2 years of domination (l = 0 and l = 1).
This again could be interpreted as a transition between SPNE’s not captured
by our theory.
35. Accordingly, expected homicide rate equals (h0 + at ) in year (1997 + t) at the
base region if d = c = xk = 0; it equals (h0 + f1 ) in year 1997 at the base region
Bertolai and Scorzafave
36.
37.
38.
39.
40.
41.
42.
43.
87
if d = xk = 0 and c = 1; and it equals (h0 + mj ) in year 1997 at region j if
d = c = xk = 0.
See, for example, Biderman et al. (2019).
The dramatic change in the estimates for f1 after considering instrumental
variables suggests that inference based OLS estimates are not reliable. We
included them in Table 1 for completeness, however.
Observe that bjt (v, q) for q 6¼ qjt (v) does not appear in problem PGp. Thus, it
is without loss of generality to assume bjt (v, q) = L also for q 6¼ qjt (v).
Only two other states (Rio de Janeiro and Pernambuco) has experienced
decline in homicide rates in the period, but these reductions were much
smaller than SP’s one.
Homicide rate’s fall in Figure 1 was driven by falling murder rates instead of
manslaughter, as shown in graph 9(i). That way, unintentional acts (like transit
accidents, for example) can be discarded as an explanation. Additionally, nonviolent crimes do not show the same sharp decline in SP.
Translating Dias (2011)’s words: With the elimination of its main rivals, the
consolidation of its domination and the conquest of hegemony, the SPPG
achieved external and internal stability and promoted a complex accommodation of relations with the government, making it possible to completely reconfigure the social relations among the prisoners, constructing a new social
order (.) characterized by a new balance of power in which physical violence ceases to be the central element of the domination relations.(.) The
reduction of the killings of prisoners is the effect of the consolidation of a
power against which there are no longer rivals. So killing is not necessary
anymore.
As a supporting evidence to our strategy in constructing the SPPG variable, it
identifies as ‘dominated prison’ 25 out of 29 prisons that participated in the
‘‘2001 mega-rebellion’’. Two of the four prisons not identified by our measure
until 2000 were created only 2 years before the mega-rebellion. This explain
the fact that we do not attribute it to SPPG domination up to 2001.
See Edlund and Machado (2019) for a similar IV strategy.
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Appendix
A. Proofs
Lemma 13. Let ra = P(0)=da. Then,
qa (tj ) = tj
ad
maxf
ra r, 0g
2B
pa (tj ) = tj B
and
qa (tj ) 2
:
tj
As a consequence, qa (t) . 0 (and, therefore, pa (t) . 0) if and only if
r\
ra .
Proof. First, the first-order condition to the monopolist’s optimization in
the trading stage is
∂p
(qa (t), a) 0
∂x
and
qa (t)
∂p
(qa (t), a) = 0:
∂x
ð21Þ
Also, ∂2 pv (x, a)=∂x2 = 2B\0 and, therefore, objective function is
strictly concave in its first argument. It follows that (21) delivers the
unique optimal solution. For qm (a) . 0, observe that ∂pv (x, a)=∂x is
strictly decreasing in x and ∂pv (0, a)=∂x is strictly positive if and only if
r(a) r).
r\
r(a) since it equals P0 (0):0 + P(0) rda = ad(
raj g, so that paj (1) . 0 for all
Lemma 14. Suppose that r\ minj f
j 2 f1, 2g. Then, there is no optimal solution for Dj ’s problem under conjecture vk = 0.
2
raj r, 0g =4B . 0. Suppose the
Proof. Observe that pa (1) = aj d maxf
conjecture vk = 0 and observe that for every choice vj . 0, we have
tj (0, 0) = 12 \1 = tj (vj , 0). Therefore, by choosing vj = 0, dealer j gets
pa (1)=2 nhbj 0 = pa (1)=2, and, by choosing any vj . 0 he gets
pa (1) nhbj vj . Therefore, any violence amount vj such that
0\vj \pa (1)=2nhbj is a response to vk = 0 strictly better than vj = 0. In
addition, any violence amount e . 0 is strictly worse than e=2 as a reply to
vk = 0.
Lemma 15. Constraint (11) can be replaced in problem (PGp) without
loss of generality by constraints ajt (v, x) = L for x 6¼ qjt (v) and
0
pjv qjt (v), ajt (v, qjt (v)) ø pL (tj (v)), 8(j, t, v) 2 f1, 2g3R4+ : (ICq )
Proof. We first observe that at the optimal solution e = (t, v, q, a, b) we
q) = L for q 6¼ qjt (v), that is, PG chooses the harshest feasible
have ajt (v, Bertolai and Scorzafave
93
punishment for dealers not complying with its recommendation for trading
after an arbitrary history (t , v). In order to see this is the case, observe that
when q 6¼ qjt (v), we have at (v, q) affecting (PGp) only through constraint
(11). Also, (11) can be rewritten as
pjv qjt (v), ajt (v, qjt (v)) ø max pjv (x, ajt (v, x))
8(j, t, v) 2 f1, 2g3R4+ :
xø0
ð22Þ
But pjv (x, a) is decreasing in a and, therefore, p(x, ajt (v, x)) ø pv (x, L) for
all feasible punishment ajt (v, x) 2 ½l, L. As a consequence, we can take
ajt (v, x) = L for all x 6¼ qjt (v) in order to minimize the right-hand side (rhs)
of inequality (22), since in doing so we relax (11) without changing any
other object in (PGp). Since maxy ø 0 pjv (y, L) = pL (tj ) we have established
0
(ICq ).
0
Lemma 16. Constraints (ICq ) and (12) can be replaced in problem (PGp)
without loss of generality by constraints ajt (v, qjt (v)) = bjt (v, qjt (v)) = L
and qjt (v) = qL (tj (v)) for v 6¼ vt and all (j, t) 2 f1, 2g3R2+ , and
pjvt qjt (vt ), ajt (vt , qjt (vt )) ø pL (tj (vt )), (IC 00 q )
0
Uj (vt , qjt (vt ), ajt vt , qjt (vt ) , bjt vt , qjt (vt ) ) ø max Uj ((x, vkt ), qL ½tj (x, vkt ), L, L) (ICv )
xø0
for all (j, t) 2 f1, 2g3R2+ :
Also, PG chooses for cooperating dealers the lightest punishment
ajt (vt , qjt (vt )) = bjt (vt , qjt (vt )) = l and the most profitable trading
qjt (vt ) = ql (tj (vt )).
Proof. Next, it is optimal to set ajt (v, qjt (v)) = bjt (v, qjt (v)) = L for v 6¼ vt ,
that is, PG chooses the harshest feasible punishments if someone refuses to
comply with its recommendation for violence after
history t.38 As a conse0
j
quence, constraint (ICq ) for v 6¼ vt becomes pv qjt (v), L ø pL (tj (v)) and
binds since the unique implementable trading in this case is qjt (v) = qL (tj ),
as implied by lemma 13. In order to see that the harshest punishment is
optimal for v 6¼ vt , observe that (12) for Dj can be rewritten as
max Uj (x, vkt ), qjt (x, vkt ), ajt ½(x, vkt ), qjt (x, vkt ), bjt ½(x, vkt ), qjt (x, vkt ) xø0
Uj (vt , qjt (vt ), ajt vt , qjt (vt ) , bjt vt , qjt (vt ) )
ð23Þ
94
Rationality and Society 33(1)
and that qjt (v) and ajt (v, qjt (v)), for v 6¼ vt , affect problem (PGp) only
0
through the left-hand side (lhs) of (ICq ) and the lhs of
(23). In particular,
0
they affect (ICq ) and (23) only through profits pjv qjt (v), ajt ½v, qjt (v) .
0
Therefore, by minimizing the lhs of (ICq ) until it binds, we relax (12) without changing the objective function and the remaining constraints.
Therefore, ajt (v, qjt (v)) = L and qjt (v) = qL (tj ). Finally, from definition (2),
it is clear that the lhs of (23) is decreasing in bjt (v, qjt (v)). Then, choosing
the maximum feasible punishment bjt (v, qjt (v)) = L relax (12) without
changing the objective function and the remaining constraints.
We now observe that for the equilibrium path (t, vt ), it is optimal
to
minimize bjt (vt , qjt (vt )) and to maximize pjvt qjt (vt ), ajt (vt , qjt (vt )) , that
is, PG chooses for cooperating dealers the lightest punishment
ajt (vt , qjt (vt )) = bjt (vt , qjt (vt )) = l and the most profitable trading
qjt (vt ) = ql (tj (vt )). Such property comes from the fact that by maximizing
equilibrium profits and minimizing
punishments, it is possible
equilibrium
0
to relax constraints (PC), IC 00 q , (ICv ) and (13) without changing objective function in (PGp). To make this point clear, observe that (13) can be
rewritten as
n
o
Uje (t j ) t j ø max Uje (x) x ,
8j 2 f1, 2g:
xø0
From
definition of Uje (x) and (2), it can be seen that
j
pvt qjt (vt ), ajt (vt , qjt (vt )) and
bjt (vt , 0qjt (vt )) appear only in the lhs of
constraints (14), (PC), IC 00 q , and (ICv ). They are relaxed by minimizing
punishment and maximizing profits in the equilibrium path.
Lemma 9. Constraint (14) binds at the optimal solution e and, therefore,
PG’s objective function equals
2
X
½pl (ti (vt )) nhvit sup Uie (xi ) xi : 0 xi 6¼ t i ,
ð15Þ
i=1
feasibility constraint ti ø 0 becomes
pl (ti (vt )) nhvit ø sup Uie (xi ) xi : 0 xi 6¼ t i :
where supff (x) : x 2 Ag is the supremum of f (A) when f is a function.
ð16Þ
Bertolai and Scorzafave
95
0
Proof. We first establish that if (14) and (ICv ) are satisfied, then (PC)
0
holds. In effect, observe that the rhs of (ICv ) is nonnegative for all t, since
Dj can always get nonnegative payoff by choosing no violence:
max Uj ((x, vkt ), qL ½tj (x, vkt ), L, L) ø Uj ((0, vkt ), qL ½tj (0, vkt ), L, L)
xø0
ø pL (tj (0, vkt )) ø 0:
0
Because we are assuming that (ICv ) holds, its lhs must also be nonnegative.
0
In particular, (ICv ) for t = (0, tk ) implies Uje (0) ø 0. It follows that, if (14)
0
and (ICv ) hold, then (PC) is satisfied since
Uje (t j ) t j ø max Uje (x) x ø Uje (0) 0 ø 0,
xø0
ð24Þ
where the first inequality comes from (14) and the second one has been just
established.
Now, suppose that the optimal extortion mechanism e = (t, v, q, a, b) features (14) slacking for at least one dealer j 2 f1, 2g, that is,
pl (tj (vt )) nhlvjt tj . Uje (x) x
8x 2 R+ nftj g:
ð25Þ
Then, consider an alternative extortion mechanism ^e = (^t , ^v, ^q, ^a, ^b) that
requires payments ^t = (tj + e, t k ) for e . 0 and prescribes for each feasible
(v, q)
8
q), bjt (v, q), if x = ^t
< vt , qt (v), ajt (v, (^vx , ^
qjx (v), ^
ajx (v, q), ^bjx (v, q)) = v^t , q^t (v), aj^t (v, q), bj^t (v, q) , if x = t
:
vx , qx (v), ajx (v, q), bjx (v, q) ,
otherwise:
That way, we assure that Uj^e (^tj ) = Uje (t j ), Uj^e (t j ) = Uje (^tj ), and
Uj^e (xj ) = Uje (xj ) for all x 62 ft, ^tg.
Since PG’s objective function is strictly increasing in extortion pay0
ments, mechanism ^e is strictly better than e. Also, because constraints (ICv )
0
and (ICv ) are satisfied by e, they also hold under ^e. This is so because both
mechanisms share the same violence and trading recommendations. For ^e
to be shown implementable, it remains to verify that (14) is satisfied under
^e. First,
96
Rationality and Society 33(1)
(
n
o
^e
sup Uj (x) x : 0 x 6¼ ^t j = max Uj^e (t j ) t j ,
x
(
sup
0x62f^t j , t j g
Uje (^t j ) t j ,
= max
(
ø max
Uje (^t j ) ^t j ,
sup
n
o
Uj^e (x) x
n
0x62f^t j , t j g
sup
0x62f^t j , t j g
Uje (x) x
)
o
)
n
o
Uje (x) x
n
o
= sup Uje (x) x : 0 x 6¼ t j
)
ð26Þ
x
where the inequality comes from ^tj . tj and the second equality comes
from Uj^e (tj ) = Uje (^tj ), and Uj^e (xj ) = Uje (xj ) for all x 62 ft, ^tg.
If (26) holds equality, then from (25) we have that
n
o
n
o
sup Uj^e (x) x : 0 x 6¼ ^tj = sup Uje (x) x : 0 x 6¼ tj \Uj^e (^tj ) ^tj
x
x
holds for sufficiently low e and, therefore, (14) is satisfied by ^e. On the
other hand, if (26) n holds with
strictly inequality, then
o
Uj^e (tj ) t j ø sup0x62f^tj , tj g Uj^e (x) x . In effect, suppose this is not the
case. Then,
n
o
sup
Uj^e (x) x . Uj^e (t j ) t j . Uje (^tj ) ^tj )
0x62f^t j , t j g
sup
0x6¼^t j
n
o
Uj^e (x) x =
sup
0x62f^t j , t j g
n
o
n
o
Uj^e (x) x = sup Uje (x) x ,
0x6¼t j
a contradiction
holding
with strict inequality. It follows that,
n
o
sup0x6¼^tj Uj^e (x) x = Uj^e (tj ) tj and, therefore,
n
o
n
o
sup Uj^e (x) x Uj^e (t j ) t j + sup Uje (x) x (Uje (^t j ) ^t j )
0x6¼^t j
= e + sup
0x6¼t j
n
o
Uje (x) x ,
0x6¼t j
n
o
where the inequality comes from sup0x6¼tj Uje (x) x ø Uje (^tj ) ^tj , as
implied by the last equality in (26). We conclude that
Bertolai and Scorzafave
Uj^e (^tj ) ^tj sup
0x6¼^tj
n
97
o
Uj^e (x) x ø Uj^e (^tj ) ^tj = Uje (tj ) tj sup
0x6¼tj
n
o
!
n
o
e + sup Uje (x) x
0x6¼tj
Uje (x) x 2e . 0
holds for sufficiently low e when (25) holds. Again, we have shown that
(14) is satisfied by ^e.
Proposition 10. (Property rights’ optimality) Suppose pa (ti ) . 0 for all
i. The optimal extortion mechanism for PG features no violence in the
equilibrium path (i.e. vt = (0, 0)) if and only if
pl (1) ø 2pL (1):
ð17Þ
0
Proof. Note that (ICv ) is satisfied for t = t and vt = (0, 0) if and only if
pl (1=2) = pl (tj (0, 0)) nhl:0 = Uj (vt , qjt (vt ), l, l)
ø sup Uj ((x, 0), qL ½tj (x, 0), L, L)= sup ½pL (tj (x, 0))nhLx=pL (1)
xø0
x.0
which is equivalent condition (17). Because we have already established
that (IC 00 )q holds for t = t, it remains to show that (16) holds for all t and
that (IC 00 )q holds for t 6¼ t. Off-equilibrium (IC 00 )q is trivially satisfied by
ajt (vt , qjt (vt )) = L and qjt (vt )) = qL (tj (vt )). Feasibility (16) holds in the
equilibrium path because under no violence
Uje (t j ) = pl (1=2) =
pl (1)
pL (1)
ø
ø sup Uje (x) x ,
2
4
0x6¼tj
ð27Þ
where second equality comes from pL ( ) being homogeneous of degree 1
and the first inequality is implied by p() (1) being decreasing. The last
inequality in (27) comes from the following reasoning. We argue that offequilibrium (t 6¼ t) recommendation ajt (vt , qjt (vt )) = bjt (vt , qjt (vt )) = L,
qjt (vt ) = qL (tj (vt )) and vjt = vL [ pL (1)=4nhL is incentive compatible
0
since it satisfy (IC 00 )q and (ICv ) with equality. In effect, (IC 00 )q trivially binds
0
and (ICv ) binds for vt = (vL , vL ) because
98
Rationality and Society 33(1)
sup Uj ½x, vL , qL (tj ½x, vL ), L, L = Uj ½vL , vL , qL (tj ½vL , vL ), L, L
0x6¼vjt
= pL (1)
LpL (1)
LpL (1) + LpL (1)
2
=
pL (1)
4
as implied by lemma 1. Such off-equilibrium
recommendation,
however,
n
o
does not necessarily minimize sup Uje (x) x : 0 x 6¼ tj as desired to
increase PG’s objective function (15) and to relax (16). In other words, the
unique SPNE of the game when treatment inside prison system is homogeneous among dealers and equal to L is not necessarily the harshest implementable punishment after a deviation from t. Thus, the rhs of (16)
satisfies
pL (1)
:
sup Uje (x) x sup (Uj ½vL , vL , qL (tj ½vL , vL ), L, L x) =
4
0x6¼tj
0x6¼tj
B. Some historical and econometric details
São Paulo (or SP), the richest state in Brazil, experienced a crime dynamics
in the 1990’s and 2000’s that motivates our theory. The state experienced
the strongest homicide rate decline in Brazil between 2001 and 2013. This
evolution is remarkably different from the experience of other Brazilian
states in the same period, making SP case an intriguing one.39 What factors
could have driven this evolution? According to Economics of Crime literature, factors like unemployment and real wages could be responsible for
this evolution. However, SP numbers in these two indicators reveals a pattern similar to ones found in other states. For example, while unemployment rate fell 4.8 percentage points between 2001 and 2012 in SP, it
decreased by 4.3 p.p. in the remaining states of Southeast region in Brazil.
In terms of per capita GDP, SP’s growth between 2001 and 2012 was
smaller than neighbor states rates (2.6% per year against 3.1% per year). In
effect, because wages and employment were increasing throughout the
country in the 2000’s, one can expect falling homicide rates as the rule in
Brazil. However, Figure 2a shows that SP’s experience was atypical for
the Brazilian context. In the period considered in the figure, homicide rate
in SP fell on average 6.28% per year, while the average among the remaining states grew on average 2.4% per year.40
It seems that the question on the causes for such decline must rest on
specific features that differentiate SP from other Brazilian states.
Consistent with the theoretical framework presented in subsection ‘‘A drug
Bertolai and Scorzafave
99
dealing game’’ and section ‘‘Property rights in illicit markets’’, we argue
that two factors makes SP’s experience unique in Brazil: (i) imprisonment
policy in SP, which has traditionally been stricter than the other states’,
has become even more aggressive in the 1990’s and (ii) a single group of
criminals has consolidated control over prison life and has propagated it
throughout the prison system.
Facts (i) and (ii) have been briefly discussed and illustrated in Figures
2a and 2b in subsection ‘‘Empirical motivation’’, but fact (ii) deserves a
more detailed discussion.
The hegemonic gang inside SP’s prison system is called Primeiro
Comando da Capital, the SP counterpart of the PG in our model (hereafter
SPPG). The origins of SPPG remounts to 1991, but the first known action
of the group is its ‘‘formal’’ foundation as a gang during a rebellion in
1993. In this event, they impose their domination over a prison unity in the
city of Taubaté (in SP), which was known by its strict discipline. One of
the main claims in the rebellion (attended by the state government) was
the transfer of the leaders in the rebellion to the largest Brazilian prison,
Casa de Detenc
xão, located in SP’s capital, São Paulo. The domination of
Casa de Detenc
xão was consolidated in 1995, when SPPG members killed
other prison gang leaders in July, 23rd (Christino and Tognolli, 2017).
SPPG expanded across prison units in SP in a similar fashion: transfers of
members to other facilities and subsequent rebellions in which opponents
were killed (generally by decapitation).
The 1990’s was a period of growing violence and strong instability
inside the prison system (prison escapes, rebellions, and murders were frequent) and growth in the number of prisoners. According to Dias (2011),
the murders inside the prison system presented in Figure 2b for the period
between 1996 and 2001 resulted from SPPG members eliminating rival
gangs inside prison. The sharp decline in murders inside the prison system
after 2001, on the other hand, is identified by Dias (2011) as a definite evidence that SPPG becomes an hegemonic gang in SP prison system.41
Another important piece of evidence about SPPG domination is the
series of rebellions that took place in 2001 (hereafter ‘‘mega-rebellion’’).
They simultaneously took place in 29 out 63 prison units and were the first
public signal of SPPG domination of life matters inside the prison system.
The mega-rebellion happened in February, 18th, during the traditional visiting of relatives on Sundays. Up to this date, Sunday was a ‘‘sacred’’ day
inside the prison according to an informal conduct code among prisoners
that prohibited fights, rebellions, etc. This code was then broken with 25
thousand prisoners in 29 prisons rebelling against the transfer of some
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Rationality and Society 33(1)
SPPG leaders from Casa de Detencxão. Relatives of prisoners were taken
as hostages and the rebellion started in all prisons in the same hour, suggesting that prisoners were communicating with each other through cell
phones.
The expansion of SPPG throughout the system between 2001 and 2006
took place with few episodes of violence inside prisons. The pattern was
shaped by the construction of new prison facilities across the state.
According to Dias (2011), as a strategy to difficult the type of coordination
observed during the mega-rebellion in 2001, the new prison unities were
built in smaller sizes and were located in a dispersed way across the state.
The implied transfers of prisoners from old facilities to the new ones made
it possible to SPPG to disseminate across the state without using violence
against rival gangs. As a piece of evidence of such dissemination, SPPG
organized in May (12th) of 2006 riots in 74 out of 130 prisons. As the
main demand, they protested against the transfer of its leader to a maximum security prison at Presidente Venceslau (a city in SP). Endowed with
such power over the prison system, SPPG coordinated attacks outside the
prison system at the same time of the rebellions: 55 police stations were
attacked and 30 policemen were murdered.
Once we have this brief history of the origin, growth and hegemony of
SPPG, we are able to describe our main empirical exploration in detail.
We investigate the impact of the emergence of this group inside SP’s
prison system into the correlation between drug dealing and homicides.
The objective is to empirically explore the main testable implication of our
theory: correlation between violent crimes (homicides) and drug dealing
changes from a positive level before the channel of influence is established
(from inside to outside prison system) to no correlation after that.
The main challenge in such analysis is how to define a variable that
incorporates the power of the SPPG inside the prison system. We capture
the growing power of SPPG over time and space by studying every rebellion in SP since 1993 and identifying those that shows murders with decapitation as a clear evidence of SPPG domination. We assume that once
SPPG dominates a prison inside a micro-region (a small set of neighbor
municipalities), its influence cover every city in this micro-region. Figure
A1b shows the fraction of cities dominated by SPPG according to this variable, denominated ‘‘PG’s control C’’. As can be seen, there is an stability
in the number of cities under SPPG domination after 2001.
This is our conservative measure of PG’s control, since it ignores that
new prisons were created after this 2001.42 As a robustness check, we
Bertolai and Scorzafave
101
Figure A1. Fraction of cities with SPPG’s influence. (a) PG’s control A. (b) PG’s
control C.
construct an alternative SPPG variable assuming that prisoners in all new
prisons starting operation after 2001 were under domination of SPPG (and,
consequently, so were all cities in new prisons’ micro-region). The proportion of cities dominated by SPPG under this measure is presented in Figure
A1a and is denominated ‘‘PG’s control A’’.
B.1 Data and Econometric Model. A panel data of municipalities in SP,
encompassing the period between 1997 and 2006, is used in the empirical
exploration presented in subsection ‘‘The empirical exploration’’. This
period is characterized by the growth and posterior fall of homicides in
SP, as well as is the period of growth and consolidation of SPPG in the
incarceration system.
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Rationality and Society 33(1)
In our most sophisticated empirical exercise, equation (20) is estimated
in order to investigate the relationship between drug traffic and homicides,
after and before SPPG’s influence in municipalities. Equation (20) is reproduced here for convenience.
hit = (a0 + f0 + m0 ) + d0 dit + (f1 + d1 dit )cit
+
10
X
s=1
as yst +
J
X
j=1
mj rij +
K
X
uk xkit + nit :
ð20Þ
k=1
Variable hit is the intentional homicide rate (per 100 thousand inhabitants)
of the municipality i in year (1997 + t). Variable dit is the drug traffic rate
(occurrences per 100 thousand inhabitants) in municipality i in year
(1997 + t). The dummy variable cit indicates the period municipality i
becomes influenced by SPPG. As discussed above, there is two alternative
measure of cit : ‘‘PG’s control A’’ and ‘‘PG’s control C’’. The list of variables (x1it , x2it , x3it ) is composed by the Gini index of formal sector wages
(k = 1), population density (k = 2) and urbanization rate (k = 3), all of them
referring to municipality i in year (1997 + t). As explained in the main text,
yst is a dummy variable indicating if year t equals s and rij is a dummy
variable indicating if region j equals i. Finally, nit is am error term with
mean equal to zero across i and t.
Because homicides and drug traffic are simultaneously determined and
drug traffic is subject to measurement error, OLS estimation of (20) is
clearly biased. In order to handle this problem, an instrumental variable
(IV) approach is implemented. In particular, two variables are used as
instruments for drug traffic. The first one is the number of people covered
by cell phones in the micro-region. For each municipality of SP state, we
use information about the exact date the first cell phone operator has
installed its telecommunication infrastructure (its antenna). There exists
variability in this variable both in time and in space. Also, the period considered in the empirical exercise encompass exactly the beginning of cell
phone adoption in Brazil. The reasoning for using this as an instrumental
variable is that cell phone is an exogenous innovation that facilitates SPPG
functioning and drug traffic activity. On the other hand, there is no plausible reason to think that cell phone adoption can be related to homicide
rates except by drug traffic rates.43 The other instrument is formal sector
median wage of the municipality. The idea is that richer cities exhibit a
more profitable drug market, leading to higher drug traffic, and this is
unrelated to nit .
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All models that includes the SPPG variable, models (19) and (20), were
estimated with cluster of micro-regions, since SPPG variable is aggregated
at this level. Data covered 443 cities in the period between 1997 and 2007.
C. Figures
(e)
(h)
(d)
(g)
(h)
(f)
(c)
Source: SAP/SP.
Figure A2. General overview on crime types I. (a) Drug dealing. (b) Robbery followed by death. (c) Robbery (except vehicles). (d)
Vehicle robbery. (e) Theft (except vehicles). (f) Vehicle theft. (g) Murder. (h) Attempted murder. (i) Murder and manslaughter: whole
state.
(b)
(a)
104
Rationality and Society 33(1)
(e)
(d)
(f)
(c)
Source: SAP/SP.
Figure A3. General overview on crime types II. (a) Detentions. (b) Confiscated guns. (c) Rapes. (d) Bank robbery. (e) Cargo
robbery. (f) Kidnapping.
(b)
(a)
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