Inducing Deterrence through Moral Hazard in Alliance Contracts

Abstract

Do military alliances foster aggressive behavior in allies to the point of undermining the security goal of the alliance? Like others, we find that alliance commitments may cause moral hazard because allies do not fully internalize the costs of actions that can lead to war. But unlike others, we show that the effect of moral hazard can improve security. Moral hazard can be the driving force behind generating deterrence and avoiding costly conflict. Aggressors may refrain from initiating crises if their target enjoys additional resources from its ally and so is more willing to fight back. So rather than incurring costs, moral hazard may be the very key to deterring potential aggressors and minimizing the risk of conflict. This behavior allows alliance partners to capture a “deterrence surplus,” which are the gains from avoiding conflict.

Keywords

alliances moral hazard bargaining game theory

Do military alliances cause allies to act so aggressively that their behavior undermines the security goal of the alliance? If so, then leaders may be cautious about joining an alliance, doing so only if they can select safe alliance partners or design the terms of the treaty in a way that captures the deterrence benefits while managing the dangers of overaggression. We show that while alliance commitments may cause alliance partners to behave aggressively, under some conditions the added aggressiveness actually enhances deterrence. From this perspective, the effectiveness of an alliance is related both to the structure and to the content of the treaty.

This view stands in contrast to the standard explanation that the content of an alliance agreement is designed to balance the benefit of deterring an enemy against the risk of emboldening an ally (Fearon 1997; Snyder 1997; Yuen 2009). Our approach builds on existing models of alliance formation (Morrow 1994; Smith 1995) but adds intra-alliance contracting over the benefits from successful deterrence. Specifically, avoiding a conflict that they otherwise might expect creates a surplus equal to the foregone cost of war, which may be divided among alliance partners.

To facilitate an understanding of the intuition, we suggest that military alliances share many similarities with standard insurance contracts. Much as an auto insurance policy stipulates how much a policy holder will receive if he or she is in an accident, an alliance agreement likewise often describes how much aid an ally will provide to the attacked party if there is a war. For example, the 1656 Treaty of Defensive Alliance between Brandenburg and France enumerates precisely the amount and form of aid each ally would provide to the other if it was attacked: Brandenburg pledged 2,400 men and 600 horses to France while France promised 5,000 men, 1,200 horses, and artillery to Brandenburg. In the 1893 Franco-Russian alliance, France promised to supply 1.3 million troops and Russia pledged to contribute 700,000 to 800,000 troops in a conflict against Germany.¹

Furthermore, in an insurance contract, the size of the insurance premium the insured pays usually depends on the amount of risk being indemnified by the insurance provider: the more risk for the insurer, the higher the premium. Our explanation of the content of alliance commitments likewise ties the level of support to the amount of security risk alliance partners face. That is, leaders of threatened countries may look to team up with each other, and the amount of support they promise one another may depend on the amount of threat each faces. However, insurance against risk carries with it the potential problem of moral hazard, which occurs when the guarantee of indemnity distorts the insured's behavior because the insurance policy insulates her from the risks of her actions (Pauly 1968, 1974; Shavell 1979). Just as insured motorists may exercise less caution in their driving, states insured by alliance treaties have an incentive to behave more aggressively in negotiating with other states.

Generally, scholars of alliances take the position that moral hazard creates potentially harmful effects. Most notably, Snyder (1984, 1997) and Christensen and Snyder (1990) claim that alliances “embolden” state leaders to “entrap” unwilling allies in wars that they would prefer to avoid. Yuen (2009) shows that moral hazard increases allied states’ level of aggression in crisis bargaining, and this added aggression may heighten the risk of war or affect the bargaining settlements. As a result of the potential harmful effects of moral hazard, scholars argue that leaders may either avoid alliances, screen alliance partners based on their likelihood of behaving recklessly, or attempt to design treaties carefully so as to balance their dueling goals of deterring external threats while restraining alliance partners (Snyder 1984, 1997; Jervis 1994; Zagare and Kilgour 2003, 2006; Yuen 2009).

A more subtle side effect of moral hazard, however, is that states may be attracted to alliances because the tendency of an ally to behave aggressively actually enhances deterrence. The possibility that an allied state will negotiate aggressively may cause third-party adversaries to refrain from initiating a crisis. Likewise, a defensive alliance might make an alliance partner more willing to retaliate if challenged because it benefits from its ally’s support in war; this may cause a prospective adversary to be reluctant to initiate a challenge targeting the alliance partner (Smith 1995).

In cases where moral hazard advances the deterrence objective of an alliance, there is little cost to entrapment, because the third-party adversary calibrates its hostility toward the allies based on its expectation about its likelihood of winning a conflict if the target of its challenge does not capitulate. The combination of added resources from an ally and the increased willingness of the target to fight back encourages the third party to refrain from initiating violence. When encouraging an alliance partner to fight back if it is attacked enhances deterrence, then the goal of the contract is to induce a maximal amount of moral hazard so as to deter potential aggressors to such an extent that the risk of conflict is negligible. In this case, allies are not called upon to expend costly resources in support of their partners, as no conflict occurs. Thus, a priori, it seems equally likely that moral hazard will deter would-be challenges or increase the likelihood of conflict. Therefore, an important challenge for a theory of alliances is to identify the conditions under which moral hazard serves the deterrence purpose of the alliance rather than causing harmful effects that undermine the alliance’s objective.

In conceptualizing alliances as insurance contracts, our approach differs from other theories of alliance formation. The literature on this topic is diverse, from the theory of alliances as producers of the public good of security (Olson and Zeckhauser 1966; Sandler 1993; Sandler and Hartley 2001) to the idea that states take advantage of the free-rider problem inherent in alliance maintenance to reduce incentives among prospective allies to compete over a disputed good (Garfinkel 2004). These theories pay little attention to the effects of moral hazard on states’ behavior, however. In contrast, our approach considers how and when moral hazard impacts the effectiveness of an alliance and dictates its content and structure.

Our approach also differs from previous theories in that we make fewer assumptions about the political relationship between the alliance partners. For instance, scholars have shown that one motivation for alliance formation is that alliances can be used as signals to establish commitment to extended deterrence (Morrow 1994; Huth 1991; Smith 1995, 1998; Fearon 1997). Many of these studies also assume defenders intrinsically value the security of their ally. While signaling and commitment problems are important to alliance theory, we abstract away from signaling and commitment problems, because these effects are well established in the literature. Our approach enables us to focus on the effect of moral hazard on deterrence and intra-alliance bargaining when commitments are credible. Additionally, we do not require that prospective allies value one another’s security to motivate the formation of their alliance.²

Our theory also relates to the literature on moral hazard in crises. The problem of moral hazard in international relations has a long tradition. Many scholars have observed that committing aid to another state may cause that state to behave more aggressively than it otherwise would (Snyder 1984, 1997; Jervis 1994; Fearon 1997; Crawford 2001, 2003). This effect occurs not only in alliances and extended deterrence agreements but humanitarian intervention as well. Crawford (2005), for instance, argues that humanitarian intervention may incite unintended rebellions, and Kuperman (2008) provides evidence from Bosnia and Kosovo to show that humanitarian intervention can cause citizen rebellions that trigger retaliation by the state.

While many scholars have highlighted the dangers of moral hazards in commitments and intervention, scholarship has pointed out that intervention can be calibrated to balance the costs of moral hazard with the benefit of increased security (Wagner 2005). In this vein, Snyder (1997) argues that flexibility and ambiguity in alliances often reflect the intention of one or more countries to restrain an alliance partner because of fears of entrapment. Zagare and Kilgour (2003, 2006) create a formal model to capture the deterrence-versus-restraint phenomenon in alliances, finding a pooling equilibrium in mixed strategies in which an ally creates some uncertainty about whether it will intervene on the behalf of its alliance partner in a conflict. The authors interpret this equilibrium behavior as a kind of ambiguous alliance designed to restrain overly aggressive behavior, although they do not model the alliance formation stage. And in her model of third-party intervention with moral hazard, Yuen (2009) shows that alliances not only can strike a balance between deterrence and an ally’s overaggression but, when the ally’s costs for fighting are sufficiently high, the alliance can actually induce the ally to make small concessions to the challenger to avoid conflict. These models, however do not capture the potential benefits of moral hazard. For example, Zagare and Kilgour (2006) do not include contracting over the terms of the alliance prior to the initial moves by the protégé or the challenger. In their model, the defender decides whether to support the protégé after the protégé decides whether to concede to a demand. They show that in equilibrium randomization by the defender can emerge when there is learning about the types of the other players. This illustrates that randomization or “ambiguity’’ can mitigate the costs of emboldening. But these approaches are silent on the possibility of structuring alliances in a manner that uses emboldenment (or moral hazard) as a means to deter aggression by challengers and thus avoid the costs of supporting a protégé.

Also missing from the literature is a theory that explicitly formalizes the negotiating environment in which prospective allies, anticipating that moral hazard may both enhance deterrence and provoke aggression, bargain over the terms of their alliance. The theory offered by Snyder (1997) comes closest to what we have in mind. In his theory, prospective allies bargain over the terms of aid and the distribution of benefits of the alliance. A country’s bargaining power grows as the ratio of its valuation of the alliance to its valuation of its alternative options decreases. Bargaining power is also affected by a country’s relative valuation of the alliance compared to the valuations by its prospective allies. Relative valuation is determined by three factors: (1) the level of threat each ally faces from a prospective adversary, (2) how much each ally expects to gain from the other’s aid, and (3) the cost each ally pays for sacrificing some autonomy by joining the alliance. Moreover, a country’s bargaining power depends on its value of remaining unaligned or allying with another country. A country’s bargaining power matters because different countries have different fears and, therefore, desire different alliance structures. If a country’s predominant concern is that its alliance partner will entrap it in a war, then it will use its bargaining power to insist on a flexible or ambiguous alliance. On the other hand, if it is primarily worried about its ally abandoning it if conflict occurs, then it will negotiate for a firm and unambiguous alliance.

In our approach, alliance partners negotiate with one another directly about the content of the alliance, and some of Snyder’s ideas related to bargaining power are also relevant in our model. However, rather than negotiate over the relative flexibility or transparency of an alliance agreement, the alliance partners in our model bargain specifically over how much assistance they are willing to transfer to one another, as well as the division of the deterrence surplus created by forming a successful alliance. This is an important piece of the puzzle because prospective allies seem to care about the terms of the alliance: the level of their obligation to their allies and of the aid they will receive in return are directly relevant to the likelihood that deterrence will succeed. Moreover, if concerns about moral hazard can be satisfied by striking a deal on just the right level of assistance with just the right alliance partner, then mechanisms designed to restrain alliance partners are not always required even when entrapment fears prevail. This approach should be contrasted with existing work (e.g., Zagare and Kilgour 2006) that limits the choice of the defender to a decision of whether or not to offer a prescribed level of support after a conflict has begun.

Another advantage of our theory is that it avoids the criticisms leveled by Rauchhaus (2005, 2009) and others against scholars who have misunderstood or misused the concept of moral hazard. Rauchhaus’s main criticism is directed toward studies that invoke moral hazard as an explanation for the outbreak of conflict without justifying why the adversary would not back down as a result of a more aggressive rebel group or ally. We agree with this point, and our model demonstrates that this is precisely the mechanism by which deterrence is achieved. In response to Wagner's (2005) concern that many studies blame overinsurance for conflict, we show that, under certain conditions, overinsurance enhances deterrence rather than causing conflict.³

In what follows we provide a model of alliance formation where the presence of security threats and the cost of fighting create incentives to form agreements of “mutual-help” in the case of war. We close by considering an extension in which alliance contracts involve transfers that actually change warfighting and the probability of victory. Although the logic behind the analysis of this extension is consistent with the baseline model, it is easier in this case to generate alliances. Here their characteristics depend on the technology governing military transfers.

Model

Consider a world with three countries, a challenger and two potential targets. With probability $π_{i}$ , target $i \in {1, 2}$ has a crisis with the challenger. We assume that $π_{1} + π_{2} = 1$ . Once a crisis starts, the challenger decides whether or not to escalate by threatening target $i$ . If the challenger chooses the status quo instead of making a threat, the crisis ends peacefully and there is no change in the stakes controlled by either side. We therefore normalize the payoff for the status quo to 0 for the challenger and 1 for the target.

If the challenger makes a threat, on the other hand, the target country can choose to resist the threat and fight to keep the status quo, or capitulate and give in to the challenger’s threat. If the target fights the dispute is settled by war. The challenger wins a war against target $i$ with probability $p_{i}$ and pays a cost $k_{i}$ . For simplicity we assume that the challenger’s costs of fighting are known. We assume that the target countries have private information regarding their costs of war in the crisis. Each target has a cost of war $c_{i} \in [0, \overset{ˉ}{c}]$ . We let $F (c)$ denote the prior on this cost and assume it has a continuous density. If the challenger issues a threat, the target can avoid war by capitulating, but then the target must forfeit the “stakes” of the crisis, $x_{j}$ , keeping the fraction $1 - x_{j}$ for itself. The game is depicted in Figure 1.

Figure 1.

Crisis game

We imbed this crisis game into a larger ex ante bargaining game in which the two potential targets can make an agreement regarding the amount of aid they will provide to each other if one is engaged in war. Discussion of the bargaining game follows our discussion of the alliances.

In general, the agreement (or alliance) will constitute a transfer from one target (say, $i$ ) to the other (say, $j$ ) of an amount $θ_{j} \geq 0$ if there is a war. We can think of the ex ante alliance agreement as a form of decentralized insurance. Much as family members might help each other financially if one loses a job or experience an accident or illness, these countries are agreeing to help each other through the transfer of resources from one country to the other in the case of war. An alliance agreement then is a pair of transfers, $θ = (θ_{1}, θ_{2}) \in R_{+}^{2}$ . Security alliances are made ex ante in the sense that the players do not know their costs of war at the time of agreement, and they do not know which state will be involved in a dispute, though they have beliefs about the distribution of these attributes.

We are interested in settings in which war is possible in the absence of alliances. This requires that with positive probability $c_{j}$ is low enough so that $j$ prefers to fight (getting payoff $1 - p_{j} - c_{j}$ ) to capitulating (getting $1 - x_{j}$ ). This necessitates that $p_{j} < x_{j}$ for both targets. We maintain this assumption throughout the analysis.

Given an agreement, θ, we let $u_{i} (θ)$ denote the expected payoff to country $i$ from this agreement. These values will be pinned down by equilibrium play in the crisis subforms. Naturally, if the parties do not consent to an agreement, then their payoffs are given by $u_{i} (0, 0)$ . Specifying players’ payoffs in both the treaty and null treaty environments facilitates comparisons that ultimately enable us to determine which treaties will emerge from equilibrium bargaining. In some situations, it will be particularly useful to distinguish between alliance agreements that are Pareto Efficient and those that are not.

Definition 1: An agreement θ Pareto dominates agreement $θ^{'}$ if $u_{i} (θ) \geq u_{i} (θ^{'})$ for $i = 1, 2$ with a strict inequality for at least one of the players. An agreement is Pareto efficient if no agreement Pareto dominates it. Finally, a treaty, θ, is Pareto dominant if for all other agreements, $θ^{'}$ , one of the following is true: θ Pareto dominates $θ^{'}$ or $u_{i} (θ) = u_{i} (θ^{'})$ for $i = 1, 2$ .

The two target countries in our model reach an agreement by bargaining over the levels of support

θ_{1}

and

θ_{2}

that they will provide to each other in a war. We focus on a particularly canonical bargaining protocol: the alternating offers procedure with an exogenous risk of break down (Rubinstein 1982; Binmore, Rubinstein, and Wolinsky 1986). In period 1, country

1

makes a proposal that

2

may accept or reject. If

2

accepts, bargaining ends and the crisis subform is played. If

2

rejects the proposal, no agreement is reached and the game continues with a lottery that determines whether bargaining resumes or players proceed to the crisis subform without an agreement. With probability

z

the crisis subform is reached and the game ends with payoffs,

u_{i} (0, 0)

. With probability

1 - z

, there is no crisis in this period and the bargaining phase of the game proceeds to period

t + 1

. If a subsequent bargaining period is reached then the roles are reversed:

2

makes an offer that

1

either rejects or accepts. The process continues until either an agreement is reached or bargaining is interrupted by a crisis where one of the countries faces the challenger without an agreement.

A standard representation of this game form involves nature first randomizing over the costs faced by states $1$ and $2$ if they enter into a conflict. These costs are unknown by both players and are only revealed to the relevant state if a crisis involving that state occurs. Next the two countries bargain over an alliance agreement, which ends with an agreement or the arrival of a crisis, at which point one of the nations enters into the crisis subform with the challenger. In an equilibrium to the alliance formation game, the bargaining is predicated on continuation values from the behavior in possible crisis subforms that is sequentially rational. During bargaining there is no room for meaningful signaling, as the states do not yet know their costs of conflict. Given the dynamic nature of the game and the information environment, we will apply the equilibrium concept of perfect Bayesian equilibrium and impose the condition known as “no signaling what you don’t know,” which in this case requires that posterior assessments of $c_{j}$ are equivalent to the prior at any history (both on and off the path) in which country $j$ has not yet learned its type. Thus, players treat $c_{j}$ as drawn from the prior at all histories except for one in which $j$ is in a crisis situation and has therefore observed its cost. We thus use the term equilibrium to mean Perfect Bayesian Equilibrium (PBE) with this additional condition. A consequence of no signaling what you do not know, and the structure of this game, is that there is no interesting learning. Accordingly the analysis consists of checking conditions equal to those for subgame perfection at the alliance formation stage. We do not discount, and thus, there is no content to assuming that a crisis occurs immediately following an agreement between allies, but occurs with possible delay if no agreement is reached. What matters is that failure to agree on a treaty leads to the possibility of a crisis without a treaty, and agreement at period $t$ ensures that the contract is present if a crisis occurs subsequently. Formally, an equilibrium is a strategy profile which calls upon each state to play sequentially rational strategies in any crisis subform (which is dependent on the alliance reached at that history) as well as bargaining strategies that are sequentially rational given that the payoffs from any settlement or delay correspond to the payoffs from the specified play in the crisis subforms.⁵

Aside from the costs to fighting there are two types of exogenous random variables in this game. The first is whether failure to reach an agreement in any period will result in another chance to negotiate or whether the states will be confronted by a challenger without formalizing a treaty, and the second is whether a crisis will involve state $1$ or state $2$ . Uncertainty about the first is captured by $1 - z$ , and uncertainty about the second is captured by the parameters $π_{1} {, π}_{2}$ . For convenience we assume that a crisis will occur with probability 1. What is in question is just whether it will occur before a treaty is agreed to and which state will be involved. Because the model does not involve discounting, the exact timing of a crisis is not crucial–all that matters is whether an agreement has been formalized and which state is involved. An alternative approach would be to assume that $π_{1} {+ π}_{2} < 1$ with the interpretation that a crisis may never occur. The results here extend immediately to a model of this form since the payoff to a terminal history in which there is no crisis is just a constant shock to the payoffs of the players. Since the set of equilibria is unaffected by affine transformations, this extension will not affect the predictions of the model.⁶

Results

To analyze the incentives that the countries face in the bargaining phase of the game, we begin by analyzing the crisis subforms taking the alliance agreement as fixed. Given a pair of agreements ${θ = (θ}_{1} {, θ}_{2})$ , the target’s decision to go to war is well defined. Specifically, target $j$ will capitulate in equilibrium only if

\begin{array}{c} 1 - x_{j} \geq 1 - p_{j} - c_{j} + θ_{j} \\ c_{j} \geq x_{j} - p_{j} + θ_{j} . \end{array}

Perfect Bayesian equilibrium, or more precisely sequential rationality, implies that if $θ_{j}$ is greater than $c_{j} - x_{j} + p_{j}$ , then $j$ will choose to go to war to maintain the status quo. This observation makes it clear that target countries that anticipate some chance of war have a utility that is increasing in the level of commitments they extract from their ally. For the alliance partner, however, the alliance commitments create different incentives. Because alliances make wars more attractive, the targets will fight back more often; thus, the more the alliance partner promises to its allies, the more frequently it will need to make the transfer. The effect of $θ_{j}$ on a target country’s behavior is analogous to moral hazard in insurance markets: the fact that a player is being indemnified in the case of war may make it choose to fight wars that it would otherwise avoid. This model captures the extent to which an alliance can influence the behavior of the challenger in a particular parsimonious way, as illustrated in the next definition.

Definition 2: We will call an alliance agreement $(θ_{1} {, θ}_{2})$ deterrent for $j$ if it is the case that the challenger would make a threat against $j$ if $θ_{j} = 0$ , but it does not make a threat against $j$ given this alliance agreement. We say an agreement is deterrent if it is deterrent for at least one player.

Importantly, both allies are (weakly) better off if deterrence is achieved. The target in the crisis is never challenged, and the ally never has to follow through on its agreement to transfer resources because war does not happen. How these various security possibilities work out and how the incentives they create for prospective alliance partners shape behavior are at the center of our theory of alliance agreements. We break up the analysis into three natural cases. First, we consider settings in which there exist treaties that are deterrent for both

1

and

2

. Then, we consider the case where there is no treaty that is deterrent for either player. Finally, we consider the case where there is a treaty that is a deterrent for 1 but no treaty is a deterrent for 2.

Large Targets

We start by considering the case where the targets can form alliance agreements that change the behavior of the challenger in some circumstances. For our purpose, the term large means that the alliance decisions of a target can influence the threat decision of the challenger.⁷ The case of two large targets corresponds to the case where there are agreements that are deterrent for both players. In particular, consider the case where

p_{j} < k_{j} a n d F (x_{j} - p_{j}) (p_{j} - k_{j}) + (1 - F (x_{j} - p_{j})) x_{j} > 0

hold for both targets. Under these conditions, if the challenger believes that the target will definitely choose war if faced with a threat, the challenger will choose to keep the status quo. On the other hand, the second inequality states that the challenger will find aggression worthwhile if the target does not have an alliance agreement because the odds that the target will choose to fight are sufficiently small. Thus, under the null treaty, the challenger would threaten either target. This is a situation where the challenger is potentially deterrable.

In this situation there exist treaties, $(θ_{1} {, θ}_{2})$ , that induce target countries to fight regardless of their costs. In particular, whenever

θ_{j} \geq {\overset{ˉ}{c}}_{j} - x_{j} + p_{j},

all types of country $j$ will fight if challenged, and thus the challenger will keep the status quo rather than fight regardless of which target country it is paired with in a crisis. Let ${\underline{θ}}_{i}$ be the smallest amount of support that target $i$ needs to receive from target $j$ to deter a threat.

In the case of two large targets the equilibrium set has a stark structure.

Lemma 1: When $p_{j} < k_{j} a n d F (x_{j} - p_{j}) (p_{j} - k_{j}) + (1 - F (x_{j} - p_{j})) x_{j} > 0$ for both targets (i.e., both targets are large) every equilibrium alliance is deterrent for 1 and 2. In other words there is no perfect Bayesian equilibrium which places positive probability on ending with a treaty $(θ_{1}^{*} {, θ}_{2}^{*})$ with $0 < θ_{j}^{*} < {\underline{θ}}_{j}$ for $j = {1, 2}$ .

In this scenario, the targets get their maximal possible payoff, because they do not receive a challenge and do not need to transfer resources to each other or to the challenger. Thus, the agreement generates a deterrence surplus, because the target countries save the expected expense of being challenged, which include either conceding the stakes or fighting a costly war.

To complete our analysis, we show that agreements that are deterrent for both targets are reached without delay.

Lemma 2: Suppose that for both targets $p_{j} < k_{j} a n d F (x_{j} - p_{j}) (p_{j} - k_{j}) + (1 - F (x_{j} - p_{j})) x_{j} > 0$ (both targets are large). Then in every perfect Bayesian equilibrium it is the case that alliance agreements are reached without delay.

These two lemmas pin down what is possible in equilibrium. By adding a constructive argument of existence we can establish the following result for alliance agreements between large states.

Proposition 1: Suppose that $p_{j} < k_{j} a n d F (x_{j} - p_{j}) (p_{j} - k_{j}) + (1 - F (x_{j} - p_{j})) x_{j} > 0$ for both targets (both targets are large). Then in every perfect Bayesian equilibrium alliance agreements are reached without delay and they completely deter challenges.

Proof

The second part of the result follows from the lemmas. To establish existence, consider the strategy profile in which crisis bargaining follows the form described earlier, and at any history in which $j$ is the proposer it offers $({\underline{θ}}_{1} {, \underline{θ}}_{2})$ and at any history in which $j$ is given the chance to accept or reject an offer, he or she accepts any offer that is weakly preferred to the continuation payoff from the null alliance in the next period and agreement to the vector $({\underline{θ}}_{1}, {\underline{θ}}_{2})$ if the game does not end in the next period. Given the proposal strategies, this acceptance rule is clearly sequentially rational, and, given the acceptance strategies, the proposal strategy implies that at every period in which $i$ is the proposer, his or her proposal is accepted and a Pareto efficient treaty is reached. Since this treaty yields the maximal payoff to each player in the crisis bargaining game, this proposal strategy is sequentially rational.

Small Targets

Next we consider the case where it is more advantageous for the challenger to fight either target when it has an alliance. That is, we assume that $p_{j} > k_{j}$ . In this case, we know that for each $i$ , the targets’ expected utility as a function of $θ_{1}$ and $θ_{2}$ is given by

\begin{aligned} u_{i} (θ_{1}, θ_{2}) = & π_{i} (F (x_{i} - p_{i} + θ_{i}) (1 - p_{i} - {\hat{c}}_{i} (θ_{i}) + θ_{i}) \\ + (1 - F (x_{i} - p_{i} + θ_{i})) (1 - x_{i})) + (1 - π_{i}) (1 - (F (x_{j} - p_{j} + θ_{j}) θ_{j})), \end{aligned}

where ${\hat{c}}_{i} (θ_{i}) = E [c | c < x_{i} - p_{i} + θ_{i}]$ denotes the expected cost of of fighting for state $i$ conditional on war given that $i$ decides whether or not to fight according to the sequentially rational rule of capitulating if and only if $c_{i} \geq x_{j} - p_{j} + θ_{j}$ .

An essential fact for small targets is that the null treaty $(0, 0)$ is efficient.

Lemma 3: Assume that $p_{j} > k_{j}$ . The null treaty is Pareto efficient.

In other words every treaty that makes one country better off makes the other country worse off relative to the null treaty. Why is this true? The argument relies on the standard interpretation of moral hazard. Any treaty other than

(0, 0)

involves a distortion such that moral hazard induces some types of at least one of the countries to resist a threat and fight a war that it would not absent the treaty. But fighting a war that should not be fought imposes a cost. The fact that the treaty compensates the fighting country represents a redistribution of this inefficiency between the treaty parties and proves that the

(0, 0)

agreement is Pareto efficient. Most importantly for our analysis, the ally that is committed to making the transfer internalizes the inefficiency and will not want to accept such a contract.

A feature of a perfect Bayesian equilibrium to the alternating-offers bargaining game is that a weak participation or individual rationality constraint must be satisfied in equilibrium. If $i$ is the proposer in period $t$ , then $j$ will not accept an offer that does not give it a payoff in the crisis game that is at least as high as the payoff from the null treaty. And clearly the proposer would also never propose a treaty that gave it a lower payoff. Thus, we see that every treaty $θ$ , that is reached in an equilibrium must satisfy the constraint $u_{i} (θ) \geq u_{i} (0, 0)$ for $i = 1, 2$ . Gaining $θ_{i}$ for $i$ comes at a cost for $j$ , and for at least one player this cost is not justified as the contract induces wars that should not be fought with positive probability.

Proposition 2: Assume that $p_{j} > k_{j}$ . Then in every perfect Bayesian equilibrium the alliance agreement is null alliance, $(0, 0)$ .

It is instructive to contrast the situation where there are two large states with the case of two small states. With the latter, the only effect of an alliance is that it increases the probability of war through moral hazard, which is detrimental for both parties. In contrast, we saw that with two large states, an alliance agreement deters the challenger from making a threat, which stops war altogether. This generated a deterrence surplus, a source of “rents” from contracting that are directly accrued by the target when the challenger changes its decision from “threaten” to “accept the status quo.” From these results, it is clear that if both states are small, having an alliance does not deter the challenger, which means there is no increase in the total welfare of the targets, and hence, no value of having an alliance at all. We now explore the intermediate case where it is possible to distort how the challenger interacts with one state but not the other.

One Large and One Small Target

Consider the situation where 1 is large enough so that if $θ_{1} \geq {\underline{θ}}_{1}$ initiation against 1 will be deterred, but 2 is sufficiently small such that no treaty will deter a challenger from threatening it. To keep the meaning clear, we will use the label L for country 1, the large country, and s for country 2, the small country. In interpreting this case it is useful to realize that whether the challenger is willing to fight with a target depends both on the prize at stake and the details of how war would unfold. The assumption that state L is large then says something about the value to the challenger of forcing a change in the status quo vis-á-vis state L as well as something about the likely outcome and costs of fighting state L.

We begin the analysis of this mixed case by showing that in every perfect Bayesian equilibrium the large country ends up with a deterrent agreement.

Lemma 4: Suppose that $p_{j} < k_{j} a n d F (x_{j} - p_{j}) (p_{j} - k_{j}) + (1 - F (x_{j} - p_{j})) x_{j} > 0$ for country $L$ but not for country $s$ . Then in every perfect Bayesian equilibrium the alliance agreement has

θ_{L}^{*} \geq {\underline{θ}}_{L} .

From this lemma we see that the gains from an alliance for the large country are always captured. The question that remains is how these gains might be distributed. To answer this question it is sufficient to fix

θ_{L}^{*} \geq {\underline{θ}}_{L}

and define

{\overset{ˉ}{θ}}_{s} = {θ_{s} | u_{L} (θ_{L}^{*}, θ_{s}) = u_{L} (0, 0)},

as the maximal transfer to s that $L$ is willing to make in exchange for entering the crisis game with an alliance that deters initiation against it. Recall that because the challenger is deterred from threatening $L$ , it is “free” for $s$ to join the alliance — it will not need to make its promised transfer — whereas supporting $s$ involves actual costs to $L$ . Thus, the two target countries bargain over how much $L$ will pay in support of $s$ so that $L$ obtains deterrence. This reduces conceptually to bargaining between $L$ and $s$ over a pie of size ${\overset{ˉ}{θ}}_{s} > 0$ . For the remainder of this section we focus on the problem of how $L$ and $s$ divide this pie. Though an alliance is an agreement of the form $({\underline{θ}}_{L}, θ_{s})$ , it is natural to think of them bargaining over the set

Y = {(y_{L}, y_{s}) \in R^{2} | y_{L} + y_{s} = {\overset{ˉ}{θ}}_{s} a n d y_{i} \geq 0 f o r i = L, s} .

This agreement results in the alliance that we compactly write as the vector, $({\underline{θ}}_{L}, y_{s})$ .

The game structure of alternating offers with the risk of breakdown has been well studied and our analysis consists mostly of showing that our particular application satisfies a known set of sufficient conditions for existence and uniqueness of equilibria. The results will hold when $z$ , the probability that negotiations exogenously end after an alliance proposal is rejected, is sufficiently large. The interpretation here is that $z$ captures the risk to failing to reach an agreement in the current round of negotiations. When $z$ is large there is a high chance that after walking away from the negotiation table without an alliance one of the states will find itself in conflict with the challenger. In contrast when $z$ is small there is little chance that a delay will have any real consequences. Following the notation of Osborne and Rubinstein (1990), we denote an alliance bargaining game with the risk of breakdown at a probability $z$ as $Γ (z)$ .

Equilibrium conditions in the bargaining game involve finding offers that are maximal for the proposer and minimally acceptable for the target that will get to propose in the next round (if it is reached). Accordingly, we need to be able to refer to an offer that makes a player indifferent to another offer that is accepted one period later. Let $v_{i} (θ_{s})$ be the amount of support given to the small country by the large country that would make country $i$ indifferent between this particular agreement in the current period and (possibly) getting the settlement that gives the small country $θ_{s}$ one period later. Recall that under both $v_{i} (θ_{s})$ and $θ_{s}$ , country L gets a deterrent transfer. In other words this amount $v_{i} (θ_{s})$ satisfies the equivalence

u_{i} ({\underline{θ}}_{L}, v_{i} (θ_{s})) = z u_{i} (0, 0) + (1 - z) u_{i} ({\underline{θ}}_{L}, θ_{s}) .

Lemma 5: (Existence and Uniqueness) If

max_{i} {sup_{θ_{s}} | \frac{u_{i}^{'} ({\underline{θ}}_{L}, θ_{s})}{u_{i}^{'} ({\underline{θ}}_{L}, v_{i} (θ_{s}))} |} < \frac{1}{1 - z},

then there is an essentially⁸ unique perfect Bayesian equilibrium to the alliance agreement game.

Although the sufficient condition for the lemma may be quite difficult to verify, the result is useful; our next proposition will use the result to provide a sufficient condition with an immediate substantive interpretation of when $z$ , the risk of conflict in the immediate period, is close to $1$ . First, we highlight an important implication of the lemma, which illustrates that the alliance agreement can be described in a tractable way. We can describe stationary proposal strategies with 2 variables. Let $x$ denote the offer that L makes if he or she is the proposer, and let $y$ denote the offer that s makes if he or she is the proposer.

Corollary 1: Assume the condition of lemma 5. The unique equilibrium solution solves two indifference conditions.

u_{L} ({\underline{θ}}_{L}, y^{*}) = z u_{L} (0, 0) + (1 - z) u_{L} ({\underline{θ}}_{L}, x^{*})

u_{s} ({\underline{θ}}_{L}, x^{*}) = z u_{s} (0, 0) + (1 - z) u_{s} ({\underline{θ}}_{L}, y^{*}) .

And country s accepts any proposal such that

x \geq x^{*}

, and country L accepts any proposal such that

y \leq y^{*}

From the corollary, we may use the implicit function theorem to obtain comparative statics on the equilibrium transfers to the small state. To do so, we write the two indifference conditions as the following system of equations:

u_{L} ({\underline{θ}}_{L}, y^{*}) - [z u_{L} (0, 0) + (1 - z) u_{L} ({\underline{θ}}_{L}, x^{*})] = 0

u_{s} ({\underline{θ}}_{L}, x^{*}) - [z u_{s} (0, 0) + (1 - z) u_{s} ({\underline{θ}}_{L}, y^{*})] = 0,

where the individual terms are given by

\begin{aligned} u_{L} ({\underline{θ}}_{L}, y^{*}) = π_{L} + (1 - π_{L}) (1 - F (x_{s} - p_{s} + y^{*}) y^{*}) \\ u_{L} (0, 0) = π_{L} (F (x_{L} - p_{L}) [x_{L} - p_{L} - {\hat{c}}_{L} (0)] - x_{L}) + 1 \\ u_{L} ({\underline{θ}}_{L}, x^{*}) = π_{L} + (1 - π_{L}) (1 - F (x_{s} - p_{s} + x^{*}) x^{*}) \\ u_{s} ({\underline{θ}}_{L}, x^{*}) = (1 - π_{L}) [F (x_{s} - p_{s} + x^{*}) (x_{s} - p_{s} - {\hat{c}}_{s} (x^{*}) + x^{*}) - x_{s}] + 1 \\ u_{s} (0, 0) = (1 - π_{L}) [F (x_{s} - p_{s}) (x_{s} - p_{s} - {\hat{c}}_{s} (0)) - x_{s}] + 1 \\ u_{s} ({\underline{θ}}_{L}, y^{*}) = (1 - π_{L}) [F (x_{s} - p_{s} + y^{*}) (x_{s} - p_{s} - {\hat{c}}_{s} (y^{*}) + y^{*}) - x_{s}] + 1. \end{aligned}

In the discussion that follows, we will refer to the system (5) and (6) as $H_{1} = 0$ and $H_{2} = 0$ , respectively, where the dependence on parameters and endogenous values $x^{*}$ and $y^{*}$ are clear. For what follows, we will assume that the costs are drawn from the uniform distribution on support $[0, 1]$ implying ${\hat{c}}_{s} (y^{*}) = \frac{x_{s} - p_{s} + y^{*}}{2}$ , ${\hat{c}}_{L} (x^{*}) = \frac{x_{L} - p_{L} + x^{*}}{2}$ , and $F (w) = w$ if $w$ is in $[0, 1]$ .

Before we find the comparative statics for solutions to this system, it is instructive to think about the equilibrium conditions. We see from the corollary that an equilibrium is characterized by a pair of indifference conditions. As second proposer, s needs to be indifferent between accepting the offer that is made by the first proposer or rejecting it, and L needs to be indifferent between accepting and rejecting the offer that s would make if it got the chance in the next round of bargaining. This indifference, of course, hinges on the offers that each player would make in the next round if they rejected their partner’s offer in the current round. The amount of support that $L$ pledges to $s$ on the equilibrium path is the minimal amount that $s$ is willing to accept given anticipation of equilibrium play following a rejection of L’s proposal. Accordingly, the comparative statics on the amount of support promised to, s denoted $x^{*}$ , depend on an analysis of the indifference condition for player s. This condition is equation (6). If there is an exogenous increase in $π_{L}$ , which is the probability that L faces a crisis with the challenger, we observe that this unambiguously increases all three terms in equation (6). If s is less likely to be involved in a crisis, then s is better off when it has an alliance agreement with L because deterrence occurs. On the other hand, when L is less likely to be in a crisis, it is better off if there is no agreement between the two states because it will be more likely to make transfers to support s in a war. But since the left-hand side of (6) is the difference of the current and continuation utilities, it is ambiguous whether $x^{*}$ needs to increase or decrease to offset this exogenous shock.

In contrast, the indifference condition for L, equation (5), is less ambiguous. For $z$ close enough to 1, the primary effects occur in the first and second terms of the left-hand side of equation (5). Here an increase in $π_{L}$ is good for L if an agreement is reached, since it is deterrent for L and bad if no agreement is reached. Thus, the left-hand side of equation (5) increases so to maintain the equality with $0$ , $y^{*}$ needs to decrease. This latter effect — that $y^{*}$ or the amount of support offered by s to L decreases — is not picked up on the equilibrium path if L is the first proposer, because in equilibrium s accepts L’s offer when L proposes. It would, however, be picked up in a different game in which $s$ moved first in the bargaining game.

We now apply the implicit function theorem to analyze the comparative statics. Importantly, we must capture both direct and indirect effects of the parameters of interest. For example, a change in the parameter $p_{s}$ will have a direct effect on the equilibrium values $x^{*}$ and $y^{*}$ , but, since $x^{*}$ and $y^{*}$ are also related in equilibrium, the effect of $p_{s}$ on $x^{*}$ will have an indirect effect on $y^{*}$ that is weighted by the effect that a change in $x^{*}$ has on $y^{*}$ . To capture all of these direct and indirect effects we treat the left-hand side of the system (5) and (6) as a two-by-one vector $H (y^{*}, x^{*}, γ)$ and the right-hand side as a vector of two zeros. Here, we let $γ$ be a vector capturing the exogenous parameters $(x_{s}, x_{L}, p_{s}, p_{L}, π_{L})$ . The implicit function theorem tells us that as long as a technical condition known as the transversality condition is satisfied, then the derivative of $y^{*} (θ)$ with respect to a coordinate of $θ$ is given by total differentiation.

We focus on the comparative statics on $x^{*}$ since this is the offer that is made and accepted on the equilibrium path. Using the implicit function theorem, we obtain the following qualitative comparative statics for equilibrium values of $x^{*} (x_{L}, x_{s}, p_{s}, p_{L}, π_{L})$ . We focus on the effects of changes in $p_{s}, p_{L}$ , and $π_{L}$ .⁹

Proposition 3: Assume $z$ is sufficiently close to 1. Then there is an essentially unique equilibrium and the offer that is made in period 1 and accepted, $x^{*}$ , is increasing in $p_{s}$ and $p_{L}$ . For any fixed values of $p_{s}$ and $p_{L}$ , $x^{*}$ is monotone in $π_{L}$ , but it is either always decreasing in $π_{L}$ or (depending on parameters) increasing in $π_{L}$ for low values of $p_{s}$ and then decreasing in $π_{L}$ for higher values of $p_{s}$ .

We pause to interpret the comparative statics. In equilibrium we should see the level of support offered to the state that is actually likely to wind up fighting respond negatively to the odds that either target would win in conflict. So as either party gets stronger relative to the challenger the magnitude of the promise to the state that is likely to fight will go down. Moreover, how the alliance terms respond to the relative risk that either party will be involved in a conflict is complicated. There is a monotonic effect if the other parameters are held fixed, but variation in several parameters can yield nonmonotonic results. This last point suggests that attempts to correlate risk of crisis and alliance terms need to be nuanced.

Let us consider how a calibrated version of this model might explain the 1893 Franco-Russian alliance. The setting of the alliance corresponds to our analysis in several ways. First, France and Russia had little in common except for a shared threat in the Triple Alliance. Second, our characterization of an alliance between a small target and a large target bears some similarities to the Franco-Russian alliance. Russia felt vulnerable after Germany refused to renew the German-Russian Reinsurance Treaty in 1890. It turned to France to gain security from Germany. The Russians were satisfied that the existence of a nominal alliance with France was sufficient to deter Germany from attacking Russia (Snyder 1997, 116). However, neither France nor Russia believed an alliance would be as effective at preventing war between France and Germany. France’s insecurity led it to insist on specific promises of military assistance in the agreement. Russia worried about moral hazard; they feared the alliance agreement could lead to war between France and Germany over Alsace-Lorraine. Third, there was extensive bargaining over the terms of the alliance, with France pushing aggressively for Germany to be named the primary threat to the alliance, for a mutual promise to mobilize immediately and simultaneously against Germany, and for a specific amount of military support to be committed by each alliance member. In the event of a conflict against Germany, France promised to dedicate 1.3 million troops and Russia committed to contribute 700,000 to 800,000 troops, which was approximately half of Russia’s fighting force (Snyder 1997, 117).

According to Snyder (1997), the final agreement is puzzling because the treaty terms, when everything from troop commitments to conditionality are taken into account, favored France in spite of the fact that it was at greater risk from Germany and appeared to need the alliance more than Russia. The comparative statics for our small-large target case may provide some guidance. Since Russia could easily gain security through its alliance with France, there was little contention over what France would supply to Russia in the event of a conflict between Germany and Russia. The main point of negotiation was the amount Russia would commit to supply to France given that there remained a risk of a war between Germany and France after the alliance was signed. Germany was stronger than Russia and France in pairwise comparisons, and its relative share of military resources was growing (Snyder 1997, 110). Germany’s growing military strength likely reflected a corresponding increase in the probability it would win a conflict with either France or Russia. The comparative statics, therefore, suggest that Russia’s promise of assistance to France was positively associated with Germany’s rising relative strength.

The comparative statics also provide some additional insight about how the treaty terms may have responded to the the relative risk that France and Russia would be involved in a conflict with Germany. Because France was at greater risk of being involved in a conflict and Germany was more likely to win, the amount Russia had to pay France to get a treaty were relatively high. Thus, our analysis helps explain why the Franco-Russian alliance favored France in spite of the fact that France appeared to be in greater need of the alliance. Russia needed the alliance to get deterrence, but because its security was achievable through the alliance and France remained insecure, Russia needed to make sizable promises to France and those promises were large due to Germany’s relative strength advantages and the greater risk faced by France.

Alliances that Change the Probability of Victory

Up to this point, we have treated alliance agreements as a cost-sharing measure. This is a natural approach if we think of alliances as obligating countries to share the burden of war. Our analysis shows that even when an alliance between two potential targets does not change the payoffs to the challenger directly, the resulting changes in the targets’ incentives and actions can change the challenger’s incentives to make a threat. In other words, by distorting the behavior of target countries, alliances can distort the behavior of attackers even when the alliance does not directly alter the payoffs of the attacker.

We might, however, think that alliances enhance alliance partners’ strength relative to the challenger by aggregating capabilities. In this case, agreements change the incentives the target and the challenger. A modeling strategy for alliances that have this effect has an alliance agreement $ρ = (ρ_{1}, ρ_{2})$ that increases the probability that $i$ wins a war by the quantity $ρ_{i}$ . In this model the set of possible alliances would involve $ρ \in [0, r_{1}] [0, r_{2}]$ with $r_{i} \leq 1 - p_{i}$ . Let $c_{i} (r_{i})$ denote the strictly increasing cost function capturing the cost to $i$ ‘s alliance partner of increasing the probability that $i$ wins by $ρ_{i}$ . Like before, we assume the costs are only borne if target $i$ ends up at war.

Much of the intuition from the case in which an alliance only affects the targets’ costs of war extend here. The concept of a “large” country in the “paying costs” model translates here to the case where it is possible to increase the probability that $i$ wins enough to deter initiation by the outsider. Thus, the case of two large countries involves the assumption that for each $i$ there exists a $ρ_{i}$ such that $ρ_{i} \geq p_{i} - k_{i}$ . As a result, there are alliances that fully deter war and end up costing the alliance partners nothing. Given alliances of this form exist, and if the second part of condition 1 holds, then every equilibrium must involve an alliance of this form.

The natural extension of our concept of small countries involves the assumption that $r_{i} < p_{i} - k_{i}$ . In this case, it is not possible for $j$ to make $i$ strong enough so that an outsider believes $i$ will fight rather than acquiesce. In other words, $i$ prefers $0$ over the lottery between $x$ and war. It is then not possible to deter the initiator but, in contrast to the case of private values, a treaty that changes the probability that $i$ wins need not be inefficient. In particular, consider when $ρ_{i}$ satisfies the condition $p_{i} - ρ_{i} > k_{i}$ . The promise $ρ_{i}$ has two effects on $i$ ’s payoffs. It expands the set of costs-types — the set of targets with different war costs — that will reject the challenger’s threat (and thus fight), and it increases the payoff to $i$ if it fights. Conditional on fighting, the payoff to $i$ increases by $ρ_{i}$ times the stakes of war (which are 1) and conditional on $j$ transferring $c_{i} (ρ_{i})$ to $i$ the value gained by $i$ is exactly $ρ_{i}$ . Accordingly, an alliance between two small countries can be efficient if and only if $c_{i} (ρ_{i}) \leq ρ_{i}$ .

Thus, in the case of two small countries, a treaty is possible in equilibrium. In particular, contrasted with the first model, 1 and 2 now bargain over treaties in which a treaty $ρ$ results in the gain to $i$ of $π_{i} F_{i} (x_{i} - p_{i} + ρ_{i}) ρ_{i}$ , and it results in a cost to $j$ of $π_{j} F_{j} (x_{j} - p_{j} + ρ_{j}) c_{j} (ρ_{j})$ . For i to be willing to accept (or propose) a treaty of this form in equilibrium, the treaty must provide weakly positive gains to $i$ . Moreover, as this condition must be satisfied for both players, any treaty that is accepted with positive probability in an equilibrium must simultaneously satisfy the inequalities

\begin{aligned} π_{1} (F_{1} (x_{1} - p_{1} + ρ_{1}) ρ_{1} \geq π_{2} (F_{2} (x_{2} - p_{2} + ρ_{2}) c_{2} (ρ_{2}) \\ π_{2} (F_{2} (x_{2} - p_{2} + ρ_{2}) ρ_{2} \geq π_{1} (F_{1} (x_{1} - p_{1} + ρ_{1}) c_{1} (ρ_{1}) . \end{aligned}

An immediate observation is that in the case where $c_{1} (ρ) = c_{2} (ρ) = ρ$ , either both equations are satisfied with equality or neither is satisfied. Here small countries may form an alliance, but the payoffs are the same as in the trivial treaty, $ρ = (0, 0)$ . We can then conclude that if a technology of war takes a transfer $ρ$ from small country $i$ and turns it into an increase in the probability of victory for $j$ that is exactly equal to the cost of the transfer, then there is no incentive for $i$ and $j$ to form an alliance, even if $j$ ’s chances of winning a war are improved. The case where $c_{i} (ρ_{i}) > ρ_{i}$ for $i = 1, 2$ will also not support nontrivial treaties, as these treaties involve inefficiencies.

We are left with the case where the technology of war makes the probability of winning increase at a faster rate than the cost of transfers to the ally. Here, small countries will have an incentive to increase their collective payoffs by supporting each other during fighting. For example, consider a $c_{i} (ρ_{i}) = α ρ_{i}$ for $i = 1, 2$ with $α < 1$ , and $x_{1} = x_{2} = x, p_{1} = p_{2} = p, π_{1} = π_{2}$ , with uniform distributions on the players costs. Then the preceding system of inequalities becomes

\begin{aligned} \frac{x - p + ρ_{1}}{x - p + ρ_{2}} \geq \frac{α ρ_{2}}{ρ_{1}}, \\ \frac{ρ_{2}}{α ρ_{1}} \geq \frac{x - p + ρ_{1}}{x - p + ρ_{2}} . \end{aligned}

Now a set of alliances that are Pareto superior to the trivial treaty exists. For example, alliances with $ρ_{1} = ρ_{2}$ are in the interior of the set of treaties satisfying these constraints. Under our bargaining protocol, some agreement in which $ρ_{i} > 0$ for at least one player (and both in protocols that are not too extreme) would emerge. Even without characterizing the equilibrium to such a model here, we can make two observations concerning the conditions that lead to productive alliances in international relations. First, alliances are attractive when it is possible for a transfer between target country 1 and target country 2 to alter the behavior of a potential challenger. This is what makes alliances between a large country and a small country viable in the private values model, and it is also what makes an alliance between two small countries supportable, under certain conditions, in the common values model. However, it is also important to observe that the formation of an alliance in the common values model relies on a technological benefit of the treaty; without such a technology, small countries would not find alliances beneficial.

Conclusion

By viewing security alliances as a form of decentralized insurance and focusing on alliance commitments in which aid may cause moral hazard, we make four key findings. First, when two large countries are threatened, they form alliances that exhibit what international relations scholars might call threat balancing. Each side commits to aid the other to a degree that the challenger is deterred from escalating a dispute. Second, the ability of large states to manipulate the incentives of the challenger by making allies more aggressive is a key element to explaining how security commitments arise. In contrast, an alliance of two small country targets fails to deter the challenger, with the result that moral hazard increases the incidence of war and the alliance serves to redistribute the war’s costs from the party at war to its ally. But since the cost of war is completely internalized by the target who is not at war, these social welfare–decreasing agreements cannot arise in equilibrium.

Third, when analyzing the asymmetric alliance case, we see that alliance agreements always generate private benefits through deterrence for the large country. The formation of an alliance in this environment then turns on the bargaining between the now-safe large country and the still-threatened small country regarding how much of this benefit will be returned in the form of transfers to the small country. Finally, we see that when the risk of a crisis is severe, that is, when $z$ is sufficiently large, there is unique equilibrium and we can make strong prediction regarding the bargaining outcome between targets. This may explain, in part, why some actual alliance agreements in history have been forged on the eve of a crisis and why they describe in detail the conditions of activation as well as the amount and form of aid.

This article has sought to explain why countries form alliances and what terms they choose when they do so. Our analysis employed a basic agreement structure in which allies exchange security guarantees, which represents a foundational class of alliances agreements in international relations. Although empirically alliance agreements take many forms, the basic environment examined here addresses fundamental questions concerning the role of moral hazard in producing deterrence and thus in alliance formation. As suggested by the empirical evidence, a more complete understanding of the role of alliances in international politics will require further investigation into why and what kind of agreements are written in this basic environment.

Footnotes

Appendix

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Notes

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