Download Intimidation: Linking Negotiation and Conflict

Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
Intimidation: Linking Negotiation and Conflict
Sambuddha Ghosha
Gabriele Grattonb
Caixia Shenc
a. Boston University
b. UNSW
c. SHUFE
U of Montreal, 24 November 2014
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
Long Conflicts and Negotiations
Why can’t we avoid conflict even when:
negotiation is efficient
commitment is possible
and conflict is long and painful
leaders are irrational/have different preferences?
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
Intimidation
asymmetric information: each player is tough with positive
probability
players have an incentive to mimic tough type to intimidate
opponent
terrorist carry out attacks to raise fear of further attacks
workers go on strike to raise fear of further strikes
generate long conflict only if sizable probability of being tough
do not link nature of conflict to why negotiations fail
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
Preview
fully dynamic model of conflict driven by reputation
delaying a concession carries the risk of great losses
long conflict even as uncertainty vanishes
pre-conflict negotiation
equilibrium offers are rejected with positive probability
all offers but the very last are unacceptable (brinkmanship)
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
(Very Brief) Literature Review
Models of terrorism:
Lapan and Sandler (1988): exogenous beliefs
Hodler and Rohner (2012): 1 period: attacks only when prob.
terrorist is committed is very large
More general game theory:
Abreu and Gul (2000), Kornhauser et al. (1989), Kreps and
Wilson (1982), and Ponsatì and Sakovics (1995): reputational
effects in bargaining/war of attrition
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
A Model of Conflict
2 players, Challenger and Defender
Each period t ∈ N has two stages
1
2
Challenger decides whether to attack
Defender decides whether to concede
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
A Model of Conflict
Each player has two types: tough and normal
Challenger is tough with probability µ0
Defender is tough with probability π0
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
A Model of Conflict
tough Challenger attacks until Defender concedes
tough Defender never Concedes
normal payoffs, with discount rates δ C and δ D :
If Defender has not conceded yet, Defender enjoys rent d > 0
if Defender concedes, Challenger enjoys rent c > 0 in current
and subsequent periods
For each attack, Challenger pays cost A > 0, Defender suffers
loss L > 0
Note: tough players can be thought of Defender with
L < d /δ D and Challenger with A < 0
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
Assumption
Assumption
δ D L > d ; A < c/(1 − δ C ).
“the loss from one attack in the next period exceeds the
enjoyment of the contested resource in the current period.”
“the cost of attacking is strictly less than the discounted value
of getting the contested resource forever”
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
Threshold Beliefs
we define threshold values π̄ and µ̄ (simple functions of
primitives)
if πt > π̄ , then Challenger concedes immediately even if
Defender would concede next
if µt > µ̄, then Defender concedes immediately even if
Challenger would concede next
−1
1
D L
µ̄ :=
1+ 1−δ
, and
d
δD
A
.
π̄ := 1 − 1 − δ C
c
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(1)
(2)
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
I: No Tough Play
Lemma
In any equilibrium, normal types of both players concede with
strictly positive probability in all periods, except possibly Challenger
in period 1.
First: if opponent is not conceding in the interim, the value of
concession can only go down→if I ever concede, I concede now
Second: never conceding cannot be equilibrium
Then: no concession at stage k⇒then concession w.p. 1 at
k − 1. But if k − 1 does not concede, then reputation jumps
to 1 and k must concede!
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
II: End Game
Lemma
In equilibrium (i) πt < π̄ and µt+1 < µ̄ implies πt+1 ≤ π̄ ; (ii)
µt < µ̄ and πt < π̄ implies µt+1 ≤ µ̄.
if πt > π̄ ≥ πt−1 and µt ≤ µ̄ then µt = µ̄
Suppose not
notice that if µt = µ̄ and Challenger concedes at t + 1, then
Defender is indifferent
t + 1: Challenger concedes at time t + 1
t: if µt < µ̄, Defender strictly prefers not to
concede⇒ πt = πt−1 < π̄!
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
III: Mixing Throughout
Lemma
If Challenger is indifferent between conceding and attacking at
times t > 1 and t + 1 in any equilibrium, then Defender’s
equilibrium concession probability and the public beliefs about him
are
1 − π̄
πt−1
σ̃ D (πt−1 ) :=
; and πt =
(3)
1 − πt−1
π̄
respectively. Similarly, if Defender is indifferent between conceding
and not conceding at times t > 1 and t + 1 in any equilibrium, then
Challenger’s probability of conceding and the public beliefs about
her type are
σ̃ C (µt ) :=
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1 − µ̄
µt
, and µt+1 = .
1 − µt
µ̄
(4)
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
Commitment and Order
Definition
A conflict is of commitment order n ∈ N ∪ {0} if n is the largest
non-negative integer such that µ0 < µ̄n and π0 < π̄ n .
Definition
In a conflict of order n, Challenger is (relatively) more committed if
µ̄n+1 < µ0 < µ̄n and π0 < π̄ n+1 ; Defender is at least as committed
if µ0 < µ̄n and π̄ n+1 < π0 < π̄ n .
Assumption
The quantities ln π̄/ ln π0 and ln µ̄/ ln µ0 are not integers.
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
Equilibrium I
Proposition
If Defender is at least as committed as Challenger,
(i) at t = 1 Challenger attacks w.p. µ0 /µ̄n < 1;
(ii) for t = 2, ..., n, (normal) Defender and Challenger
play σ̃ D and σ̃ C (strict mix) and concede from n + 1;
(iii) if Challenger attacks at time t ≥ 2 and µt−1 ≤ µ̄, we
have µt = µt−1 /µ̄; and
(iv) if Defender does not concede at t ≥ 2 and πt−1 ≤ π̄,
then πt = πt−1 /π̄.
The ex-ante probability of an attack in period t ∈ [1, n] is
t−1
µ0
.
µ̄n (µ̄π̄)
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
Equilibrium I in belief-space
(0,1)
(1,1)
Def. wins
π̄
π1 =
π0
π̄
π0
Chall. wins
(0,0) µ0 µ1 = µ̄3
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µ3 = µ̄
(1,0)
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
Equilibrium II
Proposition
If Challenger is more committed than Defender
(i) in period 1, Challenger attacks w.p. 1 and Defender
does not concede with total probability π0 /π̄ n ;
(ii) thereafter same as before: play σ̃ D and σ̃ C until stage
1 of period n + 1 (conceding with probabilities strictly
less than 1) and concede from stage 2 of period
n + 1; beliefs from the same way.
The unconditional probability of an attack in period t ∈ [1, n] is
π0 t−1 t−2
π̄ . (Only one factor different.)
π̄ n µ̄
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
Equilibrium II in belief-space
(0,1)
(1,1)
Def. wins
π2 = π̄
π1 = π̄ 2
π0
Chall. wins
(0,0)
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µ 0 = µ1
µ̄
(1,0)
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
Limit
Definition
A sequence of conflicts is balanced if
lim
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µk /µ̄n(k)
∈ (0, 1) .
µk /µ̄n(k) + πk /π̄ n(k)
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Introduction
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Negotiation and Brinkmanship
Comparative Statics and Applications
Limit
Proposition
In a balanced sequence of conflicts such that Defender is at least as
committed as Challenger for all k ∈ N. Then
lim inf Pr (attack at t) > 0 for all t ∈ N.
Proposition
In a balanced sequence of conflicts such that Challenger is more
committed for all k ∈ N. Then lim inf Pr (attack at t) > 0 for all
t ∈ N.
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Introduction
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Comparative Statics and Applications
Negotiation before Conflict
limit as interval ∆ between offers goes to zero: δ := e −r ∆ ,
∆ → 0.
at t = 0 Defender has single opportunity to offer a fraction of
the resource
Challenger decides immediately—if reject, then Challenger can
initiate conflict at some future time T
(benchmark case) at t = 0, Challenger privately observes her
type and Defender does not know either type; at t = T ,
Defender privately observes his type
focus on case in which Challenger gets strictly positive utility
from conflict at given prior (i.e. with no offer)
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
Negotiation Failure
Proposition
The optimal offer has an acceptance probability β ∗ ∈ (0, 1 − µ0 )
independent of T > 0.
key tradeoff:
offers that have a higher probability of being accepted increase
the utility of Defender (if beliefs fixed before and after offer,
Defender could avoid conflict with the normal Challenger by
offering her expected value of entering conflict)
because they are more likely to be accepted, rejecting a bigger
offer moves beliefs further against Defender in the conflict
phase
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
Negotiations and Welfare
Defender and both types of Challenger benefit from
negotiation
Defender could make a 0 offer
normal Challenger indifferent between accepting and rejecting,
but rejecting raises probability of immediate victory
tough Challenger rejects (higher probability of immediate
victory)
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Introduction
Conflict
Equilibrium
Negotiation and Brinkmanship
Comparative Statics and Applications
Brinkmanship
K offers made in period {0, ∆0 , . . . , (K − 1)∆0 }
let βk∗ | k = 0, 1, . . . , K − 1 .
Proposition
All equilibria of the K -offer game must satisfy the following for any
prior µ0 :
β0∗ = . . . = βK∗ −2 = 0 < βK∗ −1 = β ∗ .
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Introduction
Conflict
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Negotiation and Brinkmanship
Comparative Statics and Applications
Comparative Statics
Corollary
Conditional on conflict at time t > 1, the probability of an attack
at t 0 > t increases in c, d , δ C and δ D , and decreases in A and L.
Corollary
Fix the likelihood π0 that Defender is tough. The probability that
Challenger initiates conflict is increasing in µ0 (as Challenger is
more likely to be tough). It is strictly increasing if and only if
Defender is at least as committed as Challenger.
Corollary
Let Defender be at least as committed as Challenger. Then, the
probability that Challenger begins to attack is decreasing in π0 .
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Introduction
Conflict
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Negotiation and Brinkmanship
Comparative Statics and Applications
Application: Strikes and Macro Conditions
good macro conditions: µ0 large, π0 small, L large
strikes are more likely
shorter strikes
fits relevant stylized facts in Kennan and Wilson (1990)
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Introduction
Conflict
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Negotiation and Brinkmanship
Comparative Statics and Applications
Application: Terrorism
secular (µ0 low) vs religious (µ0 high) terrorist groups
religious terrorist groups more likely to start conflict
religious groups fight shorter wars
broadly fits ETA/IRA vs Hamas/LeT
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