CDDRL WORKING PAPERS

Number 145

October 2013

The Logic of Violence in Drug Wars: Cartel-State Conflict in Mexico, Brazil and Colombia

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.

D P

S D

a D −ay y [y , y] D

y P D a y P

b D B H F

D H F P s

c S sO

O ∈ H,F D

P sF = s sH = s (1− c ) ≤ sF c ∈ [0,1)c → 0 D

sH → sF

D h(·) ∈ [0,1] f (·) ∈ [0,1]h (sH ) > 0 f (sF ) > 0 sH = s (1 − c )

D h (c ) < 0 f (c ) = 0 D

D P

P P

O ∈ H,F D o ∈ h, f

c = 1

h (a) = 0

f (a) > 0 P D 0 P

−φ(·) φ(a) > 0 φ(s ) < 0

h f φ sH sF a

D y b a P b

λ

λ < y λ s c

λ λ

λ

s c

D

D

∃ a > 0 : a ≥ a⇒ f (a, sF , ·) ≤ h(sH , ·) Ca

λ b λb = 0

Ca

y h f φ

D b y D a

b D

a ≥ a

D a ∈ (0, a)

a∗h = 0;

H y ∈ [y , bh(·) )

B y ∈ [ bh(·) , y]

a∗f ≥ a;

F y ∈ [y , bf (·) )

B y ∈ [ bf (·) , y]

a∗ = a

P

b ∗(a, ·) =

b ∗h a < a

b ∗f a ≥ a

P

D a∗h = 0 a∗f ≥ aa∗ ∈ a∗h , a∗f

D

D

D

a∗h a∗f

o ∈ h, f b ∗o

O ∈ H,F

PrO =b∗

o(·) − yy − y

b ∗h b ∗f P

b ∗h = argmaxb

1−

bh(·) − yy − y

(b − λ) +

bh(·) − yy − y · 0 = λ+ h(·)y

2

b ∗f = argmaxb

1−

bf (·) − yy − y

(b − λ)−

bf (·) − yy − y φ(·) = λ+ f (·)y −φ(·)

2

D P φ

φ

2

D o ∈ h, f bmo = o(·)y bM

o = o(·)y P

o ∈ h, f b ∗o > bMo P bM

o b

b < bMhb ∗o < bm

o P bmo b

P b > bMo

b = bMo o ∈ h, f P b ∗o = max[min[b ∗o , bM

o ], b mo ]

λ

b ∗h =

h(·)y λ > h(·)yh(·)y λ < h(·)

y − (y − y )

λ+h(·)y2

b ∗f =

f (·)y λ−φ(·) > f (·)yf (·)y λ−φ(·) < f (·)

y − (y − y )

λ+ f (·)y−φ(·)2

P

D −φ(·) P

P

D P

P

a∗h = 0

a < a b ∗h

λ s c λ s

c

b ∗h

λ

s

D b

c

P D

∃ cNB ∈ (0,1) : c > cNB =⇒ b ∗h ≥ bMh a = 0

D y < y = b∗ff (·) D

a∗f = argmaxa

minmaxy ,y, y

yy (1− f (·)) 1

y − y d y + yminmaxy ,y, y(y − b ∗f )

1y − y d y − a

f φ b ∗f a

a∗f

s a

f φ

f (·) ≡ sa+s

f ∈ [0,1] D y (1− f )

y ∗ fP S

P

D −φ(·) ≡ −ϕ aa+s

ϕ P

h(·) ≡ sHη+sH

η

η

sH ≡ s (1− c ) < s Ca a > η sFsH= η

1−c ≡ a⇒ f > h

y

y ∼U12µ, 3

2µ

y

µ > ϕ4

a∗f

s ∈ (s , s ) a∗f > 0 s > s a∗f < 0 s ≤ s a∗f

a∗f ≥ a a∗f = max[a∗f , a]a∗f a

∃ sI > 0 s > sI

b ∗f

λ s 1 − PrF λ

s ϕ > λ ϕ < λ

ϕ = λ

ϕ > λ P D

λ

D U Df

a = a∗f U Dh a = a∗b = 0

s c

D

D a U Dh

a = 0 U Df

a∗fD

a D a = 0 P D

a = a∗f D b ∗

D s 1 b D

a = a∗f

D a b

µ(.44− .28) = 2.3

s c U Df > U D

h

PrF

s a b

U Df +U

Pf <U

Dh +U

Ph

s s

s

b x D

s ∗ D

c = 0

∃ s ∗ ∈+ s > s ∗ ⇒U Dh >U

Df

s

s b x

c s

z D s

D s ∗

s ∗ D

∃ c ∈ (0,1) c > c D a∗ = 0 P

b ≥ bMh D

s

λ

D s c P

D P s

c

y µ

y

µ

µ

U Dh U D

f µ

µ µ

∃ µ∗ ∈+ µ > µ∗ ⇒U Df >U

Dh

s

µ

µ

µ

y PrH = 1 s

c P

a

am aM am aM D a ≥ am a ≥ aM

P bmf bM

f PrF = 0 PrF = 1 am aM am

ϕ > λ

a∗f >maxam, aM

s ∈ [0, sI ] P

ϕ λ

s < sI s > sI

am a∗f D

Pr f > 0

λ > ϕ D

η c

D

s

aω P D

aν P D a = aν+aω

−aνaω

aω < a∗f D

a∗f

a ∗ν = a∗f −aω Pr F

D aωU D

f =U Df + aω

U Dh >

U Df

aω a∗ω ≡ U Dh −U D

f D aν = 0

aν = a∗f − aω

λ µ

s c

s c

D a ∈ (0, a)

a∗h = 0;

H y ∈ [y , bh(·) )

B y ∈ [ bh(·) , y]

a∗f ≥ a;

F y ∈ [y , bf (·) )

B y ∈ [ bf (·) , y]

P

b ∗(a, ·) =

b ∗h a < a

b ∗f a ≥ a

D a ∈ (0, a) D

ao ∈ (0, a) a < a D a P D

P b ∗o a ∈ [0, a) D

a D ao a = 0

a∗ a < a D h(·)y > y−by = b

h(·) a > a b ∗ P b

D F H a

o ∈ h, f b ∗o > bMo P bM

o b

b < bMhb ∗o < bm

o P bmo b

b ∗o > bMo D b ≥ bM

o P b

b ≤ bMo bM

o !−EU P

f (bMf )= λ−φ(·)− f (·)y

f (·)(y−y )

λ−φ(·) > f (·)y P b < bMf

b ∗o < bmo : P b > bm

o D

b ≤ bmo P b < bm

o

a < a b ∗h

λ s c λ s

c

b ∗h =λ+h(·)y

2λ h(·)

s c

1 − PrH = 1 − λ+h(·)(y−2y )2h(·)(y−y ) λ

0− ! PrH!h(·) =

λ

2(y − y )h(·)2 > 0

h(·) s c

∃ cNB ∈ (0,1) : c > cNB =⇒ b ∗h ≥ bMh a = 0

h(·)|sH=0 = 0 sH = s (1− c ) limc→1b ∗h = λ2 > lim

c→1bMh = 0

s ∈ (s , s ) a∗f > 0 s > s a∗f < 0 s ≤ s a∗f

b ∗f (a) =s32µ+λ−a(ϕ−λ)

2(a+s )

D

a∗f = argmaxa

s 32µ+λ−a(ϕ−λ)

2s

µ2

y

aa + s

1µd y + 3µ

2

s 32µ+λ−a(ϕ−λ)

2s

y −

s ( 32µ+ λ) + a(λ−ϕ)

2(a + s )

1µd y − a

a∗f = s

8sµ− (λ−ϕ)2 23µ2 + 12µϕ − 4ϕ2

28sµ− (λ−ϕ)2 − 1

Ω ≡ 8sµ − (λ − ϕ)2 Υ ≡ 23µ2 + 12µϕ − 4ϕ2

µ > ϕ4

Υ > 0 Ω s > s ≡ (λ−ϕ)28µ

Ω s

Ω s ∈ [0, s ). a∗f ∈ s > s a∗f s s > s

lims→s+a∗f = +∞ lim

s→∞a∗f = −∞ a∗f = 0 s = 0 s = s ≡ Υ

32µ+ s

s a∗f > 0 (s , s ) s > s

∃ sI > 0 s > sI

b ∗fa∗f > a b ∗f

a∗f= λ−ϕ

2+32µ+ϕ Ω·ΥΥ

!b∗f! s= 2µ(3µ+2ϕ)

Ω·Υ

a∗f < a b ∗f (a) = − 2η(ϕ−λ)−s (1−c )(2λ+3µ)

4(s (1−c )+η)!b∗f! s= η(3µ+2ϕ)(1−c )

4(s (1−c )+η)2

s ϕ > λ ϕ < λ

a∗f > a

PrF (a∗f ) = 14+ ϕ

2µ+ (λ−ϕ)

4µ

Ω·ΥΩ

! PrF! s= (ϕ − λ)

Ω·ΥΩ2

a∗fΩ·ΥΩ> 1 ! PrF

! s(φ− λ)

a∗f < a

PrF (a) =14+ λ

2µ+ (λ−ϕ)

2µη

s (1−c )! PrF! s= (ϕ − λ) η

2µs (1−c )

(φ− λ)

∃ s ∗ ∈+ s > s ∗ ⇒U Dh >U

Df

s ≥ s ⇒U Dh >U

Df s ∗

s D a = a∗f ( s ) = 0 a∗f < a

η > 0⇒ b ∗fa=0> b ∗h D a = a

D

a = a∗F = 0 s = s U Dh >U

Df s > s

∃ c ∈ (0,1) c > c D a∗ = 0 P

b ≥ bMh D

limc→1

U Dh = µ

c > cNB P D

U Dh

c∈cNB ,1 = µ η

η+(1−c ) c → 1 µ

U Df = PrF (µ

aa+s) + (1− PrF )(µ− b )− a µ a > 0 U D

f

c c : c > c =⇒U Dh >U

Df

∃ µ∗ ∈+ µ > µ∗ ⇒U Df >U

Dh

µ→ +∞ µ > 2λ3

s (1−c )+ηs (1−c )

a∗f µ

µ a∗f a∗f

U Df

a∗f= s +µ+

3(ϕ − λ)8− ϕ(ϕ − λ)

4µ−Ω ·Υ8µ

U Dh =

18µ· λ

2(s (1− c ) + η)s (1− c ) +µ

932s (1− c ) + ηs (1− c ) + η −

3λ8

limµ→∞

U D

f

a∗f−U D

h

= +∞

am aM am aM D a ≥ am a ≥ aM

P bmf bM

f PrF = 0 PrF = 1 am aM am

ϕ > λ

b ∗f b mf bM

f a

am ≡sλ+ 1

2µ

ϕ − λ ; aM ≡sλ− 3

2µ

ϕ − λ .

λ < y = 32µ aM am ϕ > λ

b ∗f b mf a d

d a(b ∗f − bm

f ) < 0 am

a ≥ am ⇒b ∗f ≤ bmf b ∗f = bm

f a < am b ∗f > bmf am b ∗f > bm

f

a dd a(b ∗f − bM

f ) > 0 a > aM ⇒ b ∗f = bMf aM < 0⇒b ∗f < bM

f a

sI ≡ s a∗f ≤maxam, aM=

(ϕ−λ)2(3µ+2ϕ)(µ+2ϕ)2

ϕ > λ

4µ(λ−ϕ)2(3µ−2ϕ)2 ϕ < λ

sI