CDDRL WORKING PAPERS

Number 143

October 2013

A Hole at the Center of the State: Prison Gangs and the Limits to Punitive Power

Benjamin Lessing Stanford University

Center on Democracy, Development, and The Rule of Law Freeman Spogli Institute for International Studies

Additional working papers appear on CDDRL’s website: http://cddrl.stanford.edu.

Center on Democracy, Development, and The Rule of Law Freeman Spogli Institute for International Studies Stanford University Encina Hall Stanford, CA 94305 Phone: 650-724-7197 Fax: 650-724-2996 http://cddrl.stanford.edu/ About the Center on Democracy, Development and the Rule of Law (CDDRL) CDDRL was founded by a generous grant from the Bill and Flora Hewlett Foundation in October in 2002 as part of the Stanford Institute for International Studies at Stanford University. The Center supports analytic studies, policy relevant research, training and outreach activities to assist developing countries in the design and implementation of policies to foster growth, democracy, and the rule of law.

S

τ S C τ

D S π

�ππ �π

S π τ

�π � π S

j > 0

S − j − jα

α > 1

y S

βy

β ≥ 1

γ ≥ 0 δ ≥ 0S τ S

�π < π

�π �π

α − ( j −α)

πδ

S

τ∗ ≡ j (π− �π/α) + y [β(1− �π)− (1−π)] + γπ+δ(1−π)

τ∗ �π π �π < π

�π π τ∗

S �π � πτ∗

τ∗

α,β,γ δ τ∗

τ∗

y τ∗

β > 1−π1−�π �π � π β ≥ 1

j π �π

ρ j � (ρ) π � (ρ)

�π � (ρ)

ρS

j

Sτ∗ S

j � (ρ) π � (ρ) �π � (ρ)ρS + 0

ρC 0 +

ρOC + +

ρ KC − +

S

ρS τ∗

α >�ππ

CS

CS�ππ

�π � π �ππ≈ 1 CS

ρC

ρC

ρOC ρ K

C

τ ρ

�π π ϕ(ρ) ≡ �π �(ρ)π �(ρ)

�π π ρ �π � (ρ) > 0 π � (ρ) > 0

ϕ(ρ) ∈�+

ϕ(ρS )

ϕ

ϕ ≈ 1ρC

ρC ϕ(ρC ) =

1 τ∗

j − jα− (βy − y ) + γ −δ > 0.

j− jα

βy− y γ−δ

a

a

ϕ∗(ρ) ϕ(ρ) < ϕ∗(ρ)⇒ d τ∗

dρ> 0

ϕ∗(ρC ) =j+y+γ−δj/α+βy

α

ϕ < 1

γ β δ y

ρOC

j π �πϕ = ϕ∗(ρC )

ϕ∗(ρOC ) > ϕ

∗(ρC ) ρOC ρC

τ∗

b

c ϕ∗

ρ KC

CS j

ϕ∗�ρ KC

�<

ϕ∗�ρC�

ρC k ∈ �+ ρ KC

j ��ρ KC

�< −k τ∗

k π � �π � α

ϕ�ρ KC

�<

j + y + γ −δj π�π +βy

CK

k

d e

π � α

CK α

ϕ∗ k CK

[y , y] y

ϕ k ϕ∗

π �(ρ)

C

π �π �π � πα

β = 1

D γ = δ = 0

y O yo

πo = 0 D O

yi C D O

y ∗= j π−�π/α�π−π : yi < y ∗⇔ C � DyC= yo+ j �π/α

1−�π : yi < yC⇔O � CyD= yo+ jπ

1−π : yi < yD⇔O � D

y ∗ CS α > �ππ

�ππ

CS

[y , y]

y ∗ > yC ⇔ yC < yD

y < yD < y ∗ < y y ∈ [yC , y ∗] y ∗

NR

NR <� y∗

yDF (·) d y CR

yD

yD

y ∗

yD y ∗

! y∗

!α> 0

! yD

!α= 0 ! yD

! yo> 0

! y∗

! yo= 0 CR

ρS yD

CS y ∗

π � (ρ) > 0 yD

α > �ππ

C �R :� y∗yDF (·)d y < NR <

� y∗yCF (·)d y

C ��R :� y∗yCF (·)d y < NR

yi

yD

ρC

ϕ∗(ρC ) =�ππ

�ππ> 1

ϕ(ρ) = ϕ∗(ρC )

! yD

! j> 0

ϕ∗�ρ KC

�< ϕ∗�ρC�

ρC k ∈ �+ ρ KC

j ��ρ KC

�< −k y ∗ k π � �π �

α

α

ρ KC ϕ�ρ KC

�= ϕ�ρC� d yD

dρ KC< d yD

dρCd yD

dρ KC> 0 k < π �

�ρC� yo+ jπ(1−π)

α

> 50%> 90%

�× 100� †

∗ †

http://www.albany.edu/sourcebook/csv/

S

τ∗ ≡ j (π− �π/α) + y [β(1− �π)− (1−π)] + γπ+δ(1−π)

τ∗ �π π �π < π

τ∗ S C

D D PG τ = τ∗ τ∗, C �S D S C PG

τ τ > τ∗ 0 PG

PG �τ �= τ∗ S

τ ∈ (0,τ∗) τ > τ∗ �τ > τ∗ PG 0 τ ∈ (0,τ∗)�τ < τ∗ �τ + �

τ∗ lim�π→πτ∗ = jπ(1− 1/α) + y (β− 1)(1−π) + γπ +δ(1−π)

�π < π

π, �π j

ξ ρ

ξ ρ j � (ρ) π � (ρ) �π � (ρ) d jdρ

dπdρ

d �πdρ

ρS

ρ j � (ρ) > 0 π � (ρ) = �π � (ρ) = 0 ρC

ρ τ∗

d τ∗

dρ= π � (ρ) [ j + y + γ −δ]− �π � (ρ)

� jα+βy�+ j � (ρ)�π− �πα

�τ∗

j � (ρ) π � (ρ) �π � (ρ)ρS + 0

ρC 0 +

ρOC + +

ρKC − +

ρS τ∗

α >�ππ

CS

τ∗

ρC ϕ(ρC ) = 1

τ∗

j − jα− (βy − y ) + γ −δ > 0.

π � (ρ) = �π � (ρ) τ∗

d τ∗

dρC= π � (ρ) [ j − j/α− y (β− 1) + γ −δ]

ϕ∗(ρC ) =j+y+γ−δj/α+βy

α γ

β δ y

τ∗

d τ∗

dρC= π � (ρ) [ j + y + γ −δ]− �π � (ρ) [ j/α+βy]

π �(ρ)�π �(ρ) <

j+y+γ−δj/α+βy

α,β,γ , δ !ϕ∗(ρC )! y

=

α j (1−αβ)−αβ(γ−δ)( j+αβy )2

γ > δ

ϕ∗(ρOC ) >

ϕ∗(ρC ) ρOC ρC τ∗

d τ∗dρC= 0 d τ∗

dρ OC= 0+ j � (ρ)�π− �π

α

�α > �π

πj � (ρ) > 0 d τ∗

dρ OC> 0

ϕ∗�ρKC�<

ϕ∗�ρC�

ρC k ∈ �+ ρKC

j ��ρKC�< −k τ∗

k π � �π � α

ϕ�ρKC

�<

j + y + γ −δj π�π +βy

CK

τ∗

ϕ∗�ρKC

�=

j + y + γ −δj/α+βy

+ j �(ρ)�

π− �π/απ �(ρ)( j/α+βy )

�

= ϕ∗�ρC�+ j �(ρ)�

π− �π/απ �(ρ)( j/α+βy )

�

j ��ρKC�< 0 ϕ∗�ρKC�< ϕ∗�ρC�

k d τ∗dρ K

C= π � (ρ) [ j + y + γ −δ] − �π � (ρ) [ j/α+βy] + j � (ρ)

�π− �π

α

�

j � (ρ) < k ≡ π �(ρ)[ j+y+γ−δ]−�π �(ρ)[ j/α+βy]π−�π/α

k

k k π �(ρ) �π �(ρ) α

!k!α=�π

(aπ− �π)2��π �(ρ)�jπ

�π +βy�−π �(ρ)( j + y + γ −δ)

�

CK

d y∗

dρ= π �(ρ)

�π j�1− 1

α

�

(π− �π)2

+ �π �(ρ)−π j�1− 1

α

�

(π− �π)2

+ j �(ρ)

π−�πα

�π−π

y∗

d yD

dρ= π �(ρ)�

yo + j(1−π)2�+ �π �(ρ) [0] + j �(ρ)

� π1−π�

yD

ρS yD CS

y∗

y∗ yD

π � (ρ) > 0 yD

yD

ρC

ϕ∗(ρC ) =�ππ

y∗d y∗

dρC> 0 �π �(ρ)

π �(ρ)> �ππ

ϕ∗�ρKC�<

ϕ∗�ρC�

ρC k ∈ �+ ρKC

j ��ρKC�< −k y∗ k ϕ

�ρKC�

α

y∗

ϕ∗�ρKC

�=�ππ+ j �(ρ)� �π−ππ �(ρ)π

· απ− �πj (α− 1)

�

CS j ��ρKC�< 0 ϕ∗�ρKC�< ϕ∗�ρC�

k y∗d y∗

dρ KC

j � (ρ) < k ≡ �π−ππ− �π/α

j (1− 1/α)(π− �π)2� �ππ �(ρ)−π �π �(ρ)�

�π �(ρ)π �(ρ)

< �ππ

k!k!π= j (α−1)(�π−π)(απ−�π)

��π−π �π �(ρ)

π �(ρ)

�CS

�π �(ρ)π �(ρ)

< �ππ

!k! �π < 0

α !k!α= j� �ππ �(ρ)−π �π �(ρ)�

(�π−απ)2

ρKC ϕ�ρKC�= ϕ�ρC� d yD

dρ KC< d yD

dρCd yD

dρ KC> 0 k < π �

�ρC� yo+ jπ(1−π)

d yD

dρ KC= π �(ρ)�

yo+ j(1−π)2�+ j �(ρ)�π

1−π

� d yD

dρCj �(ρK

C ) < 0 d yD

dρ KC> 0

− j �(ρKC ) = k < π �(ρ)�

yo+ j(1−π)2�1−ππ= π �(ρ) yo+ j

π(1−π)