គោលការណ៍នៃការបង្រួបបង្រួមក្នុងការចរចាក្នុងសម័យសង្គ្រាម

 The Principle of Convergence in Wartime Negotiations

BRANISLAV L. SLANTCHEV University of California, San Diego

If war results from disagreement about relative strength, then it ends when opponents learn enough about each other. Learning occurs when information is revealed by strategically manipulable negotiation behavior and nonmanipulable battlefield outcomes. I present a model of simultaneous bargaining and fighting where both players can make offers and asymmetric information exists about the distribution of power. In the Markov perfect sequential equilibrium, making and rejecting offers has informational value that outweighs the one provided by the battlefield. However, states use both sources of information to learn and settle before military victory. The Principle of Convergence posits that warfare ceases to be useful when it loses its informational content and that belief in defeat (victory) is not necessary to terminate (initiate) hostilities. Thus, the standard puzzle in international relations that seeks to account for prewar optimism on both sides may not be that relevant.

Why do wars end? War is an instrument of policy and its goal is to achieve peace through violent diplomacy. Attaining military victory is central for policy planners and military commanders and is explicitly stated as a goal in U.S. strategic doctrine (C. Powell 1992). However, total disarmament and complete overthrow of the opponent are quite rare (Pillar 1983). Wars most often terminate in negotiated settlements short of military collapse.

So why do opponents agree to terminate hostilities? War can be viewed as an organized coercive process through which opponents attempt to persuade one another to concede whatever is demanded by the other (Schelling 1960). Since this process is extremely costly, both sides have strong incentives to end it as soon as possible while conceding as little as they can. How long they hold out depends not only on their capabilities but also on the military situation and on what they expect to happen in the future.

Expectations are, in fact, central to explanations of rational war termination. Blainey (1988, 54) argued that the only surprise in war is that at least one side that expected to win actually lost, concluding that mutual optimism about military victory was necessary to start a war. The leading explanation of war as bargaining failure demonstrates how such optimism is possible when both players are rational (Fearon 1995; R. Powell 1996).

The emphasis on expectations about the military outcome, however, is misleading. War is coercive bargaining and ends because opponents succeed in coordinating their expectations about what each is prepared to concede.

How do they coordinate these expectations? Although fighting can result in complete military victory, its more important function is coercive: to convince the opponent to accept a settlement. This happens after opponents learn enough about their prospects in war to decide that continuation is unprofitable. Warfare is transmission of information about these prospects. The Principle of Convergence states that once expectations converge sufficiently, war loses its informational content, and hostilities can terminate with a negotiated settlement.

I derive this principle from a formal model that treats war as a costly instrument of policy and allows unlimited diplomatic exchange. I analyze how rational players learn from two sources of information: negotiating behavior and the battlefield. These sources are subject to different degrees of strategic manipulability, and may possibly provide contradictory information.

Wagner (2000) emphasizes that fighting provides a means of revealing information that is “not available in the standard bargaining models” and is in some ways superior to inferring private information from negotiation behavior. But he may have overstated his case because (a) the “fog of war” makes interpretation of any event notoriously difficult (Iklé 1971), and (b) strategic bargaining behavior remains an important source of information (Iklé 1964). The model takes all these issues into account and speaks directly to the relevance of negotiations in warfare.

States are uncertain about the probability of winning battles. As the war progresses, they observe the outcomes on the battlefield in addition to the bargaining history and evaluate their prospects for the future. Provided that states value the future sufficiently, they delay agreement in order to accrue enough information about their prospects and avoid settling prematurely on worse terms. This type of uncertainty can only be resolved ex post: The risk of war always exists and cannot be eliminated through prewar bargaining. The timing of the settlement depends on the rate at which information accrues but its terms also depend on the military position at the time agreement is struck, which explains the common last-minute jockeying for advantage prior to an armistice.

Fighting does reduce uncertainty, but the battlefield is a noisy source of information and not the only one. The strategic behavior of states at the bargaining table can be very revealing. Because the model constrains players to make counteroffers upon rejecting offers, information can accrue rather quickly and more precisely from their negotiating behavior. Since readiness to talk can be so revealing, it may provide a good rationale for delaying explicit diplomacy until after an armistice.

Military developments may provide information that contradicts the explicit bargaining behavior of the opponent. For example, making an unreasonable demand signals strength but defeat in battle reveals weakness. I explain how opponents interpret information coming from a manipulable source (negotiating table) and reconcile it with one from a nonmanipulable source (battlefield). Any theory of war termination has to account for the process of convergence of expectations that ends fighting and, so, must explain how opponents interpret new information.

The substantive findings that emerge from this analysis have theoretical implications and empirical significance. In equilibrium, as in the historical record, total military victory is a rare occurrence. A common explanation of why wars end is that both sides agree as to who the eventual winner will be. This is not necessary: War can end when both sides agree on the relative likelihood of various outcomes. This is a much weaker requirement and explains cases where wars were settled before it became clear that one side would emerge victorious for sure.

Thus, belief in eventual military defeat is not necessary for war termination. This implies that belief in eventual victory is not necessary for war initiation. The standard puzzle about how both sides can be optimistic about their chances of military victory is therefore less relevant.

Why do weak states sometimes attack stronger ones even when it is clear that they have no chance of victory? One argument forces us to assume irrational expectations or resolve. This is not necessary. As long as the stronger state believes that its opponent is a little stronger than it actually is, the weaker state can benefit from fighting a short war and settling. Or as von Clausewitz ([1832] 1984, 92) put it, if the weaker states succeeds in giving the stronger one “doubts about the future,” it can hope to profitably exploit its fear of prolonged conflict.

RELATED WORK

Three models explore the interdependence of bargaining and fighting in different environments. Powell’s

(2001) formalization is based on the Rubinstein (1982) bargaining model with inside options (Muthoo 1999) and one-sided incomplete information. Only the uninformed player can make offers, and every time an offer is rejected, players can fight. Fighting may result in the collapse of either state, and the probability of collapse is exogenously specified. The equilibrium exhibits the “skimming property,” where the uniformed state screens out its opponent by making progressively larger offers.

Filson and Werner (2002) offer a richer battlefield environment where fighting can shift the relative military advantage of each state because every time a battle is fought, states expend resources of which they have limited amounts. The bargaining protocol is also onesided and only the uninformed state can make offers. They analyze a two-period special case by assuming severe resource constraints and find a logic analogous to the skimming property.

A. Smith and Stam (2001) embed the one-sided bargaining protocol in a random walk model of warfare based on work by A. Smith (1998). In each period where states disagree, a costly fight ensues and improves one state’s chances for military victory. With time, disagreement over the probability of eventual victory disappears as both sides update their beliefs using the information revealed by battlefield outcomes. Expectations converge on stalemate and war ends with a settlement. However, since players are nonrational in their model, it is difficult to relate their results to others.

My model is closely related to this formalization but makes several crucial modifications and extensions. First, the bargaining protocol is richer than any of the available models: Both sides can make unlimited numbers of offers, allowing for screening and signaling behavior. Second, players discount the future and suffer per-period costs instead of only paying fixed costs. This is not a trivial improvement because the results hinge upon how much players value the future. Third, players use all the information available, not only the battlefield. The uninformed state learns about its opponent by observing both its strategic behavior at the bargaining table and the nonmanipulable battlefield performance.

THE MODEL

Two players, i ∈ {1, 2}, bargain over a two-way partition of a flow of benefits with size π. An agreement is a pair (x, y), where x is player 1’s share, and y is player 2’s share. The set of possible pairs is X = {(x, y) ∈ R^2: x + y = π} and 0 ≤ x, y ≤ π. Players have strictly opposed preferences and each is concerned only with the share of benefits it obtains from the agreement. Because a share x identifies a distribution uniquely, let x be equivalent to the pair (x, π − x), and y be equivalent to the pair (π − y, y). The status quo distribution of benefits is (s1, s2) with s1 + s2 = π.

The two players bargain according to the alternating-offers protocol (Rubinstein 1982). Players have a common discount factor δ ∈ (0, 1), and act in discrete time with a potentially infinite horizon and periods indexed by t (t = 0, 1, 2, . . .). In even-numbered periods, player 1 proposes a division x ∈ X to player 2. If player 2 accepts that proposal, an agreement is reached, and the game ends with players receiving their shares in (x, π − x).3 If player 2 rejects the proposal, then players fight a costly engagement, which may improve the relative military position of a player, and the period ends. Player 2 makes a counteroffer y ∈ X in the next period. If player 1 accepts, the game ends and players receive their payoffs from the agreement (π − y, y); if player 1 rejects, they fight another military engagement. The game continues until an agreement is struck or until one of the players is decisively defeated. If a player decisively defeats the other, then it obtains the entire flow of benefits π. Each military engagement is costly, and states suffer a constant per-period loss of utility, reducing their instantaneous per-period wartime payoffs to bi < si (and so b1 + b2 < π).

War is modeled as a stochastic process of attrition. It is a homogeneous Markov chain with two absorbing states: victory and loss.4 The current military position of a player at time t captures the player’s relative overall success from all engagements that have occurred up to time t. Let N ≥ 2 denote the finite number of military objectives and let k be the number of objectives achieved by player 1. The set of possible states is K = {0, 1, . . . , N}. At time t, player 1’s current military position, kt ∈ K, is the difference between the total number of its victories and that of its losses in battles that have occurred in periods (0, 1, . . . , t − 1). The state variable kt is an indicator of relative military advantage at time t and summarizes the whole history of the war up to that point in time; k0 is the position at the outset of war.

One battle over one objective occurs in each period. Player 1 wins the fight with probability p and loses with probability 1 − p.6 If player 1 wins the battle at time t, then k_{t+1} = k_t + 1, and if it loses, then k_{t+1} = k_t − 1.

k0 = 0, player 1 is militarily defeated and the game ends with player 2 imposing the settlement (0, π). If kt = N, player 2 is militarily defeated and player 1 imposes the settlement (π, 0). Players maximize the time-averaged discounted sum of per-period payoffs, (1 − δ) ∑_{t=0}^∞ δ^t r_i^t, where r_i^t is player i’s instantaneous payoff in period t and equals b_i if players disagree, 0 if player i loses the war, π if it wins, and i’s share of benefits if players terminate the war with a settlement.

This model avoids some common pitfalls. Unlike the costly lottery approach, it does not reduce war to a single-shot event and permits analysis of dynamics. Unlike the infinitely repeated game approach, it does not go against the intuition that the process does not last indefinitely, or even a large number of periods (Rubinstein 1991, 918). Instead, this model captures the dynamic nature of the process without either fixing an arbitrary number of periods or allowing it to extend indefinitely, while incorporating the time dependence of each state. Finally, war is completely instrumental: It only serves as advancing players closer to victory or defeat, but players do not benefit from fighting itself; they only care about the political deal. This differs from A. Smith 1998, and the results reflect this.

COMPLETE INFORMATION

Let W^k_i: K -> R denote player i's expected payoff from fighting to the finish starting in state k. For 0 < k < N, this function is defined recursively as

W^1_0 = W^2_N = 0,

W^1_N = W^0_N = π,

W^k_i = (1 - δ) b_i + δ [ p W^{i+1}_k + (1 - p) W^{i-1}_k ].

The functions W^i are second-order linear recurrence relations and have closed forms. It can be shown that for all k, W^i_k ∈ [0, π] and W^i_k < π. That is, there exists no state where players lack incentives to bargain.

The set of Nash equilibria of the negotiation game is very large (Slantchev 2002). The set of equilibrium payoffs consists of all payoffs that are at least as good as fighting to the end from the starting state:

[ W^1_k0, π - W^2_k0 ].

  This is common to these types of models and assumes away possible enforcement problems. In particular, if a player accepts a share that leaves it worse off and military capability is derived from the share, then the other player is facing a commitment problem because it will be unable to credibly promise not to extract further concessions in the future. This can only make settlement more difficult today.

See Grimmett and Stirzaker 1992 for random processes.

Until a settlement is reached (or a military victory obtains), there is no explicit relationship between the distribution of (nonmilitary) benefits and the military position. One way to think about k₀ is in terms of the relative strategic circumstances, that is, how much effort would be necessary to expend to achieve military victory while holding the probability of winning individual battles constant. For example, the military position of Germany vis-à-vis Czechoslovakia in 1937 favored Germany less than the post-Munich one following the surrender of the Sudetenland along with its impressive fortifications. In terms of the model, k₀(1937) < k₀(1938), making military victory more likely in the latter case even if the probability of winning battles remained the same. In this case, Germany's benefits also increased.

It is possible to relax this assumption in two ways: The probabilities of victory and defeat need not sum to 1; the probabilities of victory and defeat vary according to the current military position. It is not clear a priori how probabilities should vary with battlefield success. For example, sometimes early failure mobilizes the will to fight (e.g., Britain after Dunkirk in May 1940) but other times it does not (e.g., France in June 1940).

This implies that players only pay costs while fighting continues and war does not permanently shrink the resource base. This can be justified empirically by the Phoenix factor: It does not take states that long to recover from war. Allowing for a longer finite cost-decay period will not alter the results. I conjecture that assuming permanent destruction of resources will make players more reluctant to prolong the learning process.

The formal description of strategies and payoff functions is cumbersome because the payoffs reflect the fact that bargaining may be ended by the exogenous stochastic process. The formalization, along with several computer programs (in C⁺⁺ and Gauss) for numerical computations are available from the author. Merlo and Wilson (1995) provide a general n-player infinite-horizon complete information model, in which the identity of the proposer and the size of the pie follow a general Markov process. Their results cannot be used here because in my case the Markov process eventually terminates the bargaining.

Because these equilibria rely on incredible threats to sustain optimal behavior, I require that the strategies be subgame perfect (Selten 1975). In addition, given the structure of warfare, it is natural to restrict attention to a class of strategies where behavior depends on the military position. Strategies that condition behavior on payoff-relevant history are called Markov. Since the only variable that influences future payoffs is the military position, Markov strategies depend only on kₜ, to determine optimal behavior at time t. A subgame perfect equilibrium in Markov strategies is called Markov perfect (MPE).⁹ A stationary MPE is one where players always make the same state- dependent offers and responses. That is, offers differ by state but are time-invariant. A no-delay MPE is one where players' optimal offers are immediately accepted.

Proposition 1. The stochastic bargaining game with complete information has a unique stationary no-delay Markov perfect equilibrium, in which player 1's first state-dependent offer is immediately accepted by player 2, and no fighting occurs.

For every possible military position, there exists a unique optimal offer a player can make that will be accepted by its opponent in equilibrium. This offer is calibrated to make the other player indifferent between accepting it and delaying for one period in order to have its optimal offer accepted then. To anticipate how this result will be used in the sections that follow, the terms of the offer also depend on p in the intuitive way: Player 1's optimal proposal is strictly increasing in p, while player 2's optimal proposal is strictly decreasing.

This result establishes an important result: Modeling warfare as a probabilistic process does not lead to inefficient behavior. It is worth contrasting this result with the one obtained by A. Smith (1998), where both players prefer fighting in some states. The difference stems from the assumption that players derive direct utility from their military position, which is not the case here. When war is instrumental, the stochastic element is not sufficient to produce inefficiency.

ASYMMETRIC INFORMATION

We now assume that player 1 is uncertain about the distribution of power. While player 2 knows the true value of p, player 1 is asymmetrically informed and believes that p may be low (pL), medium (pM), or high (pH), with 0 < pL < pM < pH < 1. That is, player 2 is weak, denoted 2w, when p = pH; moderately strong, denoted 2m, when p = pM; and very strong, denoted 2s, when p = pL. Let T = {w, m, s} denote the set of type indicators. Player 1 initially believes that player 2 is weak with probability qʷ > 0, strong with probability

 A pair of strategies forms MPE if, and only if, each player’s strategy maximizes its intertemporal payoff at any time t, given kₜ and assuming that henceforth each player conforms to its strategy (Fudenberg and Tirole 1991, ch. 13).

qˢ ∈ (0, 1 − qʷ), and moderately strong with probability qᵐ = 1 − qʷ − qˢ > 0. 

The set of Bayesian equilibria (Harsanyi 1968) is very large, as is the range of payoffs that can be supported in equilibrium. Like Nash equilibrium, this solution concept ignores the dynamic structure of the game. Bayesian equilibria may rely on noncredible threats to sustain optimal behavior because strategies are only required to be best responses at the beginning of the game, and players do not learn from history. An equilibrium refinement that overcomes these shortcomings is necessary. Learning in the stochastic negotiations model is complicated because there exist two sources of information: (i) the battlefield, which is nonmanipulable, and (ii) the strategic behavior of the opponent, which is highly manipulable. The uninformed player must take both into account when updating its beliefs about the distribution of power.

Involuntary Revelation of Information

Suppose that all types of player 2 reject some proposal. Following such rejection, the support of player 1’s beliefs will remain the same, that is, if it believed that it might be facing each type with positive probability, it will continue to do so. However, depending on the outcome of the battle, player 1 will update the probability associated with each type. Intuitively, winning a battle should make player 1 more optimistic about the chance of facing a weak opponent, while losing a battle should make player 1 more pessimistic.

To formalize this intuition, let Iₜ be an indicator that equals 1 if player 1 wins the battle at time t and 0 otherwise. Suppose that all types of player 2 reject the initial offer. Using Bayes’ Rule, the posterior belief is then, for τ ∈ T,

Pr(2τ | I₀) = ( Pr(2τ) Pr(I₀ | 2τ) ) / ( Σᵢ∈T Pr(2ᵢ) Pr(I₀ | 2ᵢ) ).

Since the number of victories, v, in n battles is a binomially distributed random variable given the probability of winning, we have

Pr(v, n | p) = ( n choose v ) pᵛ (1 - p)ⁿ⁻ᵛ,

and thus, the posterior q₁ˢ ≡ Pr(2ₛ | I₀) is

( qˢ pₗI₀ (1 - pₗ)¹⁻I₀ ) / ( qʷ pₕI₀ (1 - pₕ)¹⁻I₀ + qᵐ pₘI₀ (1 - pₘ)¹⁻I₀ + qˢ pₗI₀ (1 - pₗ)¹⁻I₀ )

Algebraic manipulation shows that extending the argument to v victories after n battles allows the posterior belief qₙˢ(v), where v = I₀ + I₁ + ... + Iₙ₋₁, to be expressed directly in terms of the initial belief and the Because player 1 updates its beliefs through a non-strategic mechanism that is beyond the control of its opponent, the possibilities for strategic dissimulation by player 2 are limited. Even if weak opponents try to imitate the strategic behavior of their stronger counterparts, they cannot do so for long because their poor performance on the battlefield will reveal information that will gradually convince player 1 that it is facing a weak opponent, not a strong one. On the other hand, the battlefield is a noisy source, and in the absence of strategic behavior, player 1 will never be able to eliminate the possibility that it is facing a particular type of opponent. That is, although the probabilities associated with some type can become arbitrarily close to zero, they never equal it; the support consists of all three types. Strategic bargaining, however, changes that.

 This stylization of incomplete information reflects closely the discussions in the informal literature that stress disagreements about the relative strength of opponents (Blainey 1988). Usually, this is taken to mean that countries disagree about the eventual military outcome of war (Kecskemeti 1958; Pillar 1983).

Properties of Sequential Equilibria

The usual solution concept for dynamic games of incomplete information is Kreps and Wilson's (1982) sequential equilibrium, which elevates beliefs to the level of strategies and specifies a method for updating these beliefs. This solution concept requires that strategies are sequentially rational (that is, they are consistent with the beliefs), and beliefs are derived from strategies and updated via Bayes' Rule whenever possible. The vagueness of the last requirement stems from existence of zero-probability events (events that should never occur in equilibrium), where the rule cannot be applied.

Sequential equilibrium assigns explicit but unrestricted by theory conjectures that players use to update their beliefs following zero probability events. These conjectures are crucial because the equilibrium outcomes are quite sensitive to their specification (Rubinstein 1985b). Following Grossman and Perry (1986a), who call their solution concept perfect sequential equilibrium, I postulate the following credible conjectures for player 1: If there exist some types of player 2, for which a deviation would be profitable, player 1 updates to believe that player 2 is among these types.

This requirement eliminates sequential equilibria in which the credibility of behavior is established through incredible beliefs. Also, as in Rubinstein (1985a), I require that if player 1 becomes convinced that its opponent is of some type with probability 1, then it never revises this belief. I shall therefore look for perfect sequential equilibria in Markov strategies (MPSE), where player 1 updates its beliefs credibly based on possible optimality of deviations of its opponent, and where beliefs and offers depend on the outcome of fighting during disagreement periods.

"Once all information is revealed, the game becomes equivalent to the one analyzed in the previous section, with starting state set at the military position at the time that no more private information remains. This means that we can use the solution to the complete information model for the subgames of the asymmetric information model that follow full revelation."

Recall that by Proposition 1, there exists a unique vector of state-dependent proposals that make players indifferent between accepting the proposal and delaying agreement by one period in order to make a proposal themselves. Let (V¹ₛ(k₀), V²ₛ(k₀)) denote the unique complete information MPE offers from Proposition 1 when player 1's opponent is of type 2ₛ. V¹ₛ(k) is player 1's complete information MPE payoff when player 1 is the proposer, the current state is k, and the opponent's type is 2ₛ. Equivalently, V²ₛ(k) is the payoff to player 2ₛ when it is the proposer in state k. Note that V¹(k) is strictly increasing in p, which implies that V¹ₛ(k) is strictly larger than the analogous optimal offers made if player 2's type is moderately strong or weak. No sequential equilibrium outcome can be better (worse) for player 1 than the MPE outcome in the complete information game with 2w(2ₛ).

From Proposition 1, the best that player 1 can obtain when its opponent is 2ₛ, in the complete information MPE is V¹ₛ(k). Since it does not pay to delay such agreement, it follows that if after some history player 1 believes with probability 1 that its opponent is 2ₛ, then it will always offer V¹ₛ(k), which player 2ₛ accepts. As the following lemma shows, this implies that all types of player 2 will accept such an offer.

Lemma 1. If 2ₛ accepts some offer x in a sequential equilibrium, then 2w and 2m accept that offer also.

Proof. Suppose that in equilibrium 2ₛ accepts x but one or both of the other types reject it. Following that rejection player 1 concludes that the probability of facing 2ₛ is 0 and will, therefore, offer player 2 at most π − V¹ₘ(k) < π − V¹ₛ(k) ≤ π − x. Both 2ₘ and 2w are better off by accepting x instead. 

Lemma 2. If in a sequential equilibrium player 1 offers some x and player 2 rejects it, then player 2 either makes a unique acceptable offer or counters with an unacceptable one.

Proof. Suppose that some type of player 2 counters with y₁ and another counters with y₂ ≠ y₁. If player 1 accepts both offers, then the type which made the higher offer can profitably deviate by offering the other. 

The set of sequential equilibria is large, and many of these equilibria can be supported with optimistic conjectures, in which player 1 threatens player 2 that if it deviates from the proposed strategy, 1 will conclude that its type is 2w. However, these equilibria are eliminated by the perfectness restriction on conjectures used in the construction of the unique MPSE in the next section.

Markov Perfect Sequential Equilibrium

To construct the MPSE, I first solve for the equilibrium in which player 1 has good information about its opponent and then solve for equilibrium in games where its information is slightly worse. Since information only improves over time, we shall use the results from the first step in the solution for the next one.¹¹ Since there are only three types of opponents that player 1 might be facing, and because it does not pay to delay once the type is known, I shall look for equilibria with the most rapid disclosure of information. To solve for this type of equilibrium, we induct backward on both beliefs and strategies, starting with the last period, where, if players adhere to their equilibrium strategies, player 1 knows with probability 1 that it is facing 2ₛ. This belief holds regardless of the intermediate losses and victories because in such equilibrium the weaker types have screened themselves out earlier. The construction of this equilibrium is quite involved and is relegated to the Appendix.

Proposition 2. When players are sufficiently patient, there exists a generically unique MPSE with the following structure. In period t=0, player 1 makes an initial offer that only 2w accepts; in period t=1, player 2m makes an offer which player 1 accepts, and player 2s makes a non-serious offer, which player 1 rejects; and in period t=2, player 1 makes an offer which all types of player 2 accept.

Finding the equilibrium involves searching for a strategy for player 1 that can profitably induce the three types of player 2 to separate in equilibrium, as well as a strategy for the strong type that induces it to signal its strength by making a nonserious offer. The separating equilibrium exists only for sufficiently high discount factors. When δ is lower, then semiseparating and pooling equilibria appear as well.

Conditioning behavior on the battlefield outcomes presents players with a complicated mix of incentives. Both players prefer to settle as early as possible, but player 1 does not want to offer more than strictly necessary to induce its opponent to accept. This requires that it makes some offers that might be rejected, resulting in inefficiency in the process.

Equilibrium play is strongly conditioned by the military position at the time offers are made, accepted, or rejected. It is preferable to win a battle prior to sitting at the negotiating table. Victories make player 1 more optimistic about its chances even though at t = 1 it knows for sure that it is facing either a moderately strong or a strong opponent. Consequently, if it wins the fight, it demands more than it does if it loses it.

Players can deal with contradictory information. By rejecting player 1’s initial offer, its opponent signals its strength, but if it then loses the battle, player 1 becomes more optimistic again. The value of the discount factor necessary to induce player 2ₛ to signal its strength in period 1 depends on player 1’s belief about the likelihood of facing the strong type, which in turn depends on the battlefield outcome. This belief is more optimistic if player 1 wins the battle and so the discount factor has to be smaller than the one required to sustain separation after defeat.

There exist discount factors such that 2ₛ pools with 2ₘ after player 1 is defeated but makes a nonserious offer after player 1 wins. A strong state may refuse to settle after a victory for its opponent while still preferring to settle after its defeat. The existence of these equilibria is a good reason for the often observed behavior of attempting to gain a military advantage immediately prior to making a proposal and of strong states refusing to settle if the opponent has gained a miliary advantage.

The level of uncertainty about the type of opponent can be expressed as a combination of each type's probability of winning and beliefs about the three types. For simplicity, define the level of uncertainty as the variance of the probabilities of winning. When the three types are more or less equally likely to prevail in battle, then player 1 is quite certain about the strength of its opponent. Conversely, when the three probabilities are very different, player 1 can be said to be quite uncertain about the type of opponent it is facing.

With little uncertainty, the discount factor required to sustain separation increases quite dramatically. This implies that we are more likely to see partially or fully pooling equilibria, where players agree either immediately or, at most, with one period delay, shortening the expected duration of war. Conversely, when there is a lot of uncertainty, separation becomes easy, and wars should be longer. This result follows from the requirement that stronger types must have incentives to signal, and that player 1 must have incentives to screen. When the types are comparatively equal in strength, the necessary gains are negligible, and so the incentive to delay disappears. More uncertainty about the distribution of power results in longer duration of conflict.

DISCUSSION

Warfare as Information Transmission

Stating that war is the pursuit of political objectives, although better than regarding it as the “untrammelled manifestation of violence,” is still imprecise. In its most common form, this approach states that war is a way to secure political objectives by force. As Hobbs (1979, 46) notes, “The war aim of strategy is to clinch a political argument by force instead of words.” This is further elaborated by Wagner (1994, 603), who claims that “the primary function of force in bargaining is to improve one’s bargaining position by increasing the costs of disagreement for one’s adversary.”

The analysis here, as in the three related models, departs from this approach. The ability to increase the costs of disagreement may entail improvement of one’s position but does not explain why fighting occurs. With complete information, states still settle immediately. On the other hand, the presence of asymmetric information does not simply generate risk of war, as in Fearon 1995. In this model war does not arise as a result of breakdown of bargaining. Instead, war occurs when players are sufficiently patient and prefer to engage in strategic screening and signaling, that is, in transmission of information. War is bargaining, and bargaining is transmission of information.

Warfare only has informational content while uncertainty about the opponent persists. Fighting becomes irrelevant (in the informational sense) once players learn enough about their opponents.¹² The uncertainty reduction interpretation has become quite prominent in recent formal work, whose common theme is that fighting provides information about the relative strength of opponents and allows them to arrive at more congruent estimates of their chances of success, thus enabling them to conclude bargains (Wagner 2000). As Reiter (2003) points out, this contradicts the earlier interpretation which treated fighting as detrimental to bargaining because of the expectation that the winning side inevitably expands its demands (Wittman 1979).

R. Powell's model cannot address this because it is impossible for the uninformed player to become optimistic with time, which is a consequence of the static distribution of power (probability of collapse is exogenous and fixed). My model, along with the other two, shows that these trends are not mutually exclusive. The current military situation influences the proposals and responses. It is preferable to have won the last round of fighting prior to concluding an agreement.¹³ However, reduction of uncertainty has enormous implications for equilibrium behavior. The model demonstrates quite clearly that both effects are at work at the same time. Although the eventual bargain does depend quite significantly on the current military position, the important information transmission happens through the strategic behavior of opponents which provides more precision than the crude fighting mechanism. Combat itself may reveal less than the willingness of opponents to engage in it.

"This is a strong qualification of the existing arguments because it shifts the emphasis of war termination back to the political realm. That negotiating behavior has such dramatic implications is recognized by many diplomats and practitioners, for example, Nicolson (1954) and Ikle (1964), but has been neglected in the study of war termination.14 Focusing squarely on military developments as the most important source of information, as Wagner (2000) does, may not be helpful because of the 'fog of war' that makes for wildly divergent estimates of battlefield performance (Gartner 1997; Ikle 1971). Diplomatic exchange remains an important tool to influence expectations of the opponent, which is probably one reason governments are reluctant to engage in it while the war continues.

Unfortunately, sometimes force is necessary to convey sufficient information to induce a revision of beliefs. This is not because force itself convinces, but because" the willingness to use it, and suffer its costs, distinguishes between different types of opponents. It is not clear that there is a way to avoid this because weak players always have incentives to dissemble as being strong and only a costly delay may persuade their opponents otherwise.

The Impact of Uncertainty

It is relatively straightforward conceptually (but a bit demanding technically) to extend the analysis to any finite number of types. This will result in additional delay in the separating equilibrium because it will take more fighting to distinguish among them.

Except for A. Smith and Stam (2001), who ignore strategic behavior altogether, all other models exhibit the screening property. However, since neither R. Powell nor Filson and Werner allow the informed player to make counteroffers, it is impossible to analyze signaling behavior.¹⁵ This leads to questions about this player's willingness to delay agreement if a richer communication tool is at its disposal. However, as the results show, the ability to send more complicated messages only sometimes translates into ability to terminate the war. Although a moderately strong state will make an acceptable counteroffer, a strong one will still make an unacceptable one to demonstrate its strength.

Uncertainty benefits the weak and hurts the strong. The asymmetry of information is always detrimental to the uninformed party, who tries to screen out its opponent when under complete information it would settle immediately. A strong opponent also does worse because it has to engage in costly delays to signal its strength and separate itself from weak types who have incentives to claim they are strong but who cannot afford a prolonged fight. For both, the ex ante payoff under uncertainty is lower than the payoff under complete information.

Weaker opponents, on the other hand, do better, sometimes significantly so, than they would have done under complete information. This is because the offers player 1 must make are conditioned on its estimate that it might have to make more concessions should player 2 turn out to be strong. Because delay is costly, player 1 will not engage in it forever and, so, will make concessions that balance the need to minimize the current offer and the possibility of having it rejected for a lower settlement. Invariably, this exceeds what it would have offered had it known for sure that its opponent was weak.