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: –U A (x) increasing, U B (x) decreasing –For simplicity, assume risk neutrality for most examples: U A (x) = x and U B (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: w A, w B 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 c A, c B denote the expected costs Then, w A = p – c A w B = 1 – p – c B

6. The Reservation Value Reservation points are then x such that U A (x) = p – c A and U B (x) = 1 – p – c B With example utility functions, 01 p – c A p + c B A will accept B will accept

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 + c B > p – c A or c A + c B > 0. Note: This means that one actor could have negative costs for conflict, as long as w A, w B < 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: 1. Asymmetric information about p, c A, and/or c B, combined with incentives to misrepresent. 2. 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 3. 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: w A, w B unknown –Common assumption: p known, c A, c B unknown

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

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 If F(x) has a “monotone hazard rate,” which ensures that there exists solution to the first-order condition. The optimal offer, x*, solves 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 1.The TILI bargaining framework does not allow counter-offers artificially imposes a final move. 2.Most conflicts are preceded by efforts to signal resolve through threats and escalatory efforts.

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

16. Assumptions 01 D’s capital S’s capital q Existing border 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 w D = p – d w S = 1 – p – s

17. 01 D’s capital S’s capital q If both states are known to be satisfied, then neither will ever attack, and no serious bargaining will take place: p – dp + s 01qp – dp + s 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: Assumptions

18. 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  (1−q)+(1−  (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. q = 0.5 q = 0.33

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

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

25. Main Result If  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  →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 B Offer A Accept Reject A Message

28. A Simple Signaling Game A Challenge Status Quo B A Acquiesce Resist Stand Firm SQ A SQ B ACQ A ACQ B BD A BD B WAR A WAR B Back Down Assumptions: 1.ACQ A >SQ A, BD A 2.BD B >ACQ B 3.WAR A has cdf F 4.WAR B has cdf G

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 WAR B > ACQ B, then B has a dominant strategy to Resist –If WAR B < ACQ B, 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(BD A ) 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 BD A < SQ A for some types. –Otherwise, ACQ A > SQ A ensures that a challenge weakly dominates the status quo for all types.

32. Types of Signaling 1.“Slippery slope”: challenge creates an exogenous risk of war 2.“Tying hands”: challenge creates an “audience cost” for backing down 3.“Sunk costs” or “burning money”: A must pay an up-front cost to challenge

33. Slippery Slope A Challenge Status Quo B A Acquiesce Resist Stand Firm SQ A SQ B ACQ A ACQ B BD A BD B WAR A WAR B Back Down WAR A WAR B N  1 – 

34. Tying Hands A Challenge Status Quo B A Acquiesce Resist Stand Firm SQ A SQ B ACQ A ACQ B BD A = SQ A – a BD B WAR A WAR B Back Down

35. Sunk Costs A Challenge Status Quo B A Acquiesce Resist Stand Firm SQ A SQ B ACQ A – m ACQ B BD A = SQ A – m BD B WAR A – m WAR B Back Down

36. Equilibrium In general, for fixed , m, or a, the equilibrium strategies are defined by a set of cutpoints in the continuum of types: WAR A WAR B Challenge Stand Firm Challenge Back Down Status Quo Back Down ResistAcquiesce

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 A Acquiesce Resist Stand Firm (0,1) (1,0) (– a, 1) (w A, w B ) Back Down

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

40. Comparing Complete and Asymmetric Information Probability of a challenge –CI: A only challenges when w A > – a –AI: A challenges when w A > – 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. Magnitude of constraint, d A Probability in Equilibrium 0 1B Resists|Challenge A Challenges War Outcomes as a Function of d A

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

43. Predictions of the Two Views of Democracy Predicted effect on probability of... If democracy in A means... ConstraintsInformation Decrease in w A Complete Information Increase in a A Challenges--+ B Resists| Challenge +-- War+/--

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 Non-Democratic Initiator Democratic Initiator Not Reciprocated617 (49.2)219 (56.9) Reciprocated637 (50.8)166 (43.1) Pearson  2 = 6.95 Pr = 0.008

46. Initiator-Target Power Status Non-Democratic Initiator Democratic Initiator Major Power- Major Power 0.340.26 Major Power- Minor Power 0.340.25 Minor Power- Major Power 0.420.33 Minor Power- Minor Power 0.430.34 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