Interstate wars cost more than thirty million lives during the twentieth century, and nine new interstate wars have begun since the end of the Cold War.Footnote 1 Finding ways to avoid interstate disputes and to manage those that arise short of war, therefore, is a major priority for policy-makers and for scholars of international relations. It is well recognized that military alliances play a role in conflict prevention by deterring attacks on their members, but less attention has been paid to the role of allies in conflict management. We argue that allies, as coercive actors with a stake in the dispute, are sometimes especially well positioned to help resolve disputes short of war.
We examine the influence of allies to a target state on crisis bargaining. Not only do allies have an interest in the outcome of crisis bargaining but they also may have the ability to make credible threats to both the challenger and the target that influence the bargaining stances of both disputants. A target's ally can make it clear to the challenger that if the challenger demands too much, the ally will intervene in the war on behalf of the target. But a target's ally may also be able to make a credible threat to the target—that if the target refuses to concede to a demand that the ally finds reasonable, the ally will not intervene in the war on the target's behalf. The ally may therefore have a moderating effect not only on the challenger's demand, but also on the target's response, making peaceful settlement more likely. We refer to the former as the deterrence effect of alliances, and the latter as the restraining effect of alliances.
To study this process, we develop a three-player game-theoretic model. Most recent game-theoretic models of crisis bargaining assume that a challenger and a target bargain without influence from any outside states.Footnote 2 Yet, allies who would be obligated to intervene in a war that develops between a challenger and target will often cast a shadow over crisis negotiations. Dyadic bargaining models that assume a challenger and a target bargain without influence from any outside states are inadequate for our understanding of conflict dynamics in such situations.
We derive a number of propositions about the relationships between such factors as the costs of war, the value of the stakes, and the value of an alliance and specific outcomes. The demands challengers make, the conditions under which targets concede demands without fighting, and the conditions under which targets resist demands militarily are all influenced by the presence of an ally. Given the richness of our formal results, we make no attempt to test all of the implications of the model. We do, however, discuss a few stylized cases that highlight different outcomes, and we provide a large-N empirical test of one proposition drawn from the model. We find support for the claim that when a target depends heavily on an alliance for security, that target is less likely to resist a challenge militarily.
Our research suggests that military alliances have an influence in international politics well beyond collaborative war fighting and deterrence. Alliances deter conflicts, which in itself is a force for peace, but even when challengers are not deterred from making demands, allies can facilitate peaceful settlement. Alliances can be important institutions for conflict management, not only among their members, but between their members and outside states as well. As such, alliances can be broad institutions for peace that play an important role in maintaining the stability of the international system.Footnote 3
Alliances and Crisis Bargaining
When leaders consider making a demand of another state, they take into account many factors, but chief among them is the probability that their state will win a war that results from resistance to the demand. One of the most important factors affecting the probability of prevailing in war is whether a third party will assist the target state. Alliance commitments serve as costly signals that provide information to potential challengers about which states are likely to receive assistance in war. Early models of alliances and war predicted that alliance commitments to target states reduce conflict by deterring potential challengers from making demands through the threat of outside intervention.Footnote 4 More recently, scholars have suggested that alliances may not only deter some challengers from making demands at all, but may affect the kinds of demands challengers make: demands made of targets with allies should be smaller than those of targets without allies.Footnote 5 Empirical studies show that states with allies committed to defend them are less likely to be targets of militarized interstate disputes.Footnote 6
Alliances should have an effect not only on the behavior of challengers, however, but also on the behavior of targets during crises. The conventional wisdom is that a target with allies will be emboldened and more likely to escalate disputes.Footnote 7 Because this set of targets has a higher probability of winning wars, fighting may be a better option than conceding. This potential effect of alliance commitments suggests that when general deterrence fails, disputes involving targets with allies may be associated with higher risks of deadly conflict.Footnote 8
Existing theoretical models, however, do not explicitly consider interactions between the target and its ally. Given an ally's obligation to join a war involving the target, an ally may have a strong desire to resolve the conflict short of war. Allies may also have an ability to influence the bargaining stance of the target through a credible threat to weaken or terminate the alliance.Footnote 9 In fact, some scholars view the potential opportunity to restrain (or manage, in their parlance) a partner as one of the main attractions of alliances.Footnote 10 Gelpi provides evidence that great power allies are particularly effective mediators of disputes, and he suspects that this is attributable to the major powers' ability to restrain their allies.Footnote 11 Yet, the possibility of alliance restraint has rarely been included in formal theoretic models of crisis bargaining involving alliances.Footnote 12
In most cases, allies will have an opportunity to consult with targets regarding their response to a demand. In fact, many alliance treaties require this.Footnote 13 Although sometimes the ally may recommend that the target resist the demand and reassure the target of allied support, in other cases the ally may recommend that the target concede to the demand. A recommendation to concede is not cheap talk if the ally can credibly threaten not to come to the target's aid if the target disregards the ally's recommendation and ends up in a war. Although the target and the ally would not be allied if they did not share some common interests, and thus, the ally should generally prefer for the target to achieve as beneficial an outcome as possible from bargaining, the ally may not value a particular good as much as the target does, and thus might prefer to see a target make some concessions rather than escalate a dispute to war. In such instances the ally might prefer to preserve resources for other potential conflicts.
If an ally recommends concession, the target then must determine whether the alliance is valuable enough to warrant conceding to the challenger's demand to preserve positive relations with the ally.Footnote 14 Even if the alliance does not end as a result of the policy disagreement between the allies, its future deterrence potential might be compromised if potential future challengers witness policy divisions in the alliance.Footnote 15 When states are facing multiple potential threats, they may be unwilling to compromise the deterrent potential of their alliance in other situations over the current disagreement.
We can certainly think of cases in which an ally's advice seems to have encouraged a target to concede to a challenger, and we can also think of cases in which target states did not accept the advice of their allies to concede demands.Footnote 16 In 1938, when Adolph Hitler made clear Germany's intention to annex the Sudetenland, Czechoslovakia preferred to resist the demand. Czechoslovakia's most important ally was France, however, and French leaders not only recommended that Czechoslovakia accept the Munich agreement and cede the Sudetenland to Germany, but also informed Czech leaders that if they did not comply, they would not receive French support against a German military attack.Footnote 17 Czechoslovakia conceded the Sudeten territories to Germany, and the result was a temporary peace. In other instances, however, an ally's recommendation for peaceful settlement goes unheeded. In 1911, French attempts to exert greater control over Morocco led to demands for colonial concessions from Germany. France's ally, Russia, argued strongly that France should concede to the demands.Footnote 18 Despite this, France took a much harder-line stance than Russia preferred, unduly risking war in the Russian view. The crisis was eventually settled in compromise terms that were favorable to France.
We present a crisis bargaining model that we believe captures important features of alliance relationships and provides insight into these cases. There are three actors in the model — a potential challenger, a potential target, and the target's ally. If the challenger makes a demand, the ally recommends whether the target should concede or resist before the target responds to the challenger. In equilibrium, whether the ally recommends concession depends on the ally's value for the issue at stake, the ally's expected costs of war, and the ally's value for alliance with the target. Whether the target obliges an ally's recommendation to concede depends on the target's value for these same parameters.Footnote 19 Our formalization allows us to be precise in explaining the conditions under which targets with allies are likely to concede to demands and the conditions under which targets resist demands, allowing disputes to escalate.
Our model shows the important role that allies can play in conflict management. When an ally can credibly threaten to intervene in a conflict if a challenger demands too much and to stay out of the conflict and damage the alliance if the target concedes too little, the ally can have a significant effect on the resulting distribution of goods and the probability of war. Not only can alliances deter the initiation of disputes, as prior formal models of alliances emphasize, but they can also sometimes prevent disputes from escalating by helping to craft peaceful bargains.
Model and Equilibrium Results
Suppose that there is a dispute over a unit of good between a challenger state, C, and a target state, T. The target has an ally, A, with whom the target has a formal military alliance.Footnote 20 The status quo is that the target owns the disputed good. Let each player's valuation of the good be v i > 0, where i ∈ {c,t,a} corresponds to the challenger, the target, or the ally. The challenger demands a division of the good, x ∈ [0,1], where x is the challenger's share, and 1 − x is the target's share. If there is a war between the challenger and the target without the ally's assistance, the target wins with probability p ∈ (0,1); if the ally intervenes, then the alliance wins with probability q ∈ (0,1), where q > p. When the target or the alliance wins the war, the target keeps the whole unit of the good and both the target and the ally derive benefit from the target's possession; if the challenger wins the war, then the challenger alone enjoys the benefit of the good. War is costly, and the cost of fighting for each player is c i > 0, where i ∈ {c,t,a}. Moreover, the ally and the target may incur additional costs from damaging the alliance relationship if they disagree on the appropriate response by the target. In particular, if the ally recommends that the target accept the challenger's demand, but the target refuses, then the target pays k t > 0 and the ally pays k a > 0. See Table 1 for a summary of the notation.
Table 1. Model notation
The players' strategies and the sequence of the game are as follows (Figure 1). The challenger proposes a division of the good to the target. The ally advises the target to accept or reject the demand. The target then decides to accept or reject the demand. If the target accepts the demand, then the status quo is changed to reflect the challenger's demand; if the target rejects the demand, then the challenger has a choice of escalating the crisis to war or backing down from the demand. If a war occurs as a result of both the ally and the target rejecting the challenger's demand, then the alliance will fight against the challenger. On the other hand, if the ally advises acceptance but the target rejects the challenger's demand, then the target will fight the challenger alone. The actors who participate in a war will pay the costs of war; in addition, if the allies disagree on their response to the challenger's demand, each of them pays a cost for damaging the alliance relationship.Footnote 21
Figure 1. The Sequence of the Game
We first analyze a version of the game in which the players have complete information about all aspects of the game. Analyzing the complete information game allows us to see exactly what factors drive the outcome that interests us, namely, the outcome that the ally advises the target to concede to the challenger's demand and the target does so. We then relax the assumption to allow for the possibility that the challenger has incomplete information about the target's cost of war, which leads to other possible outcomes that we may observe in the real world.Footnote 22
To understand the equilibrium results, we first consider the challenger's decision whether to escalate a crisis at the end of the game after the target rejects its demand. Suppose (1 − q)v c − c c ≤ 0, that is, the challenger is not willing to fight against the alliance when its demand is rejected by both A and T. Then, there is a unique equilibrium to the game in which C makes no demand (x = 0), and the status quo is preserved. Consider what the equilibrium means substantively. The condition, (1 − q)v c − c c ≤ 0, can be transformed equivalently to 1 − q ≤ c c /v c . The condition says that the relative cost of fighting—the ratio between the cost of war and the value of the good—for the challenger, c c /v c , is at least as great as the expected benefit of war, 1 − q, if the challenger is fighting against the alliance. The challenger will not be willing to engage in such a war and will at most be willing to fight the target when the target is alone. Understanding this incentive, the ally will always make a credible commitment to the target about coming to the target's aid in war, and consequently, the challenger will be deterred from making a demand at all.
We consider the more interesting case in which (1 − q)v c − c c > 0, that is, the challenger is willing to fight against the alliance when its demand is rejected by both the ally and the target. If the challenger is willing to fight both, then it follows immediately that the challenger will be willing to fight the target in a bilateral war, (1 − p)v c − c c > 0, because the probability of winning against the target alone is greater than against the alliance. This means that in equilibrium the challenger fights whenever the target rejects its demand.
Next, we analyze the target's response in an equilibrium. Consider the history of the game when the ally advises the target to accept a demand by the challenger. If the target accepts x, then the target receives (1 − x)v t ; if it rejects x, then the target fights a war against the challenger by itself and receives a payoff of pv t − c t − k t . This is a case in which the target not only fights the challenger alone and pays the cost of war but also pays the cost of not listening to the ally's advice. The target accepts the demand if (1 − x)v t ≥ pv t − c t − k t , or
$$x \le \lpar 1 - p\rpar + \displaystyle{{c_t + k_t } \over {v_t }}$$
Let
$x^{\lpar 1\rpar } = \lpar 1 - p\rpar + \displaystyle{{c_t + k_t } \over {v_t }}$
. Now consider the history of the game when the ally advises the target to reject a demand by the challenger. If the target accepts the demand, then it gets (1 − x)v
t
− k
t
; if the target rejects the demand, then it gets qv
t
− c
t
because the allies fight the challenger together and the alliance relationship is intact. The target accepts the demand if (1 − x)v
t
− k
t
≥ qv
t
− c
t
, or
$$x \le \lpar 1 - q\rpar + \displaystyle{{c_t - k_t } \over {v_t }}.$$
Let
$x^{\lpar 2\rpar } = \lpar 1 - q\rpar + \displaystyle{{c_t - k_t } \over {v_t }}$
. Intuitively, x
(2) < x
(1); that is, when the target has the support of the ally in a war against the challenger, the threshold level for the target to accept a demand by the challenger is smaller. In other words, the target will be encouraged by the ally's support and reject a wider range of demands by the challenger. Note, however, the challenger understands this effect of the alliance, and its equilibrium demand will be adjusted accordingly so that it does not risk fighting a war with the alliance unnecessarily.
Given this analysis, we now know that the target will respond to the challenger's demand in three ways depending on the size of the demand, x. If x ≤ x (2), then the demand is sufficiently small that the target will always accept regardless of what the ally advises. On the other hand, if x > x (1), then the demand is so big that the target will reject regardless of what the ally advises. In these two extreme cases, the ally's position on the demand does not matter to the target. The most interesting case is when the demand falls in the middle range, that is, x (2) < x ≤ x (1), such that the target's behavior will be contingent on the ally's advice. More specifically, the target will accept the demand if the ally recommends acceptance and reject the demand if the ally recommends rejection. We define an equilibrium in which the target follows the ally's advice as a coordination equilibrium. In this equilibrium, if the ally advises the target to accept a demand and preserve the peace, then the alliance has a restraining effect on the target; on the other hand, if the ally advises the target to reject the challenger's demand, then the alliance has an emboldening effect on the target.
Having analyzed the target's equilibrium behavior in a coordination equilibrium, we now turn to the analysis of the ally's strategy in such an equilibrium. Specifically, when will the ally advise the target to accept a demand and when will it advise otherwise? If the ally recommends acceptance, then the ally's payoff is (1 − x)v a from a peaceful resolution of the conflict; if the ally recommends rejection, then it means that the ally will join the target to fight against the challenger in war and receive a payoff of qv a − c a . So the ally recommends acceptance if (1 − x)v a ≥ qv a − c a , or
$$x \le \lpar 1 - q\rpar + \displaystyle{{c_a } \over {v_a }}$$
Let
$x^{\lpar 3\rpar } = \lpar 1 - q\rpar + \displaystyle{{c_a } \over {v_a }}$
. We can see that the ally is more likely to advise restraint when the ally expects high costs from war and/or places low value on the issue at stake.
Because the challenger anticipates the effect of the alliance, for the coordination equilibrium to emerge, it has to be in the challenger's best interest to make a demand that falls in the range (x (2), x (1)]. In other words, in the equilibrium the payoff for the challenger from making a demand such that x (2) < x ≤ x (1) has to be at least as good as making a demand such that x ≤ x (2), or x > x (1). Combining all the equilibrium analysis thus far, we have the following proposition that characterizes the unique coordination equilibrium.
Proposition 1 (coordination equilibrium)
If
$\displaystyle{{c_a } \over {v_a }} \ge \displaystyle{{c_t - k_t } \over {v_t }}$
, then there is a unique equilibrium to the complete information game. In the equilibrium, the challenger demands
$x^{\lpar 3\rpar } = \lpar 1 - q\rpar + \displaystyle{{c_a } \over {v_a }}$
if
$\displaystyle{{c_a } \over {v_a }} \le \displaystyle{{c_t + k_t } \over {v_t }} + q - p$
, and
$x^{\lpar 1\rpar } = \lpar 1 - p\rpar + \displaystyle{{c_t + k_t } \over {v_t }}$
if
$\displaystyle{{c_a } \over {v_a }} \gt \displaystyle{{c_t + k_t } \over {v_t }} + q - p$
. For the target and the ally, if the challenger demands x ≤ x
(2), then the target accepts the demand regardless of the ally's recommendation, and the ally recommends acceptance; if x
(2) < x ≤ x
(1), then the target follows the ally's recommendation, and the ally recommends acceptance if x ≤ x
(3)
and rejection if x > x
(3); if x > x
(1), then the target rejects regardless of the ally's recommendation, and the ally recommends acceptance if
$\displaystyle{{c_a } \over {v_a }} \ge q - p + \displaystyle{{k_a } \over {v_a }}$
and rejection if
$\displaystyle{{c_a } \over {v_a }} \lt q - p + \displaystyle{{k_a } \over {v_a }}$
.
The equilibrium outcome for either offer by the challenger is that the ally recommends acceptance and the target follows the ally's recommendation. Two points are worth noting. First, a sufficient condition for the coordination equilibrium to emerge is that the ally's relative cost of war, c a /v a , is greater than the target's relative cost of war, c t /v t . Stated another way, the lower the value of the contested good for the ally (v a ), and the higher the ally's cost of war (c a ), the more likely it is that the ally will be able to influence the target's decision. Moreover, when the target attaches a high value to the alliance relationship (k t ), the coordination equilibrium is even more likely to emerge. Importantly, it can be plausibly argued that in the real world an ally often does have a higher relative cost of war than the target, which means that the coordination equilibrium may be a common occurrence.
Second, although the coordination equilibrium allows for both the restraining effect and the emboldening effect of the alliance, the equilibrium outcome in Proposition 1 reflects only the restraining effect: the target accepts the demand conditional on the ally advising it to accept.Footnote 23 This is because in the complete information setup, the challenger is able to make a precise demand in the equilibrium such that the ally will recommend acceptance and the target accepts. The complete information game thus informs us what type of uncertainty is necessary to bring about the emboldening effect of alliances: an emboldening effect may occur if the challenger does not have accurate information about the ally's value of the good (v a ) or the ally's cost of war (c a ) such that it makes a demand that is larger than x (3), the threshold demand that makes the ally indifferent between recommending acceptance and rejection to the target in the coordination equilibrium.Footnote 24
The next proposition characterizes the unique equilibrium for the parameter range that complements that for Proposition 1.
Proposition 2 (unconditional appeasement equilibrium)
If
$\displaystyle{{c_a } \over {v_a }} \lt \displaystyle{{c_t - k_t } \over {v_t }}$
, then there is a unique equilibrium to the complete information game. In the equilibrium, the challenger demands
$x^{\lpar 2\rpar } = 1 - q + \displaystyle{{c_t - k_t } \over {v_t }}$
. For the target and the ally, if the challenger demands x ≤ x
(2), then the target accepts the demand regardless of the ally's recommendation, and the ally recommends acceptance; if x
(2) < x ≤ x
(1), then the target follows the ally's recommendation, and the ally recommends rejection; if x > x
(1), then the target rejects regardless of the ally's recommendation, and the ally recommends acceptance if
$\displaystyle{{c_a } \over {v_a }} \ge q - p + \displaystyle{{k_a } \over {v_a }}$
and rejection if
$\displaystyle{{c_a } \over {v_a }} \lt q - p + \displaystyle{{k_a } \over {v_a }}$
.
The equilibrium outcome is that the target accepts the challenger's small demand no matter what the ally recommends; consequently, the ally recommends acceptance to avoid damaging the alliance relationship. Combining the two propositions, we find that the challenger's demand varies with the ally's relative cost of war, c
a
/v
a
, which is most clearly demonstrated in Figure 2.Footnote
25
The demand has an upward trend as the ally's relative cost of war increases. An increasing relative cost of war for the ally means that the ally is increasingly reluctant to fight alongside the target; as a result, the challenger can make an increasingly larger demand without provoking a war with the alliance. This raises an interesting question about whether the target is worse off by having an alliance in terms of the concession it has to make to the challenger. If a target values an alliance relationship, the logic of the coordination equilibrium suggests that a reluctant ally could force the target to accept a demand that it would not accept if there were no alliance. Note that without the alliance, the challenger's best offer to the target is x
0 = (1 − p) + c
t
/v
t
, which will be accepted by the target. Comparing this demand with those that the target will receive when there is an alliance, we find that if
$c_a /v_a \lt \displaystyle{{c_t } \over {v_t }} + q - p$
, then the challenger's demand is smaller when the target has an ally than it would have been if the target did not have an ally, that is, x
(3) < x
0; therefore, the target benefits from having the alliance in its negotiation. On the other hand, if
$c_a /v_a \gt \displaystyle{{c_t } \over {v_t }} + q - p$
, then the challenger demands more than the challenger would have demanded if the target did not have an alliance. The highest demand that the challenger can make in the presence of the alliance is
$x^{\lpar 1\rpar } = \lpar 1 - p\rpar + \displaystyle{{c_t + k_t } \over {v_t }}$
, and it is greater than
$x_0 $
by k
t
/v
t
, which is the target's costs of damaging the alliance relative to the target's value for the disputed good. So indeed, if the ally's relative cost of war is sufficiently large, then the target is worse off because of the restraining effect of the ally. Thus, an alliance does not uniformly decrease the challenger's demand as the existing literature argues.Footnote
26
Once we introduce a rather minimal assumption that an alliance relationship is valued by its members, more interesting dynamics emerge between the challenger and the alliance.
Figure 2. The Complete Information Game: The Challenger's Equilibrium Demand as a Function of the Ally's Relative Cost of War
We want to be careful about drawing strong inferences here about the (dis)utility of alliances for targets. Keep in mind that alliance formation is exogenous to our model. This simplifying assumption helps to keep the model tractable. Yet, equilibrium outcomes in which a challenger demands more from an allied target than the challenger would demand from a nonallied target require the ally's relative costs of war to be higher than c t /v t + q − p. This condition is more likely to be satisfied if the ally's contribution to the probability of the alliance defeating the challenger, that is, q−p, is low. Alliances in which the ally has high relative costs of war and a low contribution to the probability the alliance defeats the challenger may form rarely, and if they do form, the value of the alliance to the target (k t ) might be low. When k t is low, even the challenger's maximum demand under the coordination equilibrium differs little from the demand the challenger would make of a nonallied target.
Having fully discussed the causal mechanisms underlying the equilibrium outcomes in the complete information game, we now proceed to introduce uncertainty into our model to more fully capture various alliance dynamics in the real world. What we have learned in our analysis of the complete information game will prove to be central to our understanding of the incomplete information game as well. There are many ways of introducing uncertainty to the model, because any of the parameters in the model could potentially be unknown to some actors. We adopt a common approach for introducing uncertainty into crisis bargaining models and assume that the challenger does not know the target's costs of war, c t .Footnote 27 As well articulated by Schultz, challengers are most likely to be uncertain about the resolve of the target—captured by the costs of war relative to the value of the disputed good, c t /v t —because resolve is determined in large part by the domestic conditions of the target state.Footnote 28 These can be difficult to assess for foreign leaders. Moreover, we add uncertainty on the target's costs of war because in this study we want to maintain our focus on the bargaining dynamics between the challenger and the target while under the shadow of an alliance, even if there is nothing uncertain about the ally's preferences.Footnote 29
Suppose it is common knowledge that the challenger knows that
$c_t $
is uniformly distributed on
$\lsqb \underline T \comma \; \overline T \rsqb $
, where
$\overline T \gt \underline T \gt 0$
, while the ally knows c
t
.Footnote
30
We maintain the earlier assumption, (1 − q)v
c
− c
c
> 0, so that the challenger will always fight once it makes a demand and the target rejects. Given that the challenger's strategy is the same as before when the target rejects a demand, the target's best response is also the same as before, thus x
(1) and x
(2) are still the cutpoints that condition the target's behavior. That is, if x ≤ x
(2) then the target will always accept; if x > x
(1) then the target will always reject; if x
(2) < x ≤ x
(1), then the target's behavior will be contingent on the ally's reaction. Furthermore, the ally's best response is the same as before given that the target and challenger's subsequent optimal actions are the same as those in the complete information game. What is different is the challenger's optimal demand at the beginning of the game given that the challenger does not know the target's cost of war.
We present two main results from the incomplete information game that allow us to analyze the relationship between the kinds of equilibrium outcomes that we will observe and the ally's costs of war, its valuation of the contested good, and its valuation of the alliance. Recall that under the complete information game we cannot observe entrapment—the ally is dragged into war by the target, or abandonment — the ally leaves the target to fight the challenger alone, both of which have been discussed in historical accounts of wars. In the incomplete information game, it is much more difficult for the challenger to make a demand that will ensure acceptance by both the ally and the target; consequently, many kinds of war outcomes can occur.
Proposition 3 (restraint or entrapment)
If
$\displaystyle{{c_a } \over {v_a }} \lt q - p + \displaystyle{{k_a } \over {v_a }}$
, then there is a unique equilibrium to the incomplete information game. In the equilibrium, the challenger demands
$x_1^{\ast} = \displaystyle{1 \over 2} + \displaystyle{1 \over {2v_c }}\lpar \lpar 1 - q\rpar v_c - c_c \rpar $
if
$\displaystyle{{c_a } \over {v_a }} + \displaystyle{{c_c } \over {2v_c }} \gt \displaystyle{q \over 2}$
, and
$x^{\lpar 3\rpar } = \lpar 1 - q\rpar + \displaystyle{{c_a } \over {v_a }}$
if
$\displaystyle{{c_a } \over {v_a }} + \displaystyle{{c_c } \over {2v_c }} \le \displaystyle{q \over 2}$
; the challenger maintains its prior belief F(c
t
). For the target and the ally, if the challenger demands x ≤ x
(2), then the target accepts the demand regardless of the ally's recommendation, and the ally recommends acceptance; if x
(2) < x ≤ x
(1), then the target follows the ally's recommendation, and the ally recommends acceptance if x ≤ x
(3)
and rejection if x > x
(3); if x > x
(1), then the target rejects regardless of the ally's recommendation, and the ally recommends rejection.
The condition in the proposition shows that when the ally's relative cost of war is lower than a threshold value, or if the ally values the alliance relationship highly, then there can be peace or a multilateral war because of an entrapment effect. The peace may come under two scenarios. The first is that the challenger makes a small demand, x ≤ x (2), so that the target accepts no matter what the ally says; the second is that the challenger makes a moderately large demand, x (2) < x ≤ x (1), and the target accepts only if the ally advises restraint (acceptance) in equilibrium. Therefore, the restraining effect contributes to a peaceful outcome. In the same equilibrium, war occurs only if the challenger makes a high demand, x > x (1). In this case the target will reject the demand no matter what the ally says, and the ally supports the target. As a result, war occurs between the challenger and the alliance rather than between the challenger and the target. This is a case in which the ally joins the war because it knows that the target will fight the challenger with or without its assistance, and the ally wants to avoid damage to the alliance relationship. Perhaps, for instance, the ally believes that good relations with the target are essential for dealing with other security issues, or perhaps the ally is getting support for other foreign policy priorities from the target, or perhaps the ally judges the reputational costs of abandoning an alliance partner to be too high. The dynamic underlying Proposition 3 is consistent with the entrapment effect that appears in historical accounts. In this equilibrium the alliance does not increase the probability of war, although it does expand the war.
Proposition 4 (restraint or abandonment)
If
$\displaystyle{{c_a } \over {v_a }} \ge q - p + \displaystyle{{k_a } \over {v_a }}$
, then there is a unique equilibrium to the incomplete information game. In the equilibrium, the challenger demands
$x_2^{\ast} = 1 - \displaystyle{p \over 2} - \displaystyle{{c_c } \over {2v_c }}$
if
$\displaystyle{{c_a } \over {v_a }} + \displaystyle{{c_c } \over {2v_c }} \gt q - \displaystyle{p \over 2}$
, and
$x^{\lpar 3\rpar } = \lpar 1 - q\rpar + \displaystyle{{c_a } \over {v_a }}$
if
$\displaystyle{{c_a } \over {v_a }} + \displaystyle{{c_c } \over {2v_c }} \le q - \displaystyle{p \over 2}$
; the challenger maintains its prior belief F(c
t
). For the target and the ally, if the challenger demands x ≤ x
(2), then the target accepts the demand regardless of the ally's recommendation, and the ally recommends acceptance; if x
(2) < x ≤ x
(1), then the target follows the ally's recommendation, and the ally recommends acceptance if x ≤ x
(3)
and rejection if x > x
(3); if x > x
(1), then the target rejects regardless of the ally's recommendation, and the ally recommends acceptance.
Proposition 4 characterizes the complementary case to Proposition 3 where the ally does not value the alliance relationship too highly. Again, there can be peace or war. The peace may come about in similar circumstances as those for Proposition 3, that is, because of a very small demand, x ≤ x (2) by the challenger, or a restraining effect from the alliance partner even when the demand is moderately large, x (2) < x ≤ x (1). On the other hand, war breaks out because of different alliance dynamics from those for Proposition 3; only bilateral wars are possible. If the challenger makes a high demand, x > x (1), then the target will reject no matter what the ally says, but contrary to the previous case where the ally values the alliance relationship highly, in this case the ally will not come to the target's aid when the target fights the challenger. In other words, the ally fails at restraining the target and abandons the target in war. These are cases in which we observe the violation of alliance commitments in war.
The equilibrium results for the incomplete information game are depicted in Figure 3.Footnote
31
We observe again that the challenger's demand shows an upward trend as the ally's relative costs of war (c
a
/v
a
) increases. When the ally values the alliance relationship highly such that its relative cost of war falls below
$R^{\ast} = q - p + \displaystyle{{k_a } \over {v_a }}$
, the challenger demands either
$x^{\lpar 3\rpar } = \lpar 1 - q\rpar + \displaystyle{{c_a } \over {v_a }}$
, which is an increasing function of the relative cost, or
$x_1^{\ast} = \displaystyle{1 \over 2} + \displaystyle{1 \over {2v_c }}\lsqb \lpar 1 - q\rpar v_c - c_c \rsqb $
. Because of the challenger's uncertainty about the target's cost of war, either demand could potentially fall outside of the threshold for the target to accept a demand, x
(1). Consequently war is a possible equilibrium outcome. When the ally does not value the alliance relationship too highly, such that R
* is smaller than its relative cost of war, the challenger demands either x
(3) or
$x_2^{\ast} = \displaystyle{1 \over 2} + \displaystyle{{1-p} \over {2}} - \displaystyle{{c_c} \over {2v_c}}$
. Again, the uncertainty about the target's cost may result in either demand being larger than what is acceptable to the target.
Figure 3. The Incomplete Information Game: The Challenger's Equilibrium Demand as a Function of the Ally's Relative Cost of War
So what can we say about the relationships between observable factors and the probability that a target escalates a dispute rather than conceding a demand? Some of the factors that appear to be important in this regard are familiar to crisis bargaining models—for example, the actors' costs of war, the values the actors place on the disputed goods, and the probability that the challenger can defeat the target (or the target plus the ally) in war. The unique predictions of our model, however, have to do with the effect of the value that the target and ally place on the alliance. The most straightforward result is that the ally's ability to restrain the target increases as the target values the alliance relationship more; consequently, we expect that the probability of war decreases.
Specifically, Propositions 3 and 4 show that in equilibrium the challenger can make one of the three demands: x (3), x 1 *, and x 2 *. Moreover, we have established that war occurs if the equilibrium demand is greater than x (1). Hence, we can express the probability of war as Pr(war) = Pr(x > x (1)), where x takes one of the three values. Proposition 5 summarizes the relationship between the target's valuation of the alliance and the probability of war, which supports our intuition.
Proposition 5 (probability of war)
In the equilibrium of the incomplete information game, the probability of war decreases as the target's valuation of the alliance relationship (k t ) increases.
Now we revisit the two cases we introduced earlier in light of our equilibrium analysis. In both cases allies recommended to targets that they concede demands, but in one case the target followed the ally's advice (Czechoslovakia 1938) and in one case the target did not follow the ally's advice (France 1911). Our model's predictions suggest that the restraining effect exists in equilibrium; moreover, the effect is more likely to exist if the ally's cost of war is high, the ally's value for the contested good is low, and the target's value for the alliance is high. The result provides insights into the different outcomes of these two cases.
In 1938, France saw war with Germany as very costly. Thomas suggests that two new considerations made France believe war with Germany over Czechoslovakia was less feasible than French leaders had previously believed. First, the Austrian Anschluss of March 1938 made defending Czechoslovakia much more difficult, because Germany could launch a secondary attack through Austria.Footnote 32 Second, France was facing serious financial difficulties; pressure on the franc was threatening the government's ability to maintain its defense spending.Footnote 33 Although assistance from Britain would have made war easier, the burden of defending Czechoslovakia would still fall primarily on the French military. French army leaders reported to the cabinet in September 1938 that Czechoslovak defenses might well collapse in a matter of weeks, and it would be at least six months before Britain could provide meaningful assistance in Czechoslovakia.Footnote 34
How much did France value maintaining the integrity of its Czechoslovak ally in 1938? France cared a great deal about limiting German power, and a cornerstone of its strategy for accomplishing that involved alliances with Germany's eastern neighbors. But by 1938, it was clear that the eastern Locarno pact idea had failed; Germany would not be completely encircled by hostile forces. Given that, France had some preference for Germany focusing its expansionist goals in the east if it kept Germany from turning west. France had a number of other allies, some of whom were stronger and could offer more assistance than Czechoslovakia could in a future war. France also had other allies on Germany's eastern borders. Thus, an intact Czechoslovakia was not crucial to France's security policy. This is not to say that French leaders took the decision to abandon their ally lightly; they saw costs associated with violating the commitment.Footnote 35 Yet given the high costs of war, in the end France recommended that Czechoslovakia concede to Germany's demand.
Czechoslovakia heeded French recommendations and conceded to Germany. The alliance with France was extremely important to Czechoslovakia. Czechoslovakia's only other great power ally was Russia, and a protocol to the alliance with Russia made Russian obligations contingent on French support. A French company had majority control of the Czech arms production company, France was the third-largest investor in the Czech economy (after Germany and Britain), and France was one of Czechoslovakia's major creditors.Footnote 36 Czechoslovakia was quite dependent on good relations with France and on French military support. Thus, the Czechs dared not defy their ally.
In the second case (the Morocco crisis of 1911), the disparity between the expected costs of war and the value placed on the issue at stake for the ally were even larger. Russian leaders saw disputes over control of territories in Africa as completely outside their interests. In conversation with French Prime Minister Joseph Caillaux, Russian Ambassador Alexander Isvolsky reportedly referred to the dispute with Germany as “a more or less trivial question of compensation.”Footnote 37 Russian leaders also viewed war with Germany as very costly. In fact, Russian leaders insisted to the French that they were not yet prepared to fight and needed at least two more years to prepare their military.Footnote 38 The costs of war were high and the value for the issue low, and Russia recommended that France concede to German demands.
Yet French leaders escalated their dispute with Germany in spite of Russian warnings in 1911. France certainly valued the Russian alliance in 1911 but had other sources of support. Although France and Britain were not formally allied, Britain was providing support for France in this particular crisis, and the Entente Cordiale gave hope of further support in the future. It is certainly possible that France saw the damage to the Russian alliance as a result of their disagreement over Morocco as a reasonable cost to pay to avoid concession.
Importantly, in these examples, it is reasonable to think of alliance formation as exogenous to the immediate crisis, as it is in our model. In both cases, the alliance had existed for many years before the outbreak of the given dispute, and important aspects of the international and domestic landscapes had changed since alliance formation. The simplifying assumption that alliance formation is exogenous to crisis bargaining is not unreasonable for explaining an important set of crisis interactions.
Below we turn to a large-N analysis of a hypothesis that follows directly from Proposition 5. Although this analysis does not test all of the implications of our model, it does allow us to gain confidence in the empirical relevance of our model.
An Empirical Test of Target Resistance
Proposition 5 suggests that the probability of war decreases as the target's valuation of the alliance relationship increases. The basic intuition of the result is that as targets expect higher costs from damaging alliances, they are more likely to comply with their allies' recommendations to concede demands. Therefore, we test whether targets with high costs of damaging their alliance relationships are less likely to respond to interstate disputes with militarized actions than targets with low costs of damaging their alliance relationships.Footnote 39
To test this hypothesis we need a sample of observations in which a challenger initiated a dispute against a target that had at least one ally committed to defend it in a conflict with this challenger. To generate such a sample we combine data from the Correlates of War (COW) project and the Alliance Treaty Obligations and Provisions (ATOP) project. Ideally we would like to include only cases in which the ally recommended that the target concede to the challenger. However, not only have such data not been collected for a large sample of cases, but it is not clear that it would ever be possible to know exactly what an ally recommended to a target from the historical record in many cases. Alliance members may often have incentives to hide their internal bargaining, especially when it involves potential policy disagreements. Thus, our data pool together observations in which allies may and may not have recommended concessions, which means it will be harder to find support for our hypothesis in this sample. In other words, this is a conservative test.
First, we identify a set of observations in which a challenger initiated a dispute against a target in a given year. To do this we use the COW Militarized Interstate Dispute (MID) data set and, specifically, Maoz's dyadic version of the data set.Footnote 40 In this sample, one state, the challenger, threatened, displayed, or used force against another state, the target. These data include 2,354 directed dyad-years from 1816 to 2000 in which a challenger initiated a dispute against a target.Footnote 41
Second, we identify the subset of disputing directed dyad-years in which the target of the dispute had allies committed to defend it against the challenger of the dispute. Simply because the target is an alliance member does not mean its allies agreed to defend it against the challenger. Many alliances do not require defensive military support and those that do are often invoked only under specific conditions. Therefore, to determine whether the target in the observation had an alliance applicable to a dispute with the challenger, we utilize the coding of Johnson and Leeds.Footnote 42 Their coding is based on the ATOP data set, which has detailed information about states' promised alliance obligations.Footnote 43 When a target's alliance agreement would require an ally to intervene in a dispute against that particular challenger, Johnson and Leeds code the target as having an applicable defensive alliance. These data indicate that in 1,085 of the 2,354 disputing directed dyad-years the target had at least one alliance applicable to the dispute.
After identifying the sample, we code the dependent variable. The hypothesis we are evaluating suggests that targets that have high costs of damaging their alliance relationship will be less likely to resist the demands of their challengers militarily. Therefore, to code our dependent variable we use information from the COW MID data regarding how the target responded to the challenger. If the target responded with any form of militarized action (that is, responded with a threat, display, or use of force) we code the target as resisting the demands of the challenger.Footnote 44 The target resisted in 493 of the 1,085 directed dyad-years (45 percent).Footnote 45
The key independent variable in the empirical analysis is our measure of the target's cost of damaging its alliance relationship. To operationalize this concept we measure how important the alliance relationship is to the target's security. Some states rely a lot on a particular alliance relationship for security whereas others do not. The more important the alliance relationship is to the target's security, the more costly it will be for the target to damage the relationship. To capture how important an alliance is to a target's security we use the following expression:
$$\displaystyle{{cap_A } \over {cap_A + cap_T }}$$
which is the ratio of the alliance's military capabilities to the sum of the alliance's and target's military capabilities. This variable ranges from 0 to 1, with values closer to 1 indicating that the alliance is more important to the target's security and that damaging the relationship would be costly.Footnote 46 As this variable increases, the target should be less likely to resist the challenger, so we expect the coefficient associated with this variable to be negative.
To generate this variable we need to measure the military capabilities of each target and its allies. We use the composite index of national capabilities (CINC) scores from the COW project to measure the actors' military capabilities.Footnote
47
To identify a target's allies we refer to the alliances that the Johnson and Leeds coding indicate are applicable to the dispute, being careful to include only states that were members of these alliances during the dispute and had defensive obligations to the target.Footnote
48
After identifying the allies, we sum up all of their CINC scores to generate
$cap_A $
. If a target had only one bilateral alliance, then
$cap_A $
is a function of one state's military capabilities, but if the target had a multilateral alliance or multiple alliances, then
$cap_A $
is a function of multiple states' military capabilities.Footnote
49
Although the target's capabilities are a component of this measure, the measure is not simply a proxy for target capabilities. Even strong states are weak in comparison with the combined capabilities of a large multilateral alliance, and thus even strong states can get significant security benefits from some kinds of alliances. To be certain our results are not being driven simply by the power of the target, however, we also analyze our empirical model controlling for the target's raw capabilities. Our interpretation of results is unchanged.
We recognize that this is a coarse measure for two reasons. First, the measure does not take into account things that might make an ally valuable other than its military and economic capabilities, for example its strategic position, rare natural resources, etc. Second, the measure assumes that an ally would invest all of its capabilities in defending its partner. Of course, this is unrealistic except in rare circumstances, but a proportional discount would not be problematic for our measure. Variance in what proportion of an ally's military capabilities are at stake in an alliance could introduce some noise into our measure, but without a theoretically informed discounting function, we believe that the full capability measure is our best estimate. The measure does have some face validity as well. The value of this variable for Czechoslovakia in 1938 in its dispute with Germany is well above the mean, and the value of this variable for France in 1911 in its dispute with Germany is below the mean.Footnote 50
To estimate the effect of our key independent variable, we include several other variables in our empirical model that may also influence the target's decision to resist the challenger militarily. We include two dichotomous variables that code whether the challenger had any offense or neutrality pacts that were applicable to the dispute. These alliances would make the target less likely to resist the challenger because they provide information that other states will aid the challenger or that other states will not aid the target. The coding for these variables comes from Johnson and Leeds.Footnote 51 We also include a variable that captures the challenger's probability of defeating the target in a bilateral war. As this probability increases, we expect the target to be less likely to resist the challenger. We measure this variable using the ratio of the challenger's military capabilities to the sum of the challenger's and target's military capabilities.Footnote 52 As before, the actors' CINC scores are used to measure their military capabilities.Footnote 53
We also take into account factors that may influence selection into the sample. States do not become involved in disputes randomly, and if this process is not modeled then our estimates can be biased. To account for the selection process, we estimate our results using a censored probit model in which the dependent variable for the selection stage is a dichotomous variable that codes whether the potential challenger initiates a dispute against the potential target in the directed dyad-year.Footnote 54 The selection equation is estimated using the sample of 585,467 directed-dyad years from 1816 to 2000 in which the target had at least one alliance applicable to a potential dispute with the challenger.
We include a number of variables in the selection equation to model dispute initiation. First, we include the three control variables that are in the outcome equation: whether the potential challenger has an applicable offensive alliance, whether the potential challenger is a member of an applicable neutrality pact, and the potential challenger's probability of winning in a bilateral war. Challengers that have applicable offensive or neutrality pacts should be more likely to initiate disputes. This is because these challengers expect to receive assistance in war or that the target will not receive assistance in war, and thus these challengers expect to be more likely to succeed in accomplishing their goals through the threat or use of force. Additionally, when a challenger has a high probability of defeating the target in a bilateral war it should be more likely to initiate a dispute. We use the same operationalizations of these variables as mentioned earlier.
We also include several variables in the selection equation that do not appear in the outcome equation. These variables influence the challenger's decision to initiate a dispute but not the target's decision to resist. These variables are used to identify the censored probit model.Footnote 55 The first variable is the natural log of the capital-to-capital distance between the challenger and target. As the distance between the challenger and target increases, the challenger should be less likely to initiate a dispute because conflict with more distant states is more costly, and states further apart have fewer issues to dispute. Yet because the conflict is likely fought on the target's territory, and once a dispute has been initiated we know there are already issues in conflict, distance should be much less relevant to the target's decision to resist. The data on distance are obtained from the EUGene data generation program.Footnote 56 Second, we control for whether the challenger and target are jointly democratic. A large body of research suggests that pairs of democratic states are less likely to enter into disputes.Footnote 57 Yet there is less consistent theoretical expectation about the escalation of jointly democratic disputes once they have begun. We consider the challenger and target to be democratic if they have scores of 6 or higher on the Polity 2 variable from the Polity IV data set.Footnote 58 Third, we include the challenger and target's S-score, which is used to capture the similarity of the two states' interests.Footnote 59 As this variable increases, the two states' interests are considered to be more similar, and thus they should be less likely to have disagreements that could prompt militarized disputes. If a dispute occurs, however, general foreign policy similarity should not affect how it evolves. Finally, we take into account any temporal dependence in the data related to dispute initiation using the strategy suggested by Carter and Signorino.Footnote 60 That is, we include a variable that codes the number of years since the last conflict in the dyad as well as the square and the cube of that variable.Footnote 61
Table 2 reports our empirical results. Given that we estimate our results using a censored probit model, we have two sets of coefficients. The top section reports the coefficients that are associated with the variables in the target resistance equation, and the bottom section reports the coefficients that are associated with the variables in the dispute initiation equation.
Table 2. Censored probit analysis of dispute initiation and target resistance, 1816–2000
Notes: Standard errors are in parentheses. peaceyears, (peaceyears)2 , and (peaceyears)3 included in dispute initiation estimation stage. Two-tailed tests: ** p < .01; * p < .05.
The key variable in our analysis is the first variable in the table—our measure of the target's cost of damaging its alliance relationship. Our theoretical model leads us to expect that as this variable increases, targets should be less likely to resist their challengers. This is because as this variable increases, targets' allies will be able to restrain them if they choose and convince them to concede to demands and avoid war. We make no assumption that allies always prefer to restrain their allies; whether they do depends on their values for the issues at stake and their costs of war. What our model does suggest, though, is that when allies do wish to restrain their partners, they will be more successful at doing so as the costs of damaging the alliance increases for the target. The coefficient associated with this variable reported in Table 2 supports this implication from our model. As targets' costs of damaging their alliances increase, they are less likely to resist demands of challengers.Footnote 62
This can be seen clearly in Figure 4, which graphs the predicted probability of target resistance given dispute initiation with 95 percent confidence intervals across different costs for the target damaging its alliance relationship. To generate this graph we apply the approach suggested by King, Tomz, and Wittenberg to the censored probit model.Footnote
63
The graph shows that when it is not very costly for the target to damage its alliance relationship, the target is more likely to resist a challenger's demand than when it is costly for the target to damage its alliance relationship. The predicted probability of target resistance given dispute initiation for the lowest cost is .55 [.41, .68] but as the costs of damaging an alliance increase, the predicted probability decreases to .41 [.31, .51] for the average level of costs and .37 [.27, .47] for the highest level of costs.Footnote
64
This results in a 10 percent (
$ = \displaystyle{{.04} / {.41}}$
) decrease in the probability of target resistance given an increase in the costs of damaging an alliance from the average to the maximum and a 33 percent (
$ = \displaystyle{{.18} /{.55}}$
) decrease in the probability of target resistance given an increase in the costs from the minimum to the maximum.
Figure 4. The Effect of the Target's Cost of Damaging its Alliance on Resistance
In addition to our key independent variable having the expected effect, many of the other variables in our censored probit model are consistent with our expectations and past research. First, when the challenger has an applicable offense or neutrality pact it is more likely to initiate a dispute and the target is less likely to resist. Additionally, challengers are less likely to initiate disputes against targets that are further away and targets that have similar interests. Finally, the negative and significant correlation parameter, rho, suggests that there are important unmeasured factors that make dispute initiation more likely and target resistance less likely. This also suggests that the censored probit model is appropriate for these data.
Two variables that are not consistent with our expectations are our measures of joint democracy and the challenger's probability of winning in a bilateral war. First, joint democracy is in the expected direction but does not reach conventional levels of statistical significance. Given that this variable is significant in other studies that use all dyad-years, our finding suggests that there may be a relationship between joint democracy and the presence of outside alliances, which is a condition we used to constrain our sample. However, even in our sample, if we employ a lower threshold for joint democracy (5 or higher on the Polity2 variable), it is statistically significant and negatively related to dispute initiation as expected.
Second, our measure of the challenger's probability of winning in a bilateral war is insignificant and in the opposite direction than we expected. We considered the possibility that the correlation between this variable and our key independent variable could be affecting our analysis, because both measures include the target's capabilities as a component. The correlation between these two variables is 0.41. When we estimate the model removing our key independent variable, the coefficient for the challenger's probability of winning remains statistically insignificant. In addition, controlling for the target's raw capabilities in our full model does not change our interpretation of the results. We conclude, therefore, that it is much more likely that this statistically insignificant coefficient is explained by the difficulty in measuring the challenger's probability of winning in our sample. Our sample consists of directed dyad-years where the target has outside alliances. Given that we are unable to measure the challenger's expectations about the level of allied participation in war, we are unable to fully capture the challenger's expected probability of winning either by examining the target's capabilities alone or by summing the target's capabilities along with its allies' capabilities. In sum, the fact that these two common variables from dispute initiation models (joint democracy and the challenger's probability of winning in a bilateral war) behave slightly differently in our sample of directed dyad-years involving targets with allies does not make us question the relationships we have observed about our primary variables of interest.
We readily acknowledge that our test does not capture some of the key elements of our theoretical model. For example, we are not able to measure the size of the challenger's demand or know for certain whether an ally advised a target to resist or concede in every case in our analysis. Yet our theoretical model informs us that the basic relationship we test for here should exist. To the extent that the comparative static result derived from the model is consistent with empirical evidence over a broad spatial-temporal domain, this increases our confidence in the empirical relevance of the model, even if we have not (and could not without an enormous new data collection project) tested other implications of the model in a large-N context.
We already know that states with allies committed to defend them are less likely to be targeted in militarized disputes.Footnote 65 Past research has also demonstrated that targets with allies are less likely to resist disputes than targets without allies.Footnote 66 Here, we find that in the small sample of cases in which challengers initiate disputes against targets with allies, targets that depend most on their allies for their security (that is, targets with allies who enhance their military power significantly) are less likely to escalate disputes than those whose alliances may be less valuable to them. Taken together, these findings suggest that enhancing the security of states through powerful and valuable alliances can be a force for peace.
From a target's point of view, one might wonder whether a peace that involves concessions is desirable. Keep in mind, however, that often targets with allies concede demands that are smaller than the demands that a challenger would have made if the target did not have an ally. Therefore, the deterrence and restraining effects of alliances must be understood together. In general, powerful defensive alliances have a stabilizing effect that encourages maintenance of the status quo, but sometimes allies assist in producing peaceful transfers of benefits as well.
Conclusion
Creating institutions that effectively manage interstate conflict is a priority for policymakers. Recently scholars have invested great efforts in analyzing how factors such as mediation, international courts, peacekeeping troops, conflict management treaties, and memberships in international organizations affect the initiation and resolution of interstate disputes. We hope that this research will provide useful guidance for policymakers who wish to reduce bloodshed and encourage stability in the world.
In this regard, game-theoretic crisis-bargaining models have helped scholars focus on the factors that can cause war between challengers and targets. Conflict management scholars have taken insights from this literature to find ways in which international institutions and outside actors can change disputants' incentives and help craft peaceful bargains without war. Many of these approaches have focused on neutral third players who can provide information or resolve commitment problems, two factors identified as causes of war by the crisis bargaining literature. Instead we bring in a third party, a military ally, with a stake in the conflict and the power to coerce.
We explicitly model the effects that a military ally has on crisis bargaining between a challenger and target. Because a military ally has an interest in the conflict and might intervene to assist the target, the existence of an ally can reduce the demand a challenger makes. At the same time, if the ally prefers a peaceful settlement to war given the demand the challenger has made, and if the target values the alliance, the ally may be able to encourage the target to settle the dispute peacefully to preserve good relations with the ally. Thus, the shadow of the alliance influences the bargaining stances of both challengers and targets, and our equilibrium results suggest that for a large range of parameter values that are likely to be common in the real world, the influence reduces the demands challengers make and decreases the probability that a target escalates a crisis. We find support for the latter claim in a large-N empirical test.
Our research sheds new light on military alliances and highlights their role as conflict management devices. Scholars have long argued that alliances deter dispute initiation, but past game-theoretic research also suggested that alliance commitments would embolden targets and make dispute escalation more likely. This line of reasoning overlooked the possibility that allies may have incentives to restrain their partners, a common insight in qualitative research. Explicitly modeling this possibility leads us to the conclusion that even if general deterrence has failed, defensive alliances to the target may prevent disputes from ending up in war rather than leading to escalation.
This research has important implications not only for conflict management, but also for alliance formation. For example, there is significant debate about whether further NATO expansion, especially to former Soviet states such as Georgia, would increase the likelihood of conflict with Russia. Our research suggests that in debating this question, policy-makers should not only consider the possibility that Georgia would be emboldened by NATO membership, but also the possibility that a formal alliance commitment would increase the ability and willingness of NATO members to restrain Georgia and manage Russian-Georgian disputes short of war. Our research suggests that alliances can give outside states both an interest in peace and a credible punishment to impose on an ally who chooses war.
Finally, we point to insights to be gained from moving beyond dyadic models of crisis bargaining. Although certainly understanding the interactions between challengers and targets is key to predicting the outcomes of disputes and prescribing good policies to respond to conflicts, it is rarely the case that challengers and targets bargain in a vacuum. Although allies are not the only relevant third parties, they are frequently important ones. In 46 percent of the militarized interstate disputes between 1816 and 2000, the target of the dispute had at least one ally committed to defend it in the event of conflict.Footnote 67 For those interested in managing and resolving interstate conflicts, understanding interactions among allies may be nearly as important as understanding the interactions between adversaries.
Supplementary material
Data for replication and online appendix with supplemental information that includes all additional analysis referenced in the text and footnotes are available http://dx.doi.org/10.1017/S0020818314000137 and at http://www.ruf.rice.edu/~leeds/.
Appendix
Proposition 1 (Coordination Equilibrium)
In this proof we assume (1 − q)v c − c c > 0, that is, C is willing to fight against the alliance when its demand is rejected by both A and T, and thus (1 − p)v c − c c > 0, that is, C is willing to fight T alone (see the main text for a discussion of the condition). The condition means that C's equilibrium strategy is fight whenever T rejects.
Next, we solve for T's best response in the equilibrium. If A accepted a demand x, then T also accepts the demand if (1 − x)v t ≥ pv t − c t − k t , or
$$x \le \lpar 1 - p\rpar + \displaystyle{{c_t + k_t } \over {v_t }}$$
Let
$x^{\lpar 1\rpar } = \lpar 1 - p\rpar + \displaystyle{{c_t + k_t } \over {v_t }}$
. If A rejected x, then T accepts if (1 − x)v
t
− k
t
≥ qv
t
− c
t
, or
$$x \le \lpar 1 - q\rpar + \displaystyle{{c_t - k_t } \over {v_t }}.$$
Let
$x^{\lpar 2\rpar } = \lpar 1 - q\rpar + \displaystyle{{c_t - k_t } \over {v_t }}$
. Note that x
(2) < x
(1).
Now we turn to A and C's best responses.
Case 1: If x ≤ x
(2), then T accepts x no matter what A does. So A compares (1 − x)v
a
with (1 − x)v
a
− k
a
, and its best response is accept. Moreover, given that A and T both accept any x ≤ x
(2), C's optimal demand is x
(2), receiving a payoff of
$v_c \Big( 1 - q + \displaystyle{{c_t - k_t } \over {v_t }}\Big) $
.
Case 2: if x (2) < x ≤ x (1), then T accepts if A accepts, and rejects if A rejects. Given T's best response, A compares (1 − x)v a with qv a − c a , and will accept if (1 − x)v a ≥ qv a − c a . That is, A accepts x if
$$x \le \lpar 1 - q\rpar + \displaystyle{{c_a } \over {v_a }}$$
Let
$x^{\lpar 3\rpar } = \lpar 1 - q\rpar + \displaystyle{{c_a } \over {v_a }}$
.
-
1. Suppose x (3) < x (2), that is,
$\displaystyle{{c_a } \over {v_a }} \lt \displaystyle{{c_t - k_t } \over {v_t }}$
. Then x
(3) < x, and it will be rejected by A as well as by T. In this case, C gets (1 − q)v
c
− c
c
. -
2. Suppose x (2) ≤ x (3) ≤ x (1), that is,
$\displaystyle{{c_a } \over {v_a }} \ge \displaystyle{{c_t - k_t } \over {v_t }}$
and
$\displaystyle{{c_a } \over {v_a }} \le \displaystyle{{c_t + k_t } \over {v_t }} + q - p$
. If x ≤ x
(3), then it will be accepted by A and T, and the best demand by C is x
(3), which gives it a payoff of
$v_c \lpar 1 - q + \displaystyle{{c_a } \over {v_a }}\rpar $
. If x > x
(3), then it will be rejected by A and T, and the payoff for C is v
c
(1 − q) − c
c
. Clearly, C's best response in this case is demanding x
(3), receiving
$v_c \lpar 1 - q + \displaystyle{{c_a } \over {v_a }}\rpar $
. -
3. Suppose x (2) < x (1) < x (3), that is,
$\displaystyle{{c_a } \over {v_a }} \gt \displaystyle{{c_t + k_t } \over {v_t }} + q - p$
(which implies
$\displaystyle{{c_a } \over {v_a }} \ge \displaystyle{{c_t - k_t } \over {v_t }}$
). Then x < x
(3), and it will be accepted by A and T. In this case, the best demand C can make is
$x^{\lpar 1\rpar } = \lpar 1 - p\rpar + \displaystyle{{c_t + k_t } \over {v_t }}$
, and its payoff is
$v_c \lpar 1 - p + \displaystyle{{c_t + k_t } \over {v_t }}\rpar $
.
Case 3: If x > x (1), then T rejects no matter what A does. So A compares pv a − k a with qv a − c a , and accepts x if pv a − k a ≥ qv a − c a . That is, A accepts x if
$$\displaystyle{{c_a } \over {v_a }} \ge q - p + \displaystyle{{k_a } \over {v_a }}$$
Given A and T's strategies, by making a demand x > x
(1), C gets (1 − p)v
c
− c
c
if
$$\displaystyle{{c_a } \over {v_a }} \ge q - p + \displaystyle{{k_a } \over {v_a }}\comma \; {\rm and} \,\lpar 1 - q\rpar v_c - c_c \, {\rm if}\, \displaystyle{{c_a } \over {v_a }} \lt q - p + \displaystyle{{k_a } \over {v_a }}.\ $$
So far, we have found A's best response given any x, and we also know C's optimal demand for different regions of x. Next we find C's overall optimal demand, so that we can characterize the equilibrium for the entire game. Moreover, we are interested in a coordination equilibrium for Proposition 1. In cases 1 and 3, the target does not condition its equilibrium strategy on what A does; therefore, there is no coordination equilibrium if C's equilibrium demand is either x * ≤ x (2), or x * > x (1). A coordination equilibrium can only exist if x (2) < x * ≤ x (1) (case 2). So below we find out the conditions under which C's optimal demand is some x * such that x (2) < x* ≤ x (1). Because C's payoff depends on the location of x (3) for case 2, again we analyze three cases to find C's optimal demand.
-
1. If x (3) < x (2), that is,
$\displaystyle{{c_a } \over {v_a }} \lt \displaystyle{{c_t - k_t } \over {v_t }}$
, then C's payoff from demanding x ∈ (x
(2), x
(1)] is (1 − q)v
c
− c
c
because it will be rejected by both A and T. The strategy is strictly dominated by demanding x = x
(2), so it is not an equilibrium strategy. -
2. If x (2) ≤ x (3) ≤ x (1), that is,
$\displaystyle{{c_a } \over {v_a }} \ge \displaystyle{{c_t - k_t } \over {v_t }}$
and
$\displaystyle{{c_a } \over {v_a }} \le \displaystyle{{c_t + k_t } \over {v_t }} + q - p$
, then C's optimal demand for x ∈ (x
(2), x
(1)] is
$x^{\lpar 3\rpar } = 1 - q + \displaystyle{{c_a } \over {v_a }}$
, which brings a payoff of
$v_c x^{\lpar 3\rpar } = v_c \lpar 1 - q + \displaystyle{{c_a } \over {v_a }}\rpar $
. It is easy to see that this payoff is higher than that from demanding x ≤ x
(2) (case 1). If v
c
x
(3) is also higher than the payoff from x > x
(1) (case 3), then we will have found a coordination equilibrium. If
$q - p \gt \displaystyle{{c_a - k_a } \over {v_a }}$
, then the payoff from x > x
(1) is (1 − q)v
c
− c
c
, which is smaller than
$v_c \lpar 1 - q + \displaystyle{{c_a } \over {v_a }}\rpar $
, so the coordination equilibrium exists. On the other hand, if
$q - p \le \displaystyle{{c_a - k_a } \over {v_a }}$
, then the payoff from x > x
(1) is (1 − p)v
c
− c
c
. To have
$v_c \lpar 1 - q + \displaystyle{{c_a } \over {v_a }}\rpar \ge \lpar 1 - p\rpar v_c - c_c $
, it must be that
$q - p \le \displaystyle{{c_a } \over {v_a }} + \displaystyle{{c_c } \over {v_c }}$
, which is satisfied given that
$q - p \le \displaystyle{{c_a - k_a } \over {v_a }}$
. Therefore, there is indeed a unique coordination equilibrium when x
(2) ≤ x
(3) ≤ x
(1). The equilibrium outcome is that C demands
$x^{\lpar 3\rpar } = 1 - q + \displaystyle{{c_a } \over {v_a }}$
and it will be accepted by both A and T. The conditions for the equilibrium are:
$\displaystyle{{c_a } \over {v_a }} \ge \displaystyle{{c_t - k_t } \over {v_t }}$
and
$q - p \ge \displaystyle{{c_a } \over {v_a }} - \displaystyle{{c_t + k_t } \over {v_t }}$
. -
3. If x (2) < x (1) < x (3), that is,
$\displaystyle{{c_a } \over {v_a }} \gt \displaystyle{{c_t + k_t } \over {v_t }} + q - p$
(which implies
$\displaystyle{{c_a } \over {v_a }} \ge \displaystyle{{c_t - k_t } \over {v_t }}$
), then C's optimal demand for x ∈ (x
(2), x
(1)] is
$x^{\lpar 1\rpar } = \lpar 1 - p\rpar + \displaystyle{{c_t + k_t } \over {v_t }}$
, which brings a payoff of v
c
x
(1). It strictly dominates demanding either x ≤ x
(2) and x > x
(1), thus there is a unique coordination equilibrium for this case. The equilibrium outcome is that C demands
$x^{\lpar 1\rpar } = 1 - p + \displaystyle{{c_t + k_t } \over {v_t }}$
and it will be accepted by both A and T.
To summarize, if x
(3) < x
(2), that is,
$\displaystyle{{c_a } \over {v_a }} \lt \displaystyle{{c_t - k_t } \over {v_t }}$
, then there is no coordination equilibrium; if x
(2) ≤ x
(3), that is,
$\displaystyle{{c_a } \over {v_a }} \ge \displaystyle{{c_t - k_t } \over {v_t }}$
, then there is a unique coordination equilibrium, but the demand made by the challenger is different depending on a further condition: whether
$\displaystyle{{c_a } \over {v_a }} \gt \displaystyle{{c_t + k_t } \over {v_t }} + q - p$
.
Proposition 2 (Unconditional Appeasement Equilibrium)
From case 1 in the proof of Proposition 1, we know that if x
(2) is demanded, then T will accept it no matter what A recommends. Given that T always accepts x
(2), A will also accept x
(2) because (1 − x
(2))v
a
> (1 − x
(2))v
a
− k
a
. Moreover, we know that demanding x
(2) is C's optimal demand for the range of x ≤ x
(2), and it also strictly dominates any x
(2) < x ≤ x
(1) when
$\displaystyle{{c_a } \over {v_a }} \lt \displaystyle{{c_t - k_t } \over {v_t }}$
. What is left to check is whether C has an incentive to deviate to demanding some x > x
(1). We only need to check two cases.
-
1. If
$q - p \le \displaystyle{{c_a - k_a } \over {v_a }}$
, then x
(2) is C's best response if x
(2)
v
c
≥ (1 − p)v
c
− c
c
, which implies
$q - p \le \displaystyle{{c_t - k_t } \over {v_t }} + \displaystyle{{c_c } \over {v_c }}$
. This is always true given
$q - p \le \displaystyle{{c_a - k_a } \over {v_a }}$
and
$\displaystyle{{c_a } \over {v_a }} \lt \displaystyle{{c_t - k_t } \over {v_t }}$
. -
2. If
$q - p \gt \displaystyle{{c_a - k_a } \over {v_a }}$
, then x
(2) is C's best response if x
(2)
v
c
≥ (1 − q)v
c
− c
c
, which implies
$\displaystyle{{c_t - k_t } \over {v_t }} \gt - \displaystyle{{c_c } \over {v_c }}$
. This is always true given
$\displaystyle{{c_a } \over {v_a }} \lt \displaystyle{{c_t - k_t } \over {v_t }}$
.
Proposition 3 (Restraint or Entrapment)
Suppose it is common knowledge that C only knows c
t
~F(c
t
), where F(c
t
) is a uniform distribution on
$\lsqb \underline T \comma \; \overline T \rsqb $
(
$\overline T \gt \underline T \gt 0$
), while A knows the value of c
t
. In solving the game, we impose further restrictions on
$\underline T $
and
$\overline T $
so that the game is not trivial. Specifically, we require x
(2) ≥ 0, which implies c
t
≥ k
t
− (1 − q)v
t
; otherwise, T will never have an incentive to accept an offer when the ally recommends rejection. On the other hand, we require x
(1) ≤ 1, which implies c
t
≤ pv
t
− k
t
; otherwise, T will never have an incentive to reject a demand when the ally recommends acceptance. Without the two restrictions on c
t
, the game will be trivial: the ally will have a total control of the target's decision. Thus, we assume
$\underline T = k_t - \lpar 1 - q\rpar v_t $
and
$\overline T = pv_t - k_t $
.
Now we turn to the solution of the game. We maintain the earlier assumption, (1 − q)v
c
− c
c
> 0, so that C will always fight if T rejects a demand. Given C's strategy, T's best response is the same as before, and consequently A's best response is the same as those in the complete information game. To briefly recap, T always accepts
$x \le x^{\lpar 2\rpar } = \lpar 1 - q\rpar + \displaystyle{{c_t - k_t } \over {v_t }}$
, always rejects
$x \gt x^{\lpar 1\rpar } = \lpar 1 - p\rpar + \displaystyle{{c_t + k_t } \over {v_t }}$
, and follows A's advice for x
(2) < x ≤ x
(1). For A, if x ≤ x
(2), then A accepts; if x
(2) < x ≤ x
(1), then A accepts only if
$x \le x^{\lpar 3\rpar } = \lpar 1 - q\rpar + \displaystyle{{c_a } \over {v_a }}$
; if x > x
(1), then A accepts only if
$q - p \le \displaystyle{{c_a - k_a } \over {v_a }}$
.
Suppose
$\displaystyle{{c_a } \over {v_a }} \lt q - p + \displaystyle{{k_a } \over {v_a }}$
. The condition implies that if the challenger demands x > x
(1), then the ally will join the target to fight the challenger. The challenger's best response is a demand that maximizes its expected utility. Again, from the complete information game, we know that if x ≤ x
(2), then C gets v
c
x; if x
(2) < x ≤ x
(1), then C's payoff also depends on where x
(3) locates in relation to x
(2) and x
(1). In particular, if x
(2) < x
(3) < x
(1), then both A and T will accept if x ≤ x
(3), and both reject if x > x
(3). Finally, if x > x
(1), then C gets (1 − p)v
c
− c
c
if
$\displaystyle{{c_a } \over {v_a }} \ge q - p + \displaystyle{{k_a } \over {v_a }}$
, and (1 − q)v
c
− c
c
otherwise.
-
1. Suppose the challenger demands x ≤ x (3). Then,
$$\hskip -30pt\eqalign{&Pr\lpar x \le x^{\lpar 2\rpar } \rpar v_c x + Pr\lpar x^{\lpar 2\rpar } \lt x \le x^{\lpar 1\rpar } \rpar v_c x + Pr\lpar x \gt x^{\lpar 1\rpar } \rpar \lpar \lpar 1 - q\rpar v_c - c_c \rpar }$$
$$= Pr\lpar x \le x^{\lpar 1\rpar } \rpar v_c x + Pr\lpar x \gt x^{\lpar 1\rpar } \rpar \lpar \lpar 1 - q\rpar v_c - c_c \rpar$$
$$\eqalign{& = Pr\lpar c_t \ge \lpar x + p - 1\rpar v_t - k_t \rpar v_c x + Pr\lpar c_t \lt \lpar x + p - 1\rpar v_t - k_t \rpar \lpar \lpar 1 - q\rpar v_c - c_c \rpar }$$
$$\hskip -27pt\eqalign{& = \displaystyle{{\overline T - \lpar x + p - 1\rpar v_t + k_t } \over \Delta }v_c x + \displaystyle{{\lpar x + p - 1\rpar v_t - k_t - \underline T } \over \Delta }\lpar \lpar 1 - q\rpar v_c - c_c \rpar \comma \; }$$
where
$\Delta = \overline T - \underline T $
. The first-order condition (FOC) is:
$\displaystyle{{\overline T - \lpar x + p - 1\rpar v_t + k_t } \over \Delta }v_c - \displaystyle{{v_t v_c x} \over \Delta } + \displaystyle{{v_t } \over \Delta }\lpar \lpar 1 - q\rpar v_c - c_c \rpar = 0$
, which gives us the solution:
$x_1^{\ast} = \displaystyle{{\overline T } \over {2v_t }} - \displaystyle{p \over 2} + \displaystyle{1 \over 2} + \displaystyle{{k_t } \over {2v_t }} + \displaystyle{1 \over {2v_c }}\lpar \lpar 1 - q\rpar v_c - c_c \rpar $
. Substituting
$\underline T = k_t - \lpar 1 - q\rpar v_t $
and
$\overline T = pv_t - k_t $
into the expression, we have
$x_1^{\ast} = \displaystyle{1 \over 2} + \displaystyle{1 \over {2v_c }}\lpar \lpar 1 - q\rpar v_c - c_c \rpar $
. If x
1
* is an interior point, it must satisfy x
1
* < x
(3), that is,
$\displaystyle{1 \over 2} + \displaystyle{1 \over {2v_c }}\lpar \lpar 1 - q\rpar v_c - c_c \rpar \lt \lpar 1 - q\rpar + \displaystyle{{c_a } \over {v_a }}$
. Thus, the condition for x
1
* to be an optimal interior solution is
$\displaystyle{{c_a } \over {v_a }} + \displaystyle{{c_c } \over {2v_c }} \gt \displaystyle{q \over 2}$
.
-
2. Suppose the challenger demands x > x (3). Then,
$$\eqalign{&Pr\lpar x \le x^{\lpar 2\rpar } \rpar v_c x + Pr\lpar x^{\lpar 2\rpar } \lt x \le x^{\lpar 1\rpar } \rpar \lpar \lpar 1 - q\rpar v_c - c_c \rpar + Pr\lpar x \gt x^{\lpar 1\rpar } \rpar \lpar \lpar 1 - q\rpar v_c - c_c \rpar }$$
$$\hskip -42pt\eqalign{& = Pr\lpar x \le 1 - q + \displaystyle{{c_t - k_t } \over {v_t }}\rpar v_c x + Pr\lpar x \gt 1 - q + \displaystyle{{c_t - k_t } \over {v_t }}\rpar \lpar \lpar 1 - q\rpar v_c - c_c \rpar }$$
$$\hskip -20pt\eqalign{ &= Pr\lpar c_t \ge \lpar x + q - 1\rpar v_t + k_t \rpar v_c x + Pr\lpar c_t \lt \lpar x + q - 1\rpar v_t + k_t \rpar \lpar \lpar 1 - q\rpar v_c - c_c \rpar }$$
$$\hskip -50pt\eqalign{& = \displaystyle{{\overline T - \lpar x + q - 1\rpar v_t - k_t } \over \Delta }v_c x + \displaystyle{{\lpar x + q - 1\rpar v_t + k_t - \underline T } \over \Delta }\lpar \lpar 1 - q\rpar v_c - c_c \rpar .}$$
The FOC is
$\displaystyle{{\overline T - \lpar x + q - 1\rpar v_t - k_t } \over \Delta }v_c - \displaystyle{{v_t v_c x} \over \Delta } + \displaystyle{{v_t } \over \Delta }\lpar \lpar 1 - q\rpar v_c - c_c \rpar = 0$
, therefore,
$x_1^{\ast \ast } = \displaystyle{{\overline T } \over {2v_t }} - \displaystyle{q \over 2} + \displaystyle{1 \over 2} - \displaystyle{{k_t } \over {2v_t }} + \displaystyle{1 \over {2v_c }}\lpar \lpar 1 - q\rpar v_c - c_c \rpar $
, or
$x_1^{\ast \ast } = \displaystyle{1 \over 2} + \displaystyle{1 \over {2v_c }}\lpar \lpar 1 - q\rpar v_c - c_c \rpar - \displaystyle{{k_t } \over {v_t }} - \displaystyle{{q - p} \over 2}$
. If x
1
** is an interior point, then x
1
** > x
(3), that is,
$\displaystyle{1 \over 2} + \displaystyle{1 \over {2v_c }}\lpar \lpar 1 - q\rpar v_c - c_c \rpar - \displaystyle{{k_t } \over {v_t }} - \displaystyle{{q - p} \over 2} \gt \lpar 1 - q\rpar + \displaystyle{{c_a } \over {v_a }}$
. Thus, the condition for x
1
** to be an optimal interior solution is
$\displaystyle{{c_a } \over {v_a }} + \displaystyle{{k_t } \over {v_t }} + \displaystyle{{c_c } \over {2v_c }} \lt \displaystyle{p \over 2}$
.
The conditions found in 1 and 2 cannot be satisfied simultaneously, therefore we can at most have one optimal point in one of the two ranges. We consider each case separately.
Suppose
$\displaystyle{{c_a } \over {v_a }} + \displaystyle{{c_c } \over {2v_c }} \gt \displaystyle{q \over 2}$
, then x
1
* is an interior optimal solution for x < x
(3), while there is no interior solution for x > x
(3). Because x > x
(3) is half open and half closed, and the boundary point x = 1 is never optimal,
$x_1^{\ast} = \displaystyle{1 \over 2} + \displaystyle{1 \over {2v_c }}\lpar \lpar 1 - q\rpar v_c - c_c \rpar $
is optimal for all x ∈ [0,1] for this case.
Suppose
$\displaystyle{{c_a } \over {v_a }} + \displaystyle{{k_t } \over {v_t }} + \displaystyle{{c_c } \over {2v_c }} \lt \displaystyle{p \over 2}$
, thus x
1
** is optimal interior solution for x > x
(3), while no interior solution exists for x ≤ x
(3). We need to compare the boundary point x
(3) with x
1
**. They bring the same payoff for C except when x
(3) ∈ (x
(2), x
(1)], in which case demanding x
(3) is strictly better because
$\lsqb \lpar 1 - q\rpar + \displaystyle{{c_a } \over {v_a }}\rsqb v_c \gt \lpar 1 - q\rpar v_c - c_c $
. Hence, x
(3) weakly dominates x
1
** and C will demand x
(3) in equilibrium. If neither
$\displaystyle{{c_a } \over {v_a }} + \displaystyle{{c_c } \over {2v_c }} \gt \displaystyle{q \over 2}$
nor
$\displaystyle{{c_a } \over {v_a }} + \displaystyle{{k_t } \over {v_t }} + \displaystyle{{c_c } \over {2v_c }} \lt \displaystyle{p \over 2}$
holds, then there is no interior solution for x < x
(3) and x > x
(3), and the boundary point between the two regions, x
(3), is the optimal solution.
In sum, if
$\displaystyle{{c_a } \over {v_a }} + \displaystyle{{c_c } \over {2v_c }} \gt \displaystyle{q \over 2}$
, then C demands
$x_1^{\ast} = \displaystyle{1 \over 2} + \displaystyle{1 \over {2v_c }}\lpar \lpar 1 - q\rpar v_c - c_c \rpar $
; otherwise C demands x
(3). In terms of the equilibrium outcome, if C's equilibrium demand, whether it is x
(3) or x
1
*, is greater than x
(1), then there is multilateral war with the ally joining the target to fight the challenger; otherwise, the equilibrium outcome is peace due the ally's restraining effect.
Proposition 4 (Restraint or Abandonment)
The proof for the proposition is very similar to that for Proposition 3. Because of space considerations, the details are provided in the online appendix.
Proposition 5 (Probability of War)
In Propositions 3 and 4, there are three types of optimal demand that C could make: x (3), x 1 *, and x 2 *. Moreover, we have established that war occurs if C's equilibrium demand is greater than x (1). Note that k t does not appear in any of the possible demands, but it enters positively in x (1); therefore, x (1) increases in k t , which decreases the probability of war. As an example, suppose C demands x (3), then
$$Pr\lpar war\rpar = Pr\lpar x^{\lpar 3\rpar } \gt x^{\lpar 1\rpar } \rpar = Pr\lpar 1 - q + \displaystyle{{c_a } \over {v_a }} \gt 1 - p + \displaystyle{{c_t + k_t } \over {v_t }}\rpar$$
$$\hskip -58pt= Pr\lpar c_t \lt \lpar \displaystyle{{c_a } \over {v_a }} - q + p\rpar v_t - k_t \rpar$$
Let
$H = \Big( \displaystyle{{c_a } \over {v_a }} - q + p\Big) v_t - k_t $
, and
$\displaystyle{{\partial H} \over {\partial k_t }} = - 1$
.
The proof when the demand is x 1 * or x 2 * is similar.
