COERCIVE EFFECTS OF MILITARIZED CRISIS BEHAVIOR

Perhaps the main problem that leaders face in a crisis is credibility: how does a leader persuade an opponent that his threat to use force is genuine? That he would follow up on it should the opponent fail to comply with his demands? The decision to carry out the threat depends on many factors, some or all of which may be unobservable by the opponent. The leader has to communicate enough information to convince her that he is serious. If the opponent believes the message and wants to avoid war, she would be forced to make concessions. However, if there exists a statement that would accomplish this, then all leaders—resolved and unresolved alike—would make it, and hence the opponent would have no reason to believe it. The problem then is to find a statement that only resolved leaders would be willing to make.

Jervis (1970) studies signals, which do not change the distribution of power, and indices, which are either impossible for the actor to manipulate (and so are inherently credible) or are too costly for an actor to be willing to manipulate. In modern terms, he distinguishes between “cheap talk” and “costly signaling,” even though he prefers to emphasize psychological factors that influence credibility.

It is well known that the possibilities for credible revelation of information when talk is cheap are rather limited and depend crucially on the degree of antagonism between the actors (Crawford and Sobel 1982; Morrow 1994b).³

One can think of such tactics in terms of expected benefits from war and expected costs of avoiding it: anything that increases one relative to the other could commit an actor by tying his hands at the final stage. Fearon (1994) noted that domestic political audiences can generate costs for leaders who escalate a crisis and then capitulate, creating an environment in which a leader could tie his hands and, thus, signal resolve to foreign adversaries. Even though leaders pay the costs only if they back down, their willingness to risk escalation to a point where each of them would be irrevocably committed to not backing down can reveal their resolve.

This contrasts with another signaling mechanism that relies on sinking costs; that is, incurring expenses that do not directly affect the expected payoffs from war and capitulation (Spence 1973). Only actors who value the issue sufficiently would be willing to pay these costs, turning them into a credible revelation of resolve by separating from low-resolve actors through their action. When the last clear chance to avoid war comes, these costs are sunk and cannot affect the decision to attack, hence they cannot work as a commitment device and their function is purely informational.

What is the role of military actions, such as mobilization, in a crisis? Fearon (1994, 579) notes that the “informal literature on international conflict and the causes of war takes it as unproblematic that actions such as mobilization ‘demonstrate resolve’,” and argues that “if mobilization is to convey information and allow learning, it must carry with it some cost or disincentive that affects low-resolve more than high-resolve states.” He then goes on to dismiss the financial costs of mobilization as being insufficient to generate enough disincentive to engage in it and concludes that we should focus on an alternative mechanism—domestic political costs—that has a hands-tying effect.

Although one may quibble with the notion that mobilization is not costly enough, the more important omission is that the argument treats mobilization (and similar militarized crisis activities) as costly actions that are unrelated to the actual use of force. However, one can hardly wage war without preparing for it, and the primary role of mobilization is not to incur costs but rather to prepare for fighting by increasing the chances of victory. But improving one's prospects in fighting increases the value of war relative to peace and can therefore have a hands-tying effect. In fact, it is difficult to conceive of pure sunk costs in this context. Perhaps military exercises away from the potential war zone could qualify as such, but almost anything countries can do in terms of improving defenses or enhancing offensive capability affects the expected payoff from fighting quite apart from the costs incurred in doing it. Even though he does not analyze it, Fearon (1997, n. 27) does recognize this and notes that “insofar as sunk-cost signals are most naturally interpreted as money spent building arms, mobilizing troops, and/or stationing them abroad … the probability of winning a conflict … should increase with the size of the signal.”

Underestimating mobilization's role as a commitment device beyond its immediate costliness leads one influential study to conclude that “the financial costs of mobilization rarely seem the principal concern of leaders in a crisis” (Fearon 1994, 580), implying that these costs are insufficient to generate credible revelation of resolve. As I will show, this is true only if mobilization functions solely as a sunk cost; if we consider its hands-tying function, mobilization does acquire crisis bargaining significance.⁴

Empirically then, it seems that military actions which states take during a crisis—mobilizing troops, dispatching forces—entail costs that are paid regardless of the outcome, and in this sense are sunk; however, they also improve one's expected value of war relative to peace, and in this sense they can tie one's hands. Militarized coercion involves actions with these characteristics, but existing theories of interstate crisis bargaining have not analyzed their consequences properly.

In the formal literature, the issue has been almost completely side-stepped in favor of models that incorporate only one of the two functions: the probability of winning is exogenously fixed instead of being determined endogenously by the decisions of the actors.⁵

In other words, nearly all existing models cannot seriously investigate the impact of military moves in crisis situations because they ignore the hands-tying effect they may have. This is an important shortcoming because, in these models, the probability of winning determines the expected payoff from war, which in turn determines the credibility of threats and, hence, the actor's ability to obtain better bargains. As Banks (1990) demonstrates, the higher the informed actor's expected payoff from war, the higher his payoff from setting the dispute peacefully, and the higher the probability of war in equilibrium. All crisis bargaining models that treat the probability of winning as exogenous would produce this dynamic. However, as I argued, this crucial variable that essentially generates optimal behavior in crisis bargaining models should be part of the process that depends on it. If deliberate actions influence its value, which in turn affects the informational content of these actions, how are we to interpret mobilization decisions? To what extent are costly military actions useful in communicating in crisis: do they make crises more or less stable? What levels of military mobilizations should we expect and what is the price of peace in terms of maintenance of military establishment by defenders?

To answer such questions, the model must have the following features: (a) both actors should be able to choose the level of military mobilization as means of tacit communication, (b) an actor's mobilization should be costly but should increase its probability of winning if war breaks out, (c) mobilization may not necessarily increase the expected utility from war (even though it makes victory more likely, a positive impact, its cost enters negatively), (d) at least one of the actors should be uncertain about the valuation of the other, and (e) actors should be able to make their deliberate attack decisions in light of the information provided by the mobilization levels. Consequently, the model I construct in the next section incorporates all of these.

THE MODEL

Two players, S₁ and S₂, face a potential dispute over territory valued at v₁ ∈ (0, 1) by the status quo power S₁, who is currently in possession of it. Although this valuation is common knowledge, the potential revisionist S₂'s valuation is private information.⁷

Initially, S₁ decides on his military allocation level, m₁ [ges ] 0. Choosing m₁ = 0 is equivalent to relinquishing the claim to the territory and ending the game with payoffs (0, v₂). Otherwise, the amount m₁ > 0 is invested in possible defense. The costs of mobilization are sunk and incurred immediately. After observing his choice, S₂ either decides to live with the status quo or makes a demand for the territory by starting a crisis. S₂ can escalate by choosing a level of mobilization, m₂ > 0, or can opt for the status quo with m₂ = 0, ending the game with the payoffs (v₁ − m₁, 0). After observing S₂'s level of mobilization, S₁ can capitulate, ending the game with payoffs (−m₁, v₂ − m₂); preemptively attack, ending the game with war; or resist, relinquishing the final choice to S₂. If he resists, S₂ decides whether to capitulate, ending the game with payoffs (v₁ − m₁, − m₂), or attack, ending the game with war.

If war occurs, each player suffers the cost of fighting, c_i ∈ (0, 1). Victory in war is determined by the amount of resources mobilized by the players and the military technology. Defeat means the opponent obtains the territory. The probability that player i prevails is λm_i/(λm_i + m_−i), where λ > 0 measures the offense–defense balance.⁸

, and, if i is attacked, it is

. It is easy to show that

. If defense dominates, then the expected value of war is higher when one is attacked than when one attacks first.⁹

The solution concept is perfect Bayesian equilibrium (or simply “equilibrium”), which requires that strategies are sequentially rational given the beliefs, and that beliefs are consistent with the strategies, and derived from Bayes rule whenever possible (Fudenberg and Tirole 1991). The model incorporates the empirically motivated features I identified in the preceding section. It is complicated by the continuum of types and actions, and so it trades an ultimatum “bargaining” protocol for rich mobilization possibilities in letting both actors choose the level of forceful persuasion.

THE MOBILIZATION OF THE REVISIONIST STATE

It will be helpful to analyze the signaling game beginning with S₂'s allocation decision given some allocation m₁ > 0. In any equilibrium, the strategies would have to form an equilibrium in this continuation game, and since S₁ is uninformed, his initial decision reduces to choosing (through his allocation) the equilibrium that yields the highest expected payoff.

By subgame perfection, S₂ would attack at her final decision node if, and only if, her expected payoff from war is at least as good as capitulating:

. That is, v₂ [ges ] c₂ + c₂m₁/(λm₂) ≡ γ(m₁, m₂) > 0, where γ(m₁, m₂) is the highest type that would capitulate if resisted at the allocation level (m₁, m₂). All types v₂ < γ(m₁, m₂) capitulate, and all types v₂ [ges ] γ(m₁, m₂) attack when resisted. Note that γ(m₁, m₂) > 0 implies that the lowest-valuation types never attack even if they are certain to win. For any posterior belief characterized by the distribution function G(γ(m₁, m₂)) that S₁ may hold, resisting at the allocation (m₁, m₂) yields S₁ the following expected payoff:

. If S₁ attacks preemptively, he would get

. Since

, it follows that

regardless of S₁'s posterior belief. Therefore, if defense dominates, then in equilibrium S₁ never preempts; he either capitulates or resists.

Suppose that S₁ capitulates for sure if he observes an allocation

. There can be at most one such assured compellence level in equilibrium. To see that, suppose that there were more than one. But then all S₂ types who allocate the higher level can profit by switching to the lower one. Obviously,

is an upper bound on any equilibrium allocation by S₂. Furthermore S₂ would never mobilize m₂ [ges ] 1 in any equilibrium. This is because the best possible payoff she can ever hope to obtain is v₂ − m₂ if S₁ capitulates, and this is nonpositive for any m₂ [ges ] 1, for all v₂ [les ] 1.

All of this suggests that S₂'s equilibrium behavior would be determined by the relationship among the payoffs she can obtain from optimal offensive war, assured compellence, and capitulation. That is, S₂'s strategy can be characterized by a series of cut-points that divide her types into subsets who behave the same way. To this end, I now derive these cut-points and then show that only two configurations can occur in equilibrium.

Let β(m₁) denote the type that is indifferent between optimal war and assured compellence; that is,

, where

is the optimal allocation by type v₂ if she expects to fight for sure some m₁. That is,

maximizes

subject to the constraint that

. Substituting and solving for β(m₁) yields β(m₁)= \smash

. The following lemma establishes the S₂'s preference between optimal war and assured compellence (all proofs are in the Appendix).

Assured Compellence

In an assured compellence equilibrium, all types of S₂ that mobilize do so at a level just enough to make S₁ capitulate with certainty. Intuitively, if S₁ has mobilized at a low level, it is relatively easy for S₂ to countermobilize such that S₁'s payoff from war becomes sufficiently low. This undermines S₁'s incentive to resist even if there still exists a chance that S₂ is bluffing. Despite S₁'s certain capitulation, not all low-valuation types will be willing to bluff because of the inherent costliness of mobilization. Hence, we shall look for an equilibrium in which all low-valuation types capitulate, and the rest allocate the assured compellence level. S₁ resists all allocations smaller than this level (because only low-valuation types that would capitulate if resisted would fail to allocate the higher level) and capitulates otherwise.

Suppose α [les ] δ and α < 1. By Lemma 2, S₂'s optimal strategy must take the following form: all v₂ < α capitulate immediately, and all v₂ [ges ] α mobilize at the compellence level

. By definition,

, and therefore

. If

, then the assured compellence level is feasible because there exists a type of S₂ that could choose to allocate

optimally, and so S₁ is potentially compellable. Otherwise, he is uncompellable.

Subgame perfection implies that, if

, all types

capitulate if resisted (bluffers) and all

fight if resisted (genuine challengers). If

, only genuine challengers mobilize in equilibrium. Given S₁'s prior belief F(.), his posterior belief that S₂ would capitulate when resisted conditional on

is

if

), and 0 otherwise. S₁ capitulates whenever R₁(m₁, m₂) [les ] −m₁. Because R₁ is strictly decreasing in m₂ and because excess mobilization by S₂ is pointless if S₁ is sure to capitulate, it follows that in equilibrium

must solve

, or

Let

be the unique solution to equation (1).¹¹

Risk of War

When S₁'s mobilization level increases, S₂'s countermobilization required to achieve assured compellence increases as well. As ensuring that outcome gets costlier, risking optimal war becomes more attractive. In particular, if the type who is indifferent between war and capitulation has a lower valuation than the type who is indifferent between assured compellence and capitulation, all intermediate-valuation types would rather fight than ensure the exceedingly costly capitulation by S₁ or give up themselves. Increasing m₁ even further eliminates all possibility that some type would be willing to attempt compellence, reducing S₂'s choice to capitulation or optimal war.

Turning to the formal statement of this result, suppose δ < α. By Lemma 3, only one possible configuration exists: δ < α < β. Since all v₂ > δ prefer optimal war to capitulation, all challenges in this equilibrium are genuine, and G = 0 simplifies equation (1) yielding an analytic solution to the compellence level

. This is also the solution to equation (1) for the assured compellence equilibrium when

. Substituting for

yields β = (1/4λm₁)(λc₂ − m₁v₁/c₁)².

Assured Deterrence

Finally, if S₁ mobilizes at a very high level, then he can become uncompellable and no types would be willing to challenge him given that he is certain to resist. In other words, S₁ can achieve assured deterrence. This can happen when there is no type that is willing to fight even an optimal war, and when the assured compellence level is not feasible. The following proposition states the necessary and sufficient conditions for this equilibrium.

THE DEFENSE OF THE STATUS QUO STATE

Collectively, the three mutually exclusive equilibria exhaust all possible configurations of the cut-points and, therefore, provide the solution for the continuation game for any set of the exogenous parameters and any m₁ > 0. I now turn to S₁'s initial mobilization decision. Since S₁ is the uninformed actor, his choice boils down to selecting which type of equilibrium will occur in the continuation game. It is not possible to derive an analytic solution to this problem because of the nonlinearities involved in the optimization at the second stage. Still, because we can generally establish the order in which the continuation game equilibria occur as a function of m₁, we can say what type of choices S₁ will face if he increases his mobilization level. With the help of computer simulations, we can derive precise predictions for interesting ranges of the exogenous variables too.

The compellence equilibrium always exists regardless of the values of the exogenous parameters because, for m₁ small enough, the necessary and sufficient condition form Proposition 1 are satisfied. What happens once m₁ begins to increase? As the derivations in the previous section suggest, two cases are possible. First, as m₁ increases, the conditions for deterrence can be satisfied, and the continuation game has only two possible solutions, both involving peace. Second, as m₁ increases, the existence conditions can successively satisfy the risk of war and deterrence equilibria.

To see how S₁ would choose his initial mobilization, if any, we must consider his expected payoffs in each of the possible continuation game equilibria. To conduct comparative statics simulations and analyses, I impose the additional assumption that F is the uniform distribution. This also allows me to reduce the expected payoffs for S₁ to manageable expressions.

In the compellence equilibrium, S₁ obtains the prize with probability Pr(v₂ [les ] α) = α by the distributional assumption and concedes it without fighting with complementary probability. His expected payoff is EU

. In the risk of war equilibrium, S₁ obtains the prize with probability Pr(v₂ [les ] δ) = δ, fights a war with probability Pr(δ < v₂ [les ] β) = β − δ, and concedes the prize with probability Pr(v₂ > β) = 1 − Pr(v₂ [les ] β) = 1 − β. His expected payoff is

where we used

. Finally, in the deterrence equilibrium, S₁'s payoff is:

. In equilibrium there can be only one assured deterrence allocation level by S₁ because, if there were two, then S₁ could profitably deviate to the lower one.

I now provide two numerical examples that will facilitate the substantive discussion. Assume the uniform distribution for S₂'s valuations, and set the parameters v₁ = 0.6, c₁ = 0.2, and λ = 0.99. In the simulation in Figure 1(a), S₂'s costs of fighting are high, c₂ = 0.35, and in the simulation in Figure 1(b), her costs of fighting are low, c₂ = 0.01. The solid line shows the range of values for m₁ for which the various equilibria exist. The dotted vertical line shows S₁'s valuation for reference, and the solid vertical line shows S₁'s equilibrium mobilization level.

Examples of Equilibria as Function of S1's Mobilization

In the first example, the equilibrium outcome is peace: one of the actors will capitulate. S₁ mobilizes

and takes his chances that S₂ may be a high-valuation type that would compel him to capitulate. The assured compellence level is

. The probability that S₁'s low mobilization level would be able to deter S₂ is Pr(v₂ < α) = 33%, and so the risk of having to concede is 67%. All types v₂ < α quit and S₁ gets to keep the territory. On the other hand, all types v₂ [ges ] α allocate

, after which S₁ relinquishes the territory without a fight.

In the second example, the outcome can be either capitulation by one of the actors or war. S₁'s optimal mobilization increases to

. What follows depends on just how high the challenger's valuation is. If it is v₂ < δ = 0.36, then S₂ would be deterred from mobilizing, and the outcome would be peace. If it is v₂ [ges ] β = 0.55, then S₂ would mobilize at the assured compellence level

, S₁ would capitulate, and the outcome would be peace again. However, if v₂ ∈ [0.36, 0.55), then S₂ would allocate her optimal fighting level

, and the outcome would be war. The ex ante probability of war is 19%, but conditional on S₂'s mobilization it is 30%, with war being certain if S₂'s mobilization level is less than

.

S₁'s expected payoff in this equilibrium is 0.02, which is much less than the 0.13 he would have expected in the previous example. This is not surprising, because as S₂'s costs of fighting decrease, so does S₁'s equilibrium payoff: to wit, his opponent is able to extract a better deal because going to war is not as painful, and so the threat to do it is much more credible.

These dynamics clearly demonstrate that establishing a credible commitment by tying one's hands can avoid war only if it also makes fighting sufficiently unpleasant to the opponent. A credible threat to fight cannot buy peace by itself, and a perfect commitment can virtually guarantee war if the opponent's valuation is misjudged. It is worth noting that crises that are peacefully resolved may involve higher military allocations than those that end in war: either S₁ mobilizes a large enough force to deter S₂, or S₂ mobilizes a large enough force to compel S₁. These allocations are higher than the optimal war allocations that either state would make if they expect to fight for sure. In other words, arms buildups are not necessarily destabilizing in a crisis. In fact, they appear positively related to peace when it comes to threatening the use of force.

Bluffing with Implicit Threats

The most obvious difference that is of great substantive interest is that actions involving each mechanism separately result in equilibria where bluffing is not possible.¹²

In Fearon's hands-tying model, bluffing cannot occur because actors with high valuations can generate arbitrarily high costs for backing down, which they never have to pay because their opponent would submit. Maximizing the payoff of high-valuation types reduces to maximizing the probability of capitulation by the opponent. This does not work in a model where hands-tying is inherently costly because now maximizing the probability of capitulation by the opponent must be balanced against its costs, which may put a cap on worthwhile mobilization levels, and that in turn can induce lower-valuation types to bluff because it makes it affordable. In addition to its costliness and impact on one's own war payoff, an actor's mobilization also affects the expected war payoff of its opponent. This separates mobilization from the audience-cost models where one's actions have no direct bearing on the opponent's payoffs. In other words, the actors' ability to generate high signals is constrained both by the costliness of the military instrument and by the actions of their opponent.

Take, for example, the assured compellence equilibrium in Figure 1(a). There are bluffers here: all v₂ ∈ [α, γ) ≡ [.33, .42) would not attack should S₁ decide to resist. The ex ante probability of a bluffer is Pr(α [les ] v₂ < λ) = 9%, which increases to 13% after S₂ mobilizes. However, even though S₁ is now far more likely to be facing a bluffer, he is also far more likely to be facing a genuine challenger (87% versus an initial 58%), and so he chooses not to resist. The small mobilization has successfully screened out low-valuation types and S₁ is unwilling to run a risk of war at this stage given how much S₂'s mobilization has reduced his payoff from war. Note that S₁ could have eliminated all bluffers if he wished to do so by allocating approximately m₁ = 0.28 (this is where γ = α), but doing so is not optimal because of the costs involved. Hence, not only is bluffing possible in equilibrium but S₁ would not necessarily attempt to weed out such challengers. Further, S₂'s countermobilization has essentially untied S₁'s hands by lowering his expected payoff from war to the point where capitulation is preferable.

On the other hand, bluffing is impossible in equilibria that involve genuine risk of war. Consider Figure 1(b): there can be no bluffing here, for a bluffer would have to mobilize at the assured compellence level—otherwise she would be forced to back down when S₁ resists and suffer the costs of mobilization—and this level is too high given S₁'s initial mobilization.

Hence, bluffing is possible only in equilibria that do not involve much revelation of information and involve no danger of war. This corresponds to results of Brito and Intriligator (1985), who also find that in the pooling (no signaling) equilibrium bluffing is possible but the probability of war is zero. Preventing bluffing involves precommitment to a positive probability of war, and the willingness to run this risk does transmit information.

The model reveals a subtle distinction in the conditions that permit bluffing. Bluffing is only optimal when S₁ is expected to capitulate, but his willingness to do so depends on how likely S₂ is to fight, which in turn depends on S₂'s costs of fighting and S₁'s mobilization level. Paradoxically, bluffing by S₂ is possible only when her costs of fighting are relatively high (she is “weak”). The reason is the effect this has on S₁'s decision: because S₂ is weak, and therefore not very likely to be willing to mobilize at a high level, S₁ reduces his own costly allocation and thereby exposes himself to the possibility of having to concede. It is this low mobilization that makes bluffing an option: one must choose to expose oneself to bluffing. It is always possible to eliminate that possibility by making it too dangerous a tactic. When S₂ has relatively low costs of fighting (she is “strong”), S₁ knows that low mobilization would virtually ensure his capitulation, and so he ups the ante, eliminating bluffing possibilities in the process. Essentially, bluffing becomes too expensive even if it is certain to succeed. For this result to obtain, mobilization must both be inherently costly and increase probability of victory.

Fearon buttresses his no-bluffing results by quoting an observation by Brodie (1959, 272), who states that “bluffing, in the sense of deliberately trying to sound more determined or bellicose than one actually felt, was by no means as common a phenomenon in diplomacy … it tended to be confined to the more implicit kinds of threat.” I have emphasized the distinction between verbal threats and implicit threats because it is very important. Reputational concerns may eliminate the incentives to bluff with words (Guisinger and Smith 2002; Sartori 2002) but may not work for implicit threats like the ones in this model. As Iklé (1964, 64) observes, “whether or not the threat is a bluff can be decided only after it has been challenged by the opponent's noncompliance.” But probing an implicit threat is too dangerous because by its very nature, and unlike words, it influences the expected outcome of war. In equilibrium, these types of bluffs are never called, and hence S₂ is never revealed as having made an incredible threat.¹³

Military coercion is a blunt instrument because its intent is not to reveal the precise valuation of the informed party but rather to communicate one's willingness to fight. Although much nuance is possible if actors had in mind the former goal, the latter is, of necessity, rather coarse. That one must resort to tacit bargaining through implicit threats cannot improve matters. Historians have emphasized the difficulty in clarifying “the distinction between warning and intent” (Strachan 2003, 18). Perhaps it is precisely because mobilization has such a crude signaling role, which is hard to disentangle from preparation for war, that mobilization has traditionally been considered very dangerous.

Endogenous Distribution of Power

Military coercion has a somewhat peculiar dynamic completely lost to models that ignore the war-fighting implications of military measures. For example, it is now generally accepted that the stronger the actor, the more willing he is to risk war to obtain a better bargain. The risk–return trade-off then resolves itself in higher equilibrium probability of war and a better expected negotiated deal (Banks 1990; Powell 1999).

Generally, a strong actor is one with a large expected war payoff. Valuation of the issue (high), costs of fighting (low), probability of winning (high), and military capabilities (large) can be lumped together to produce an aggregate expected payoff from fighting (high), which in turn defines the actor's type (strong). Potential opponents can then be indexed by their war payoffs, which are taken to be exogenous to the model. Bargaining essentially involves attempts to discern just how much concessions the opponent is prepared to make, and that in turn depends on how much he expects to obtain by fighting. When the distribution of power is fixed, the only way weak types can be discouraged from mimicking the behavior of strong types and demanding too much is for strong types to run a higher risk of war. Mobilization endogenizes the payoff from fighting, and its costliness provides another way to discourage weak types without necessarily running a higher risk of war.¹⁴

This now means that we need to pay closer attention to the effects of short-term mobilizations because they help determine, at least in part, the expected payoff from war endogenously.¹⁵

The model shows that the expected payoff from the crisis does increase in the actors' valuation of the issue, but not necessarily at the cost of a higher risk of war. In other words, the risk–return trade-off does not necessarily operate in this context, where the relevant trade-off is between signaling cost and expected return. To see that, consider the risk of war equilibrium. All low-valuation types capitulate immediately and so face zero probability of fighting. All midvaluation types mobilize their optimal fighting allocations, and the probability of war jumps to one. On the other hand, high-valuation types manage to scrape together the assured compellence level, which resolves the crisis with S₁'s capitulation, and the probability of war drops back to zero. In other words, although these types do spend more during the crisis, they obtain the surrender of their opponent without risking war. The possibility to compel S₁ arises out of the latter's initial decision: he could have mobilized enough resources to make himself uncompellable by even the highest valuation type but, because of uncertainty, it is not optimal to do so. This is not to say that technology, war costs, and capabilities are not important—indeed, the two examples show the impact of S₂'s war costs—but rather that the commonly accepted crisis dynamics based on incentive-compatibility arguments dependent on a fixed distribution of power may not hold when that distribution of power is endogenous.

Furthermore, S₁'s optimal mobilization is not monotonically related to either his fighting costs or those of his opponent. For example, recall that, when

in the risk of war equilibrium in Figure 1(b). Increasing S₂'s costs to c₂ = 0.25 produces

in an assured compellence equilibrium with no bluffers (figure not shown). Increasing them further to c₂ = 0.35 produces

in the compellence equilibrium with bluffers in Figure 1(a). Note the distinction between the last two outcomes. When S₂'s costs are intermediate, S₁ eliminates all bluffers and practically ensures that he would obtain S₂'s capitulation (the probability of him having to concede instead is less than 1%). When S₂'s costs increase further, S₁ responds by drastically slashing his own military spending, even exposing himself to bluffing by doing so. Although he is now quite likely to concede (67%), his loss in this case is not too drastic because of the savings from the low allocation. In the previous case, on the other hand, even though he was nearly certain to win, the cost of doing so was quite high, making this tactic no longer profitable. In expectation, S₁'s payoff does increse in c₂, and he obtains 0.13 in the latter case as opposed to 0.11 for the intermediate costs case. Perhaps counterintuitively, the status quo power is more likely to concede when his opponent is weaker (has higher costs of fighting) but equilibrium mobilization levels will be lower.

The Price of Peace

Figure 2 illustrates the impact of varying S₁'s costs. It shows the ex ante probability of war, S₁'s optimal allocation, and his payoff in equilibrium for various values of c₁. The parameters are set to v₁ = .999 (so that high costs do not become immediately prohibitive), λ = .99, and c₂ = 0.10.

Probability of War and Optimal Allocations by S1

The nonmonotonicity is again evident. Because of his extremely high valuation, S₁ cannot be compelled if his costs are relatively low. It is only at intermediate costs (c₁ > 0.30) that compellence becomes feasible again. However, S₂ will not attempt it in equilibrium, and hence, up to c₁ ≈ 0.35, war is certain if S₂ mobilizes. The ex ante probability of war declines across this range but

increases. That is, seemingly aggressive mobilization behavior can be seen as S₁ compensating for the relative weakness in war occasioned by somewhat high costs: because war is more painful, he is prepared to pay more to decrease the chances of having to fight it. Nothing, of course, can help S₁ overall in the sense that the costlier the fighting, the less must he accept in expectation.

Continuing the increase of c₁ makes assured compellence not just feasible but also desirable, and from c₁ ≈ 0.35 no equilibrium outcome will involve war because S₁'s high costs make fighting quite unattractive for him. Peace can be had in two ways: either S₁ can deter his opponent, or S₂ can compel her opponent. S₁'s behavior in the intermediate cost range is rather intriguing. While he can afford it, his strategy is to deter S₂ or, failing that, to ensure that the probability of a challenge (to which he will surely concede) is relatively low. Note that, until c₁ ≈ 0.45, the outcome is either assured deterrence or assured compellence but with extremely high mobilization levels by S₁. Even after it becomes impossible to deter all types of S₂, the status quo power persists in very high allocations that minimize the probability of having to concede in the compellence equilibrium (less than 0.1%). This is where peace can be very expensive.

Finally c₁ becomes prohibitively high, and S₁ drastically revises his strategy: maintaining a low probability of concession becomes too expensive. The trade-off between the costs of mobilization and expected concessions kicks in, and S₁ precipitously decreases his allocation, exposing himself to ever increasing possibilities for bluffing as his costs go up.

As Figure 1(b) made clear, S₂ types with high valuations must spend substantially more to compel S₁ to capitulate than to fight him. This is, perhaps, not very surprising: given the initial mobilization by the status quo power, it may take a lot of threatening to persuade him to relinquish the prize peacefully. Still, it does go to show that peace can be expensive. This conclusion receives very strong support once we investigate the initial decision itself, as we did earlier. Peace may involve mobilizations at levels that are substantially higher than mobilizations that precede the outbreak of war. The price of peace can be rather steep either for the status quo state or for the potential revisionist.

As war becomes costlier, S₁ minimizes the probability of having to wage it, even when this requires skyrocketing mobilization costs. The goal of avoiding war transforms into the goal of avoiding concessions, and S₁ spends his way into successful deterrence until that, too, becomes too expensive. When this occurs, S₁ simply “gives up” and switches to having a permanent, but small, military establishment. That is, he mobilizes limited forces he does not expect to use, and whose impact on the potential revisionist's behavior is rather minimal. These “useless” mobilization levels do serve to weed out frivolous challenges but generally do not work as a deterrent to genuine revisionists or to more determined bluffers.

Peace need not be expensive if either actor has very high costs of fighting. Its price rises steeply, however, when these costs go down. Powell (1993) finds that the peaceful equilibrium in a dynamic model where states redistribute resources away from consumption toward military uses also involves nonzero allocations, which sometimes can be quite substantial. The results here underscore his conclusions and provide a nuance to their substantive interpretation and empirical implications. These findings further imply that the common assumption of a costless status quo outcome in formal models may be quite distorting because it fails to account for the resources states must spend on mutual deterrence to maintain it.¹⁶

It is worth emphasizing that peace does not depend solely on the credibility of threats. In fact, when war occurs in equilibrium, both actors possess perfectly credible threats and both know it. However, their prior actions have created an environment where neither finds war sufficiently unpleasant compared to capitulation. This illustrates the danger of committing oneself without ensuring that the opponent is not similarly committed (Schelling 1966). Although this may happen easily when actors move simultaneously, it is perhaps surprising that it can also happen when they react sequentially and seemingly have plenty of opportunity to avoid it.¹⁷

There may exist circumstances where, although peace is, in principle, obtainable, the cost of guaranteeing it is so high that the actors are unwilling to pay it. Peace in this model requires the successful compellence of S₁ or deterrence of S₂. In a situation where the value of war is determined endogenously, each actor can potentially be coerced into capitulation. The interesting question becomes why sometimes one or both of them choose not to do it. There are, of course, the trivial cases where the cost of doing that exceeds one's valuation so that it is not worth it (assured deterrence), but, more intriguingly, there are the cases where the necessary allocation costs less than one's valuation. In the second example, all types v₂ ∈ (α, β) fight optimally even though allocating

would ensure S₁'s capitulation.

Creating Commitments and Communicating Them Credibly

Consider the notion of credibility in the common rational deterrence models.¹⁸

Although superficially analogous, these models are very different from the one presented here because they assume that actors are unable to change the bargaining situation: one either has a credible threat or does not. However, in addition to their informational role, strategic moves can have a functional one (O'Neill 1991). They may alter the physical environment and restructure incentives altogether. That is, bargaining can create commitments because actors can manipulate their expected payoffs from following through on a threat and failing to do so.

Consider a stylized scenario where an actor makes a demand and issues a threat to go to war if the opponent does not concede. If that actor restructures the situation such that fighting becomes more attractive than ending the crisis without obtaining the concession, then it has effectively created a commitment not to back down. Imagine scales with the expected payoff from backing down (peace) on one side and the expected payoff from fighting (war) on the other. As long as the peace payoff outweighs the war payoff, the actor has no credible threat. Subtracting weight from the peace side (by making public statements that engage the national honor) or adding weight to the war side (by mobilizing troops) alters the balance, and eventually the war payoff may outweigh the peace payoff: at this point, the actor has created a credible commitment to fight.

Fearon (1994) offers a commitment model of this type. In it, leaders who choose to continue the crisis incur ever increasing audience costs; that is, the longer they escalate, the costlier it is for them to back down. If they prolong the crisis sufficiently, they will become locked into positions from which neither would recede, and the inevitable outcome will be war. The basic mechanism that enables them to tie their hands relies on progressively decreasing the benefit of peace until at some point war becomes the more attractive option.

Despite its popularity, the audience cost mechanism has several shortcomings. First, we have had limited success accounting for its microfoundations; that is, the domestic politics that would generate these costs (Schultz 2001b; Slantchev n.d.; Smith 1998). Second, audience costs are not inherently costly because leaders only pay them if they back down without obtaining concessions from their opponents. As Fearon (1997, 80) notes, leaders can generate arbitrarily high audience costs if they want because there are no physical constraints on doing so. Third, the mechanism requires the demanding assumption that leaders incur sufficiently high audience costs; so high, in fact, that peace becomes worse than war. When one considers something as vague and as amorphous as “national honor” and compares it to the destruction of lives and property, and the psychological scars a war inflicts on participants, this assumption becomes heroic indeed.

Military moves are a suitable candidate for coercive bargaining behavior that has both informational and functional aspects, and they do not suffer from the empirical implausibility of other commitment tactics. To gain some intuition about the workings of the military instrument, consider the other side of the decision-for-peace equation—the expected payoff from war—and actions such as mobilizing troops and sending them to the likely war zone. These are costly activities but they do improve one's chances should war actually break out. Imagine the precrisis situation of insufficient fighting preparedness with the attending prospect of having to spend the resources to “get there.” Compare this with the situation in which one has already paid the costs, and one's troops are ready to go on a short notice. Clearly, the latter situation would afford one a better bargaining position because one's expected payoff from war is now so much higher. If one succeeds in improving that expectation sufficiently, war can become more attractive than peace under the new circumstances, thereby enabling one to commit credibly to fighting. One's military moves can create a credible commitment. Unfortunately, the process of creating and communicating such a commitment may lead to war.

To see how this logic operates, let's examine the example in Figure 1(b) with complete information. Suppose v₂ = 0.5; that is, she is one of the types that would end up in a war under incomplete information. It is easy to verify that in the unique subgame perfect equilibrium war does not occur. Instead, S₁ allocates

, and S₂ capitulates immediately. The outcome is successful deterrence by S₁. What is especially striking about this result is that S₁ achieves deterrence even though his best war-fighting payoff (−0.02) is worse than immediate capitulation (0). In other words, in a regular deterrence model, this actor does not possess a credible threat, and so one should not expect it to prevail under complete information. Why does this work here? Because sinking the mobilization cost makes capitulation costlier than before: if S₂ resists, the new choice S₁ has is between quitting (which now yields a payoff of −0.37, the sunk cost of mobilization) and fighting. The payoff from fighting at m₁ = 0.37, assuming S₂ mobilizes at her optimal level

, would be at least −0.05. Thus, S₁ has tied his hands by sinking the mobilization costs at the outset, and he will certainly fight if challenged now even though at the outset he would have capitulated rather than fought even under the best circumstances. Because of S₁'s rather high mobilization level, fighting becomes too painful for S₂ and so she capitulates. In this way, the military instrument has enabled S₁ to create a credible commitment, and, because there is no uncertainty, the crisis is resolved in his favor.

Under complete information, communicating a commitment is not an issue. Consider now the analogous situation under asymmetric information where S₁ is uncertain about S₂'s valuation. In this case, S₁ allocates

. First, this is less than what is required to get S₂ with valuation v₂ = 0.5 to capitulate m₁ > rapprox 0.37). Second, it is more than the maximum mobilization at which S₂ would bother getting S₁ to capitulate m₁ [lap ] 0.23). S₁'s mobilization level is too high for him to backtrack once S₂'s valuation is revealed given what S₂ is willing to do, but it is too low to get S₂ to capitulate either. The outcome is war: S₁'s actions have now created a situation where neither opponent is prepared to back down. This situation arises because of uncertainty and would not have occurred had S₁ known his opponent's valuation from the beginning. Signaling for S₂ is pointless even though it perfectly reveals her valuation, and so her mobilization is simply preparation for war, not a warning.

In the rational deterrence context, the results show that uncertainty drives actors to choose mobilization levels that may change the bargaining context and render capitulation unpalatable to either side despite complete revelation of information. The model demonstrates how this can occur in a two-step fashion: actors fight because they create a situation where they have incentives to do so, and this situation arises because of the actors' crisis behavior under uncertainty. In other words, asymmetric information causes actors to risk committing too much (so that they would not want to back down if resisted) but not quite enough to force their opponent to back down (and so the opponent resists). Military moves may enable one to create and communicate commitments credibly, but, because they are costly and because they can be countered, there are limits to how effective they will be.

The notion of a commitment lock-in under complete information must be tempered: in the model, war occurs without residual uncertainty because the game form does not allow actors to bargain. Hence, the model does not speak to the inefficiency puzzle with complete information (Fearon 1995; Powell 2004). Rather, it provides a rationale for taking the military instrument seriously. Incorporating it in a flexible bargaining context must remain an avenue for future work.