Property rights’ emergence in illicit drug markets

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 privately-provided 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

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 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 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.

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 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:

imprisonment policy in SP, which has traditionally been stricter than that of other states, has become even more aggressive in the 1990’s and;

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.

Figure 1.

Homicide rate – São Paulo and Brazilian regions (SP excluded).

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).

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 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 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 = ϕ_{0} + δ_{0} d + (ϕ_{1} + δ_{1} d) c + Θ X + ν

(1)

where the channel of influence (c) from inside prison was taken into account, X is a vector of covariates and $(ϕ_{0}, ϕ_{1}, δ_{0}, δ_{1})$ and $Θ$ are vectors of parameters. According to our model, correlation between h and d is positive before the channel has been established ( $δ_{0} > 0$ since $c = 0$ ), and it is nonexistent after the channel takes place ( $δ_{0} + δ_{1} = 0$ 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.

Table 1.

Econometric estimations of relationship (20).

Homicide rate (h)		PG’s control A		PG’s control C
Homicide rate (h)		OLS	IV	OLS	IV
Drug (d)	$δ_{0}$	$+ 0.016$ $(0.043)$	$+ 2 . 013^{* *}$ $(0.828)$	$- 0.015$ $(0.034)$	$+ 2 . 831^{* * *}$ $(0.954)$
Drug^* PG ( $d . \bar{c}$ )	$δ_{1}$	$- 0.001$ $(0.048)$	$- 1 . 216^{* * *}$ $(0.278)$	$+ 0.038$ $(0.036)$	$- 1 . 536^{* * *}$ $(0.430)$
PG ( $\bar{c}$ )	$ϕ_{1}$	$0.117$ $(1.950)$	$36 . 58^{* * *}$ $(8.12)$	$- 4.643$ $(1.790)$	$+ 48 . 96^{* * *}$ $(12.18)$
Gini on wages ( $x^{1}$ )	$θ_{1}$	$- 0 . 814^{* * *}$ $(0.179)$	$+ 0.126$ $(0.577)$	$- 0 . 782^{* * *}$ $(0.171)$	$+ 0.345$ $(0.673)$
Population density ( $x^{2}$ )	$θ_{2}$	$- 0 . 026^{* * *}$ $(0.007)$	$- 0 . 029^{*}$ $(0.015)$	$- 0 . 024^{* * *}$ $(0.005)$	$- 0 . 039^{* *}$ $(0.016)$
Urbanization rate ( $x^{3}$ )	$θ_{3}$	$+ 0 . 649^{* * *}$ $(0.165)$	$+ 0.098$ $(0.583)$	$+ 0 . 493^{* * *}$ $(0.169)$	$+ 0.370$ $(0.543)$
Year dummies ( $y_{st}$ )	$α_{s}$	Yes	Yes	Yes	Yes
p-value (joint test)	$δ_{0} + δ_{1} = 0$	0.434	0.074	0.190	0.299
Number of observations	N	4873	4873	4873	4873

***

Significant at 1%; ** Significant at 5%; * Significant at 10%.

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.

Figure 3.

Decision tree for the Skarbek (2011)’s model.

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 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 $τ$ , 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.

Figure 4.

Decision tree in Lessing (2017b)’s model.

By refusing to comply with behavior M, S is arrested under probability $π$ . The probability of being arrested under behavior M, denoted $\tilde{π}$ , is assumed greater than or equal to $π$ . Punishment inside prison, denoted $j > 0$ , is lessened by a factor $1 / α \in (0, 1]$ if the arrested criminal has complied with M outside prison. Behavior M can also increase expected criminal profit y by a factor $β \geq 1$ . Additive utility punishments are imposed on S when it refuses to comply with M. It amounts to $γ \geq 0$ inside prison and to $δ \geq 0$ outside prison.

LS interpret the the maximum taxation $τ$ consistent with S complying with $(M, τ)$ 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 $\tilde{π} / π$ they entail. LS’s analysis shows that policies with sufficiently negative impact on j and sufficiently positive impact on $\tilde{π} / π$ 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 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 $j \in {1, 2}$ as $D_{j}$ . 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 $ν \in [0, 1]$ , which we assume to be independent among criminals, and it arrests dealer $j \in {1, 2}$ for selling drugs according to the probability $ρ \in [0, 1]$ .¹⁹

Figure 5.

Timing of the two-stage game.

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 $v = (v_{1}, v_{2}) \in R_{+}^{2}$ , where $v_{i}$ stands for the violence amount chosen by $D_{i}$ . Actions (violence) in this stage are simultaneously chosen and determine the distribution of monopoly power over consumers/territory: $D_{j}$ gets monopoly power over a proportion $t_{j} (v)$ of the aggregate demand for drugs.

With probability $ν$ dealer $i \in {1, 2}$ is arrested and suffers punishment $h_{i}$ 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 $h_{i}$ is composed of a punishment common to all prisoners, h, and a specific one, $β_{i}$ , by supposing that $h_{i} = β_{i} h$ for $β_{i} > 0$ .²⁰

The trading stage

Suppose turf war has taken place under violence profile v. Dealer $i \in {1, 2}$ trade drugs under monopoly in his territory: $D_{i}$ decides how much drug to sell, $q_{i} \geq 0$ , in order to fulfill its market demand $t_{i} (v) Q (p)$ ,²¹ so that $q_{i} = t_{i} (v) Q (p)$ . We assume that there is no cost in producing and supplying drugs and, therefore, profits equal revenues.²²

With probability $ρ$ , dealer $i \in {1, 2}$ is arrested and suffers punishment $d_{i} = α_{i} d$ for each unit of drug sold, where $α_{i} > 0$ . This is again the present value of costs associated to being in the jail and is composed by common and specific factors, d and $α_{i}$ , respectively.

Expected payoffs and the contest success function

After turf war $v = (v_{1}, v_{2})$ has taken place and the distribution of consumers/territories $(t_{1} (v), t_{2} (v))$ has being established, the expected payoff the monopolist dealer gets after selling $q_{i}$ units of drug and facing punishment factors $α$ and $β$ is

\begin{matrix} U_{i} (v, q_{i}, α, β) \equiv P (q_{i} / t_{i} (v)) q_{i} \\ - {(1 - ν) ρ d_{i} q_{i} + (1 - ρ) ν h_{i} v_{i} + ρ ν (h_{i} v_{i} + d_{i} q_{i})} \\ = P (q_{i} / t_{i} (v)) q_{i} - ρ d α q_{i} - ν h β v_{i} \end{matrix}

(2)

where $P (q)$ denotes the individual (inverse) demand function $Q^{- 1} (q) = A - Bq$ , in which $A = X / Z > 0$ and $B = 1 / Z > 0$ .

The relation between market shares $(t_{1}, t_{2})$ and violence efforts $(v_{1}, v_{2})$ 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:

\frac{t_{i}}{t_{k}} = \frac{v_{i}}{v_{k}}, if v_{k} > 0 .

(3)

If $v_{k} = 0$ , we have $t_{k} = 0$ if $v_{i} > 0$ and $t_{k} = t_{i}$ if $v_{i} = 0$ . Using (3) and the fact $t_{1} + t_{2} = 1$ , we have the same Contest Success Function²⁴ studied in Hirshleifer (1989) and Burrus (1999):

t_{j} (v) = {\begin{matrix} \frac{v_{j}}{v_{1} + v_{2}}, & if (v_{1}, v_{2}) \neq (0, 0) \\ \frac{1}{2}, & otherwise . \end{matrix}

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.

Comparing environments

Compared to SK’s model, only the left-hand-side 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 $v_{i}$ and drug dealing $q_{i}$ . 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, $d_{i} q_{i}$ , and violence acts, $h_{i} v_{i}$ . In addition, LS’s punishment distortions inside prison ( $α$ and $γ$ ), is assumed here as proportional to punishment: the factor is $α_{i}$ for drug dealing punishment and $β_{i}$ 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 $τ$ . For simplicity, though, we assume that arrestment probabilities $ρ$ and $ν$ are invariant to cooperating with the prison gang: in LS’s notation, in our environment $\tilde{π} = π$ .

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 ${\bar{ρ}}_{α} = P (0) / d α$ and $α_{i} = α$ , $D_{i}$ ’s optimal quantity and its implied optimal expected profit are given by

q_{α} (t_{i}) \equiv \arg max_{x \geq 0} π_{v}^{i} (x, α) = t_{i} \frac{α d}{2 B} max {{\bar{ρ}}_{α} - ρ, 0}

(4)

π_{α} (t_{i}) \equiv π_{v}^{i} (q_{α} (t_{i}), α) = t_{i} B {(\frac{q_{α} (t_{i})}{t_{i}})}^{2},

(5)

as established by Lemma 13 in Appendix A.

This shows that drug dealing is profitable ( $q_{α} (t_{i}) > 0$ ) if society’s ability to enforce property rights (drug dealing prohibition) is not high enough ( $ρ < P (0) / d α$ ). Also, it is made explicit how such ability depends on the ability to detect and convict those dealing drugs ( $ρ$ ) and the magnitude of the expected punishment ( $d α$ ).

The turf war equilibrium

Taking as given the equilibrium payoffs in the last stage, $π_{α} (t)$ in (5), $D_{i}$ ’s expected payoff in (2) equals $π_{α} (t_{i} (v)) - ν h β_{i} v_{i}$ . More specifically, it is given by

t_{i} (v_{1}, v_{2}) B {(\frac{q_{α} [t_{i} (v_{1}, v_{2})]}{t_{i} (v_{1}, v_{2})})}^{2} - ν h β_{i} v_{i} = \frac{v_{i}}{v_{1} + v_{2}} \frac{{(α d max {{\bar{ρ}}_{α} - ρ, 0})}^{2}}{4 B} - ν h β_{i} v_{i} .

(6)

Then, for each conjecture $v_{k}$ about his opponent choice on violence, $D_{j}$ chooses $v_{j} \geq 0$ in order to maximize (6).²⁵

No violence is optimal when drug dealing is not profitable ( $ρ \geq ρ_{α})$ : 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 ( $v_{i} = 0$ ) is not a best response for the no-violence conjecture ( $v_{k} = 0$ ). The reasoning behind this result is that, absent property rights, the only punishment $D_{i}$ gets choosing $v_{i} = ε > 0$ , with $ε \to 0$ , is $h β ε$ under probability $ν$ , whereas the expected reward $π_{α} (t_{i} (ε, 0)) = π_{α} (1)$ is strictly positive and invariant to $ε$ . 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 $H > π_{α} (1)$ , 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 $v_{k} > 0$ , so that the objective function (6) is strictly concave in $v_{i}$ . Then, first-order optimality condition and constraint $v_{i} \geq 0$ imply the following $D_{i}$ ’s best response function:

v_{i} (v_{k}) = {\begin{matrix} 0 & if v_{k} > π_{α_{i}} (1) / ν h β_{i} \\ \sqrt{\frac{π_{α_{i}} (1)}{ν h β_{i}}} \sqrt{v_{k}} - v_{k} & otherwise \end{matrix} .

(7)

Observe that no violence is optimal to $D_{i}$ if $D_{k}$ ’s violence ( $v_{k}$ ) is sufficiently high. The cutoff violence beyond which violence is not worth ( $π_{α_{i}} (1) / ν h β_{i}$ ) is lower when $D_{i}$ ’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 $v_{k}$ beyond such cutoff. But, we know from Lemma 14 that this cannot constitute an equilibrium, since $D_{k}$ has no optimal response to no-violence conjecture ( $v_{i} = 0$ ) 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.

Lemma 1. Suppose that $ρ < min_{i} {{\bar{ρ}}_{α_{i}}}$ , so that $π_{α_{i}} (1) > 0$ for all $i \in {1, 2}$ . Then, turf war equilibrium, for given equilibrium payoffs in the trading stage, is

v^{*} = (v_{1}^{*}, v_{2}^{*}) = (\frac{π_{α_{1}} (1)}{ν h β_{1}} θ (α, β), \frac{π_{α_{2}} (1)}{ν h β_{2}} θ (α, β))

(8)

where $θ (α, β) = \frac{β_{1} β_{2} π_{α_{1}} (1) π_{α_{2}} (1)}{{(β_{2} π_{α_{1}} (1) + β_{1} π_{α_{2}} (1))}^{2}} \in [0, 1]$ . As a consequence, equilibrium payoffs are

U_{i}^{*} (α, β) \equiv U_{i} (v^{*}, q_{α_{i}} (t_{i} (v^{*})), α, β) = π_{α_{i}} (1) {(\frac{β_{k} π_{α_{i}} (1)}{β_{1} π_{α_{2}} (1) + β_{2} π_{α_{1}} (1)})}^{2}

(9)

Equilibrium violence $v_{j}^{*}$ is obtained by computing $v_{j}^{*} = v_{j} [v_{k} (v_{j}^{*})]$ , where $v_{j} (v_{k})$ is defined in (7), and equilibrium payoffs $U_{j}^{*} (α, β)$ can be computed using straightforward algebra and (6). When $(α_{1}, β_{1}) = (α_{2}, β_{2}) = (a, b)$ 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 $π_{a} (1) / 4$ and are strictly lower than the payoff both would get under no violence, which equals $π_{a} (1) / 2$ . 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 ( $ρ < min_{i} {\bar{ρ}}_{α_{i}}$ ), violence and trading will take place in illicit drug markets.

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, $(α, β)$ , 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 ( $α_{i} = β_{i} = 1$ 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 $λ \leq 1$ and $Λ \geq 1$ .

Throughout this section, we assume that social choice process (whatever it is) has resulted in an ability to enforce property rights such that $ρ < {\bar{ρ}}_{1}$ . That way, trading and violence occur in equilibrium in the first situation, as implied by subsection “The subgame-perfect equilibrium” when $α_{1} = α_{2} = 1$ . 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 $(α_{i}, β_{i})$ based on $D_{i}$ ’s outside-prison behavior. PG’s power to distort inside-prison payoff is not unlimited: $α_{i} \in [λ, Λ]$ and $β_{i} \in [λ, Λ]$ .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 $(v_{1}, v_{2})$ and drug dealing $(q_{1}, q_{2})$ . In addition, PG credibly promises to distort punishment inside prison $(α, β)$ 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 $e = (τ, v, q, a, b)$ such that $τ \in R^{2}$ , $v : R^{2} \to R^{2}$ , $q : R^{4} \to R^{2}$ , and $(a, b) : R^{6} \to R^{2} \times R^{2}$ .

Component functions in the extortion mechanism $e = (τ, v, q, a, b)$ has the following interpretation: $τ = (τ_{1}, τ_{2})$ specifies a payment $τ_{i}$ from each dealer $i \in {1, 2}$ ; $v_{\bar{τ}} = (v_{1 \bar{τ}}, v_{2 \bar{τ}})$ specifies a vector of violence amounts for each payment vector $\bar{τ} \in R^{2}$ ; $q_{i \bar{τ}} (\bar{v})$ specifies, for each possible $(\bar{τ}, \bar{v})$ , the quantity to be sold in $D_{i}$ ’s market; $b_{i \bar{τ}} (\bar{v}, \bar{q})$ describes for each possible $(\bar{τ}, \bar{v}, \bar{q})$ , the punishment factor $D_{i}$ gets inside the prison system when arrested for violence acts; and $a_{i \bar{τ}} (\bar{v}, \bar{q})$ specifies, for each possible $(\bar{τ}, \bar{v}, \bar{q})$ , the specific punishment $D_{i}$ 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, $(\bar{τ}, \bar{v}, \bar{q})$ . Most important, once operating in the market, they are subject to the promised punishments a and b under probabilities $ρ$ and $ν$ , respectively.

When choosing the extortion mechanism, PG is restricted to mechanisms that are feasible.

Definition 4. The extortion mechanism $e = (τ, v, q, a, b)$ is feasible if

\begin{matrix} τ_{i} \geq 0, v_{i \bar{τ}} \geq 0, q_{i \bar{τ}} (\bar{v}) \geq 0 and \\ λ \leq a_{i \bar{τ}} (\bar{v}, \bar{q}) \leq Λ, λ \leq b_{i \bar{τ}} (\bar{v}, \bar{q}) \leq Λ \end{matrix}

for each $i \in {1, 2}$ , each $\bar{τ} \in R_{+}^{2}$ , each $\bar{v} \in R_{+}^{2}$ and each $\bar{q} \in R_{+}^{2}$ .

Property $τ_{i} \geq 0$ requires payments from dealers to PG to be nonnegative, that is, PG is not able to make payments to outside dealers. Condition $v_{i \bar{τ}} \geq 0$ 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 $q_{i \bar{τ}} (\bar{v}) \geq 0$ requires trading to be nonnegative and properties $λ \leq a_{i \bar{τ}} (\bar{v}, \bar{q}) \leq Λ$ and $λ \leq b_{i \bar{τ}} (\bar{v}, \bar{q}) \leq Λ$ say that PG’s power to distort prisoners’ payoffs is limited to $[λ, Λ]$ .

Having defined what a feasible mechanism is, we are now able to define expected payoffs for PG and $(D_{1}, D_{2})$ . Let $e = (τ, v, q, a, b)$ be given. PG is assumed to derive utility from the revenues extorted from dealers and, therefore, its payoff function is

W (e) = W (τ, v, q, a, b) = τ_{1} + τ_{2} .

(10)

If $D_{i}$ expects the other dealer to completely comply with the extortion mechanism e, then its expected payoff when paying $τ' \geq 0$ and complying with the remaining recommendations in the mechanism is

U_{i}^{e} (τ') \equiv U_{i} [v_{\hat{τ}}, q_{i \hat{τ}} (v_{\hat{τ}}), a_{i \hat{τ}} [v_{\hat{τ}}, q_{\hat{τ}} (v_{\hat{τ}})], b_{i \hat{τ}} [v_{\hat{τ}}, q_{\hat{τ}} (v_{\hat{τ}})]],

where $\hat{τ} = (τ', τ_{k})$ and the function $U_{j} (\cdot)$ 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 $e = (τ, v, q, a, b)$ is said to be compatible with $D_{i}$ ’s participation if it is feasible and

U_{i}^{e} (τ_{j}) - τ_{j} \geq 0 . (PC)

Also, it is said to be incentive compatible for $D_{i}$ if it is feasible and $(τ, v, q)$ is optimal for $D_{i}$ in each possible contingency, that is, for each $\bar{τ} \in R_{+}^{2}$ and $\bar{v} \in R_{+}^{2}$ ,

q_{i \bar{τ}} (\bar{v}) solves max_{x \geq 0} π_{\bar{v}}^{i} (x, a_{i \bar{τ}} [\bar{v}, (x, q_{k \bar{τ}} (\bar{v}))])

(11)

v_{i \bar{τ}} solves max_{x \geq 0} U_{i} [{\hat{v}}_{\bar{τ}} (x), {\hat{q}}_{i \bar{τ}} (x), (a, b)_{\bar{τ}} ({\hat{v}}_{\bar{τ}} (x), {\hat{q}}_{1 \bar{τ}} (x), {\hat{q}}_{2 \bar{τ}} (x))]

(12)

τ_{i} solves max_{t \geq 0} {U_{i}^{e} (t) - t}

(13)

where ${\hat{v}}_{\bar{τ}} (x) \equiv (x, v_{k \bar{τ}})$ denotes the pair of violence level observed after payment profile $\bar{τ}$ when $D_{i}$ chooses violence amount x and $D_{k}$ comply with e. The amount ${\hat{q}}_{i \bar{τ}} (x) \equiv q_{i \bar{τ}} ({\hat{v}}_{\bar{τ}} (x))$ is the trading level after payments $\bar{τ}$ and violence levels ${\hat{v}}_{\bar{τ}} (x)$ . Finally, the mechanism is implementable if it is compatible with participation and with incentives for all dealers.

We are now able to define what is an optimal extortion mechanism for the prison gang.

Definition 6. The mechanism $e^{*} = (τ, v, q, a, b)$ is optimal for PG if it solves

max_{e} {W (e) : (PC), (11), (12), and (13)} (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 $e^{*} = (τ, v, q, a, b)$ is quite simple: it promises the minimum punishment in the equilibrium path (i.e. $a_{i τ} [v_{τ}, q_{τ} (v_{τ})] = b_{i τ} [v_{τ}, q_{τ} (v_{τ})] = λ$ for all i) and the worse treatment inside prison after any deviation from recommendations (i.e. $a_{i \bar{τ}} (\bar{v}, \bar{q}) = b_{i \bar{τ}} (\bar{v}, \bar{q}) = Λ$ for all $(\bar{τ}, \bar{v}, \bar{q}) \neq [τ, v_{τ}, q_{τ} (v_{τ})]$ and all i). Also, trading is always monopoly quantities $q_{α_{i}} (t_{i})$ 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-of-equilibrium path and in-the-equilibrium path.

Remark 8. Incentive compatibility condition (13) can be rewritten without loss of generality as

U_{i}^{e} (τ_{i}) - τ_{i} \geq max_{x \geq 0} {U_{i}^{e} (x) - x}, \forall i \in {1, 2} .

(14)

Also, under the optimal extortion mechanism

(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 $π_{v_{τ}}^{i} (q_{λ} (t_{i} (v_{τ})), λ) = π_{λ} (t_{i} (v_{τ})) \geq π_{Λ} (t_{i} (v_{τ}))$ and function $π_{α} (t_{i} (v_{τ}))$ is decreasing in $α$ . Also, the left-hand side of both (PC) and (14) becomes $U_{i}^{e} (τ_{i}) - τ_{i} = π_{λ} (t_{i} (v_{τ})) - ν h v_{i τ} - τ_{j} .$ 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

\sum_{i = 1}^{2} [π_{λ} (t_{i} (v_{τ})) - ν h v_{i τ} - sup {U_{i}^{e} (x_{i}) - x_{i} : 0 \leq x_{i} \neq τ_{i}}],

(15)

feasibility constraint $τ_{i} \geq 0$ becomes

π_{λ} (t_{i} (v_{τ})) - ν h v_{i τ} \geq sup {U_{i}^{e} (x_{i}) - x_{i} : 0 \leq x_{i} \neq τ_{i}} .

(16)

where $sup {f (x) : x \in A}$ 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 $(v_{i τ}, q_{i τ} (v_{τ}))$ in $R_{+}^{2}$ and $(a_{i τ} [v_{τ}, q_{i τ} (v_{τ})], b_{i τ} [v_{τ}, q_{i τ} (v_{τ})])$ in $[λ, Λ]^{2}$ for each $(i, τ) \in {1, 2} \times R_{+}^{2}$ in order to maximize (15) subject to feasibility constraint (16) and incentive constraints ( ${IC}_{q}^{″}$ ) and ( ${IC}_{v}^{'}$ ), 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 ( $v_{1 τ} = v_{2 τ} = 0$ ). 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 $π_{α} (t_{i}) > 0$ for all i. The optimal extortion mechanism for PG features no violence in the equilibrium path (i.e. $v_{τ} = (0, 0)$ ) if and only if

π_{λ} (1) \geq 2 π_{Λ} (1) .

(17)

Proof. See Appendix A.

Condition (17) is equivalent to no violence ( $v_{i τ} = 0$ ) being an optimal response for $D_{i}$ when $D_{k}$ is not using violence as a competition instrument ( $v_{k τ} = 0$ ) and payments were executed as demanded by PG ( $τ = (τ_{1}, τ_{2})$ ). By complying with the no-violence agreement, $D_{i}$ gets half of consumers/territories from which profit level $π_{λ} (1) / 2$ is earned. By deviating from the agreement, $D_{i}$ gets monopoly power over the whole market which provides $π_{Λ} (1)$ in profits. Because treatment inside the prison system after complying with the agreement is better than the respective treatment after deviating (i.e. $λ \leq Λ$ ) 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 $ρ = 0$ because he would then never be arrested, and that cooperation with PG would have no effect on imprisonment costs if $λ = Λ$ .

Corollary 11. (Collusion sustainability) The optimal extortion mechanism for PG features no violence in the equilibrium path if $(ρ, λ, Λ)$ is such that ${\bar{ρ}}_{Λ} \leq ρ < {\bar{ρ}}_{λ}$ , that is,

ρ λ < \frac{P (0)}{d} \leq ρ Λ (POe)

and only if police work has some effectiveness ( $ρ > 0$ ) and PG has some power inside the prison system ( $Λ > λ$ ).

Proof. When $ρ < {\bar{ρ}}_{λ}$ , we have $π_{λ} (t_{i}) > 0$ for all i. Thus, from proposition 10, we know that no-violence in the equilibrium path is optimal if and only if $π_{λ} (1) \geq 2 π_{Λ} (1)$ , that is,

\begin{matrix} Γ (ρ, λ, Λ) \equiv π_{λ} (1) - 2 π_{Λ} (1) \\ = max_{x \geq 0} [P (x) x - ρ d λ x] - 2 max_{y \geq 0} [P (y) y - ρ d Λ y] \geq 0 . \end{matrix}

For the necessity result, observe that $Γ (0, λ, Λ) = - max_{x} P (x) x < 0$ for all $λ \leq 1 \leq Λ$ , and $Γ (ρ, 1, 1) = - π_{1} (1) < 0$ for all $ρ \in [0, \bar{ρ})$ . For the sufficiency result, remember that $π_{v}^{i} (q, α)$ is assumed strictly concave in q and observe that its derivative at $q = 0$ equals $P (0) - ρ d α$ . Then, using (21) we have that $P (0) > ρ d α$ implies $q_{α} (1) > 0$ , since in this case $\partial π_{v}^{i} (0, α) / \partial q > 0$ . Also, $P (0) \leq ρ d α$ implies $q_{α} (1) = 0$ , since then $\partial π (q, α) / \partial q < 0$ for all $q > 0$ . Now, observe that under (POe) and $ρ < {\bar{ρ}}_{λ}$ , we have $ρ d λ < P (0) \leq ρ d Λ$ and, therefore, $q_{λ} (1) > 0 = q_{Λ} (1)$ . As a conclusion, $π_{λ} (1) > 0 = 2 π_{Λ} (1)$ .

The sufficiency result holds because by increasing detention probability $ρ$ and/or by increasing PG’s power to distort punishments $Λ - λ$ so that (POe) holds, the benefit in deviating from collusion is eliminated $(π_{Λ} (1) \to 0)$ , while some benefit from cooperation is still available ( $π_{λ} (1) > 0$ ).²⁸ The necessary conditions ( $ρ > 0$ and $Λ > λ$ ) 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 $ρ (Λ - λ) > 0$ .

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 out-of-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 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. $ρ < 1$ and $ν < 1$ ) and to punish them inside the prison system (i.e. $d_{i} = α_{i} d < \infty$ and $h_{i} = β_{i} h < \infty$ ).

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 $τ'$ to PG not equal to $τ_{i}$ is punished through the violence level recommended to the other dealer: $v_{k (τ', τ_{k})}$ . This is the counterpart in our model of LS’s punishment parameter for defectors, $δ$ . 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 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.

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.

Figure 6.

Drug traffic rate and homicides rate for SP.

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 $Δ h_{t} / Δ d_{t} \equiv (h_{t} - h_{t - 1}) / (d_{t} - d_{t - 1})$ , where $h_{t}$ denotes homicide rate in year t and $d_{t}$ 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:

h_{it} = α_{0} + β_{0} d_{it} + \sum_{s = 1}^{10} (α_{s} + β_{s} d_{is}) y_{st} + ε_{it},

(18)

where $h_{it}$ denotes homicide rate observed in municipality i in year $(1997 + t)$ , $d_{it}$ denotes drug traffic rate observed in municipality i in year $(1997 + t)$ . The variable $y_{st}$ equals 1 in year $(1997 + t)$ if $s = t$ and equals zero otherwise. The variable $ε_{it}$ is assumed to have zero mean across i and across t. Equation (18) implies that expected homicide rate in 1997 equals $α_{0}$ if drug dealing is absent. Similarly, expected homicide rate in year $(1997 + t)$ equals $(α_{0} + α_{t})$ if drug dealing is absent. The correlation between homicide and drug traffic in 1997 is $β_{0}$ , while the same measure for year $(1997 + t)$ equals $(β_{0} + β_{t})$ . The estimated values for $(β_{0} + β_{t})$ are reported in Figure 7a.

Figure 7.

Association between drug dealing and homicides. (a) Association evolution. (b) Lagged association.

Figure 7a shows that the pattern of association between $h_{t}$ and $d_{t}$ 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 ( $Δ h_{t} / Δ d_{t}$ ) 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 $h_{t}$ and $d_{t}$ 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):

h_{it} = γ_{0} + η_{0} d_{it} + \sum_{l = 1}^{6} [(γ_{- l} + η_{- l} d_{it}) c_{it} (- l) + (γ_{l} + η_{l} d_{it}) c_{it} (l)] + ε_{it}

(19)

where $h_{it}$ 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 $d_{it}$ denotes drug dealing in year $(1997 + t)$ at region i and has changed in the same way. The variable $ε_{it}$ is assumed to have zero mean across i and across t. The new variable $c_{it} (l)$ indicates if in year $(1997 + t)$ region i have been under PG’s control for l periods (years). Accordingly, if $(1997 + t_{i})$ denotes the year PG took control over drug dealing markets in region i, then $c_{it} (l) = 1$ if $l = t - t_{i}$ and $c_{it} (l) = 0$ otherwise.

Equation (19) implies that expected homicide rate in the year PG took control of a region equals $γ_{0}$ if drug dealing is absent. Accordingly, expected homicide rate $l \in {- 6, - 5, \dots, 0, \dots, 5, 6}$ periods from the time PG’s control took place equals $(γ_{0} + γ_{l})$ if drug dealing is absent. The correlation between homicide and drug traffic in the year PG took control of a region is $η_{0}$ , while the same measure $l \in {- 6, - 5, \dots, 0, \dots, 5, 6}$ periods from the time PG’s control took place equals $(η_{0} + η_{l})$ . The estimated values for $(η_{0} + η_{l})$ are reported in Figure 7b as a function of l and show that before PG control ( $l < 0$ ), the correlation between homicides and traffic was positive and 2 years after PG control in the region ( $l \geq 2$ ), 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 $η_{l} = η_{-}$ for $l < 0$ and $η_{l} = η_{+}$ for $l \geq 0$ in relationship (19) and reintroduce factors common to all regions in a given year (variable $y_{st}$ ). 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

h_{it} = (α_{0} + ϕ_{0} + μ_{0}) + δ_{0} d_{it} + (ϕ_{1} + δ_{1} d_{it}) {\bar{c}}_{it} + \sum_{s = 1}^{10} α_{s} y_{st} + μ_{i} + \sum_{k = 1}^{K} θ_{k} x_{it}^{k} + u_{it}

(20)

where $μ_{i}$ denotes the fixed effect for region i and $x_{it}^{k}$ 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 $u_{it}$ is assumed to have zero mean across i and across t. The variable ${\bar{c}}_{it} = max_{l > 0} c_{it} (l) \in {0, 1}$ 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 ${\bar{h}}_{0} \equiv (α_{0} + ϕ_{0} + μ_{0})$ if drug dealing and PG’s control were both absent and $x^{k}$ were zero for all k.³⁵ The correlation between homicide and drug traffic is $δ_{0}$ if PG’s control is absent, while the same measure under PG’s control equals $(δ_{0} + δ_{1})$ .

Existing empirical work on the influence of PG on violence has paid attention to $\bar{c}$ ’s effect on h without telling apart its impact through drug dealing ( $d . \bar{c}$ ).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 $ϕ_{1}$ , our empirical exploration calls attention for a neglected perspective on data: positive correlation between violence and drug dealing when property rights are not enforceable ( $δ_{0} > 0$ when $\bar{c} = 0$ ) and no correlation at all when property rights emerges as result of PG’s control ( $δ_{0} + δ_{1} = 0$ when $\bar{c} = 0$ ). Although the effect of PG’s influence on homicides ( $ϕ_{1} + δ_{1} d$ ) 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 ( $\bar{c}$ ) 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 ( $δ_{0} > 0$ ) and become much lower after it ( $δ_{0} + δ_{1} \approx 0$ ). Although the hypothesis $(δ_{0} \leq 0)$ cannot be rejected at a satisfactory level of confidence under OLS estimates, it is strongly rejected under IV estimation. Also, the hypothesis ( $δ_{0} + δ_{1} = 0$ ) 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.

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 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 $λ$ , 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 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.

Footnotes

Appendix

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, Lígia 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

Notes

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Property rights’ emergence in illicit drug markets

Abstract

Keywords

Introduction

Closely related literature

Empirical motivation 11

The model

A drug dealing game

The environment

The timing

The turf war stage

The trading stage

Expected payoffs and the contest success function

Comparing environments

The subgame-perfect equilibrium

The trading equilibrium

The turf war equilibrium

Property rights in illicit markets

The hegemonic prison gang’s problem

Property rights’ optimality and enforceability

Some insights and the empirical exploration

The empirical exploration

Final remarks

Footnotes

Appendix

Acknowledgements

Funding

ORCID iD

Notes

References

Empirical motivation¹¹