Power and Deterrence in Alliance Relationships

Abstract

I investigate the rationality of challenge and escalation when a third party, allied either to defender or challenger, is uncertain about the enemy’s power. The analysis illustrates how third-party ally power impacts general and immediate deterrence and willingness to intervene. In alliances with the defender, uncertainty about the enemy’s strength leads the third party to support the defender with a probability that decreases with the benefits that his intervention would provide although the likelihood that he is facing a strong challenger in war has increased. If the third party is allied to the challenger, ally behavior is more nuanced.

Keywords

alliance deterrence game theory power reneging war

1. Introduction

Successful deterrence requires credibility of the deterrent threat, and credibility is about having the means and being willing to make use of them (Zagare and Kilgour, 2000; Huth, 1999; Huth and Russett, 1993; Zagare, 1987, among many others). Do more powerful allies successfully deter the enemy? More power implies greater capability to implement the deterrent threat of intervention. But are more powerful allies also more reliable? Leeds (2003a) links alliance reliability to the costs of reneging on commitments. Allies who incur lower costs of reneging should renege with higher probability. She then argues that major powers suffer less if they renege because they are less dependent on alliance partners for their security. Her empirical analysis confirms that being a major power increases the probability of reneging on alliance commitments in war. While there is empirical evidence that major powers are more likely to renege, formal analyses of alliances do not focus on the impact of an ally’s power on his behavior in war.¹ In this article, I seek to explain the empirical evidence by analyzing ally behavior as it relates to his power to change the outcome to the enemy’s detriment.

The argument that minor powers find reneging on commitments more costly than major powers because they rely more on allies for security suggests that the purpose of the alliance is defensive. In the classic challenger–defender–protégé framework, the ally’s role is to defend the target, his protégé, against potential challengers. Ally intervention then acts as a deterrent for challengers who must weigh the benefits of fighting for a revision of the status quo against the costs that depend on the ally’s intervention and his power. More powerful allies diminish the enemy challenger’s prospects in war commensurately, but only if the ally intervenes. Reneging undermines the ally’s ability to deter attack, but it also discourages escalation of the conflict by the protégé. If deterrence is about avoiding conflict altogether, an ally’s role may well be ambiguous and depends on the impact of power on reliability. The link between alliance and deterrence is further complicated by the fact that the third-party ally could be partnered with the challenger.

States need not ally with the potential target of enemy aggression but can, instead, be allied to the challenger of the status quo. Instead of enhancing security, such alliances potentially increase the allied states’ autonomy or their ability to seek revisions of the status quo that better match their interests (Morrow, 1991). A more powerful third-party ally encourages challenge while deterring the enemy target from escalating the conflict. But the dynamics of third-party intervention may be quite different from those that prevail in the classic challenger–defender–protégé situation. Not only are the challenger’s prospects in war made better by ally support, but the anticipation of that support also improves his bargaining position if the enemy backs down. Third-party ally power now hurts the enemy by reducing the terms of a peaceful settlement and by diminishing his prospects in war. When considering the enemy’s incentive to escalate the conflict, third-party ally power cuts both ways: third-party intervention in war discourages escalation, but ally power increases the demands that the challenger can make on the enemy target to avoid war. But again, the ally’s decision to support or to renege on alliance commitments is the wild card. And this decision may depend on power.

The link between alliance and deterrence is clearly a complex one and it has been studied in the literature in many of its facets. Attempts to find a strong empirical relationship between alliance and peace has a long and venerable history (see for example Singer and Small, 1968; Levy, 1981; Huth and Russett, 1984; Huth, 1988). But the evidence is not clear cut. Smith (1996) argues that selection bias may be at work as challengers will tend to attack targets whose allies are unreliable. Nevertheless, Huth (1999: 39) concludes that: “Given the limited empirical evidence on alliances it is not possible to reach clear conclusions about their utility as extended deterrents.” In response Leeds (2003b) makes the point that deterrence outcomes depend on alliance type. A defensive alliance should deter aggression but an offensive alliance may well encourage it. Empirical work that uses aggregate data masks the distinction and may therefore blur the relationship between alliance and deterrence. Leeds’s empirical analysis confirms the hypothesis.

Meanwhile, formal approaches to ally behavior and its consequences are focused on the challenger–protégé–defender relationship in which the alliance partner (the defender) adopts the role of protector against potential assaults on the status quo by the challenger (see for example Morrow, 1994; Smith, 1995, 1998; Werner, 2000). As Leeds notes, the classic formal models of alliance “are concerned exclusively with the defensive scenario” in which the third party’s decision is to intervene on behalf of the target of aggression (Leeds, 2003b: 430). Yet between 1816 and 1944, more states were called upon to uphold alliance commitments with an original challenger in war than with an original target.² Despite extensive contributions to our understanding, the relationship between alliance and deterrence has not been fully unpacked. While Leeds (2003b) describes the incentives that are present in offensive alliances, she does not develop a formal model of alliance between a third party and a challenger. And the relationship between deterrence and the third-party ally’s power to affect the outcome to the enemy’s detriment remains largely unaddressed.³ The present article offers to take a first formal step by extending the game theoretic approach to alliances with the challenger while explicitly focusing on the impact of third-party power on deterrence. In order to harmonize terminology across all cases, I will, in what follows, refer to the challenger, the defender, and the third-party ally. If the enemy is the challenger, the defender is the target of attack and is allied to the third party. If the enemy is the defender, he is the potential target of an attack coming from a challenger who is allied to the third party.

The focus of game models has been on an integrative analysis of alliance formation and reliability when the potential alliance partner and adversary are uncertain about the possible third-party ally’s commitment (see for example Morrow, 1994; Smith, 1995, 1998; Fearon, 1997).⁴ Alliance formation then serves as a costly signal that should deter aggressors and reassure potential targets. Aggressors in these models are typically uncertain about the third-party ally’s costs of war. If war is too costly for the third-party ally, he will not intervene. If costs of intervention are thought of as a proxy for power, such models assume that powerful (low war cost) third-party allies always choose to intervene on the alliance partner’s behalf. More generally, if third-party allies are modeled as being powerful or not, the model must also specify, and therefore assume, the ally’s behavior as it relates to his type. Understanding why a powerful ally would choose to renege on commitments requires shifting the focus of uncertainty away from the third-party ally’s type. Inspired by the importance to policy makers of assessing the capabilities of the adversary, I focus instead on uncertainty about the enemy’s strength.

“To estimate confidently the intentions and capabilities of opponents is one of the most important yet elusive tasks of intelligence analysts and policymakers,” writes Huth (1988: 2). The enemy’s aggressive intentions calibrate what is at stake. A strong and powerful enemy will prefer to fight unless he can negotiate possession of a high share of the contested asset, while a more modest adversary will settle for less to avert war. How does this impact a third-party ally? The third party called upon to help an embattled alliance partner has a stake in the final sharing of the asset and loses more to a powerful enemy if he prevails. This provides incentive to support the embattled friend in the hope that this will deter the more dangerous enemies. On the other hand, weaker enemies, who can only inflict a modest potential loss for the alliance, may well lead the third party to renege on alliance commitments to save on the costs of war despite the reputation costs of reneging. As Werner states, albeit in a different context, “even committed third parties are unwilling to pay the costs of intervention if the aggressor’s goals are modest enough” (Werner, 2000: 730). A weak enemy will have low bargaining power at the negotiating table and this must dampen the demands that he can make. While aggressor strength and goals are not to be conflated, strength can manifest as demands that are major or modest. For this reason, third-party allies, regardless of their power, may be inclined to intervene against the strong while ignoring the weak. I adopt this behavioral premise to examine how a third-party ally can rationally deal with an embattled ally’s call for help when in doubt about the adversary’s type .

If uncertainty is about the third-party ally, as is typically assumed,⁵ the strategic question for the enemy is one of assessing the likelihood that the third party will intervene given uncertainty about the costs of intervention that define the ally’s type. If the focus is, instead, on the third-party ally’s point of view, and uncertainty is about the enemy’s strength, the third party must weigh the risk of abandoning an ally to a likely enemy victory against the cost of supporting an ally who could successfully engage the enemy without third-party help. The change in focus reveals a relationship between the third party’s power to increase the enemy’s cost of war and make his victory less likely, and the probability that he will intervene if war breaks out: more powerful third-party allies will always be less likely to intervene unless they are allied to the aggressor. Yet they are more likely to be facing a highly threatening enemy in war once the status quo has been challenged. If allied to the aggressor, the relationship between the third party’s power and the likelihood of intervention is more nuanced and depends on whether the ally is increasing enemy war costs or reducing the probability that he wins the war. The analysis also highlights relationships between third-party ally power and deterrence as they change with the identity of the enemy. Enemy challengers are deterred by the third-party ally’s power only up to a point, while strong enemy defenders remain undeterred by ally intervention unless uncertainty about the enemy’s strength has been largely resolved. These results underscore the complexity of the relationship between ally power and deterrence and might explain the difficulties encountered by scholars seeking evidence for the deterrent effect of alliances.

2. The Game Model: Players, Moves, and Payoffs

Like Smith and Morrow I consider the interaction between three states: the challenger C, the defender D, and the third-party ally A who appear on the scene in that order. In what follows, I sometimes refer to the third-party ally, A, simply as the ally. My model seeks to understand the ally’s decision to renege under uncertainty about the threat posed by the enemy, whether the enemy is the target of the attack, D, or the initiator, C. How threatening the enemy might be shows up in the probability with which he wins a bilateral war that the ally does not join. The enemy’s type determines the bargain that he is willing to make to avert costly conflict and, if the enemy is the challenger, his type also determines the circumstances in which he will want to challenge the status quo at all. There are four possible outcomes to the game:

The challenger decides not to challenge and the status quo prevails;

The challenger challenges the status quo and the defender backs down accepting compromise on terms that are the best possible for the challenger;

The defender escalates the conflict to war when challenged and the ally supports his alliance partner;

The defender escalates the conflict to war when challenged and the ally reneges on his commitments letting defender and challenger fight out a bilateral war.

The order of play and the basic structure of the game are the same whether the enemy is the challenger or the defender. The status quo divides a contested asset, whose value is normalized to 1, between challenger and defender. The challenger has share q and the defender holds the rest. The ally values the share that falls to his partner and therefore has a stake in the outcome of a possible conflict between the challenger and defender. The game begins if the challenger C decides to challenge the status quo. Following Werner (2000), I assume that the ally’s support in war has two effects: it increases the enemy’s cost of war by Δw and reduces the probability that he will win the war by Δp. The cost of fighting for a chance at winning the contested asset in a bilateral war is set at w for both the challenger and the defender, regardless of the enemy’s type.⁶

In combination, Δw and Δp define the ally’s power to change the outcome of war by intervening in the conflict. The two aspects of ally power are distinct, although they may not be entirely independent. For example, the ally’s decision to send troops to help his partner fight the enemy increases the enemy’s cost of war as it enhances the chances that the alliance partner will win. But the sharing of intelligence about the enemy, or assistance with the deployment of military strategy, improves the odds that the alliance partner will win without necessarily affecting the enemy’s cost of war. It turns out that the components of ally power impact the outcome of the alliance game differently depending on whether the third party is allied to the defender or to the challenger. I will return to this as I discuss the equilibria of the alliance games.

The ally incurs reputation costs if he reneges on his commitments and leaves his embattled friend with diminished prospects that directly affect his own valuation of the outcome of the crisis. If instead, the ally upholds his alliance commitments and intervenes in an escalated conflict, he incurs support costs that vary with the enemy’s type, but that are mitigated by the enhanced prospects that his support brings to his embattled partner. The enemy’s type determines the probability with which the challenger can win a bilateral war. Precise payoffs depend on the enemy’s role as challenger or defender and I describe each of these situations in turn.

2.1. The Enemy is the Challenger

If the challenger is the enemy, he wins the contested asset in a bilateral war with probability p^H if he is highly threatening to the alliance and p^L if not, with $p^{H} > p^{L}$ . If the conflict escalates to war, challenger and defender incur war costs and their prospects are affected by the ally’s decision to renege or uphold alliance commitments to the defender. In war, the challenger’s prospects are affected by ally behavior in the same way regardless of type. The extensive form of the alliance game when A is allied to D is pictured in Figure 1.

Figure 1.

Alliance Game when A is allied to D

For each of the possible outcomes of the game, payoffs to Challenger, Defender and Ally are listed in that order. For example if the game ends with the ally reneging on his commitments to the defender at A1, the Challenger’s expected payoff is $p^{H} - w$ while the Defender can expect $1 - p^{H} - w$ and the Ally $α (1 - p^{H}) - R^{h}$ .

If challenged, the defender must weigh the likelihood of ally intervention against the prospect of backing down and accepting a compromise settlement that is to the challenger’s advantage. I assume that a peaceful settlement can be secured by the defender as long as he is willing to accept the best terms available to the challenger.⁷ When the challenger is the enemy these terms, determined by his type, match the probability with which he would win the contested asset in a bilateral war. In a peaceful settlement, the defender must therefore relinquish share p^H of the contested asset to a strong challenger keeping $(1 - p^{H})$ for himself. Thus the strong enemy makes higher demands to settle peacefully than the weak. If he was sure that his ally would renege on his commitments, the defender would always prefer to compromise because fighting is costly and fighting alone does not improve his chances of winning.

If the conflict escalates to war, challenger and defender incur war costs w, but expected payoffs depend on the ally’s decision to support the defender. Support from his ally increases the defender’s expected payoff from war by Δp.⁸ The ally’s intervention is desirable to the defender if his improved prospects in war exceed the cost of fighting. Indeed, if this was not the case, the defender would not care about ally intervention at all because he would prefer to compromise rather than escalate the conflict under all possible circumstances. The issue of ally intervention would then be moot and the game would boil down to a decision by the challenger to challenge if the value of compromise exceeds the value of the status quo. Parameters of the model must therefore be calibrated to ensure that ally intervention is attractive to the defender or:⁹

Δ p > w

(1)

Stated differently, the ally must be powerful enough to make a difference.

If the ally supports the defender in war, the defender incurs war cost w but his prospects improve by $Δ p$ . By contrast, the ally’s power to change the outcome of war by increasing the challenger’s war costs and reducing the probability that he will win the war determine the challenger’s diminished prospects. More precisely, if the challenger is highly threatening, his expected payoff from a war fought against defender and ally together is $p^{H} - Δ p - w - Δ w$ . It would be $p^{H} - w$ if the third-party ally reneged on his commitments and abandoned the defender in war. The low-threat challenger’s prospects are derived identically with p^H replaced by p^L.

The ally has a stake in the defender’s fate. To reflect this I assume that the ally receives utility from his partner’s share of the contested asset which is proportional to that share. Thus the ally’s status quo payoff $α (1 - q)$ is proportional to the defender’s share of the asset, $(1 - q)$ . The ally’s intervention increases the probability that the defender will win the war but support is costly. On the other hand, the ally will suffer reputation costs if he reneges and he will not be able to affect the outcome in his alliance partner’s favor. When deciding whether to support his alliance partner in war, the ally must weigh the value of his intervention net of its costs against the reputation costs of reneging. However, when the time comes to make this decision, the ally is uncertain about the enemy’s strength and this uncertainty is consequential.

I assume that if the third party knew that the enemy was weak, he would do best by staying out of a fight that the defender is likely to win. Only if the enemy is strong is intervention the preferred course of action. If the defender is attacked by a highly threatening enemy the stakes are high, and the ally would want to intervene to protect the alliance’s interests despite the cost of intervention. This assumption about ally behavior implies the following relationship between the parameters of the model:

S^{h} - α^{h} Δ p - R^{h} < S^{l} - α^{l} Δ p - R^{l}

(2)

where $S^{h} > S^{l}$ , $α^{h} \geq α^{l}$ and $R^{h} \geq R^{l}$ .

Relationship (2) states that the costs of intervention net of its benefits are lower than the cost of reneging if the enemy is strong while the opposite holds if the enemy is weak. This does not mean that the cost of intervention is lower when the enemy is strong than when the enemy is weak. Indeed I assume that the opposite is true ( $S^{h} > S^{l})$ . Instead relationship (2) holds for two possible reasons: because the ally values his intervention more when the enemy is strong and his alliance partner is more vulnerable ( $α^{h} > α^{l}$ ); or because he suffers higher costs of reneging if he abandons his alliance partner to a powerful enemy ( $R^{h} > R^{l})$ .¹⁰ Uncertainty about the enemy’s type is the source of the ally’s strategic dilemma as enemy strength determines whether intervention is the preferred course of action. The ally’s evaluation of the impact of his intervention speaks in favor of upholding alliance commitments against a highly threatening enemy, but intervening in a war against an enemy whose aims remain modest is not worth the trouble.

2.2. The Enemy is the Defender

The ally’s priorities in deciding whether to support the challenger in a dispute that has escalated to war are the same as they would be if his alliance partner was the defender. He wants to help a challenger who is at risk of losing the war with high probability but will prefer to let him fight it out in a bilateral war if the enemy is less threatening. Thus payoffs to the ally in case of escalation to war satisfy condition (2).

The game, when the enemy is the defender, is pictured in Figure 2. Again, the payoffs to Challenger, Defender, and Ally are provided for each possible outcome of the game, in that order.

Figure 2.

Alliance Game when A is allied to C

For consistency of notation, p^H and p^L continue to represent the probability that the challenger will win in a bilateral war with the defender. And $p^{H} > p^{L}$ . But now that the defender is the enemy, he will win a bilateral war with probability $(1 - p^{L})$ if he is strong, a prospect that must be measured against the possibility that the ally will intervene on the side of the challenger increasing his war costs by $Δ w$ and reducing his chances at victory by $Δ p$ . The defender could also consider submitting to a peaceful settlement to avoid war costs. But the settlement would be on the challenger’s terms and he would have to give up $p^{L} + Δ p$ of the contested asset, or the expected share that the challenger can hope for in the best of circumstances—when his ally supports him. The ally’s power now improves the terms of a settlement for the challenger, and this affects the enemy defender’s incentive to back down. Payoffs in the case where the enemy has modest aims are worked out identically but with p^L replaced by p^H as the challenger now wins a bilateral war with high probability.

When the enemy is the defender, player incentives and strategic dilemmas are the same as when the enemy is the challenger: the ally weighs the prospect of abandoning the challenger to a strong and powerful enemy against interfering in a war against an enemy whose modest aims are not worth the cost of intervention. The alliance partner must decide whether to challenge the status quo without knowing whether he can count on ally support or enemy back-down, while the enemy must decide whether to escalate the conflict to war knowing that ally intervention would greatly diminish his prospects. Yet the logic of power and deterrence is changed now that the alliance partner is the challenger. More powerful allies will encourage challenge of the status quo. And if uncertainty leads the ally to support the challenger probabilistically, his ability to increase the odds that the challenger will win the war increases the likelihood of his support as it improves the challenger’s prospects if he backs down. I find that the opposite is true when the third party is allied to the defender. This is an important finding in view of the fact that allies must decide to uphold alliance commitments with challengers just as often as they do with defenders.

3. Equilibrium Behavior

The Perfect Bayesian Equilibria (PBE) of the alliance games described above are entirely worked out in an appendix posted at http://georgetownmeansbusiness.com/tech-docs/Power-and-Deterrence-in-Alliance-Relationships-MathematicalAppendix.pdf.¹¹ They can be mapped out according to the challenger’s share of the status quo at the outset, q, and the ally’s beliefs about the enemy threat. The mapping of equilibrium outcomes is also affected by the ally’s power to affect the outcome $Δ w + Δ p$ .¹²

3.1. The Enemy is the Challenger

3.1.1. War Outcome, Ally Power, and the Status Quo

The range of possible outcomes for the challenger depends on his type. However, ally intervention shifts the bargaining range to the defender’s advantage and could lead the challenger to prefer the status quo to the expected outcome of war. The ally’s power to shift the bargaining range in the defender’s favor is illustrated in Figure 3 for the low-threat challenger.

Figure 3.

War Outcomes for the Defender and Low-Threat Challenger

Given the positioning of the status quo in Figure 3, the challenger, C, prefers to fight than to remain with the status quo if the ally reneges on his alliance commitments. But if the ally intervenes, the challenger is worse off in war than he would be if he stayed with the status quo. Indeed, as can be seen in Figure 3, the challenger expects $p^{L} - w > q$ from war if the ally reneges and $p^{L} - w - Δ w - Δ p < q$ if not. In the situation illustrated above, the low-threat challenger might then consider a challenge if the ally is sufficiently likely to stay out of the conflict, but he would decide to stay with the status quo if he was sure that the ally would support the defender.

A mental shifting of the position of the status quo, q, in Figure 3 informs about the low-threat challenger’s likely behavior if he is rational. Indeed, if q is so low as to fall below $p^{L} - w - Δ p - Δ w$ it is in the challenger’s best interest to challenge the status quo whatever the ally might do. By contrast, if $q > p^{L}$ the low-threat challenger cannot do better than to stay with the status quo. But if $p^{L} - w - Δ p - Δ w < q < p^{L}$ the low-threat challenger’s behavior depends on his expectations about the ally’s rational behavior in case of escalation by the defender. And this range for q widens as the ally’s power increases.

The challenger could also be highly threatening, in which case the probability that he could win a bilateral war p^H exceeds p^L. The high-threat challenger’s situation with respect to the status quo, as it relates to the expected outcomes of war, is identical to that of his low-threat counterpart. It could be illustrated in a figure identical to Figure 3 but with p^L replaced everywhere by p^H. However, the ally’s power determines a relationship between the prospects of strong and weak enemies that modifies the equilibrium outcomes of the alliance game and leads to unexpected relationships between the ally’s power and his ability to deter challenge or escalation of the dispute. Specifically, equilibrium outcomes are affected by the value of $p^{H} - w - Δ p - Δ w$ as it relates to p^L or $p^{L} - w$ . A very powerful ally could damage the high-threat challenger’s prospects enough to set $p^{H} - w - Δ p - Δ w$ below $p^{L} - w$ with consequences for equilibrium outcomes. The discussion that follows highlights this interaction.

3.1.2. Equilibrium Behavior

The behavior of each of the players in equilibrium depends on ally and defender initial belief that the enemy is strong, β, and the value of the status quo to the challenger, q. Critical values for $β$ are

\underline{β} = \frac{S^{l} - R^{l} - α^{l} Δ p}{(S^{l} - R^{l} - α^{l} Δ p) + (α^{h} Δ p - S^{h} + R^{h})}

that makes the ally indifferent between supporting and reneging, and

\bar{β} = \frac{α^{h} Δ p - S^{h} + R^{h}}{(S^{l} - R^{l} - α^{l} Δ p) + (α^{h} Δ p - S^{h} + R^{h})}

beyond which a low-threat challenger will forgo challenge altogether.¹³ Range [ $\underline{β}$ , $\bar{β}$ ], within which the ally intervenes with probability in case of challenge, widens with increases in Δp.¹⁴ As expected, the challenger will remain with a good enough status quo in equilibrium. Challenge takes place in equilibrium when the status quo is sufficiently unfavorable for the risk of escalation to war to justify the attempt to revise it. Assessing the risks associated to a challenge requires understanding how defender and ally will behave. Table 1 identifies player behaviors in Perfect Bayesian Equilibrium.

Table 1.

Player Behaviors in Equilibrium when the Challenger is the Enemy

Beliefs	Status Quo Value	Player Behaviors
$β \leq \underline{β}$	$0 \leq q \leq p^{L}$	CHALLENGER: Challenges regardless of type
		DEFENDER: Compromises
		ALLY: Reneges
	$p^{L} < q \leq p^{H} - Δ w - Δ p$	CHALLENGER: High-threat challenger challenges. Low-threat challenger remains with the status quo
		DEFENDER: Escalates
		ALLY: Supports
	$q > Max {p^{L}, P^{H} - w - Δ w - Δ p}$	NO CHALLENGE: Extended deterrence succeeds
$\underline{β} < β \leq \bar{β}$	$0 \leq q \leq p^{L} - w - Δ w - Δ p$	CHALLENGER: Challenges regardless of type
		DEFENDER: Escalates
		ALLY: Supports
	$P^{L} - w - Δ w - Δ p < q \leq p^{L} - w$	CHALLENGER: High-threat challenger challenges with certainty. Low-threat challenger challenges probabilistically
		DEFENDER: Escalates
		ALLY: Supports probabilistically
	$q > Max {p^{L} - w, p^{H} - w - Δ w - Δ p}$	NO CHALLENGE: Extended deterrence succeeds
$β > \bar{β}$	$0 \leq q \leq p^{L} - w - Δ w - Δ p$	CHALLENGER: Challenges regardless of type
		DEFENDER: Escalates
		ALLY: Supports
	$p^{L} - w - Δ w - Δ p < q \leq p^{H} - w - Δ w - Δ p$	CHALLENGER: High-threat challenger challenges. Low-threat challenger remains with the status quo
		DEFENDER: Escalates
		ALLY: Supports
	$q > Max {p^{L}, P^{H} - w - Δ w - Δ p}$	NO CHALLENGE: Extended deterrence succeeds

Ally power, as measured by $Δ p + Δ w$ , affects the behavior of all parties in ways that are best appreciated graphically. Figure 4 identifies the types of Perfect Bayesian Equilibria (PBE) of the alliance game according to initial belief β, the value of the status quo to the challenger q and the ally’s power. Two extreme cases are presented. In the top figure, Figure 4A, ally power $Δ p + Δ w$ is sufficiently high for $p^{H} - w - Δ w - Δ p$ to fall below $p^{L} - w$ while, in the bottom figure, Figure 4B, ally power is low enough that $p^{H} - w - Δ w - Δ p$ > p^L. All other parameters remain the same across the two diagrams. The discussion that follows comments on the various regions of Figure 4 and the differences that emerge as ally power increases.

Figure 4.

Equilibria of the Alliance Game when the Challenger is the Enemy

The ally’s ability to deter a challenge altogether depends on his power. More precisely, it depends on the low-threat challenger’s probability of victory in war p^L as it relates to the high-threat challenger’s prospects in a multilateral war, $p^{H} - w - Δ w - Δ p$ . If $p^{L} < p^{H} - w - Δ w - Δ p$ as in Figure 4B, adversaries will challenge the status quo only ifq falls within the range $[0, p^{H} - w - Δ w - Δ p]$ , a range that gets narrower as the ally’s power increases. Thus, the ally’s power to affect the outcome to the enemy’s detriment, as measured by $Δ w + Δ p$ , can deter a challenge if it is capable against the most powerful enemy. Indeed, if $q > p^{H} - w - Δ w - Δ p$ and $p^{L} < p^{H} - w - Δ w - Δ p$ the low-threat challenger’s best outcome is the status quo. A challenge could then only come from the high-threat challenger leading both defender and ally to update their beliefs about the identity of the challenger. Knowing that the challenge comes from a high-threat enemy, the ally will support the defender and the defender will escalate the dispute if challenged. As a result, the high-threat enemy, anticipating a payoff of $p^{h} - w - Δ w - Δ p < q$ from a challenge, will prefer the status quo. In Huth’s (1988) terminology, extended general deterrence succeeds.

Within specific bounds, increased ally power favors extended deterrence success. However, if the ally is powerful enough for $p^{H} - w - Δ w - Δ p$ to fall below $p^{L} - w$ or $Δ p + Δ w \geq p^{H} - p^{L}$ , as illustrated in Figure 4A, further increases in ally power do not serve to deter further challenge. In all but the case where the enemy is very likely to be highly threatening, that is for all $β < \bar{β}$ ,¹⁵ the range for q within which extended deterrence fails has an upper bound that becomes independent of ally power. Where defender and ally believe that the enemy is unlikely to be highly threatening or $β < \underline{β}$ , the status quo will be challenged as long as $0 < q < p^{L}$ ,¹⁶ and if beliefs are in the intermediate range $\underline{β} < β < \bar{β}$ the status quo will be challenged as long as $0 < q < p^{L} - w$ . Thus if ally power $Δ w + Δ p$ exceeds $p^{H} - p^{L}$ , more powerful allies no longer narrow the range of status quo values that are vulnerable to challenge.

If beliefs fall in the intermediate range, $\underline{β} < β < \bar{β}$ , equilibrium behavior of the ally relates to his power in unexpected ways. And this range for initial beliefs widens with the ally’s ability to affect the outcome to the enemy’s detriment. As can be seen in Figures 4A and 4B, challengers will challenge regardless of type if the ally’s threat of intervention is not even capable against a low-threat enemy or $0 \leq q < p^{L} - w - Δ w - Δ p$ . In such a case any challenger is better off challenging regardless of type and regardless of the ally’s response.¹⁷ This range for q narrows as ally power increases.

Ally behavior is of particular interest in the region bounded by $\underline{β} < β < \bar{β}$ and $p^{L} - w - Δ w - Δ p < q < p^{L} - w$ , a region that expands in all of its dimensions as ally power increases. Within this region, the low-threat challenger could consider a challenge if the ally is sufficiently unlikely to support the defender by intervening in the conflict. But, as shown in the Appendix (case 1.4), the probability of ally support in this region decreases with ally power. Indeed the probability with which the ally supports the defender reads:¹⁸

p_{s} = \frac{p^{L} - w - q}{Δ w + Δ p}

(3)

which is strictly decreasing in the ally’s power. However, this is enough to partially deter the low-threat challenger who challenges the status quo with probability $p_{c}^{l} = \frac{β (S^{l} - R^{l} - α^{l} Δ p)}{(1 - β) (R^{h} - S^{h} + α^{h} Δ p)}$ which decreases with ally power.¹⁹ But the high-threat challenger is not deterred by the ally’s increasing power. As a result, as the enemy’s chances at victory are increasingly diminished by the ally, or $Δ p$ increases, the chances that the alliance is facing a highly threatening enemy in case of challenge actually increases. Indeed, in case of challenge, the probability that the challenger is high threat, given that the status quo is challenged, can be written:

{p^{h} |}_{challenge} = \frac{β}{β + (1 - β) p_{c}^{l}}

(4)

and

\frac{{\partial p^{h} |}_{challenge}}{\partial Δ p} > 0 .^{20}

In sum, when $\underline{β} < β < \bar{β}$ and $p^{L} - w - Δ w - Δ p < q < p^{L} - w$ , more powerful allies are less likely to join the fight. Nevertheless, the low-threat challenger challenges with a probability that decreases as the ally’s power increases. In this sense, ally power favors immediate deterrence success. But the paradox is that a more powerful ally is less likely to support the defender in war despite the fact that the defender is more likely to be facing a highly threatening enemy.

3.2. The Defender is the Enemy

When the defender is the enemy, his response when challenged depends on his type as well as the anticipated behavior of the ally. For example, if the defender expects the ally to support the challenger for sure if he escalates the dispute, he is better off backing down if challenged. Player behaviors in equilibrium depend, in this case as well, on initial beliefs and the value of the status quo to the challenger. Critical values for beliefs are again $\underline{β}$ at which the ally is indifferent between supporting the challenger in war and reneging and $\bar{β}$ beyond which the low-threat defender is no longer willing to escalate a dispute that will end in a multilateral war.²¹Table 2 describes player behaviors in Perfect Bayesian Equilibrium as beliefs, $β$ , and the value of the status quo to the challenger, q, change.

Table 2.

Player Behavior in Equilibrium when the Defender is the Enemy

Beliefs	Status Quo Value	Player Behaviors
$β \leq \underline{β}$	$0 \leq q \leq p^{H} + β (p^{L} - p^{H}) - w$	CHALLENGER: Challenges
		DEFENDER: Escalates regardless of type
		ALLY: Reneges
	$q > p^{H} + β (p^{L} - p^{H}) - w$	NO CHALLENGE: Extended deterrence succeeds
$\underline{β} < β \leq \bar{β}$	$0 \leq q \leq p^{H} + β (p^{L} - p^{H}) - w + p_{s} Δ p + K$ *	CHALLENGER: Challenges
		DEFENDER: High-threat defender escalates. Low-threat defender escalates probabilistically.
		ALLY: Supports probabilistically
	$q > p^{H} + β (p^{L} - p^{H}) - w + p_{s} Δ p + K$	NO CHALLENGE: Extended deterrence succeeds
$β > \bar{β}$	$0 \leq q \leq p^{H} + β (p^{L} - p^{H}) + Δ p$	CHALLENGER: Challenges
		DEFENDER: Compromises regardless of type
		ALLY: Supports
	$q > p^{H} + β (p^{L} - p^{H}) + Δ p$	NO CHALLENGE: Extended deterrence succeeds

where $K = \frac{1 - β + \underline{β}}{(1 - \underline{β)}} [Δ p (1 - p_{s}) + w]$ and is derived in the Appendix (see case 2.2).

When the defender is the enemy, the ally’s power no longer serves to deter an initial challenge. While this is not entirely surprising, this fact is obscured in extant literature by the focus on alliances that form to protect the state that is under attack. Yet alliance partners are called upon to uphold their commitments by original challengers in war more often than by original defenders. Indeed, as mentioned above, original targets called upon their allies in 26 cases between 1816 and 1944, while original challengers did so in 45 cases. The relationship between ally power and the challenge of the status quo is illustrated in Figure 5.

Figure 5.

Equilibria of the Alliance Game when the Defender is the Enemy

If the enemy is unlikely to be highly threatening and $β < \underline{β}$ , the ally will let the challenger fight it out alone. As a result the defender will escalate the dispute rather than settle on the best possible terms for the challenger. This implies that the challenger will only seek revision of the status quo if it is worth fighting for alone or if:

q \leq β (p^{L} - w) + (1 - β) (p^{H} - w)

which yields the upper bound of the region in which the status quo is challenged when $β < \underline{β}$ in Figure 5. As the ally never intervenes when the enemy is believed to be low threat, the situations in which challenge and escalation take place do not depend on ally power. Ally power enters the equation as soon as $β \geq \underline{β}$ .

If the enemy is very likely to be highly threatening ( $β > \bar{β}$ ), the ally’s commitment to support leads the challenger to challenge the status quo whenever q falls short of the expected payoff that he can get from the defender when he backs down or:

β (p^{L} + Δ p) + (1 - β) (p^{H} + Δ p) = p^{H} + β (p^{L} - p^{H}) + Δ p

This upper bound for challenge clearly increases with $Δ p$ . Thus the ally’s increasing power enables challenge although his commitment to intervene if the conflict escalates to war favors immediate deterrence success.

Of particular interest is the behavior of the parties in the intermediate range for beliefs, that is when $\underline{β} < β < \bar{β}$ , a range that widens with ally power. Within this range for beliefs, the ally supports the challenger with a probability $p_{s}$ that can be written (proof is given in Appendix case 2.2):

p_{s} = \frac{Δ p - w}{Δ p + Δ w}

(5)

Clearly, $p_{s}$ decreases unambiguously with $Δ w$ but increases with $Δ p$ .²² These two indicators of the ally’s power are not necessarily independent. So the impact of power on the probability of support may well be ambiguous when the defender is the enemy. For example, if the ally supports the challenger by providing tanks or troops, the defender’s war costs increase, and the chances that the challenger will win the war are likely to increase as well. As a result, the relationship between power and the probability that an ally will uphold his alliance commitments in war may not be empirically clear cut when the third party is allied to an original challenger in war.

At the same time, and despite the ally’s ambiguous show of support for the challenger as his power increases, the enemy is more likely to be highly threatening if the conflict escalates to war. Indeed for $\underline{β} < β < \bar{β}$ :

{p_{E}^{h} |}_{escalation} = \frac{β}{β + (1 - β) p_{E}^{l}}

(6)

and

\frac{\partial {p_{E}^{h} |}_{escalation}}{\partial Δ p} = \frac{\partial {p_{c}^{h} |}_{challenge}}{\partial Δ p} > 0 .^{23}

Thus while the challenger becomes increasingly likely to face a highly threatening enemy, the increased power of his ally need not spell a higher likelihood of support in war.

One can characterize alliances with the challenger as offensive while alliances with the defender are defensive although alliances can be both offensive and defensive. The logic of defensive agreements suggests deterrence rather than a desire for intervention while the logic of offensive alliances suggests a willingness on the part of the ally to intervene in war. It is therefore of interest to compare the probability of ally support in both these cases by comparing formulae (3) and (5). It turns out that offensive alliances are upheld more often only if the ally is powerful enough, or if:²⁴

Δ p > p^{L} - q

Thus allies are not always more likely to fight when the alliance is offensive, but they will be if they are powerful enough.

3. Conclusion

A game theoretic analysis of ally behavior when the alliance partners are unsure about the enemy’s type reveals unexpected relationships between the third-party ally’s behavior and his power to affect the outcome to the enemy’s detriment. It also highlights the impact of allies on conflict initiation and escalation of the conflict to war.

In the classic situation where the ally is called upon to defend a protégé against enemy attack, unless the ally is quite convinced that the enemy is highly threatening, the ally’s power serves to deter challenge only up to a point. Moreover, for a wide range of initial beliefs, the ally’s efforts to deter challenges serves to limit attack by weak enemies but does not deter the strong. And it does not discourage escalation of the conflict by the protégé. Overall, alliance in defense of a target encourages the target to fight even if intervention becomes less likely as the ally’s power increases. In sum, the ally’s tendency to pull back as his power increases does not favor peaceful settlements as policy makers might hope, it merely inhibits the less threatening enemy attacks.

Overall, states are better served in offensive alliance relationships than in defensive ones even if the ally must be sufficiently powerful to support his partner in war more often in an offensive alliance than in a defensive one. With the enemy as defender, the challenger benefits unambiguously from the ally’s power. A more powerful ally widens the range of status quo values that can be fruitfully contested and improves the challenger’s prospects if the enemy defender backs down. Moreover, the probability that the ally will intervene increases with his power to boost the challenger’s chances of winning the war, although this effect can be cancelled by the countervailing impact of the ally’s ability to increase the enemy’s war costs. Belligerent challenger states have reason to celebrate the ally’s power. Nevertheless, strong enemy defenders are undeterred by ally intervention unless the alliance partners’ uncertainty about the enemy’s strength is largely resolved.

The impact of third-party ally power on deterrence and alliance reliability remains an outstanding empirical issue. The present game theoretic analysis provides a precise theoretical basis for that investigation. Leeds (2003a) and Krause and Singer (1997) find that the probability of reneging on alliance commitments in war is significantly increased if the third-party ally is a major power but they do not distinguish between the elements of power, elaborate on its degree or mediate the result by identifying the role of the enemy. Indeed, the relationship may not apply if the major power is allied to the original challenger in war, making the use of aggregate data to test the impact of power on reliability problematic. Moreover, these authors attribute their result to alleged lower costs of reneging for major powers. But, as argued by Gibler (2008) reneging affects a state’s reputation for honesty, a valuable resource for major as well as minor powers. My game theoretic analysis suggests a more complex reality and a different rationale: uncertainty about the enemy’s strength leads allies to deter weak enemies in order to protect the alliance against the strong. The empirical analysis of alliance reliability in war might need to be revisited.

While the present modeling effort could serve as a basis for an empirical investigation, the model can be improved and enhanced in a number of ways. One possible enhancement introduces formal bargaining between challenger and defender if the defender chooses to avoid escalation to give bargaining a chance. Escalation would then result from a breakdown of a bargaining process in which the uninformed party attempts to screen the enemy for type. Does the enemy have any interest in revealing his type through bargaining? Or would the integration of a bargaining process reveal enemy incentives to hide weakness even if war becomes more likely? A more ambitious extension would repeat the game in time so that the ally’s reliability in war becomes the basis for his reputation. Would reputation effects inhibit the ally’s willingness to renege? And in all of these situations, how does the ally’s power impact the outcome?

Footnotes

1

Formal models typically assume that allies have high or low costs of war if they intervene. But it is not necessarily true that major powers have high war costs that would discourage intervention, and the characterization of an ally as incurring costs of war that are high or low does not focus on his power to change the outcome to the enemy’s detriment.

2

Leeds (2003a) identified the alliance partner as an original target in war in only 25 of the 148 cases in which allies were called upon to meet alliance commitments between 1816 and 1944, while the alliance partner was an original challenger in war in 46 cases. In all other cases the states joined an on-going war on one side or the other. A merge of Correlates of War data with Alliance Treaty Obligations and Provisions (ATOP) data is the source (Leeds et al., 2002).

3

The relationship between power and alliance has been discussed extensively in the literature on balancing and bandwagoning. provide a summary of the literature and new evidence of bandwagoning. I do not seek to address the relationship between third-party power and the decision to ally. Rather I propose to investigate the impact of the third-party ally’s power on behavior when called upon to uphold alliance commitments in war.

4

Uncertainty about commitment is incorporated in models of crisis bargaining as well. is an example. But these are typically two-player models. The classic alliance models of Morrow and Smith address the strategic issues of three players: challenger, defender, and third-party ally.

5

Zagare and Kilgour (2003) and are exceptions as they extend the modeling of uncertainty to the other players in their models. Zagare and Kilgour examine the ally’s strategic choices under uncertainty about the challenger’s and the protégé’s types (challenger and protégé are also uncertain about the third-party ally’s type in their model). Quackenbush also develops a three-player model (challenger, protégé, and defender) in which each player is uncertain about the other two players’ types. Zagare and Kilgour are interested in the possible realignment of the protégé if the ally reneges, a move that is made available to the protégé if the ally fails to support him, while Quackenbush is interested in the challenger’s choice to challenge protégé or defender. I focus on the alliance partners’ uncertainty about the enemy and the resulting impact on challenge, escalation, and the third-party ally’s support.

6

Costs of war could be assumed to be different for enemy and target and according to enemy type. However, this does not change the main results of the model and results in more cumbersome notation.

7

This assumption can be relaxed as long as the implied condition on the ally’s probability of support still defines a true probability (a number between 0 and 1) and ally power to reduce the enemy’s probability of winning is high enough. One could also expand the model to include screening of the challenger by the defender through successive compromise offers. This would substantially complicate the model and goes beyond the scope of the present article.

8

As the whole prize has a value of 1, p is both the increase in the probability that the defender will win the war with ally support and the expected value of that support.

9

The contested asset has a value of 1. This scales the parameters of the model. In particular it must be the case that $w < 1$ for war to even be contemplated by players who are fighting for an increase in their share of the asset.

10

Costs of reneging are associated here to the plight of the alliance partner depending on the enemy’s strength, not to the power of the ally as assumed by Leeds. Note also that relationship (2) can hold under a variety of assumptions. For example, one could assume that R^h = R^l if the ally values his intervention against a strong enemy more than he does against a weak enemy. Or alternatively, if the ally does not differentiate between the value of his intervention against a strong vs. a weak enemy but evaluates the cost of reneging when the enemy is strong above those suffered if the enemy is weak, (2) can still hold.

11

It is also available from the author upon request.

12

$Δ p$ can be interpreted as the share of the contested asset lost by the enemy when the ally intervenes. The asset’s value is normalized to 1 so that $Δ p$ and $Δ w$ are commensurate. As they can both be interpreted as costs to the enemy, their sum represents the overall costs that the ally can inflict on the adversary.

13

$\underline{β}$ and $\bar{β}$ are worked out in the Appendix and conditions for $\underline{β} < \bar{β}$ are specified.

14

Derivatives $\frac{\partial \underline{β}}{\partial Δ p} < 0, \frac{\partial \bar{β}}{\partial Δ p} > 0$ are worked out in the Appendix.

15

$\bar{β}$ , worked out in the technical appendix, is the upper bound for initial belief $β$ (for which the enemy is high threat) for a low-threat challenge to escalate with probability if $p^{L} - w - Δ w - Δ p < q < p^{L} - w$ and the ally is expected to support probabilistically.

16

If $q < p^{L}$ and $β < \underline{β}$ , the low-threat challenger can gain from a challenge if the defender prefers to compromise rather than escalate the conflict to war. This happens if the ally is sure to renege. And the ally will renege with probability 1 if the challenger challenges regardless of type, a rational move in equilibrium.

17

In equilibrium, the ally will support the defender as long as $β > \underline{β}$ so that the defender will escalate the dispute to war if challenged. If $β < \underline{β}$ , the ally will renege on alliance commitments if the defender escalates the dispute, so that the defender will compromise in equilibrium.

18

As shown in the Appendix (case 1.4), probabilistic reneging by the ally results from semi-separating behavior of the challenger followed by a decision to escalate by the defender. The defender escalates if he prefers the expected outcome from escalation to what he would get in a compromise. The expected outcome of escalation is determined by the probability of support $p_{s}$ and puts a lower bound on it. Specifically it must be the case that $p_{s} \geq \frac{w}{Δ p}$ when the compromise is the best for the challenger. If we relax this assumption and allow a compromise to give the defender $1 - p^{H} + x$ when the challenger is strong and $1 - p^{L} + x$ when the challenger is weak, escalation by the defender requires that $p_{s} \geq \frac{w + x}{Δ p}$ which is a true probability if $x \leq Δ p - w$ . For the semi-separating equilibrium to hold, the low-threat challenger must be indifferent between challenging or not, which determines $p_{s} = \frac{p^{L} - w - q}{Δ w + Δ p}$ . The lower bound on $p_{s}$ resulting from the relaxing of the assumption on the value of compromise then requires that $Δ p \geq \frac{(w + x) Δ w}{p^{L} - 2 w - q - x}$ for the equilibrium to obtain.

19

$p_{c}^{l}$ is worked out in the Appendix. See case 1.4. $\frac{\partial p_{c}^{l}}{\partial Δ p} = \frac{- α^{l} β (1 - β) (R^{h} + α^{h} Δ p - S^{h}) - α^{h} β (1 - β) (S^{l} - R^{l} - α^{l} Δ p)}{{[(1 - β) (R - S^{h} + α^{h} Δ p)]}^{2}} < 0$ following (2).

20

\frac{{\partial p^{h} |}_{challenge}}{\partial Δ p} = \frac{- β (1 - β) \frac{\partial p_{c}^{l}}{\partial Δ p}}{{[β + (1 - β) p_{c}^{l}]}^{2}} > 0

21

Critical values for $β$ are the same whether the enemy is challenger or defender but $\bar{β}$ here is the upper limit on beliefs for the probability that the low-threat defender will escalate to be a true probability (see case 2.2 in the Appendix).

22

\frac{\partial p_{s}}{\partial Δ p} = \frac{Δ w + w}{{(Δ p + Δ w)}^{2}} > 0

23

As shown in the Appendix, cases 1.4 and 2.2, it turns out that $p_{c}^{l} = p_{E}^{l}$

24

\frac{Δ p - w}{Δ p + Δ w} > \frac{p^{L} - w - q}{Δ p + Δ w} if Δ p > p^{L} - q

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