Conflict bargaining as a signal to third parties

Abstract

Within the international system, states frequently fight even when opponents have little or nothing to offer them. Yet, international relations scholars envision conflict as a means for states to acquire some amount of a desired good, and view bargaining through this lens. This paper presents a model in which war and conflict bargaining can serve as signals to potentially hostile third parties. The analysis indicates that states sometimes have incentives to bargain harder than they would otherwise, in order to conceal information from future enemies. This can lead to war, even when a peaceful settlement should be possible.

Keywords

Bargaining observation security signaling

“If, when the chips are down, the world’s most powerful nation, the United States of America, acts like a pitiful, helpless giant, the forces of totalitarianism and anarchy will threaten free nations and free institutions throughout the world.”

(President Richard M. Nixon on the war in Vietnam, 30 April 1970)

In 1954, President Eisenhower articulated a “domino theory” of foreign policy, arguing that if one state in Southeast Asia (e.g. Vietnam) fell to Communist rule, all of the region would soon follow. In accordance with this principle, the United States soon involved itself in a war between Vietnamese Communist and anti-Communist forces. Nixon’s 1970 address on behalf of the war effectively echoes Eisenhower’s claim, arguing that the United States’ decision in Vietnam would be viewed by other states as a signal of its resolve. By behaving “like a pitiful, helpless giant”, and refusing to defend Vietnam actively, the U.S. would demonstrate an unwillingness to fight, opening itself up to threats and attacks from enemies around the world.

In justifying the wars in Southeast Asia, American presidents did not argue that the United States needed to initiate or continue a war in a state to realize particular gains from the immediate conflict. Instead, these wars were defended as demonstrations of American willingness to support democratic (or quasi-democratic) allies. In other words, the American involvement in Vietnam had little (if anything) to do with Vietnam itself. Rather, it was intended as an exhibition to the rest of the world— in particular, to the Communist powers who were the United States’ chief rivals— of American power and determination.

Ultimately, this means that if not for the perceived need for such an exhibition, the United States might have avoided war in Vietnam, or at least significantly reduced its duration and cost. Instead, because the American goals in Vietnam were mostly unrelated to the prize over which the combatants were fighting, no deal short of total capitulation by the Vietnamese would have sufficed. Indeed, by 1966, Johnson had demanded the “unconditional surrender” of North Vietnam (Peake, 1986: 340), suggesting that there was no room for real negotiation. Given any opposition, the United States had to incur costs and to signal toughness, in order to achieve its broader goals on the world stage.

U.S. foreign policy in Southeast Asia during the Cold War is not the only instance in which this phenomenon occurs. Any time that states enter into a dispute, the conflict is generally observed by other states in the international system. After witnessing a state’s conflict behavior, a third party learns and adjusts its own decisions accordingly. Knowing that this will occur, a rational state is likely to act strategically, taking into account what potential rivals may learn from its actions and altering its behavior appropriately. Although the specter of subsequent conflict might not be sufficiently important to prompt strategic changes in all cases, there are many disputes, especially those involving states with enduring rivals, where it will matter greatly.

In this paper, I argue that conflict bargaining in a two-player context is only part of the story. Current models of bargaining and war are well suited to determining how states will act when concerned only with the current conflict. However, international disputes do not occur within a vacuum. Like the U.S. during the Cold War, states often concern themselves with how their actions will be perceived by likely enemies and allies. States that believe that they are being observed by a relevant hostile third party will act differently than those that believe they are not. In the models below, I show how conflict behavior changes when disputants must consider the signals that they send to outside parties. I then provide and discuss an example of an empirical case in which a third party featured prominently in the bargaining situation, in order to demonstrate both the plausibility of the model’s assumptions, and the mechanisms that underlie it.

Previous work

That a state’s decisions during a conflict may be affected by outside observers is not strictly unheard of within international relations. Indeed, Jervis (1988: 678) notes that a leader might “fight a war he knows he will lose … to impress third parties” because “[o]ther nations which had quarrels with the state might infer that it is willing to fight even when its position is weak, and such an inference might strengthen the state’s bargaining position”. Thus, fighting to demonstrate resolve can affect future interactions with outside parties. Jervis’ argument is an extension of the well-known chain-store paradox in economics (Selten, 1978; Kreps and Wilson, 1982). In addition, the audience cost literature (e.g. Schelling, 1960; Fearon, 1994; Schultz, 2001; Sartori, 2005; Tomz, 2007; as well as the special issue of Security Studies Volume 21 (August 2012))— though focused on domestic audiences— suggests that the conflict behavior of one party can affect the evaluations of others. The situation portrayed here can be likened to a form of “international audience costs” wherein third parties can update information about a potential future opponent by observing its behavior in a prior dispute. Even if a state takes pains to make itself seem strong or resolved, its behavior in a crisis can convey a different message, leading to “punishment” from potential enemies. Importantly, the microfoundations of punishment by an observing state are far simpler than those of a domestic audience,¹ as the former finds itself directly opposed to the state in question, and this punishment is merely the exploitation of a favorable scenario by this outside actor.

Despite these observations, the formal literature on conflict bargaining is replete with assumptions of myopic behavior on the part of states involved in disputes. Within many of these works, there is an implicit supposition that neither a state’s behavior during a dispute nor that dispute’s outcome will have any effect on future conflicts. By extension, then, a rational combatant will engage in behavior that will maximize its payoff during the current period, simply balancing magnitude of demands against costs of fighting, without having to worry about the strategy negatively affecting other disputes. This assumption is found— at least implicitly— in most of the rational choice work on bargaining and war, including many of the most influential pieces, and underlies much of the extant theory on the causes and duration of war (e.g. Fearon, 1995; Powell, 1999; Slantchev, 2003; Smith and Stam, 2004).

Among the rationalist explanations for war onset, private information may be the most frequently invoked.² As Powell (2004b: 231) notes, “recent formal work in international relations theory… focuses almost entirely on informational asymmetries”. The private information story behind conflict onset comes primarily from Blainey, who argued that “[w]ars usually begin when two nations disagree on their relative strength, and wars usually cease when the fighting nations agree on their relative strength” (Blainey, 1988: 293). Building on the work of Blainey and other earlier scholars (e.g. Wittman, 1979; Morrow, 1989; Bueno de Mesquita and Lalman, 1992), Fearon (1995) demonstrates quite elegantly that, in a dispute in which the participants have full information, there should always exist a range of settlements that both sides prefer to going to war.³ The corollary to this is that a lack of information may cause both states to overestimate their probabilities of victory, reducing or eliminating the bargaining range, and leading rational states to fight a costly war over the prize in question.⁴ This approach to conflict bargaining has been adopted by a number of scholars (e.g. Powell, 1999; Filson and Werner, 2002; Slantchev, 2003; Powell, 2004a; Filson and Werner, 2007) interested in modeling war from a rational perspective. While these studies build upon the basic model in a number of interesting ways, they all maintain the same premise: the parties involved in a dispute are simply interested in maximizing their payoffs within the current conflict.

If war onset is the result of a bargaining failure stemming from private information, then it follows that fighting should alleviate this problem. As battlefield outcomes are essentially non-manipulable— unless a state is “fighting to lose”, its efforts should reflect its capabilities— they can be expected to reveal information to the parties involved, eventually eliminating the problems caused by a lack of knowledge (Goemans, 2000; Wagner, 2000; Filson and Werner, 2002; Slantchev, 2003; Powell, 2004a). Slantchev (2004) refers to this as the “principle of convergence”, and argues that states should continue fighting until a sufficient amount of information has been revealed to allow the parties to locate and agree upon a mutually acceptable settlement (see also Smith and Stam, 2004). Two crucial assumptions underlie this theoretical framework: first, that private information is the exclusive cause of war onset; and, second, that states are seeking to reveal, rather than to obscure information. While battlefield outcomes may be non-manipulable,⁵ Slantchev shows formally that the manipulable information at the bargaining table tends to be the more important of the two sources. It is conceivable that a state wishing to keep information private could use wartime negotiations to make itself seem more (or less) powerful, overriding the information conveyed by fighting. The assumption that is traditionally made by scholars— that states only care about how much they get out of the conflict— makes such a strategy irrational. Relaxing this supposition, by introducing a third party, could potentially change a state’s incentives.

Although the literature on conflict bargaining has generally neglected the role played by third parties, the scholarship on reputational effects in interstate conflict has considered it quite explicitly. This literature builds upon work on entry deterrence and reputation in economics (e.g. Schelling, 1956; Selten, 1978; Kreps and Wilson, 1982; Milgrom and Roberts, 1982; Abreu and Gul, 2000; Mailath and Samuelson, 2006), and has taken a number of forms. Observation of who fights whom, for example, can allow states to make inferences about the likelihood that their interests will be congruent with others. States will behave more aggressively toward friends of the enemy, and will enjoy more peaceful relations with the friends of a friend (Crescenzi, 2007; Crescenzi et al., 2007). Similarly, research on war contagion and expansion suggests that participants’ decisions can alter the decisions of outside actors, ultimately affecting the outcome of the dispute (Huth, 1988; Siverson and Starr, 1991; Kadera, 1998; Werner, 2000; Melin and Koch, 2010). Other literature— especially within the commitment problem paradigm (e.g. Powell, 2006)— endows states with a somewhat longer-term perspective, allowing them to consider future interactions with the same opponent.

Ultimately, however, the work that has most closely has incorporated the role of outside actors into conflict bargaining has looked at whether states or leaders can acquire a reputation for “toughness”, at either the interstate (e.g. Sartori, 2002; Mercer, 2010; Sechser, 2010; Dafoe and Caughey, 2011; Slantchev, 2011) or the intrastate level (Walter, 2006a, 2009). In the tradition of the chain-store paradox, this literature has primarily looked at defenders (i.e. threatened states or leaders entertaining demands from potential rebel groups), showing that refusal to concede in one period can, in some cases, deter future threats. The need to cultivate a tough reputation provides states with an incentive to fight, even when they might otherwise prefer to accept a challenger’s demand. This is an important and intuitively appealing result, which suggests that war can potentially come about because of the expected actions of parties who are not even involved in the current dispute.

Interestingly, while previous literature has focused heavily on the role of the defender in conveying information to outside observers, the role of the attacker remains surprisingly underanalyzed. This is important because the canonical bargaining model generally endows the defender with only two alternatives: to concede to the challenger’s demands, or to go to war. It is, therefore, intuitive that the choice of whether or not to fight in one stage may provide information about whether or not a state would fight when subsequently presented with a similar demand.⁶ It is less clear, however, whether an attacker’s demand provides useful information to outside actors about whether it would make concessions when threatened.

The model presented in this paper addresses this question, building upon the extant literature in two ways. First, it demonstrates that an attacker’s demand can indeed convey information to an outsider about the likelihood that it would back down from a subsequent conflict. Thus, observation provides attackers with incentives to modify their demands by behaving more aggressively than they would otherwise. This is not a readily apparent result, and thus it significantly extends our understanding about the degree to which reputation can affect conflict bargaining. Second, because demands are made along a continuum, in contrast to the defender’s dichotomized choice, the attacker has significant latitude for fine tuning its strategy. This is a useful feature of the model, as it facilitates a detailed analysis of the effect, which can be achieved by simulating model equilibria and varying interesting parameters. In this way, it is possible to see not only whether an attacker’s behavior should be expected to change while being observed, but precisely how it should change. Analysis of the specific effect of observation on attacker demands is both interesting on a theoretical level and useful for deriving potential empirical implications.

The role of third parties

During conflict bargaining, an attacker’s primary goal may not be to maximize its payoff over that dispute, but to use the situation to its advantage elsewhere. If, as international relations theorists (e.g. Wagner, 2000; Slantchev, 2004) believe, fighting reveals information to participants, then it should also reveal information to any other observers. By carefully observing the interactions of the disputants, outside actors can acquire information costlessly, which may then affect future (or even ongoing) interactions. This is especially true for enemies or potential enemies of participants, who are likely to watch the proceedings closely, and to pounce if they perceive weakness. For example, after observing India’s defeat in the Sino–Indian War, Pakistani leaders noticed India’s “relative weakness”, and were led to believe that India would adopt a “more realistic approach to … the Kashmir dispute” (Kacowicz, 1994: 103–104). Upon learning that India was weaker than it believed, Pakistan updated its beliefs, and inferred that India would likely accept worse deals in future negotiations, making it more willing to issue demands.

Disputants are not unaware of observation by outside actors. Indeed, the strategies of participants in militarized conflicts should be conditioned not only on the expected outcome within the present conflict, but also on its implications for future disputes with observers. President Kennedy— during whose administration significant escalation in Vietnam began— noted that the United States had “a problem in making [its] power credible” and that Vietnam was “the place” to resolve that (Wiest, 2002: 21). Importantly, Kennedy did not cite particular aims or objectives. Rather, he stressed the fact that the United States needed to demonstrate its power before its rivals in the Soviet Union and Communist China. The Kennedy administration implied a willingness to take significant losses in the Vietnam conflict, in order to secure gains in outside disputes.

States that expect their actions to influence the behavior of (hostile) outside parties are likely to act differently from those that do not. If, like the United States under Kennedy, states feel the need to act tough in front of rivals in order to display strength or hide weakness, then they can be expected to adopt a tougher bargaining stance in disputes with other states. An alternative that is an acceptable bargain in a conflict with no observer may no longer be feasible if one of the parties is worried about the conclusions that a third state will draw from its acceptance. While this may not be the optimal strategy when the conflict occurs in a vacuum, the expected loss may be useful as a signaling mechanism.

By excluding the possibility of a third-party observer, traditional models of bargaining and war fail to take into account factors that may play a crucial role in challengers’ bargaining strategies. Beginning from flawed premises, the implications drawn from such models may be unreliable. Below, I present a model of conflict bargaining under observation that demonstrates, under fairly general conditions, both whether and how an attacker’s equilibrium behavior will vary when an important outside issue is at stake.

Model

I begin from a basic model of conflict bargaining. This is based largely on Fearon’s (1995) private information model, in that it is an ultimatum game with two-sided private information and war characterized as a one-shot costly lottery. Although other, more complex models of war have been proposed (e.g. Smith, 1998; Filson and Werner, 2007), it is useful to begin from a relatively simple, well-known model, in order to demonstrate the effect of incorporating outside learning. Extending the concept to more complicated models is left for future research. In the analysis, I construct a model of conflict bargaining that differs from the canonical two-state bargaining game in that it incorporates a move by a third player, who learns from the interactions of the first two. This model contains multiple subgames, and players’ beliefs are relevant. In this case, although the game is more complex, the solution concept remains a perfect Bayesian equilibrium (PBE). I show that in this model, the equilibrium from the two-player game no longer exists. The presence of the outside observer changes the state’s equilibrium behavior. Finally, I characterize a class of equilibria for the three-player game.⁷ In general, I find that as the total expected value of the second stage increases, states become more willing to pool with one another, often becoming more demanding than they would be if unobserved.

Let two states, 1 and 2, be involved in a crisis over some divisible issue, whose value is normalized to 1. Assume that the two have strictly increasing preferences over the amount of the good that they receive. In the initial stage, state 1 proposes an efficient division between the two states, $(x, 1 - x)$ , where $x \in X = [0, 1]$ is state 1’s proposed share of the deal. After receiving the offer, state 2 chooses whether to accept or reject. If 2 accepts, then the good is divided, and payoffs are realized based on the amount of the good received. If 2 rejects 1’s offer, then the two states go to war with one another over the good. The war is a one-shot costly lottery for the entirety of the good, which state 1 wins with probability $p \in (0, 1)$ , and state 2 wins with probability $1 - p$ . The states’ respective probabilities of victory are common knowledge. Following the war, the winner gets all of the prize, while the loser gets nothing. Both states pay a positive cost for fighting, $c_{i} \in C_{i} = [0, {\bar{c}}_{i}]$ for $i = 1, 2$ . Whereas the probability of victory is common knowledge, each state’s cost for fighting is private information. This assumes that both states know their relative strength, but are unaware of the opponent’s level of resolve. Let it be common knowledge that each state $i$ ’s costs are drawn from a uniform distribution, with support $C_{i}$ .⁸

This is the conventional bargaining game, with two-sided incomplete information, as found in much of the formal literature. As this game has seen significant analysis in the past, I do not explicitly solve it here. In equilibrium, state 1 will engage in a fully separating strategy, with each type making a unique offer that it uncovers by optimizing the relevant risk–reward tradeoff. On the interior, this offer will be given by $x^{*} = p + ({\bar{c}}_{2} - c 1) / 2$ . By incorporating the role of outside observers, it is possible to determine whether such a strategy still holds when the game does not end after the first conflict.

To assess the role of the outside observer, let state 1 also be in possession of some second, perfectly divisible good. Suppose that another actor, state 3, is interested in this second good, and therefore observing state 1’s conflict with state 2. State 3 is able to observe the exchange between states 1 and 2, including 1’s offer and 2’s reaction, and update its beliefs about 1’s type. If state 2 accepts 1’s offer, then state 3’s posterior belief is based on state 1’s equilibrium strategy and the action that it takes. If state 2 rejects the offer, then war reveals private information to all actors. Because state 3 observes the conflict, it also learns the combatants’ types with certainty.⁹ After updating, state 3 initiates a crisis with state 1 over this second issue. Suppose that the total expected value of this second stage is given by $α \in R_{+}$ .¹⁰ In initiating conflict, state 3 makes an offer to 1 of $(y, 1 - y)$ , $y \in Y = [0, 1]$ , in which it proposes that it keep $y$ of the prize, and to leave the remaining $1 - y$ of it to state 1. State 1 can then choose to accept or reject the offer. If it accepts, the two states receive their respective payoffs. If it rejects, then they go to war. State 3 wins a war with probability $q \in (0, 1)$ and loses with probability $1 - q$ . If war occurs, then both states pay their respective costs, which are private information, as before.

In the single stage game, in which both states care exclusively about their share of the good in the present conflict, the initiator plays a separating strategy, making an offer that is decreasing in its cost of fighting. Thus, the first question to consider when looking at the new game, which allows for the possibility of a second stage, is whether state 1 will still make the offer, $x^{*} = p + ({\bar{c}}_{2} - c 1) / 2$ (on the interior), will play a modified separating strategy (i.e. an offer that is still a function of $c_{1}$ , but with a different functional form), or will not separate at all. If either the equilibrium from the conventional conflict bargaining game still holds, or if there exists some similar equilibrium offer that is still decreasing in $c_{1}$ , then the traditional conception may retain some of its explanatory power. If, however, no such equilibrium exists, then the implications drawn from that model are likely to be flawed, indicating the need to rethink the conventional wisdom.

Proposition 1. In the three-player game, there never exists a fully separating equilibrium, such that every type of state 1 prefers to make a different offer.

A fully separating strategy gives every type its best first-stage payoff. However, it also eliminates all of state 1’s private information, allowing state 3 to offer state 1 its war payoff, which it will accept. State 1’s belief that there exists a third state who will learn from the interaction and use this information to extract additional concessions provides an incentive to deviate. Intuitively, this deviation should be to the optimal demand of a more powerful type. Because state 3 believes that it knows 1’s type with certainty, its demand will be increasing in its belief about state 1’s cost of fighting. Thus, given a separating strategy, state 1 can expect to be offered more by mimicking a lower-cost type. Given some positive value for the second stage, this means that all $c_{1} > 0$ will have some incentive to deviate from the strategy.

A formal proof for Proposition 1 is located in the technical appendix. If, as the proposition suggests, there is no incentive to separate, the amount of information that state 3 can learn from bargaining will be limited in equilibrium. It can only become completely informed about state 1’s type if there is war in the first stage. If the various types of state 1 chooses to pool and the offer is accepted by state 2, then state 3 will remain (at least somewhat) uncertain about state 1’s type, meaning that its optimal offer will entail a risk–reward tradeoff, rather than a guarantee of peaceful settlement. Thus, for all but the weakest types of state 3 (i.e. those for whom the optimal division is given by the probability of victory $(q, 1 - q)$ , which is always acceptable), the response to a pooling strategy on the part of state 1 must entail some positive probability of war.

Proposition 2. In the three-player game described here, if $alpha$ is sufficiently large, then:

all types of state 1 make the offer $(x^{*}, 1 - x^{*})$ , such that $x^{*} = \min {p + \frac{{\bar{c}}_{2}}{2}, 1}$ ;

state 2 accepts any offer such that $x^{*} < p + c_{2}$ ;

state 3 believes that state 1 is equally likely to be of any type in $μ$ , such that

μ = (\begin{matrix} C_{1} & if x = x^{*} and state 2 accepts \\ {\bar{c}}_{1} & if x \neq x^{*} and state 2 accepts \\ c_{1} & if state 2 rejects \end{matrix}

all types of state 3 offer $(y^{*}, 1 - y^{*})$ , such that if $μ$ is a singleton, $y^{*} = \min {q + μ, 1}$ , and if $μ = C_{1}$

y^{*} = (\begin{matrix} q & if c_{3} > {\bar{c}}_{1} \\ q + \frac{{\bar{c}}_{1} - c_{3}}{2} & if {\bar{c}}_{1} - 2 (1 - q) \leq c_{3} \leq {\bar{c}}_{1} \\ 1 & if c_{3} < {\bar{c}}_{1} - 2 (1 - q) \end{matrix}

Proposition 2 shows that a fully pooling equilibrium will exist when $α$ is sufficiently large. As $α$ increases, the value of the second stage increases relative to the value of the first. This makes it rational for each type of state 1 to mimic a wider range of stronger types. When $α$ is sufficiently large, the increase in the expected second stage payoff can compensate for the lower expected payoff and increased probability of war in the first stage for even the weakest types. As such, all types will be willing to mimic the strongest type, given a large enough second-stage value.

The proposition is proved formally in the technical appendix. In the fully pooling equilibrium, all types of states choose to make a riskier offer in stage one, in order to maintain uncertainty about strength in stage two. What is especially interesting about this finding is not that all types are willing to pool, but the offer that they will choose to make. No pooling strategy is acceptable unless it is the most demanding one.

Corollary 1. In the three-state game, the pooling equilibrium offer in the first stage leads to a probability of war that is at least as high as the strongest type would be willing to tolerate.

Corollary 1 follows directly from the proof of Proposition 2. In stage one of the three-state game, all types of state 1 attempt to mimic the toughest type. In equilibrium, then, all types of state 1 make the optimal two-state offer for a type who could fight war costlessly. This not only means that there remains a positive probability of war in the first stage, but that the probability increases, relative to the situation in which $α = 0$ (i.e. when a second stage is irrelevant) for every type of state 1. Types not only make the same offer, but they coordinate on the optimal offer with the highest possible probability of war.

Lemma 1. Given any positive probability of a second stage occurring, an equilibrium must satisfy the following necessary conditions:

pooling across types must occur almost everywhere on the interval $[0, 1 - q]$ ;

there must exist some ${\hat{c}}_{1} \leq {\bar{c}}_{1}$ such that a subset of types, $c_{1} \in [0, {\hat{c}}_{1})$ or $c_{1} \in [0, {\hat{c}}_{1}]$ , makes an offer such that $x^{*} = \min {p + {\bar{c}}_{2} / 2, 1}$ ;

any pooling set must include types from the interval $[0, 1 - q)$ ;

state 1’s equilibrium offer must be weakly decreasing in $c_{1}$ ;

for any pooling interval, the demand made must be the preferred offer for the infimum of the set.

The first three parts of Lemma 1 follow directly from the previous propositions. Parts four and five are fairly intuitive. First, suppose that two offers, $x$ and $x^{'}$ , $x < x^{'}$ , are made by the intervals $[c_{L}, c_{M}]$ and $(c_{M}, c_{H}]$ , respectively, where $c_{L} < c_{M} < c_{H}$ . If $c_{M}$ weakly prefers the payoff from offering $x$ to that of $x'$ , then there must be some $c_{M} + ε$ , $ε > 0$ , with higher costs, who will strictly prefer it, as the offer will increase both the first and second stage payoffs for that type. Thus, the offer must be decreasing in $c_{1}$ . To understand part five, suppose that the types $[c_{L}, c_{M}]$ make some offer, $x \neq x^{*} (c_{L})$ . Assuming the offer is accepted, state 3 believes that a state that offers $x$ is of type $(c_{L} + c_{M}) / 2$ , and makes the corresponding offer. Type $c_{L}$ will reject this offer, as its war payoff is higher, and can therefore strictly increase its payoff by deviating to $x^{*} (c_{L})$ . Therefore, the demand must be at $c_{L}$ ’s preferred offer.

A full proof of this lemma can be found in the technical appendix. If a second stage has any relevance whatsoever, any equilibrium will satisfy all five of these conditions. Note that these requirements affect the form of the equilibrium strategy, given existence. They are not necessary conditions for existence of an equilibrium, per se. The fifth part of the lemma is perhaps the most important. It generalizes the key result of the fully pooling equilibrium, showing that for any pooling interval, all types must mimic the lowest-cost type. This means that states will behave more aggressively in general, increasing the first-stage probability of war at least marginally even when $α$ is not sufficiently high to sustain a fully pooling equilibrium.

In determining what a semi-separating equilibrium would look like, it is helpful to return to Lemma 1. In particular, part two indicates that there must be a pooling interval beginning at $c_{1} = 0$ . The equilibrium can be characterized by beginning from this point, and finding the set of conditions and lowest-cost type that would lead to a deviation. This information can be used to determine subsequent pooling intervals.

Proposition 3. For $α > 0$ , a unique semi-separating equilibrium will exist, such that the number of pooling intervals is decreasing in $α$ .

A complete proof of Proposition 3 is included in the technical appendix. However, the logic behind the proof is fairly intuitive. Larger values of $α$ indicate a relatively more important second stage, which increases the range of types with which a state would be willing to pool. This dynamic can be seen in the proof of Proposition 2. This suggests that when $α$ is very low, we can expect to see many distinct offers, each of which is made by a relatively small set of types. As the value increases, smaller numbers of larger pooling intervals can be sustained in equilibrium. This pattern is expected to continue until $α$ reaches the critical value for a fully pooling equilibrium, at which point a single offer is made by all types. When $α$ is sufficiently large, there is no incentive for types to break into smaller pooling intervals.¹¹

The complete proof of Proposition 3 demonstrates that the number and size of the pooling intervals in any equilibrium will be a function of the model parameters. As $α$ varies, the intervals and offers will vary as well. An analysis of the implications of the model requires an assessment of the effects of a change in $α$ not only on the various strategies of the types, but also on the expected probabilities of war and utilities for the various actors. I turn now to an analysis of the model’s comparative statics.

Analysis

When a second stage is relevant, the model predicts a variety of different possible outcomes. As such, it is important to compare the outcomes of different parameter values. First, as the expected total value of a second stage changes, how is the probability of war affected? To answer this question, I simulate equilibria using the following parametrization: ${\bar{c}}_{1} = 1$ , ${\bar{c}}_{2} = 0.9$ , ${\bar{c}}_{3} = 1.5$ , $p = 0.4$ , and $q = 0.45$ . I then vary $α$ to examine four different strategic responses.¹² In particular, I use $α \in {0, 0.05, 0.25, 5}$ , where the fourth value of $α$ demonstrates what happens when the second stage is more important for the attacker than the first. Note that the parametrization for which $α = 0$ leads to a fully separating equilibrium, in which there is no (relevant) second stage. However, this is uninteresting. To allow for comparability, it is better to conceptualize this situation as one in which there are two completely independent conflicts (such that no information can be transmitted), causing state 1 to act as though $α = 0$ in the first stage. Under the same parameters, the states play the game described earlier, but what happens in the first stage remains concealed from state 3, meaning that state 1’s actions in the first stage do not affect its payoff in the second. In this way, it is possible to compare outcomes across the two types of games.

Figure 1 depicts the set of equilibrium offers made for various total expected values of the second stage. Most importantly, it shows the number, size, and location of the various equilibrium pooling intervals, as $α$ changes. In the case of independent conflicts, we see complete separation, with the equilibrium offer decreasing linearly in $c_{1}$ . For $α = 0.05$ , we see five pooling intervals and one separating interval. A small group of low-cost types pool on the demand made by $c_{1} = 0$ in the independent case, a slightly larger group of types pool on a lower offer, and still larger groups pool on three subsequent smaller demands. The final pooling interval contains a small subset of types of $c_{1} > 1 - q$ . All higher-cost types that are not a part of that interval simply separate, as revealing their types to state 3 will not provide any actionable information (since state 3 can demand, at most, all of the prize). Similarly, in the $α = 0.25$ case, we have two pooling intervals. The first contains a large subset of more-resolved types, who make the $c_{1} = 0$ offer from the independent setup. A second pooling interval, containing some types of $c_{1} > 1 - q$ , makes a significantly lower demand. Those high-cost types that are not a part of the latter interval simply separate. Finally, in the case of $α = 5$ , there is no separation. The second stage is sufficiently important to all types that they all pool on the maximal offer from the separating equilibrium, $x^{*} = p + {\bar{c}}_{2} / 2$ .

Figure 1.

Optimal demands in equilibrium.

What is especially important about the dynamics illustrated in the figure is the way in which equilibrium behavior changes with $α$ . For the independent conflicts, states separate into an infinite number of (singleton) groups, making similarly diverse offers. As $α$ increases, the number of groups shrinks, and the membership of each set grows. This is especially true of the pooling offer for the lowest-cost types, the growth of which is apparent as $α$ gets larger. Eventually, when the second stage is sufficiently important, relative to the first, all types collapse into a single group, making the same offer to state 2, and eliminating the ability for state 3 to learn anything. Another important dynamic is particularly apparent in the $α = 0.05$ equilibrium, for which there are five pooling intervals and one separating interval. As $c_{1}$ increases, both the size and distance of the intervals increase. This suggests an increased willingness to trade off first-stage value for a better second-stage payoff as the cost of fighting gets larger. In other words, high-cost types will mimic relatively more aggressive types than will their low-cost counterparts.

The offers depicted in the figure above are demonstrative of the higher level of aggression fostered by observation. For all positive values of $α$ , the offer made by a particular $c_{1}$ is always weakly larger than the offer the same type would make if $α$ were zero.¹³ The degree to which aggression increases depends upon the size of the pooling intervals. For small values of $α$ , these increased demands are both smaller and apply to fewer types, relative to higher second-stage values.

The probability that war occurs in the first stage is directly related to the offer made: greater demands lead to higher probabilities of war. Thus, the dynamics of the probability of war in the first stage, across values of $α$ and $c_{1}$ will be very similar to Figure 1. Intervals that make different offers will see the probability of war differ correspondingly. The notable case occurs during independent conflicts, when $c_{1} \geq 0.9$ , as all types on the interval offer $x^{*} = p$ . For this region, the probability of war would be zero, as it is an offer that is acceptable to all types of $c_{2}$ .

More interesting is the probability of war in stage two, given states’ strategies. This is depicted in Figure 2, which shows, across the four different values of $α$ , each type of state 1’s expected second-round probability of war, following its equilibrium strategy from Figure 1. Unsurprisingly, the highest probability of war comes in situations in which the two stages are independent, but a dispute still arises. This is because, by assumption, state 3 does not observe the first round, and so no information is transmitted, rendering it unable to calibrate its offer to state 1 in the second stage. In the fully pooling equilibrium, despite the fact that no information is transmitted through bargaining, the probability of war is significantly lower because of the (relatively likely) possibility that the offer is rejected. In that case, war would occur in stage one, allowing state 3 to make an acceptable offer in the second round.

Figure 2.

Equilibrium probabilities of war in the second stage.

When conflicts are not independent and a dispute arises in the second stage,¹⁴ the probability of war is weakly decreasing in the number of pooling intervals. The lowest probability of war is found in the six-interval case, where $α = 0.05$ . Because the groups are many and small, significant information is transmitted from state 1’s demands. This allows state 3 to make much more targeted offers, avoiding those that its opponent would be especially likely to reject. Among the equilibria for which $α > 0$ , two points are especially notable. First, as the size of the pooling interval(s) increases, war becomes more likely in the second stage. This is in direct contrast to the expected results from chain-store models (e.g. Walter, 2006b; Mercer, 2010), where acting tough is meant to deter challengers. Second, as the expected total value of the second stage decreases, and the number of pooling intervals expands, the slope of the probability of war steepens. This means that types for whom war is more costly tend to benefit more (in terms of the likelihood of being made an acceptable offer) from transmitting additional information, relative to types for whom war is cheap.

A final point of interest from Figure 2 can be found at the beginning of the second pooling interval ( $c_{1} \approx 0.09$ ) when $α = 0.05$ . While the probability of war in the second stage is weakly decreasing in $c_{1}$ generally, there is a small but noticeable increase early in the second interval. Simulations with other values of $α$ suggest that this is common for equilibria with many pooling intervals, in which the group pooling with the strongest type is especially small. Because the value of the second stage is comparatively low, these types are sufficiently far from the toughest type to prefer to get a better deal in the first round, even if it means a lower offer and a higher probability of war in the second.

Figure 3 shows the probability for each type of state 1 that war occurs in either stage, given a particular value of $α$ and the corresponding equilibrium behavior from Figure 1. The (weakly) highest overall probability of fighting, across all values of $c_{1}$ , occurs when types fully pool. Because all types are mimicking a state that can fight for free, the probability that war occurs is always at least one-half. The equilibrium for the independent case initially leads to a higher probability of war than either of the two semi-separating equilibria. As $α$ decreases, the portion of the type space that leads to a probability of war higher than what would be seen across two independent conflicts also decreases. The probabilities of war for the semi-separating equilibria eventually converge with that of the independent case when $c_{1}$ is very large, and types have no incentive to pool.

Figure 3.

Equilibrium probability of war in either stage.

The story told by Figure 3 is an interesting one. When the expected total value of the second stage is positive but small, states initially behave in a slightly more aggressive manner. This increases the probability that war occurs in the first round by a small amount, but the information that it transmits helps parties to circumvent armed conflict later. When concerned with the total probability of war, information acts to offset aggression. As this expected total value of stage two increases, less information is transmitted, and the degree of offset lessens. Ultimately, as $α$ gets very large, all types elect to make the same offer, increasing aggression, while simultaneously transmitting no information. In this case, the probability of war can increase for all positive values of $c_{1}$ relative to situations in which $a$ is low. However, states are, of course, concerned not simply with the probability that conflict occurs, but with what they expect to receive. Thus, it is informative to examine expected utilities as well.

Figure 4 shows the expected performance, in terms of utility, across different parametrizations. For each value of $α$ and $c_{1}$ , it is possible to compute the expected utility from equilibrium play (as depicted in Figure 1). The total utility that can be acquired by a state is $1 + α$ , if it gains everything in each stage, while paying zero costs.¹⁵ Because the total possible utility is increasing in $α$ , the figure depicts the ratio of expected to total utility, as this provides a comparable measure across all values of $α$ . This is calculated by dividing the expected utility for a given cost and strategy by $1 + α$ (or by two in the independent case).

Figure 4.

Expected proportion of utility acquired.

The results suggest that states generally tend to prefer situations in which they can provide information about those costs. In particular, states perform better in situations with a large number of small pooling intervals, relative to situations in which pooling occurs across a few large intervals. The reason for this is that it allows them to convey limited information to the future opponent, suggesting that are likely to reject particularly aggressive demands, while simultaneously keeping the adversary unsure of the exact cost for fighting. This prevents state 3 from simply making a demand at state 1’s reservation value.

The fact that the relative utility from fighting two independent conflicts of equal value is so low lends credence to this explanation. When unconcerned with a subsequent dispute, states behave less aggressively initially, but communicate no information. The placement of the four utility curves indicates that states prefer revealing more information to less, so long as something can be held back. Ultimately, the analysis suggests that a state that believes it will have to fight two conflicts would actually prefer that its opponent observe the fighting, given that some private information can be retained. For attackers, conflict bargaining is both a means of getting a good deal from a defender, as well as a show of strength to confuse or deter a future attacker. Having analyzed the theoretical model, I turn now to an assessment of an empirical case which illustrates this phenomenon nicely.

Empirical reflection

Conventional stories about private information and commitment issues would find it difficult to explain the Vietnam War, in which, based on the claims of American presidents, the United States initially chose to fight so that it could send a signal of strength and resolve to the Soviets. The actual outcome, apart from its broader geopolitical context, mattered to a much smaller degree. The U.S. pushed for a costly war in Vietnam, despite the small stakes. Such behavior is not uncommon in the international system. To further demonstrate the empirical plausibility of the model sketched above, and to illuminate the mechanisms that drive it, I discuss the Sino–Indian War, a specific case of a war that appears to have been driven by states posturing for a rival.

The rivalry between India and Pakistan is one of the most active, long-lasting, and well-known enduring conflicts. Originating shortly after the area was granted independence from Britain and partitioned into two states, the dispute is largely territorial, beginning with the First Kashmir War in 1947 (Diehl et al., 2005; Paul, 2005). Because of the intense nature of the rivalry between the two nations, each closely observes the other’s foreign policy choices, and understands that its own actions are under similar scrutiny. The 1962 war between China and India was no exception.

The relationship between India and China has also been plagued by border issues, albeit to a lesser extent than that of India and Pakistan. The two states share a northern border along the Himalayas, which includes a number of disputed regions. In particular, both China and India claimed ownership of the Aksai Chin, an area between Kashmir and Tibet. Despite the fact that the Chinese maintained a clear numerical advantage within the territory, during 1961 and 1962, India initiated a “forward policy” to try to recapture the area, leading to a series of skirmishes between Chinese and Indian troops. As the policy took effect, China demanded that India withdraw its troops from the area (see Orme, 1987). Despite China’s threats and apparent military advantage, India continued to push into the region, leading to war in October of 1962.

Although war might have been avoided, India appears to have decided to press its claim to Aksai Chin in order to signal strength to Pakistan. Indeed, Abitbol (2009: 80) notes that India was forced “to take a hard-line stance in their border dispute with China out of fear that concessions would show weakness [to Pakistan] and endanger their control of Kashmir”. Unfortunately for India, refusing to withdraw from the region led to its defeat at the hands of the Chinese, revealing its apparent weakness. Moreover, Pakistan did appear to be watching the conflict. Upon observing India’s defeat, Pakistan revised its beliefs about Indian power, leading to a second confrontation over Kashmir in 1965.

In his analysis of the case, Orme (1987: 117–118) does not mention the relationship between India and Pakistan, instead attributing the cause of the war to a miscalculation of Chinese resolve on the part of India. In particular, he argues that morale in the Chinese army was relatively low at the time and that Nehru believed the Soviet Union would restrain China (i.e. threaten to impose costs in case of war). He also points to China’s failure to follow through on deterrent threats earlier in the conflict. Based on this evidence, Orme makes the case that India’s continued insistence on expanding deeper into the Aksai Chin was due to India’s belief that Chinese threats were empty. This explanation is not inconsistent with the story told in the model above. Indeed, under the assumptions of the model, if India had been certain that China would follow through on its threats, it would have had no incentive to continue to push for more territory, as the resulting conflict would reveal India’s true costs to Pakistan. The belief that the Chinese would suffer significant costs from going to war led the Indian government to believe that war was unlikely. Because war was sufficiently unlikely, India could act tougher, in order to gain more territory and to signal its strength to Pakistan.

The case of the border dispute between India and China is interesting and informative for a number of reasons. It is clear that, although relatively weak, India attempted to act tough in order to avoid revealing this weakness to its rival. In doing so, it provoked a war, which demonstrated the very weakness it attempted to shield. Following the conflict, Pakistan was led to believe “that India had been revealed as weak and blundering, so that political and military pressure on India could lead eventually to a territorial change in Kashmir at minimum costs” (Kacowicz, 1994: 104). Most interestingly, however, is the fact that Pakistan waited 3 years before challenging India, at which point the power dynamics had changed, and the subsequent war resulted in a clear Indian victory. By attempting to appear stronger than it actually was, and to head off a threat from one of its rivals, India brought about war with both rivals. This escalated conflict likelihood is precisely the scenario about which the model above warns.

Conclusion

Modern theories of conflict bargaining make two untenable assumptions: that conflicts are independent and that countries care only about the prize over which they are fighting. Simply looking at some of the major conflicts since World War II demonstrates the flawed nature of these suppositions. It would be difficult to argue, for example, that the Vietnam War— one of the costliest conflicts in U.S. history— took place because of the great value that the United States placed on South Vietnamese regime type. It is clear that U.S. involvement in Vietnam was due to concerns about Soviet expansionism and what the Soviet Union would infer from American abstention.

Vietnam is, of course, not the only case in which a state’s actions are driven by a desire to send a signal to an uninvolved party. Understanding why wars occur requires an understanding of third-party effects. To this end, I proposed a model in which states concern themselves not only with the current conflict, but also its effects on future conflicts. The result is that, if a future conflict is sufficiently likely and the stakes are sufficiently high, then a state’s optimal behavior will change. In particular, it acquires an incentive to act tough, making greater demands, in order to appear stronger in the eyes of the future opponent. This behavior raises the probability that war will occur in a given conflict, as a state’s bargaining attitude becomes less flexible. In addition, the attempt to obfuscate reduces the information communicated to observers, raising the probability that war occurs in the subsequent dispute as well. Analysis of the Sino–Indian conflict provides evidence that this theory has some empirical plausibility. India’s behavior appears to have been driven in part by a desire to seem strong in the eyes of its rival, Pakistan.

While Vietnam and the border dispute between China and India lend some credence to the theory outlined above, more is left to be done. In particular, the model leads to a number of empirically testable implications about the likelihood of war. A broad-based statistical analysis of these implications could help to illuminate the conditions under which bargaining is likely to break down, while simultaneously prodding those studying international conflict empirically to think beyond the dyad, and to consider the greater international context. As the model here suggests, it is unlikely that conflict behavior can be fully understood from either a purely monadic or dyadic perspective. This theoretical finding is consistent with recent advances in international relations on the empirical side (Poast, 2010). The implications also suggest future theoretical avenues. This model looks only at war onset, but a similar explanation is likely applicable to war continuation. Looking at war as a costly process, which reveals a certain amount of information to third parties at each stage, may uncover the reason that certain wars tend to last longer than others. The presence of a third party is also likely to make states eager to obfuscate during a conflict, seeking out mechanisms such as mediation or the intervention of powerful allies. It seems plausible that states would be interested in these options in cases that are most likely to influence future conflicts. Explicit modeling of these situations may lead to a number of informative findings about conflict behavior.

Ultimately, the model presented in this paper is meant to suggest that context is important. In recent years, international relations scholars have learned not to ignore domestic politics when analyzing international conflict. The findings here suggest that neglecting the role of third parties may be equally devastating. Interstate conflicts are complex and strategic interactions that often have far-reaching effects. Any analysis of a particular conflict will be incomplete unless it considers these effects and their implications for the bargaining process.

Footnotes

Technical Appendix

Proof of Proposition 1: We want to show that state 1 has an incentive to deviate from any strategy in which all types make different offers. The proof of this contains two parts. First, I show that if a separating equilibrium exists, it must be one in which any type, $c_{1}$ , makes an offer of the form $x^{*} (c_{1}) = p + \frac{{\bar{c}}_{2} - c_{1}}{2}$ . Next, I show that offering $x^{*}$ cannot be an equilibrium for any $α > 0$ .

Assume a strategy such that a given type, $c_{1}$ , makes the offer $x^{s} (c_{1})$ , such that $x^{s} (c_{1}) \neq x^{s} (c_{1}^{'})$ , for $c_{1} \neq c_{1}^{'}$ . Assume further that for at least one type, $x^{s} (c_{1}) \neq x^{*} (c_{1})$ . Let $E u_{11} (x)$ denote state 1’s first stage payoff for making some offer, $x$ . In the case in which state 1 is playing a separating strategy, state 3 will either be able to deduce 1’s type from its offer,¹⁶ or learn it with certainty if it observes war. Conditional on observing that state 1 is of some type, $c_{1}$ , state 3’s optimal offer is $y^{*} = q + c_{1}$ , which will make a state of type $c_{1}$ indifferent between accepting and rejecting. If a type, $c_{1}$ , follows a separating strategy, then, it can expect to get $α (1 - q - c_{1})$ in the second round. Thus, $c_{1}$ ’s expected payoff for offering $x^{s} (c_{1})$ is given by

E u_{11} (x^{s} (c_{1})) + α (1 - q - c_{1})

By offering $x^{*} (c_{1})$ in the first round, instead of $x^{s} (c_{1})$ , state 1 receives a first stage payoff of $E u_{11} (x^{*} (c_{1}))$ . In round 2, state 3 will make an offer, $y^{*} (x^{*} (c_{1}))$ , that is contingent upon its beliefs after observing the first round play. Because state 1 has the option of rejecting this offer and going to war, by deviating to $x^{*} (c_{1})$ , it can guarantee itself a minimum expected payoff of

E u_{11} (x^{*} (c_{1})) + α (1 - q - c_{1})

We know from research on the two-state game that $E u_{11} (x^{*} (c_{1})) = \max_{x \in [0, 1]} E u_{11} (x)$ for any $c_{1}$ . Since $x^{s} (c_{1}) \neq x^{*} (c_{1})$ , it must be true that $E u_{11} (x^{*} (c_{1})) > E u_{11} (x^{s} (c_{1}))$ . Because state 1 can guarantee itself a second stage payoff from deviating that is at least as large as what it would get under the separating strategy, for any $x^{s} (c_{1}) \neq x^{*} (c_{1})$ , it will be the case that $E u_{1} (x^{*} (c_{1})) > E u_{1} (x^{s} (c_{1}))$ . Thus, there can be no separating equilibrium in which state 1 offers any $x (c_{1}) \neq x^{*} (c_{1})$ .

Now we want to determine the conditions under which state 1 has an incentive to deviate from the strategy in which it offers $x^{*} (c_{1}) = p + ({\bar{c}}_{2} - c_{1}) / 2$ . A state of type $c_{1}$ ’s expected utility from making the offer, $x^{*}$ , is given by

\begin{matrix} E u_{1} (x^{*} (c_{1}) | c_{1}) = x^{*} (c_{1}) \int_{x^{*} (c_{1}) - p}^{{\bar{c}}_{2}} \frac{1}{{\bar{c}}_{2}} d c_{2} + (p - c_{1}) \int_{0}^{x^{*} (c_{1}) - p} \frac{1}{{\bar{c}}_{2}} d c_{2} + α (1 - q - c_{1}) \\ = x^{*} (c_{1}) [1 - \frac{x^{*} (c_{1}) - p}{{\bar{c}}_{2}}] + (p - c_{1}) [\frac{x^{*} (c_{1}) - p}{{\bar{c}}_{2}}] + α (1 - q - c_{1}) \end{matrix}

The expected utility for deviating to the offer for some $c_{1}^{'} \neq c_{1}$ is given by:¹⁷

\begin{matrix} E u_{1} (x^{*} (c_{1}^{'}) | c_{1}) = x^{*} (c_{1}^{'}) \int_{x^{*} (c_{1}^{'}) - p}^{{\bar{c}}_{2}} \frac{1}{{\bar{c}}_{2}} d c_{2} + (p - c_{1}) \int_{0}^{x^{*} (c_{1}^{'}) - p} \frac{1}{{\bar{c}}_{2}} d c_{2} \\ + α (\int_{x^{*} {(c_{1})}^{'} - p}^{{\bar{c}}_{2}} \frac{1}{{\bar{c}}_{2}} d c_{2} \max {1 - q - c_{1}^{'}, 1 - q - c_{1}} + \int_{0}^{x^{*} (c_{1}^{'}) - p} \frac{1}{{\bar{c}}_{2}} d c_{2} (1 - q - c_{1})) \\ = x^{*} (c_{1}^{'}) [1 - \frac{x^{*} (c_{1}^{'}) - p}{{\bar{c}}_{2}}] + (p - c_{1}) [\frac{x^{*} (c_{1}^{'}) - p}{{\bar{c}}_{2}}] \\ + α ([1 - \frac{x^{*} (c_{1}^{'}) - p}{{\bar{c}}_{2}}] (1 - q - \min {c_{1}, c_{1}^{'}}) + [\frac{x^{*} (c_{1}^{'}) - p}{{\bar{c}}_{2}}] (1 - q - c_{1})) \end{matrix}

State 1 has an incentive to deviate from $x^{*} (c_{1})$ to $x^{*} (c_{1}^{'})$ if $E u_{1} (x^{*} (c_{1}) | c_{1}) < E u_{1} (x^{*} (c_{1}^{'}) | c_{1})$ . We can derive the necessary conditions for this to hold by subtracting the utility for deviating from the equilibrium utility:

\begin{matrix} (x^{*} (c_{1}) - x^{*} (c_{1}^{'})) (1 + \frac{p}{{\bar{c}}_{2}}) - x^{*} (c_{1}) [\frac{x^{*} (c_{1})}{{\bar{c}}_{2}}] + x^{*} (c_{1}^{'}) [\frac{x^{*} (c_{1}^{'})}{{\bar{c}}_{2}}] \\ + (\frac{p - c_{1}}{{\bar{c}}_{2}}) (x^{*} (c_{1}) - x^{*} (c_{1}^{'})) + α [1 - \frac{x^{*} (c_{1}^{'}) - p}{{\bar{c}}_{2}}] (1 - q - c_{1}) \\ - α [1 - \frac{x^{*} (c_{1}^{'}) - p}{{\bar{c}}_{2}}] (1 - q - \min {c_{1}, c_{1}^{'}}) < 0 \end{matrix}

From this, we can derive the deviation condition:

\begin{matrix} [x^{*} (c_{1}) - x^{*} (c_{1}^{'})] ({\bar{c}}_{2} + 2 p - c_{1} - x^{*} (c_{1}) - x^{*} (c_{1}^{'})) \\ + α (p + {\bar{c}}_{2} - x^{*} (c_{1}^{'})) (\min {c_{1}, c_{1}^{'}} - c_{1}) < 0 \end{matrix}

which, for $c_{1} > c_{1}^{'}$ , reduces to

(c_{1} - c_{1}^{'}) < 2 α (c_{1}^{'} + {\bar{c}}_{2})

Because $C_{1}$ is convex, for any $c_{1} > 0$ , there will exist some $c_{1}^{'} < c_{1}$ that will satisfy this inequality, as long as $α > 0$ . Thus, there can be no fully separating equilibrium in which all types of state 1 offer $x^{*} (c_{1}) = p + ({\bar{c}}_{2} - c_{1}) / 2$ .

By the first part of the proof, we know that if a fully separating equilibrium exists, it must be one in which $x^{*} (c_{1}) = p + ({\bar{c}}_{2} - c_{1}) / 2$ . By the second part, we see that if $α > 0$ , an equilibrium of this type cannot exist. Thus, if a second stage occurs with any positive probability, there can be no fully separating equilibrium.

Proof of Proposition 2: We want to show that there exists a fully pooling PBE in which all types of state 1 make the first round offer $(x^{*}, 1 - x^{*})$ , such that for some $γ \in [{\bar{c}}_{2} - 2 (1 - p), {\bar{c}}_{2}]$ , $x^{*} = p + ({\bar{c}}_{2} - γ) / 2$ .¹⁸ Given such a strategy and the distribution of types, if state 3 observes an offer of $(x^{*}, 1 - x^{*})$ , it will place equal probability on all $c_{1} \in [0, {\bar{c}}_{1}]$ . Thus, adapting the solution from the two-state game, state 3’s optimal offer after observing $(x^{*}, 1 - x^{*})$ in round 1 is to offer $(y^{*}, 1 - y^{*})$ , such that

y^{*} = (\begin{matrix} q & if c_{3} > {\bar{c}}_{1} \\ q + \frac{{\bar{c}}_{1} - c_{3}}{2} & if {\bar{c}}_{1} - 2 (1 - q) \leq c_{3} \leq {\bar{c}}_{1} \\ 1 & if c_{3} < {\bar{c}}_{1} - 2 (1 - q) \end{matrix}

If state 1 deviates to any $x^{'} \neq p + ({\bar{c}}_{2} - γ) / 2$ , let state 3 believe state 1’s type to be ${\bar{c}}_{1}$ , so that its optimal offer is $y^{*} (x^{'}) = \min {q + {\bar{c}}_{1}, 1}$ . For any $x^{'} \neq x^{*}$ , then, state 1’s payoff in the second round is independent of its offer, and will depend only on its type. So for some fully pooling strategy, $x^{*}$ , state 1’s expected stage two payoff will be

\begin{matrix} E u_{12} (x^{*} | c_{1}) = α [\int_{x^{*} - p}^{{\bar{c}}_{2}} \frac{1}{{\bar{c}}_{2}} d c_{2} ((1 - q) \Pr (c_{3} > {\bar{c}}_{1}) \\ + (1 - q - \min {\frac{{\bar{c}}_{1} - E [c_{3}]}{2}, c_{1}, 1 - q}) \\ \Pr ({\bar{c}}_{1} \geq c_{3} \geq {\bar{c}}_{1} - 2 \min {c_{1}, 1 - q}) \\ + \max {0, 1 - q - c_{1}} \Pr (c_{3} < {\bar{c}}_{1} - 2 \min {c_{1}, 1 - q})) \\ + \max {0, 1 - q - c_{1}} \int_{0}^{x^{*} - p} \frac{1}{{\bar{c}}_{2}} d c_{2}] \\ E u_{12} (x^{'} | c_{1}) = α [(1 - \min {y^{*} (x^{'}), q + c_{1}}) \int_{x^{*} - p}^{{\bar{c}}_{2}} \frac{1}{{\bar{c}}_{2}} d c_{2} \\ + \max {0, 1 - q - c_{1}} \int_{0}^{x^{*} - p} \frac{1}{{\bar{c}}_{2}} d c_{2}] \end{matrix}

It should be obvious that for $E u_{12} (x^{'} | c_{1})$ , the second stage payoff will always be at state 1’s reservation value. For $E u_{12} (x^{*} | c_{1})$ , the probabilities listed above, and the value of $E u_{12} (x^{*} | c_{1})$ will depend upon the values of the model parameters. The expected value of being offered some $y^{*}$ , following an acceptance in the first round is given by

E u_{1} (y^{*} | Acc) = (\begin{matrix} 1 - q - \frac{c_{1} ({\bar{c}}_{1} - c_{1})}{{\bar{c}}_{3}} & if c_{1} \leq 1 - q and {\bar{c}}_{1} \geq 2 c_{1} \\ 1 - q - \frac{(1 - q) ({\bar{c}}_{1} - (1 - q))}{{\bar{c}}_{3}} & if c_{1} > 1 - q and {\bar{c}}_{1} \geq 2 c_{1} \\ 1 - q - \frac{{\bar{c}}_{1}^{2}}{4 {\bar{c}}_{3}} & if {\bar{c}}_{1} < 2 c_{1} \end{matrix}

From previous work, we know state 1’s payoff in stage one for any offer, $x \in [0, 1]$

E u_{11} (x | c_{1}) = x \int_{x - p}^{{\bar{c}}_{2}} \frac{1}{{\bar{c}}_{2}} d c_{2} + (p - c_{1}) \int_{0}^{x - p} \frac{1}{{\bar{c}}_{2}} d c_{2}

Thus, the complete expected utility expressions are given by

\begin{matrix} E u_{1} (x^{*} | c_{1}) = x^{*} \int_{x^{*} - p}^{{\bar{c}}_{2}} \frac{1}{{\bar{c}}_{2}} d c_{2} + (p - c_{1}) \int_{0}^{x^{*} - p} \frac{1}{{\bar{c}}_{2}} d c_{2} \\ + α [\int_{x^{*} - p}^{{\bar{c}}_{2}} \frac{1}{{\bar{c}}_{2}} d c_{2} (E u_{1} (y^{*} | Acc)) \\ + \max {0, 1 - q - c_{1}} \int_{0}^{x^{*} - p} \frac{1}{{\bar{c}}_{2}} d c_{2}] \\ E u_{1} (x^{'} | c_{1}) = x^{'} \int_{x^{'} - p}^{{\bar{c}}_{2}} \frac{1}{{\bar{c}}_{2}} d c_{2} + (p - c_{1}) \int_{0}^{x^{'} - p} \frac{1}{{\bar{c}}_{2}} d c_{2} \\ + α [1 - q - \min {c_{1}, 1 - q}] \end{matrix}

From the two-state game, it follows that state 1’s optimal deviation is to

x^{'} = (\begin{matrix} p & if c_{1} > {\bar{c}}_{2} \\ p + \frac{{\bar{c}}_{2} - c_{1}}{2} & if {\bar{c}}_{2} - 2 (1 - p) \leq c_{1} \leq {\bar{c}}_{2} \\ 1 & if c_{1} < {\bar{c}}_{2} - 2 (1 - p) \end{matrix}

Given values for $x^{*}$ and $x^{'}$ , a PBE exists if, for all values of $c_{1}$ :

E u_{1} (x^{*} | c_{1}) \geq E u_{1} (x^{'} | c_{1})

Given these results, there are four cases to consider.

Case 1: Let $c_{1} \leq 1 - q$ and $c_{1} \leq {\bar{c}}_{1} / 2$ . Then the condition that must be satisfied will be

\begin{matrix} (p + \frac{{\bar{c}}_{2} - γ}{2}) (\frac{1}{2} + \frac{γ}{2 {\bar{c}}_{2}}) + (p - c_{1}) (\frac{1}{2} - \frac{γ}{2 {\bar{c}}_{2}}) \\ + α [(\frac{1}{2} + \frac{γ}{2 {\bar{c}}_{2}}) (1 - q - \frac{c_{1} ({\bar{c}}_{1} - c_{1})}{{\bar{c}}_{3}})) + (\frac{1}{2} - \frac{γ}{2 {\bar{c}}_{2}}) (1 - q - c_{1})] \geq \\ (p + \frac{{\bar{c}}_{2} - c_{1}}{2}) (\frac{1}{2} + \frac{c_{1}}{2 {\bar{c}}_{2}}) + (p - c_{1}) (\frac{1}{2} - \frac{c_{1}}{2 {\bar{c}}_{2}}) + α (1 - q - c_{1}) \end{matrix}

This is satisfied if $α$ is sufficiently large:

α \geq \frac{{\bar{c}}_{3} {(c_{1} - γ)}^{2}}{2 c_{1} ({\bar{c}}_{2} + γ) ({\bar{c}}_{3} - {\bar{c}}_{1} + c_{1})}

and the additional condition, ${\bar{c}}_{3} > {\bar{c}}_{1}$ is met. However, for any $γ \neq 0$ , we can always choose $c_{1} \geq 0$ sufficiently small, such that ${\bar{c}}_{3} {(c_{1} - γ)}^{2} / 2 c_{1} ({\bar{c}}_{2} + γ) ({\bar{c}}_{3} - {\bar{c}}_{1})$ is arbitrarily large. In other words, for any set of parameters, there will always be some type with incentive to deviate. Therefore, there can be no PBE with $γ \neq 0$ . What remains is $γ = 0$ . This will be satisfied so long as

α \geq \frac{{\bar{c}}_{3} c_{1}}{2 {\bar{c}}_{2} ({\bar{c}}_{3} - {\bar{c}}_{1} + c_{1})}

for all $c_{1} < 1 - q$ . For ${\bar{c}}_{3} > {\bar{c}}_{1}$ , the fraction is increasing in $c_{1}$ . Therefore, this condition is most restrictive as $c_{1}$ approaches $1 - q$ (or ${\bar{c}}_{1} / 2$ , if ${\bar{c}}_{1}$ is sufficiently small). Thus, it must be that $α$ satisfies the following inequality:

α \geq \frac{{\bar{c}}_{3} \min {\frac{{\bar{c}}_{1}}{2}, 1 - q}}{2 {\bar{c}}_{2} ({\bar{c}}_{3} - {\bar{c}}_{1} + \min {\frac{{\bar{c}}_{1}}{2}, 1 - q})}

for all $c_{1} < 1 - q$ , with the supplementary condition, ${\bar{c}}_{3} > {\bar{c}}_{1}$ . Because types sufficiently close to $c_{1} = 0$ will always exist, these conditions must always be met.

Case 2: Suppose that $c_{1} > 1 - q$ and $c_{1} \leq {\bar{c}}_{1} / 2$ . Then the condition that must be satisfied will be

\begin{matrix} (p + \frac{{\bar{c}}_{2} - γ}{2}) (\frac{1}{2} + \frac{γ}{2 {\bar{c}}_{2}}) + (p - c_{1}) (\frac{1}{2} - \frac{γ}{2 {\bar{c}}_{2}}) \\ + α [(\frac{1}{2} + \frac{γ}{2 {\bar{c}}_{2}}) (1 - q - \frac{(1 - q) ({\bar{c}}_{1} - (1 - q))}{{\bar{c}}_{3}})] \geq \\ (p + \frac{{\bar{c}}_{2} - c_{1}}{2}) (\frac{1}{2} + \frac{c_{1}}{2 {\bar{c}}_{2}}) + (p - c_{1}) (\frac{1}{2} - \frac{c_{1}}{2 {\bar{c}}_{2}}) \end{matrix}

Again, this is satisfied when $α$ is sufficiently large:

α \geq \frac{{\bar{c}}_{3} {(c_{1} - γ)}^{2}}{2 γ {\bar{c}}_{3} + 2 {\bar{c}}_{2} ({\bar{c}}_{3} (1 - 2 q) - 2 (1 - q) ({\bar{c}}_{1} - (1 - q)))}

The right-hand side is clearly increasing in $c_{1}$ , so the condition will be most stringent when $c_{1} = {\bar{c}}_{1} / 2$ . In addition, because the previous case demonstrates that only $γ = 0$ can sustain an equilibrium for sufficiently low-cost types, let $γ = 0$ . Thus, the condition that must be satisfied will be

α \geq \frac{{\bar{c}}_{3} {\frac{{\bar{c}}_{1}}{2}}^{2}}{2 {\bar{c}}_{2} ({\bar{c}}_{3} (1 - 2 q) - 2 (1 - q) ({\bar{c}}_{1} - (1 - q)))}

with the supplementary condition, ${\bar{c}}_{3} (1 - 2 q) > 2 (1 - q) ({\bar{c}}_{1} - (1 - q))$ . This also requires $q < \frac{1}{2}$ .

Case 3: Suppose that $c_{1} \leq 1 - q$ and $c_{1} > \frac{{\bar{c}}_{1}}{2}$ . Then the condition that must be satisfied will be

\begin{matrix} (p + \frac{{\bar{c}}_{2} - γ}{2}) (\frac{1}{2} + \frac{γ}{2 {\bar{c}}_{2}}) + (p - c_{1}) (\frac{1}{2} - \frac{γ}{2 {\bar{c}}_{2}}) \\ + α [(\frac{1}{2} + \frac{γ}{2 {\bar{c}}_{2}}) (1 - q - \frac{{\bar{c}}_{1}^{2}}{4 {\bar{c}}_{3}}) + (\frac{1}{2} - \frac{γ}{2 {\bar{c}}_{2}}) (1 - q - c_{1})] \geq \\ (p + \frac{{\bar{c}}_{2} - c_{1}}{2}) (\frac{1}{2} + \frac{c_{1}}{2 {\bar{c}}_{2}}) + (p - c_{1}) (\frac{1}{2} - \frac{c_{1}}{2 {\bar{c}}_{2}}) + α (1 - q - c_{1}) \end{matrix}

Again, this is satisfied when $α$ is sufficiently large:

α \geq \frac{2 {\bar{c}}_{3} {(c_{1} - γ)}^{2}}{({\bar{c}}_{2} + γ) (4 c_{1} {\bar{c}}_{3} - {\bar{c}}_{1}^{2})}

For $\frac{{\bar{c}}_{1}}{2} < 1 - q$ , the condition will be most stringent at $c_{1} = 1 - q$ , and so, for $γ = 0$ , the necessary condition will be

α \geq \frac{2 {\bar{c}}_{3} \min {{\bar{c}}_{1}, (1 - q)}^{2}}{{\bar{c}}_{2} (4 \min {{\bar{c}}_{1}, (1 - q)} {\bar{c}}_{3} - {\bar{c}}_{1}^{2})}

This requires only a weaker version of the supplementary condition above: $2 {\bar{c}}_{3} > {\bar{c}}_{1}$ .

Case 4: Finally, suppose that $c_{1} > 1 - q$ and $c_{1} > {\bar{c}}_{1} / 2$ . Then the condition that must be satisfied will be

\begin{matrix} (p + \frac{{\bar{c}}_{2} - γ}{2}) (\frac{1}{2} + \frac{γ}{2 {\bar{c}}_{2}}) + (p - c_{1}) (\frac{1}{2} - \frac{γ}{2 {\bar{c}}_{2}}) + α [(\frac{1}{2} + \frac{γ}{2 {\bar{c}}_{2}}) (1 - q - \frac{{\bar{c}}_{1}^{2}}{4 {\bar{c}}_{3}})] \\ \geq (p + \frac{{\bar{c}}_{2} - c_{1}}{2}) (\frac{1}{2} + \frac{c_{1}}{2 {\bar{c}}_{2}}) + (p - c_{1}) (\frac{1}{2} - \frac{c_{1}}{2 {\bar{c}}_{2}}) \end{matrix}

Once more, this is satisfied when $α$ is sufficiently large:

α \geq \frac{2 {\bar{c}}_{3} {(c_{1} - γ)}^{2}}{({\bar{c}}_{2} + γ) (4 {\bar{c}}_{3} (1 - q) - {\bar{c}}_{1}^{2})}

This is increasing in $c_{1}$ , yielding

α \geq \frac{2 {\bar{c}}_{3} {\bar{c}}_{1}^{2}}{{\bar{c}}_{2} (4 {\bar{c}}_{3} (1 - q) - {\bar{c}}_{1}^{2})}

This requires the supplementary condition $1 - q > \frac{{\bar{c}}_{1}^{2}}{4 {\bar{c}}_{3}}$ .

There are three possible types of fully pooling PBE, depending on the parametrization of the model: ${\bar{c}}_{1} \leq 1 - q$ , $1 - q < {\bar{c}}_{1} < 2 (1 - q)$ , or ${\bar{c}}_{1} \geq 2 (1 - q)$ . For ${\bar{c}}_{1} \leq 1 - q$ , types $c_{1} \in [0, {\bar{c}}_{1} / 2)$ are subject to Case 1, and types $c_{1} \in ({\bar{c}}_{1} / 2, {\bar{c}}_{1}]$ are subject to Case 3. For $1 - q < {\bar{c}}_{1} < 2 (1 - q)$ , types $c_{1} \in [0, {\bar{c}}_{1} / 2)$ are subject to Case 1, types $c_{1} \in ({\bar{c}}_{1} / 2, 1 - q]$ are subject to Case 3, and types $c_{1} \in (1 - q, {\bar{c}}_{1}]$ are subject to Case 4. Finally, for ${\bar{c}}_{1} \geq 2 (1 - q)$ , types $c_{1} \in [0, 1 - q)$ are subject to Case 1, types $c_{1} \in (1 - q, {\bar{c}}_{1} / 2]$ are subject to Case 3, and types $c_{1} \in ({\bar{c}}_{1} / 2, {\bar{c}}_{1}]$ are subject to Case 4. Thus, we can write down the equilibrium conditions for each possible setup:

α \geq (\begin{matrix} \frac{2 {\bar{c}}_{3} {\bar{c}}_{1}^{2}}{({\bar{c}}_{2}) (4 {\bar{c}}_{1} {\bar{c}}_{3} - {\bar{c}}_{1}^{2})} & if {\bar{c}}_{1} \leq 1 - q \\ \frac{2 {\bar{c}}_{3} {\bar{c}}_{1}^{2}}{{\bar{c}}_{2} (4 {\bar{c}}_{3} (1 - q) - {\bar{c}}_{1}^{2})} & if 1 - q < {\bar{c}}_{1} \leq 2 (1 - q) \\ \max {\frac{2 {\bar{c}}_{3} {\bar{c}}_{1}^{2}}{{\bar{c}}_{2} (4 {\bar{c}}_{3} (1 - q) - {\bar{c}}_{1}^{2})}, \\ \frac{{\bar{c}}_{3} {(\frac{{\bar{c}}_{1}}{2})}^{2}}{2 {\bar{c}}_{2} ({\bar{c}}_{3} (1 - 2 q) - 2 (1 - q) ({\bar{c}}_{1} - (1 - q))}} & if {\bar{c}}_{1} > 2 (1 - q) \end{matrix}

with supplementary conditions ${\bar{c}}_{1} < {\bar{c}}_{3}$ in all cases, and ${\bar{c}}_{3} > \frac{2 (1 - q) ({\bar{c}}_{1} - (1 - q))}{1 - 2 q}$ for ${\bar{c}}_{1} > 2 (1 - q)$ . □

Proof of Lemma 1: Parts 1 and 2 of the lemma are corollaries of the proofs above. Part 1 follows directly from the proof that types cannot fully separate for $α > 0$ , while part 2 follows from the necessary condition for an equilibrium derived above. Part 3 is a less direct extension, but simple. Suppose that all types on the interval $[1 - q, 1 - q + ε]$ , $ε > 0$ , pool on a particular offer, $x \neq p + ({\bar{c}}_{2} - c_{1}) / 2$ . Because the war payoff for all of these types is negative, the optimal offer for state 3 is $y^{*} = 1$ , which will be accepted. Because $c_{1}$ can do no worse in the second stage by deviating to $p + ({\bar{c}}_{2} - c_{1}) / 2$ , which will improve its first stage payoff, it will have an incentive to deviate. Thus, an equilibrium pooling set must contain some types who will reject some feasible offers. The proof of parts 4 and 5 follow.

We want to show that an equilibrium requires that state 1’s demand ( $x$ ) is decreasing in its cost of war. In order for an equilibrium to exist, for any types $c_{1}$ and $c_{1}^{'}$ with equilibrium offers $x^{*} (c_{1})$ and $x^{*} (c_{1}^{'})$ , respectively, it must be true that

E u_{1} (x^{*} (c_{1}) | c_{1}) \geq E u_{1} (x^{*} (c_{1}^{'}) | c_{1})

and

E u_{1} (x^{*} (c_{1}^{'}) | c_{1}^{'}) \geq E u_{1} (x^{*} (c_{1}) | c_{1}^{'})

To simplify notation, let $a$ be the probability that $x^{*} (c_{1})$ is accepted and $a^{'}$ be the probability that $x^{*} (c_{1}^{'})$ is accepted, let $x$ and $x^{'}$ be the expected first round payoffs for offering $x^{*} (c_{1})$ and $x^{*} (c_{1}^{'})$ , respectively, and let $w$ and $w^{'}$ be the war payoffs, $1 - q - c_{1}$ and $1 - q - c_{1}^{'}$ , respectively. Finally, let $θ$ denote the payoff from a second round offer following an acceptance, such that $θ (c)$ is type $c$ ’s payoff after making the offer, $x^{*} (c_{1})$ , and $θ^{'} (c)$ is type $c$ ’s expected payoff after making the offer $x^{*} (c_{1}^{'})$ . Note that the value of $θ$ is (weakly) decreasing in both $c_{1}$ and $c_{H}$ , state 3’s belief about the upper end of the range of types willing to make a given offer. The specific utility is given by

θ (c_{1} | x, c_{H}) = (\begin{matrix} 1 - q - c_{1} (\frac{c_{H} - c_{1}}{{\bar{c}}_{3}}) & if c_{1} < 1 - q \\ (1 - q) (1 - \frac{c_{H} - (1 - q)}{{\bar{c}}_{3}}) & if c_{1} \geq 1 - q \end{matrix}

An equilibrium requires that the following two conditions hold:

\begin{matrix} a (x + θ (c_{1})) + (1 - a) w \geq a^{'} (x^{'} + θ^{'} (c_{1})) + (1 - a^{'}) w \\ a^{'} (x^{'} + θ^{'} (c_{1}^{'})) + (1 - a^{'}) w^{'} \geq a (x + θ (c_{1}^{'})) + (1 - a) w^{'} \end{matrix}

These two inequalities imply

\begin{matrix} a (x + θ (c_{1})) + (1 - a) w + a^{'} (x^{'} + θ^{'} (c_{1}^{'})) + (1 - a^{'}) w^{'} \geq \\ a^{'} (x^{'} + θ^{'} (c_{1})) + (1 - a^{'}) w + a (x + θ (c_{1}^{'})) + (1 - a) w^{'} \end{matrix}

which can be rewritten as

a ((θ (c_{1}) - θ (c_{1}^{'})) - (w - w^{'})) \geq a^{'} ((θ^{'} (c_{1}) - θ^{'} (c_{1}^{'})) - (w - w^{'}))

Without loss of generality, let $c_{1} < c_{1}^{'}$ . Then it follows that $w > w^{'}$ , and because $θ (\cdot)$ is (weakly) decreasing in $c_{1}$ , for a given offer, $x$ , $θ (x) \geq θ^{'} (x)$ . Note that for any $c_{1}$ and any $x$ , $θ (x) \geq w$ . Given these conditions, in order to demonstrate that it must be true that $a^{'} \geq a$ , it suffices to show that $(θ (c_{1}) - θ (c_{1}^{'})) \leq (θ^{'} (c_{1}) - θ^{'} (c_{1}^{'})) \leq (w - w^{'})$ . Because $θ (x) \geq θ^{'} (x) \geq w$ , this condition will be true if $θ (x) - θ^{'} (x)$ is decreasing in $θ$ . The value of $θ$ is a (monotonically decreasing) function of $c_{H}$ , the highest type of state 1 that makes a given offer in equilibrium. Thus, it will suffice to show that the difference is increasing in $c_{H}$ . Depending on the values of $c_{1}$ and $c_{1}^{'}$ , there are six possible scenarios. Taking the derivative of the difference with respect to $c_{H}$ yields

\frac{\partial (θ (x) - θ^{'} (x))}{\partial c_{H}} = (\begin{matrix} \frac{c_{1}^{'} - c_{1}}{{\bar{c}}_{3}} & if 1 - q > c_{1}^{'} > c_{1} \\ \frac{1 - q - c_{1}}{{\bar{c}}_{3}} & if c_{1}^{'} \geq 1 - q > c_{1} \\ 0 & if c_{1}^{'} > c_{1} \geq 1 - q \end{matrix}

In each of these cases, $\partial (θ (x) - θ^{'} (x)) / \partial c_{H} \geq 0$ . Thus, it must either be the case that $a \leq a^{'}$ , or $\Pr (Accept | x^{*} (c_{1})) \leq \Pr (Accept | x^{*} (c_{1}^{'}))$ . Because probability of acceptance is decreasing in the offer size, this implies that $x^{*} (c_{1}) \geq x^{*} (c_{1}^{'})$ . Thus, some type will always have an incentive to deviate unless weaker types make smaller offers. Therefore, a necessary condition for an equilibrium is that $x^{*} (c_{1})$ is decreasing in $c_{1}$ .

Proving that the demand made by a pooling interval must be at the ideal one-stage offer of the minimum of the set is simple. For any equilibrium interval, $[c_{L}, c_{H}]$ , state 3 knows that $c_{1} \geq c_{L}$ , and thus will offer no more than $c_{L}$ ’s reservation value. Then for $c_{L}$ , the second stage payoff will be weakly smaller than what it receives from separating. Therefore, $c_{L}$ will strictly prefer to separate unless it makes the optimal one-stage offer. Because $C_{1}$ is convex, if the interval is open on the left (i.e. $(c_{L}, c_{H}]$ ), then for any $x \neq p + ({\bar{c}}_{2} - c_{L}) / 2$ , we can find a type sufficiently close to $c_{L}$ to prefer deviation. Thus, the offer made by all types on the interval must be $x = p + ({\bar{c}}_{2} - c_{L}) / 2$ . □

Proof of Proposition 3: From Lemma 1, we know that any equilibrium must contain some set, $[0, {\hat{c}}_{1}]$ that makes the offer, $x^{*} = \min {p, {\bar{c}}_{2} / 2, 1}$ . To identify a semi-separating equilibrium, first identify ${\hat{c}}_{1}$ . This type is indifferent between pooling with $c_{1} = 0$ and separating. Any $c_{1} < {\hat{c}}_{1}$ will strictly prefer pooling and any $c_{1} > {\hat{c}}_{1}$ will strictly prefer separating. On the interval, $[0, {\hat{c}}_{1}]$ , $c_{H} = {\hat{c}}_{1} > c_{H} / 2$ , so we must find the solution using the generalized versions of cases 3 and 4 from the proof of Proposition 2. Solving for $c_{1}$ yields

{\hat{c}}_{1} = (\begin{matrix} \frac{4 {\bar{c}}_{2} {\bar{c}}_{3} α}{2 {\bar{c}}_{3} + {\bar{c}}_{2} α} & for c_{1} \leq 1 - q \\ \frac{2 {\bar{c}}_{2} {\bar{c}}_{3} α (1 - q)}{\sqrt{{\bar{c}}_{2} {\bar{c}}_{3} α (1 - q) (2 {\bar{c}}_{3} + {\bar{c}}_{2} α)}} & for c_{1} > 1 - q \end{matrix}

Given these values, we can find the conditions for which such a type exists. For separation to occur at the first point, there must be ${\hat{c}}_{1} \leq 1 - q$ . This will exist if

{\bar{c}}_{2} \leq \frac{2 {\bar{c}}_{3} (1 - q)}{α (4 {\bar{c}}_{3} - (1 - q))}

If this condition is satisfied, then there will be a new interval beginning with $4 {\bar{c}}_{2} {\bar{c}}_{3} α / 2 {\bar{c}}_{3} + {\bar{c}}_{2} α$ . If not, then the condition for the second possibility is

{\bar{c}}_{2} \leq \frac{2 {\bar{c}}_{1}^{2} {\bar{c}}_{3}}{α (4 {\bar{c}}_{3} (1 - q) - {\bar{c}}_{1}^{2})}

If this condition is met and the previous is not, then all $c_{1} \geq 2 {\bar{c}}_{2} {\bar{c}}_{3} α (1 - q) / \sqrt{{\bar{c}}_{2} {\bar{c}}_{3} α (1 - q)}$ $\sqrt{(2 {\bar{c}}_{3} + {\bar{c}}_{2} α)}$ will separate, offering $\max {p + ({\bar{c}}_{2} - c_{1}) / 2, 0}$ . If neither of these is met,¹⁹ then all types will pool. Notably, the partial with respect to $α$ for both values of ${\hat{c}}_{1}$ is strictly positive, indicating that the size of the pooling interval is increasing in $α$ .

We know from Lemma 1 that a pooling interval must make an offer at the infimum of the set. Letting $γ = {\hat{c}}_{1}$ , we can find the type indifferent between pooling with ${\hat{c}}_{1} = c_{L}$ and separating. Again, this will use a generalized version of the case structure from the proof of Proposition 2. In each case, $c_{H} = c_{1} > c_{H} / 2$ . There are four conditions to check. We find the next pooling interval by checking them in the following order.

Condition 1: First, we look for boundaries at $c_{1} \leq 2 c_{L}$ and $c_{1} \leq 1 - q$ . The type that will be indifferent between pooling and separating is

c_{1} = γ + 2 α ({\bar{c}}_{2} + γ)

A new interval begins at this point, as long as $γ + 2 α ({\bar{c}}_{2} + γ) \leq \min {2 γ, 1 - q}$ . This will always be true if the following holds:

{\bar{c}}_{2} \leq \min {γ (\frac{1 - 2 α}{2 α}), 1 - q - \frac{γ (1 + 2 α)}{2 α}}

If this is satisfied, then a new interval can begin here. Otherwise, we check the next condition.

Condition 2: Now look for boundaries at $1 - q \geq c_{1} > 2 c_{L}$ . The indifferent type will be

c_{1} = \frac{2 {\bar{c}}_{3} γ + 2 α ({\bar{c}}_{2} + γ) ({\bar{c}}_{3} + γ) + \sqrt{2 α {\bar{c}}_{3} ({\bar{c}}_{2} + γ) (2 α ({\bar{c}}_{2} + γ) ({\bar{c}}_{3} + γ) - γ^{2})}}{2 {\bar{c}}_{3} + α ({\bar{c}}_{2} + γ)}

Separation is viable so long as this value is less than equal to $1 - q$ . If this condition is met, then a new pooling interval will begin at the value of $c_{1}$ above. If it is unsatisfied, we move to the next condition.

Condition 3: Now look for $2 c_{L} \geq c_{1} > 1 - q$ . The indifferent type under these conditions will be

c_{1} = γ + \sqrt{2 α (1 - q - γ) ({\bar{c}}_{2} + γ)}

For a separation to occur here, this value must be weakly smaller than ${\bar{c}}_{1}$ . As in condition 1, this will always be met if ${\bar{c}}_{2}$ is sufficiently small:

{\bar{c}}_{2} \leq \frac{{({\bar{c}}_{1} - γ)}^{2} - 4 γ α (1 - q - γ)}{2 α (1 - q - γ)}

If this condition is met, then all types with higher costs than the type above will offer $\max {\frac{{\bar{c}}_{2} - c_{1}}{2}, 0}$ . Otherwise, we move to the next condition.

Condition 4: Now look for $c_{1} > 2 c_{L}$ and $c_{1} > 1 - q$ . The indifferent type under these conditions will be

\begin{matrix} c_{1} = \frac{2 {\bar{c}}_{3} γ + 2 γ α ({\bar{c}}_{2} + γ)}{2 {\bar{c}}_{3} + α ({\bar{c}}_{2} + γ)} \\ + \frac{\sqrt{2 α {\bar{c}}_{3} ({\bar{c}}_{2} + γ) (2 α {\bar{c}}_{2} - γ (γ - 2 α (1 - {\bar{c}}_{2} - γ)) - 2 q α ({\bar{c}}_{2} + γ) + 4 {\bar{c}}_{3} (1 - q - γ))}}{2 {\bar{c}}_{3} + α ({\bar{c}}_{2} + γ)} \end{matrix}

Separation at this interval will occur as long as the above value is weakly less than ${\bar{c}}_{1}$ . If this condition is met, then all types with higher costs than the type above will offer $\max {({\bar{c}}_{2} - c_{1}) / 2, 0}$ . If all conditions go unmet, then no $c_{1}$ prefers separation to pooling with $c_{1} = 0$ .

These equations can be used to find each subsequent boundary point. These conditions ensure that no type of state 1 has an incentive to deviate. State 2’s strategy and state 1’s second round strategy will be identical to the strategies described in Proposition 2. State 3’s beliefs can again be given by a set, $μ$ , such that state 3 places equal probability upon all $c_{1} \in μ$ . In equilibrium, this will be $μ^{*} = [c_{L}, c_{H}]$ for any pooling interval, if state 2 accepts; $μ^{*} = c_{1}$ for any accepted offer of $0 < x < p + ({\bar{c}}_{2} - (1 - q)) / 2$ , or any rejected offer; $μ^{*} = [{\bar{c}}_{2}, {\bar{c}}_{1}]$ for an offer of $x = 0$ ; and $μ^{*} = {\bar{c}}_{1}$ for any offer not made by any pooling interval. Given its beliefs, state 3’s strategy will be to play $y^{*} = \min {q + μ^{*}, 1}$ if $μ^{*}$ is a singleton, and if $μ^{*}$ is some interval, $[c_{L}, c_{H}]$ , then

y^{*} = (\begin{matrix} q + c_{L} & if c_{3} > c_{H} \\ q + \frac{c_{H} - c_{3}}{2} & if {\bar{c}}_{1} - 2 (1 - q) \leq c_{3} \leq {\bar{c}}_{1} \\ 1 & if c_{3} < c_{H} - 2 (1 - q) \end{matrix}

This set of beliefs and strategies, along with the strategies defined above constitute a semi-separating PBE. Because the partial derivative for each boundary point is positive with respect to $α$ , this demonstrates that the average size of the pooling interval is increasing in $α$ , and the number of pooling intervals is decreasing in $α$ .

Acknowledgements

Previous versions of this paper were presented at the 2011 meeting of the Southern Political Science Association and the Wallis Political Economy Working Group at the University of Rochester. The author thanks audience participants, as well as Mark Fey, Hein Goemans, Shawn Ramirez, and Curt Signorino for comments on earlier versions of the paper.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Notes

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