Bargaining and Conflict

1. Bargaining and Conflict

2. Why Study Bargaining? • Most recent models view conflict as the result of a failure of bargaining. • The puzzle: why fight when there are deals that both sides prefer to fighting ex post? • Conflict is costly to both parties, so rational parties have a shared interest in avoiding it. • Other modeling approaches: deterministic/ mechanical (Richardson), structural

3. Bargaining occurs in many areas of politics: • IPE: Trade negotiations • Legislative: President-Congress over vetos • Comparative: Coalition formation • Models of bilateral bargaining drawn from economics predominate. • Multilateral issues–third party intervention–largely unexplored.

4. Basic Ideas of Bargaining • We assume the parties bargain over the division of something of value. • Surplus from trade in economics • Resolution of some international issue, such as territory • We normalize this range of possible deals to [0,1]. • A settlement is x  [0,1].

5. State A prefers larger values of x; state B smaller ones. • uA(x) increasing, uB(x) decreasing • For simplicity, assume risk neutrality for some examples; uA(x) = x and uB(x) = 1 - x.

6. Each party has a minimal acceptable settlement–its reservation point (or level)–the deal it sees as equivalent to no deal. • What determines an actor’s reservation point? • In economics, its value for the good, which could be determined by other possible deals • In a law suit, going to trial/judgment • In conflict, its value for fighting

7. Value for Fighting If we think of war as a lottery with known chances of victory and defeat, Expectation from war = p(victory)u(victory) + p(defeat)u(defeat) - costs of fighting • Normalize payoffs and p = p(A wins), then EA(war) = p - cA and EB(war) = 1 - p - cB

8. Reservation points are then x such that uA(x) = p - cA and uB(x) = 1 - p - cB • With example utility functions, A’s reservation point is p - cA; B’s is p + cB. A will accept B will accept 0 1 p - cA p + cB

9. Zone of Agreement • All settlements between the two reservation points constitute the zone of agreement; both sides prefer all these deals to fighting. • If the probability of each side winning is known and both sides are risk-neutral or risk-averse, then there is always a zone of agreement. • Differs from economic bargaining where a deal may not be possible.

10. Bargaining Protocols • So far, we have some assumptions about what the parties want, but none about how they bargain. • Who makes offers, in what order • Who knows what when they make and respond to offers • Do the parties have other options? • In other areas, institutions often set parts of bargaining protocols.

11. Simple Ultimatum Model The simplest bargaining protocol is an ultimatum: • A makes an offer Ω. • B accepts or rejects Ω. If B accepts, A receives Ω; B receives 1 - Ω. If B rejects, A receives p - cA; B receives 1 - p - cB.

12. Equilibrium (from backwards induction): • B accepts Ω if   p + cB. • A makes offer that gives it biggest share given B’s reply: Ω = p + cB. Comments: • A can use ultimatum power to grab all of surplus. • War never occurs because A knows exactly what B will accept.

13. Introduce Incomplete Information • Now assume that cB [0,C]. Now A no longer knows exactly what deal B will accept. For ease, assume cB uniformly distributed. • What should A offer? • Choose x to maximize p(B accepts x)uA(x) + p(B rejects x)uA(war) = [(p + C - x)(x) + (x - p)(p - cA)]/C

14. = p + (C - cA)/2 [could be limited to 0 or 1] • Complete equilibrium: • A offers p + (C - cA)/2 • B accepts any offer  such that   p + cB Comments: • Risk-return tradeoff in A’s offer • P(war) = ½(1 - cA/C) • Uncertainty/incomplete info necessary for war

15. Signaling Most crises involve an interchange of threats, rather than just a single ultimatum out of the blue. These moves could influence the bargaining by allowing the parties to signal their private information to one another. Threats and escalation could operate as signals of resolve–a side’s willingness to fight.

16. Escalation Game A simple escalation model: • A makes a threat to B or does nothing. If A does nothing, SQ is the outcome. • B either resists the threat or gives in. If B gives in, 1 is the outcome. • A either carries out the threat (goes to war) or backs down. If A backs down, the outcome is 0. War outcome already defined.

17. Types of Players • A has cost of fighting uniformly distributed on [0,C’]. • B has cost of fighting uniformly distributed on [0,C]. These costs are private information, revealed to the players at the beginning of the game.

18. Strategies and Updating Continuous types lead to cutpoint strategies; uniform distributions lead to simple updating of beliefs.

19. Find Strategies Assume parameters are such that all decisions possible in equilibrium. • A’s last move: find x such that A indifferent between war and back down: p - x = 0 x = p

20. Find y such that A indifferent between threat and no threat: Notice that y plays no role in this decision because this type of A is bluffing. Instead, this equation determines B’s cutpoint, z, making p(B resists) = 1 - q.

21. Find y and z such that B indifferent between resist and submit:

22. Full Equilibrium • A makes a threat if cA < p(p + C(1 - q)); does not otherwise. • B resists if cB < C(1 – q); gives in otherwise. • A carries out its threat if cA < p; backs down otherwise. This equilibrium requires the parameters satisfy certain conditions, such as p < C’.

23. Other Equilibria Without these conditions, this equilibrium does not hold. Exercise: Find equilibrium if • C’ < p, • p + C(1 - q) < 1, or • C’ < p(p + C(1 - q)).

24. Comparative Statics The multiplicity of equilibria creates a problem for comparative statics because we have to keep track of switching across them. I will ignore this issue for now and look at the comparative statics of the main equilibrium.

25. p(threat) [= p(p + C(1-q))/C’] increases as… p increases C’ decreases C increases q decreases p(resist|threat) [= 1-q] increases as… q decreases p(war|resist) [= 1/(p + C(1-q))] increases as... p decreases C decreases q increases p(war) [= p(1-q)/C’] increases as… p increases q decreases C’ decreases

26. How Might We Test This?

27. Abstract Objections to the Model • No bargaining; no flexibility to offers • Artificial final move • Limited parameters: reputation locked at a high level • No option for B to attack So back to bargaining models…

28. Rubinstein Bargaining Model The Rubinstein bargaining model is the classic, basic noncooperative model of bargaining. • Two players bargain over a unit ( gives 1’s share) • Alternating offers, accept or reject responses • Discounting across rounds

29. Equilibria of RBM • RBM has an infinite number of continua of Nash equilibria; any division is possible in any round. • However, it has only one subgame perfect equilibrium: • In odd numbered rounds, • Player 1 offers  = (1-2)/(1-12) in odd-numbered rounds. • Player 2 accepts   (1-2)/(1-12).

30. In even numbered rounds, • Player 2 offers  = (1-1)/(1-12) in odd-numbered rounds. • Player 1 accepts   (1-1)/(1-12). • Comments: • Bargaining is ex post efficient; they always settle in the first round. • Patience (higher discount factor) gets more. • First-mover advantage

31. Variations on RBM Exercise: Find equilibrium when players suffer fixed costs d1 and d2 after each rejection of an offer for… • d1  d2, • d1 > d2

32. Options to Bargaining • Outside option: ability to end bargaining for a set payoff. • War is typically thought of as outside option. • A player cannot receive less than its outside option. • Inside option: payoff per round while bargaining continues. • Control of territory in civil wars as inside option. • Players divide the surplus between the values of their inside options as in RBM.

33. Return to Incomplete Info Before we looked at simple ultimatum model with incomplete info. Extend this model so that uninformed party can make any number of offers. • Offers march upward in value; screening out less resolved types. • Nature of equilibrium depends on whether the players know a surplus exists. • The Coase conjecture and ex post efficiency

34. Problems and Possibilities • So why does war occur? • Possibility of collapse of one party • Passage of time as signal of resolve • Bargaining does not end when war begins.