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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 Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE 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? Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE 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 Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE 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) Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE 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 Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE 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 Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE 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 Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE 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 Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE 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” Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE 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 Ghosh, Gratton and Shen Terrorism (1) (2) BU, UNSW, SHUFE 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! Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE 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! Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE 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 ) := Ghosh, Gratton and Shen Terrorism 1 − µ̄ µt , and µt+1 = . 1 − µt µ̄ (4) BU, UNSW, SHUFE 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. Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE 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) Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE 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 Ghosh, Gratton and Shen Terrorism µ3 = µ̄ (1,0) BU, UNSW, SHUFE 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 Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE 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) Ghosh, Gratton and Shen Terrorism µ 0 = µ1 µ̄ (1,0) BU, UNSW, SHUFE Introduction Conflict Equilibrium Negotiation and Brinkmanship Comparative Statics and Applications Limit Definition A sequence of conflicts is balanced if lim Ghosh, Gratton and Shen Terrorism µk /µ̄n(k) ∈ (0, 1) . µk /µ̄n(k) + πk /π̄ n(k) BU, UNSW, SHUFE Introduction Conflict Equilibrium 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. Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE Introduction Conflict Equilibrium Negotiation and Brinkmanship 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) Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE 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 Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE 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) Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE 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 ∗ . Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE Introduction Conflict Equilibrium 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 . Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE Introduction Conflict Equilibrium 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) Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE Introduction Conflict Equilibrium 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 Ghosh, Gratton and Shen Terrorism BU, UNSW, SHUFE