Bargaining and Signaling

1. Bargaining and Signaling

2. Basic Set-Up • Two parties, A and B, bargain over the division of something of value. • Division of territory • Distribution of economic gains • Policy (e.g., taxes) • We often normalize this range of possible deals to [0,1]. • A settlement is x [0,1].

3. Basic Set-Up • A prefers larger values of x; B prefers smaller ones: • UA(x) increasing, UB(x) decreasing • For simplicity, assume risk neutrality for most examples: UA(x) = x and UB(x) = 1 – x.

4. Basic Set-Up • Each party has a minimal acceptable settlement • “reservation value” • the deal that it sees as equivalent to no deal. • The reservation value is determined by the expected value of the “outside option”: • the expected value of war • the expected value of a revolution or coup • An actor can always guarantee its reservation value by implementing the outside option

5. The Reservation Value • Most generic form: wA, wB • We sometimes assume that conflict can be seen as a “costly lottery”: • let p denote the probability that A will win • assume that the winner imposes its most preferred outcome • let cA, cB denote the expected costs • Then, wA = p – cA wB = 1 – p – cB

6. The Reservation Value • Reservation points are then x such that UA(x) = p – cAand UB(x) = 1 – p – cB • With example utility functions, A will accept B will accept 0 1 p – cA p + cB

7. Zone of Agreement • All settlements between the two reservation points constitute the “zone of agreement”: the set of deals that both sides prefer to conflict. • The zone of agreement is always non-empty if • Conflict is costly in aggregate In our example: The zone of agreement is non-empty if p + cB > p – cA or cA + cB > 0 . Note: This means that one actor could have negative costs for conflict, as long as wA, wB < 1. • The actors are not too risk acceptant

8. Fearon, “Rationalist Explanations for War” Motivation: If war is costly, there exist settlements that both sides should prefer to war. Why do states sometimes fail to reach ex post efficient bargains? Proposed mechanisms: • Asymmetric information about p, cA, and/or cB , combined with incentives to misrepresent. • Commitment problems: Deals in the zone of agreement may be non-self enforcing due to • First-strike advantages • Exogenous shifts in the power distribution • Endogenous shifts in the power distribution • The good is lumpy or indivisible.

9. Asymmetric Information • Assume that each actor is incompletely informed about the other’s value for conflict • Most generic: wA, wB unknown • Common assumption: p known, cA, cB unknown

10. “Take It or Leave It” Bargaining x, 1 – x Accept A B Offer x Reject p – cA, 1 – p – cB

11. Equilibrium Strategies • There exists a “risk-return tradeoff” in B’s decision: • Increasing x decreases the risk of war, F(p – x), but also decreases B’s return on the deal, 1 – x. • More profitable bargains can only be achieved by accepting a greater risk of war. • But it never makes sense to offer more than .

12. Equilibrium Strategies The optimal offer, x*, solves If F(x) has a “monotone hazard rate,” which ensures that there exists solution to the first-order condition. In general, the optimal offer entails a positive probability of war—i.e., .

13. Equilibrium Strategies If A’s costs are distributed uniformly, then The equilibrium probability of war is

14. Two Shortcomings • The TILI bargaining framework • does not allow counter-offers • artificially imposes a final move. • Most conflicts are preceded by efforts to signal resolve through threats and escalatory efforts.

15. Powell, “Bargaining in the Shadow of Power” Accept Accept D D S … S Reject Reject Offer Offer Attack Attack t=1 t=0

16. Assumptions D’s capital Existing border S’s capital 0 q 1 • Until an agreement or war, D gets a per-period payoff of q and S gets a per-period payoff of 1 – q. • War is a costly lottery. Let p = Pr(D wins), Let d and s denote per-period loss from having fought a war. Hence, per-period expected values of war are • wD = p – d • wS = 1 – p – s

17. Assumptions • If p – d > q, then D is dissatisfied. If 1 – p – s > 1 – q, or p + s < q, then S is dissatisfied. • It is easy to see that at most one state can be dissatisfied: D’s capital S’s capital 0 q p – d p + s 1 • If both states are known to be satisfied, then neither will ever attack, and no serious bargaining will take place: 0 p – d q p + s 1

18. Assumptions • To generate incomplete information,assume • If , then D is potentially dissatisfied. • At most one state can be potentially dissatisfied.

19. Key Result Lemma. The potentially dissatisfied state never rejects an offer in order to make a counter-offer. Hence, in equilibrium, the equilibrium outcome is the same as in the TILI bargaining game: • S offers • D either accepts or attacks

20. Intuition • Conjecture that some dissatisfied type(s) of D counters with an offer, x. Let r denote the most resolute type that does so. • Possible outcomes • War in some future period • But war now is better than a period of SQ followed by war. • D accepts some offer from S in future period • But the most S will ever offer is p−r, which is equivalent to the war payoff. War now is better for type r. • S accepts the counter-offer • But S can always reject x, leading to the SQ payoff in that period, and then offer p−r, which it knows will be accepted. S will reject any offer which gives it less than d(1−q)+(1−d(1-p+r). • But D of type r could get p−r>q immediately and in all future periods by attacking now. Hence, this type is not willing to make a counter-offer that S would accept.

21. The Relationship of Power and War

22. The Relationship of Power and War q = 0.5 q = 0.33

23. Leventoğlu and Tarar, “War and Incomplete Information” Accept Accept D D S … S Reject Reject Attack Attack t=1 t=0

24. Leventoğlu and Tarar, “War and Incomplete Information” Accept Accept D S D S D … S Reject Reject Attack Attack Attack Attack t=1 t=0

25. Main Result • If d is sufficiently high, then there exists a “no risk” equilibrium in which D rejects a low initial offer and then makes a counter-offer which is accepted. • This implies that incomplete information leads to war only when • the states are impatient, or • they fail to coordinate on the risk free equilibrium

26. Thoughts • As the time between offers shrinks to zero, or d →1, a peaceful equilibrium always exists. • Failure of bargaining is not well explained by “pure” bargaining models. • Key question: Given that the existence of an efficient deal is common knowledge, why would states ever walk away from the bargaining table?

27. Signaling Accept B A A Message Offer Reject

28. A Simple Signaling Game A • Assumptions: • ACQA>SQA, BDA • BDB>ACQB • WARAhas cdf F • WARB has cdf G Challenge Status Quo B SQA SQB Resist Acquiesce A Back Down Stand Firm ACQA ACQB BDA BDB WARA WARB

29. The Risk-Return Tradeoff • Even in this simple setting, B faces a risk-return tradeoff: • Assume BD is B’s first-best outcome • If WARB > ACQB, then B has a dominant strategy to Resist • If WARB < ACQB, then B faces a choice between • getting its second-best payoff for certain, and • a lottery between its first- and third-best payoffs. • The odds of the lottery are determined by the posterior belief that A will fight.

30. The Risk-Return Tradeoff • Let q denote B’s posterior belief that A will stand firm given that A has challenged. • Then B will Acquiesce if

31. Informative Signaling • Let p = 1 – F(BDA) denote prior probability that A will stand firm • A’s challenge is informative if q > p. • For this to happen, the probability of a challenge must be less than one. • Separation of types requires that BDA < SQA for some types. • Otherwise, ACQA > SQA ensures that a challenge weakly dominates the status quo for all types.

32. Types of Signaling • “Slippery slope”: challenge creates an exogenous risk of war • “Tying hands”: challenge creates an “audience cost” for backing down • “Sunk costs” or “burning money”: A must pay an up-front cost to challenge

33. Slippery Slope A Challenge Status Quo B SQA SQB Resist Acquiesce A Back Down Stand Firm ACQA ACQB N 1 – p p WARA WARB BDA BDB WARA WARB

34. Tying Hands A Challenge Status Quo B SQA SQB Resist Acquiesce A Back Down Stand Firm ACQA ACQB BDA = SQA – a BDB WARA WARB

35. Sunk Costs A Challenge Status Quo B SQA SQB Resist Acquiesce A Back Down Stand Firm ACQA – m ACQB BDA = SQA – m BDB WARA – m WARB

36. Equilibrium • In general, for fixed p, m, or a, the equilibrium strategies are defined by a set of cutpoints in the continuum of types: Status Quo Back Down Challenge Back Down Challenge Stand Firm WARA Acquiesce Resist WARB

37. Schultz, “Do Democratic Institutions Constrain or Inform?” • Questions: Does democracy influence crisis outcomes, and if so how? • Competing Theories • Institutional constraints: democracy increases the political costs of war • Informational: democratic institutions increase transparency and/or increase audience costs • Realism (the null hypothesis): democracy doesn’t matter • Problem: While it is relatively easy to determine whether democracy matters, it is much harder to distinguish competing arguments for why it matters.

38. The Theoretical Model A Challenge Status Quo B (0,1) Resist Acquiesce A Back Down Stand Firm (1,0) (– a, 1) (wA, wB)

39. Putting Democracy in the Model • Institutional constraints • Democracy  lower expected value for war on average • Assume wA ~ [– CA – dZA, – dZA], where dA > 0 and ZA = 1 if state A is a democracy • Information • Democracy  higher audience costs (a) • Transparency  democracy generates complete information about wA

40. Comparing Complete and Asymmetric Information • Probability of a challenge • CI: A only challenges when wA > – a • AI: A challenges when wA > – b , with b > a • Probability of resistance • CI: B never resists conditional on a challenge • AI: B resists with nonzero probability for some parameters • Probability of war • CI: Zero • AI: Nonzero for some parameters

41. Outcomes as a Function of dA 1 B Resists|Challenge A Challenges Probability in Equilibrium War 0 Magnitude of constraint, dA

42. Outcomes as a Function of a 1 A Challenges Probability in Equilibrium B Resists|Challenge War 0 Magnitude of Audience Costs, a

43. Predictions of the Two Views of Democracy

44. The Data • Dependent variable: Did the target resist? • Data set: Militarized Interstate Disputes (MIDs) • 1654 disputes over period 1816-1980 • arranged in dyads of initiator-target • RECIP = 1 if target reciprocated the initiator’s action, and RECIP = 0 otherwise. • Main independent variable: Regime type of the initiator • Data set: Polity III • DEMINIT = 1 if initiator is democratic (score of 7 or higher on 21-point composite democracy scale), and DEMINIT = 0 otherwise.

45. Bivariate Correlation Pearson c2 = 6.95 Pr = 0.008

46. Predicted Probabilities of Reciprocation

47. Summary • Use of model to • generate testable hypotheses and • identify a critical test between theories. • Convinced?

48. Summary • Use of model to • generate testable hypotheses and • identify a critical test between theories. • Potential problems • Unmeasured factors • Democracies select weak targets • Democracies make smaller demands • Observed correlation could arise from more than one causal pathway (identification problem) • Mismatch between data and model