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⁷ Technically, uncertainty is reduced by a screening process that separates the types for the uninformed player. The situation is very different if there is a commitment problem as in Fearon, ‘Fighting Rather than Bargaining’. Fearon shows that a government's use of offers to separate the weak insurgents from the strong is precluded (under certain conditions about frequency of offers) by the fact that the weak expect the government to renege. Because of this commitment problem, the government can only attempt to screen and separate the types by observing the result of fighting. If the weak hold out long enough, the cost to the government of continuing to fight outweighs the benefits of making an offer that both weak and strong will accept. The government will then make a pooling offer.

⁸ Unlike the continuous-time framework, discrete-time does not allow the players to manipulate the costs imposed by delaying tactics when using pure strategies. Using mixed strategies would allow such manipulation but would raise technical problems as well as the familiar interpretation issues that are associated with probabilistic strategies.

⁹ The standard subgame perfect (SPE) and perfect Bayesian (PBE) equilibria are merely Markov perfect (MPE) equilibria of one kind or another. All these perfectness concepts share the requirement of sequential rationality: each player maximizes his expected payoff at his turn of play given a current state of the game that summarizes his information. What distinguishes one type of equilibrium from another is how states, and the transition from state to state, are defined. In a PBE the transition is defined by Bayesian updating of beliefs. In a grim trigger SPE the transition results from observing a violation of expected play. MPEs are particularly attractive in repeated games because a few relevant states can summarize the various prior histories of play. The term MPBE refers to Markov perfect equilibria in repeated games when Bayesian updating of beliefs is involved, a necessary feature if we are to deal with uncertainty in a meaningful manner.

¹⁰ Langlois, Catherine C. and Jean-Langlois, Pierre P., ‘Does Attrition Behavior Help Explain the Duration of Interstate Wars? A Game Theoretic and Empirical Analysis’, International Studies Quarterly, 53 (2009), 1051–1073CrossRefGoogle Scholar. We briefly discuss our empirical approach in the discussion section.

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¹³ Wagner, Harrison R., ‘Bargaining and War’, American Journal of Political Science, 44 (2000), 469–484CrossRefGoogle Scholar; and Wagner, Harrison R., War and the State (Ann Arbor: The University of Michigan Press, 2007)CrossRefGoogle Scholar. Wagner discusses how the outcome of limited war can inform so that reasonable bargaining positions can emerge without a need to engage in fights to the finish. However, he does not derive equilibria of his war and bargaining model when uncertainty is present.

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¹⁶ In ‘The Dynamics of Bargaining and War’, Filson and Werner subsequently relax the two-type assumption, multiplying the number of types and number of possible battles. They solve their model numerically.

¹⁷ Filson and Werner, ‘A Bargaining Model of War and Peace’: Reiter, Dan, ‘Exploring the Bargaining Model of War’, Perspectives in Politics, 1 (2003), 27–43CrossRefGoogle Scholar.

¹⁸ A fact that is only reinforced by the assumption that resource constraints would only allow for at most two battles (Filson and Werner, ‘A Bargaining Model of War and Peace’).

¹⁹ Slantchev, ‘The Principle of Convergence in Wartime Negotiations.’

²⁰ Powell, ‘Bargaining and Learning while Fighting’.

²¹ Powell, ‘Bargaining and Learning while Fighting’.

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²³ This follows a result developed in Gul, Faruk and Sonnenchein, Hugo, ‘Bargaining with One-Sided Uncertainty’, Econometrica, 56 (1988), 601–611CrossRefGoogle Scholar.

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²⁵ The Markov process has victory by one rival or the other as the only absorbing states.

²⁶ Powell, ‘Bargaining and Learning while Fighting’.

²⁷ Letting the period separating successive turns approach zero is not equivalent to moving to the continuum. The logic of alternating turns together with the cost of waiting for one's next turn remains in the former but not in the latter. In continuous time, the timing of offers and acceptance becomes a decision variable and this erases the incentive to accept a given offer earlier rather than after the fixed and costly delay that is inherent to the alternating-turns framework.

²⁸ Perry, Motty and Reny, Philip J., ‘A Non-Cooperative Bargaining Model with Strategically Timed Offers’, Journal of Economic Theory, 59 (1993), 50–77CrossRefGoogle Scholar: Sakovics, Jozsef, ‘Delay in Bargaining Games with Complete Information’, Journal of Economic Theory, 59 (1993), 78–95CrossRefGoogle Scholar.

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³⁰ This discussion is inspired by the extensive and insightful feedback received from one of the anonymous reviewers of this article.

³¹ Slantchev, ‘The Principle of Convergence in Wartime Negotiations’; Powell, ‘Bargaining and Learning while Fighting’.

³² In a fully generalized discrete-time model, it is not clear that separating equilibria can be obtained, let alone semi-separating equilibria, which would be even harder to obtain given the non-linearity of Bayesian updating. In the continuous-time framework, by contrast, we can use the calculus of integrals, which is a far more powerful mathematical tool than the algebra of power series that must be used in discrete-time models. This is, in fact, the main motivation for our modelling choice.

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³⁶ This also assumes a_ia_j−k_ik_j > 0.

³⁷ z_i is common to all types i on side .

³⁸ We, of course, assume that the prize is divisible at will or that compensations, monetary or otherwise, allow such divisions.

³⁹ By contrast, Perry and Reny, ‘A Non-Cooperative Bargaining Model’, or Sakovics, ‘Delays in Bargaining Games’, impose reaction delays. Our choice is motivated by our criticism that set unit periods exclude by design explanations other than uncertainty for war duration.

⁴⁰ Technically, I's terms will be fully described by a function y_i of time t that can be left-discontinuous but is right-continuous with a piecewise continuous derivative.

⁴¹ We also assume that an overgenerous offer y_i(t) ≥ 1−y_j(t) by i at time t means an immediate acceptance of y_j(t) and that simultaneous acceptances of mismatched offers have no effect. Making no explicit offer is interpreted as offering y_i = 0. If the game ends at time θ in the military victory of one side the offers of the two sides are trivially set to the corresponding outcome (e.g., y_i ≡ 0 for t ≥ θ if i wins) for notational consistency.

⁴² The notation dφ_j(s) rather than reflects the fact that φ_j can be discontinuous. At discontinuities dφ_j(s) is a mass with magnitude equal to the size of the jump.

⁴³ Implicit in Equation 4 is the convention that π_i(x_i, s) ≡ π_i(x_i, θ) for s ≥ θ if a military outcome x_i is reached at time θ. Therefore, for all t ≥ θ since, also by convention, x_i ≡ 1−y_i(s) ≡ y_j(t) for all t ≥ s ≥ θ.

⁴⁴ In the case of a non-uniform initial distribution of types j, the probability φ_j would involve the beliefs about side J as well as the distribution of types.

⁴⁵ While pooling equilibria may exist, players would not learn about each other's types, convergence could not take place and the issue of convergence would be moot. And semi-separating equilibria would picture each individual type making probabilistic decisions, a feature that is unnecessary given our continuum of types and would raise interpretation problems.

⁴⁶ The positive part of y_j satisfies the boundary condition discussed in the Appendix (Definition 2).

⁴⁷ http://www.globalsecurity.org/military/ops/desert_storm.htm

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⁵⁴ Langlois and Langlois, ‘Does Attrition Behavior Help Explain the Duration of Interstate Wars?’

⁵⁵ If only types i ≥ i* can make offer y_i optimally and φ_i(t ⁻) < i* was side 's belief just before that offer, belief jumps to φ_i(t) = i* at time t.

⁵⁶ An ‘overmatching’ offer y_i(t) > 1−y_j(t) would be quickly accepted by side and would be a net loss for any i who could accept 1−y_j(t) instead.

⁵⁷ One replaces y_j by u_i(y_j), 1−y_j by u_j(1−y_j) and by in all formulae.

⁵⁸ Intuitively, offer y_j increases so fast that all types i prefer to wait until it levels off.

⁵⁹ The case allows an increasing belief φ_i while offer y_i is unacceptable. The consequences for belief φ_i are only given for the completeness of Lemma 4. But only the consequences for belief φ_j are necessary in the Main Theorem at the end.

⁶⁰ Intuitively, a jump in y_i is an ‘extremely fast’ increase, which was found unacceptable in Lemma 2.

⁶¹ Consistent beliefs would have to jump according to: . But the jump Δφ_j(t) > 0 would be incompatible with an acceptable profile starting at t. Proofs of these statements are available from the authors upon request.

⁶² The operators and are continuous in the topology of uniform convergence (or any weaker topology).

⁶³ See Earl Coddington and Norman Levinson, Theory of Ordinary Differential Equations (New York: McGraw-Hill, 1955), p.12. Bounded partial derivatives guarantee the standard Lipschwitz condition.

⁶⁴ We can define . If offers match at time t, and then becomes infinite and φ_j(t) = 1 provided . If beliefs need to be defined for s ≥ t we simply set them as the constant φ_j(s) ≡ φ_j(t).

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⁶⁷ It is entirely possible that and y_j match at some t < θ. In that case, belief φ_i may (or may not) reach the value 1 in the interval [t, θ]. Also note that y_j need not be acceptable in this argument.

⁶⁸ If the game has ended by time t ≥ θ, we have , the integrand in Equation A7 is nil and where y_j(θ) is the outcome of war or the agreed upon bargain at time θ.

⁶⁹ This is a typical ‘Optimal Control’ problem. The standard technique for solving such problems is Variational Calculus and the Pontryagin Maximum Principle. Unfortunately, this is unsuccessful here because of the non-linearity of Equation A4. Indeed, the resulting ‘costate equations’ are not solvable. However, we do use a method borrowed from Variational Calculus in Equation A19.

⁷⁰ If , then has no effect on , which remains nil.

⁷¹ can obtained by differentiating the identity i = φ_i(t_i).

⁷² We must choose t so that offers are not matching, or γ could become unbounded.

⁷³ Indeed, signalling by making any offer y_i other than the minimum acceptable one turns out to be suboptimal.

⁷⁴ Optimal t_i ≤ t here is the time at which offers match and can only increase with t.

⁷⁵ Of course, all arguments are valid in the ‘s|t’ sense and therefore hold for any current history and beliefs, thus ensuring sequential rationality.