Domestic constraints in crisis bargaining

Abstract

This paper analyzes an agency model of crisis bargaining where two states have private information about war payoffs. In the model, two leaders bargain on behalf of their own states. Importantly, owing to political bias and audience costs, a leader’s war payoff and peace payoff differ from those of her state at large. I establish general results about leaders’ bargaining strategies and the possibility of peaceful resolution. By examining incentive compatibility constraints, I show that in any equilibrium that has zero probability of costly war, a leader’s payoff net of audience costs cannot vary with their private information. After that, I identify the size of resource necessary to appease both states. If this necessary condition holds, which is affected by political bias, there exist properly specified audience costs that guarantee peaceful bargaining outcomes.

Introduction

A near-consensus in the international relations (IR) literature is that information asymmetry can give rise to war (Blainey, 1988; Fearon, 1995). Under this perspective, many crises were successfully resolved because mediators reduced the chances that states miscalculate in the face of this information asymmetry. For example, Robert Gates’s persuasive message that the war would be detrimental is often regarded as the key to his successful mediation of the 1990 Kashmir crisis (Hagerty, 1998).

But states rarely bargain in a vacuum. In reflecting on the later India–Pakistan crises between 1999 and 2016, Hagerty writes: “In all of the crises, the Indian government opted for markedly restrained, non-escalatory responses in the face of intense domestic pressures to retaliate with punishing military force” (Hagerty and Hagerty, 2020: 11). However, the elements of domestic politics seem beyond the scope of the rationalist explanations. ¹ Several prominent articles address the agency issues governing a state’s decision to go to war. Some analyze political leaders’ incentive to gamble for resurrection (Downs and Rocke, 1994; Hess and Orphanides, 1995; Goemans and Fey, 2009); others show that political leaders’ disproportional gains and losses compared with those of the citizens can be a cause of war (Bueno De Mesquita et al., 1999; Jackson and Morelli, 2007). Despite these valuable efforts, it remains unclear how the agency issues interact with information asymmetry and affect the possibility of peaceful resolution of conflict.

This paper addresses a straightforward question: under what conditions can states resolve crises peacefully in the face of information asymmetry, when crisis bargaining takes place between political leaders whose preferences for war and peace are generally misaligned with the states?

To answer this question, I study a general agency model of crisis bargaining with the game-free approach. The model assumes standard information asymmetry regarding each state’s war payoffs. Each state delegates a leader to bargain. The key innovation is introducing the wartime and peacetime preference divergences between leaders and citizens to the game-free analysis framework (e.g., Banks (1990), Fey and Ramsay (2011)). Specifically, to capture the idea that people do not benefit or suffer equally from a conflict, I introduce a cost parameter differentiating the war payoffs between leaders and citizens. Such a modeling device can be viewed as a version of “political bias” à la Jackson and Morelli (2007). To capture the payoff difference between leaders and citizens conditional on a peaceful settlement, I introduce “audience costs” that agree with the standard usage (e.g., Fearon (1994)). By focusing on leaders’ incentive compatibility constraints, I derive a set of qualitative results about their bargaining strategies and outcomes that are independent from the specifications of the game form and domestic constraints. Along the way, I provide conditions supporting peaceful resolution as an equilibrium outcome of crisis bargaining.

I show that, for crisis bargaining to achieve a peaceful equilibrium outcome, a leader’s payoff net of audience costs cannot vary with her private information. To see the intuition, suppose a higher-type leader (an informed leader who receives a higher war payoff) receives a different equilibrium payoff from a lower-type leader. Then, the type of leader who receives a strictly lower payoff can gain by unilaterally deviating to the equilibrium strategy chosen by the other type. In doing so, they obtain a higher peacetime payoff without bearing an increased risk of war (the war probability is constant zero by “peacefulness”). Thus, as the first implication of the game-free analysis, peacefulness essentially eliminates the “risk–return” tradeoff that is common in crisis bargaining. Consequently, if war must be averted, leaders tend to pool on the bargaining strategy inducing the maximum payoff net of audience costs.

For peaceful resolution to be plausible, the total size of the resource being divided should exceed the sum of two states’ expected war payoffs; otherwise, at least one side is willing to resolve the crisis via war in lieu of peace negotiation. Notably, the above condition is least likely to hold if each state has the type that receives the highest expected war payoff. Building on this observation, I identify the size of resource necessary to appease states of all types. Such a quantity is affected by political bias, because political bias alters states’ expected war payoffs. If the necessary condition about resource fails to hold, then conflicts must occur with a strictly positive probability, regardless of how audience costs are specified. Thus, we arrive at the second implication of the game-free analysis: the (violation of the) necessary condition about resource makes a case in which states cannot design audience costs to avert war.

If the necessary condition about resource holds, I prove the existence of a pair of audience costs that supports peaceful resolution. Thus, as the third implication of the game-free analysis, the necessary condition about resource may be viewed as prescribing the limits of peaceful resolution. In models where peaceful resolution is not a must, students of war may specify audience costs and examine how political leaders’ bargaining strategies respond to these incentive schemes. These models are doubtlessly important for understanding conflicts and resolution in the real world. However, as an intellectual exercise, fully specified models of audience costs cannot tell whether ending crises is plausible when states have the opportunity to design their domestic constraints. My findings fill this gap.

Related literature

This paper builds on various formal models of crisis bargaining. ² Specifically, my crisis bargaining game takes the “costly lottery” approach to war, as in Fearon (1995). The notion of audience costs is borrowed from Fearon (1994), Schultz (2001), and Ashworth and Ramsay (2017), whereas the notion of political bias is borrowed from Bueno De Mesquita et al. (1999) and Jackson and Morelli (2007). In terms of “designing audience costs” to achieve certain political ends, my model is closest to Ashworth and Ramsay (2017). They analyze the audience cost design to maximize a state’s payoff from crisis bargaining, whereas I examine the audience cost design that facilitates peaceful resolution of conflict.

Methodologically, this paper adds to the mechanism design approach to war. Banks (1990) first recognizes the necessity of “game-free” analyses for formal models of war. Fey and Ramsay (2009, 2011) study the possibility of peaceful resolution in the face of different sources of uncertainty. Hörner et al. (2015) compare the power of mediation and arbitration in minimizing the chances of war. Fey and Kenkel (2021) show that the ultimatum game is a robust protocol in studying crisis bargaining. Kenkel and Schram (2021) augment the set of actions to the canonical models. This paper explores the limits of peaceful resolution subject to domestic constraints and generalizes a set of robust results in crisis bargaining.

Model

Setup

Two states are disputing over a unit size of resource. Each state, indexed i, has two active players: a “leader” and a “citizen.” Crisis bargaining takes place between two leaders; its outcome is either war or a peaceful settlement $x=(x_1,x_2)$ specifying a division of resource $x_i$ to state i, with $x_1+x_2\le 1$ and $x_1,x_2\ge 0$ . Let $\pi \in [0,1]$ be the probability of war. Then, the bargaining outcome can be summarized as a tuple $(\pi ,x)$ : states fight with probability $\pi$ , whereas states reach a peaceful settlement $x$ with probability $1-\pi$ . War is a costly lottery, which imposes a cost on all players regardless of its outcome. The probability that state i wins the war is denoted $p_i$ , with $p_1+p_2=1$ . The winner takes all the resources.

The crisis bargaining game has the standard structure of information uncertainty about resolve; resolve affects states’ war payoffs but not their preferences for peaceful settlements. Let ${\theta }_i\in {\mathrm{\Theta }}_i:=\left[{\underset{¯}{\theta }}_i,{\overline{\theta }}_i\right]$ be state i’s resolve, or “type.” ${\theta }_i$ is drawn from a distribution with cdf $F_i\left({\theta }_i\right)$ and density $f_i\left({\theta }_i\right)>0$ . Then, the type space can be written as $\Theta ={\Theta }_1\times {\Theta }_2$ , with a typical element being $\theta =({\theta }_1,{\theta }_2)$ . ${\theta }_1$ and ${\theta }_2$ are independent. Players within state i observe the realization of ${\theta }_i$ but not ${\theta }_j$ . The war payoffs for players of state i are assumed to be strictly increasing and differentiable in ${\theta }_i$ and weakly decreasing in ${\theta }_j$ . According to Fey and Ramsay (2011), ${\theta }_i$ may represent the opposite of state i’s fighting costs (i.e., ${\theta }_i=-c_i$ ); ${\theta }_i$ may also represent state i’s relative strength, which (jointly with ${\theta }_j$ ) determines the probability that state i wins a war $p_i({\theta }_i,{\theta }_j)$ .

The decision of a leader from state i, denoted $d_i$ , comes from the set $D_i$ . $D_i$ has a variety of possibilities in crisis bargaining situations. It has a special case $D_i={\Theta }_i$ , which means that the leader’s strategy is restricted to reporting a type.

At this moment, the common game-theoretic approach is to posit an extensive form of crisis bargaining, as well as specifying the domestic constraints faced by leaders. For example, Fearon (1994) models crisis bargaining as a war of attrition and defines audience costs as the electoral cost that a leader pays after backing down from the crisis. Jackson and Morelli (2007) model crisis bargaining as a game of arm race, and define political bias as a leader’s risk/reward ratio from a war compared with that of the citizen she/he represents. As this paper aims to disentangle the role of domestic constraints in crisis bargaining, the game-theoretic approach admits a drawback foreshadowed by Banks (1990): the results derived under certain fully specified game form and domestic constraints may not necessarily extend to other specifications.

To obtain game-free results, I apply the tool of Bayesian mechanism design to the crisis bargaining game. This approach allows me to bypass the myriads of crisis bargaining game forms, and focus on conditions that are necessary for peaceful resolution. Its validity comes from the powerful revelation principle (Myerson, 1979), whose IR version is elaborated in Fey and Ramsay (2011).

The revelation principle is a statement about the outcome of a direct mechanism, which is a pair of functions mapping from the type space to the bargaining outcome, $(\pi ,x)$ . One can think of a direct mechanism as a kind of mediation procedure: first, leaders report their private types $\stackrel{\sim }{\theta }$ (not necessarily truthfully) to a powerful mediator; later, the mediator enforces an outcome $\left(\pi \left(\stackrel{\sim }{\theta }\right),x\left(\stackrel{\sim }{\theta }\right)\right)$ contingent on the report. A special class of direct mechanisms, in which leaders find it in their interests to report types truthfully, is called incentive compatible direct mechanisms. The revelation principle says that, from any crisis bargaining game between states, we can construct an outcome-equivalent incentive compatible direct mechanism.

Here, I briefly sketch the intuition behind the revelation principle. In an unmediated India–Pakistan crisis, each leader’s equilibrium strategy is a mapping from the private type ${\theta }_i$ to an optimal action $d_i^{*}\left({\theta }_i\right)$ , which induces an equilibrium outcome $\left(\pi \right(\theta ),x(\theta \left)\right)$ . Now, imagine Gates approaches two leaders and makes the following speech prior to crisis bargaining:

Instead of you two playing the equilibrium bargain strategy directly, why not let me bargain on behalf of you? All you need to do is text me about your type. I will use your reported types, together with your equilibrium strategies in the unmediated crisis bargaining game, to compute the action you are supposed to take, $d_i^{*}\left({\theta }_i\right)$ . Bear in mind that if you happen to text a wrong message ${\stackrel{\sim }{\theta }}_i\ne {\theta }_i$ , then I will accordingly play $d_i^{*}\left({\stackrel{\sim }{\theta }}_i\right)$ for you. Such an action cannot bring you a strictly higher payoff than your optimal action $d_i^{*}\left({\theta }_i\right)$ . So, please text me truthfully.

By committing to this mediation mechanism, Gates can elicit truthful reports from two sides and replicate the equilibrium strategy and outcome from the unmediated crisis bargaining game.

Thus, by the revelation principle, to explore the possibility that states may reach peaceful outcomes of crisis bargaining, I can study whether an incentive compatible direct mechanism can induce peaceful outcomes. If no such mechanism achieves “always peace” (i.e., zero probability of costly war), then always peace cannot be an equilibrium outcome of the crisis bargaining game.

Domestic constraints

Now, I formalize audience costs and political bias within a standard crisis bargaining setting.

Audience costs. Audience costs are often modeled as leaders’ electoral costs when they initiate crises but end up backing down (Fearon, 1994; Schultz, 2001; Ashworth and Ramsay, 2017). Typically, they take the form of a cost (benefit) after leaders strike a bad (good) deal off the bargaining table. I follow the standard approach: the audience cost $a_i$ is a mapping from each peaceful settlement $x_i$ to a real number $a_i\left(x_i\right)$ , including zero. Conditional on the settlement $x_i$ , citizens receive a payoff of $x_i$ , whereas leaders receive $v_i=x_i-a_i\left(x_i\right)$ . Notably, the settlement $x_i$ is likely to be affected by leaders’ bargaining strategy, which is a function of the private type $\theta$ . Thus, $v_i$ is also a function of $\theta$ .

Political bias.

Political bias is often modeled as the war payoff difference between leaders and citizens (Jackson and Morelli, 2007; Bueno De Mesquita et al., 1999; Ashworth and Ramsay, 2017). Following Ashworth and Ramsay (2017), I assume that within state i, a citizen’s war cost is $c_i$ , whereas a leader’s war cost is ${\lambda }_i{c}_i$ . Thus, their respective war payoffs are $w_i^c=p_i-c_i$ and $w_i^l=p_i-{\lambda }_i{c}_i$ . The cost parameter ${\lambda }_i$ represents the agency issues during wartime. For example, political elites may avoid serving military or suffer financially in a different way than grassroots because of war.

For expository convenience, I use capitalized letters to denote players’ interim peacetime and wartime payoffs. The interim payoffs for the citizen and the leader of type ${\theta }_i$ can be written as:

$\begin{array}{cc}\left(\mathrm{W}\mathrm{a}\mathrm{r}\right)\hspace{1em}& W_i^l\left({\theta }_i\right)={\int }_{{\underset{¯}{\theta }}_j}^{{\overline{\theta }}_j}{w}_i^l\left({\theta }_i,{\theta }_j\right)d{F}_j\left({\theta }_j\right),\hspace{1em}{W}_i^c\left({\theta }_i\right)={\int }_{{\underset{¯}{\theta }}_j}^{{\overline{\theta }}_j}{w}_i^c\left({\theta }_i,{\theta }_j\right)d{F}_j\left({\theta }_j\right),\\ \left(\mathrm{P}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{e}\right)& V_i\left({\theta }_i\right)={\int }_{{\underset{¯}{\theta }}_j}^{{\overline{\theta }}_j}{v}_i\left({\theta }_i,{\theta }_j\right)d{F}_j\left({\theta }_j\right),\hspace{1em}{X}_i\left({\theta }_i\right)={\int }_{{\underset{¯}{\theta }}_j}^{{\overline{\theta }}_j}{x}_i\left({\theta }_i,{\theta }_j\right)d{F}_j\left({\theta }_j\right).\end{array}$

For a type ${\theta }_i$ leader, denote her/his expected audience cost by $A_i\left({\theta }_i\right)={\int }_{{\underset{¯}{\theta }}_j}^{{\overline{\theta }}_j}{a}_i\left(x_i\left({\theta }_i,{\theta }_j\right)\right)d{F}_j\left({\theta }_j\right)$ and her/his expected probability of war by $\overline{\pi }\left({\theta }_i\right)={\int }_{{\underset{¯}{\theta }}_j}^{{\overline{\theta }}_j}\pi \left({\theta }_i,{\theta }_j\right)d{F}_j\left({\theta }_j\right)$ .

Participation constraints

The international system is anarchic. For example, Waltz (2001: 240) writes,

With many sovereign states, with no system of law enforceable among them, with each state judging its grievances and ambitions according to the dictates of its own reason or desire–conflict, sometimes leading to war, is bound to occur.

This means that states may back out from any signed peaceful agreement. In the crisis bargaining setting, states’ decisions to go to war are generally assumed to occur at the interim stage (Fey and Ramsay, 2011: 151). Thus, for peaceful resolution of conflicts, a necessary condition is that the signed agreement brings states higher expected payoffs than war. This requirement is known as the voluntary participation constraint in the game-free analysis.

This model considers the participation constraints for the leaders and citizens of two states. As the pivotal decision-makers of states, leaders’ voluntary participation clearly matters for the resolution of conflicts. Incorporating citizens’ participation constraint is meant to guarantee citizens to be the essential players of crisis bargaining. In a model without this constraint, the citizen’s preference becomes completely irrelevant for a state’s decision to go to war. Worse, such a model alludes to the possibility that the citizen can be coerced to accept undesirable peaceful settlements or war outcomes, which is uncommon at least in most modern democracies. Even in settings where citizens have no say in international affairs (e.g., one side of the crisis is a totalitarian regime), the game-free analysis can proceed without problem. I elaborate on this point in the Comments subsection.

Now, I formalize the participation constraints. For each ${\theta }_i\in \left[{\underset{¯}{\theta }}_i,{\overline{\theta }}_i\right]$ ,

$\begin{array}{cc}\left(\mathrm{L}\mathrm{e}\mathrm{a}\mathrm{d}\mathrm{e}\mathrm{r}\right)& W_i^l\left({\theta }_i\right)\le V_i\left({\theta }_i\right),\end{array}$

(1)

$\begin{array}{cc}\left(\mathrm{C}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{n}\mathrm{s}\right)& W_i^c\left({\theta }_i\right)\le X_i\left({\theta }_i\right).\end{array}$

(2)

Payoffs

Each player is risk neutral. Let $u_i^l,u_i^c$ be the ex post utility of the leader and the citizen of state i, which are the average payoffs from war $\left(w_i^l,w_i^c\right)$ and peace $(v_i,x_i)$ , weighted by the war probability $\pi$ . To emphasize their dependence on the type $\theta$ , $u_i^l,u_i^c$ are written as below:

$\begin{array}{cc}{u}_i^l\left(\theta \right)& =\pi \left(\theta \right)w_i^l\left(\theta \right)+\left(1-\pi \left(\theta \right)\right)v_i\left(\theta \right),\\ u_i^c\left(\theta \right)& =\pi \left(\theta \right)w_i^c\left(\theta \right)+\left(1-\pi \left(\theta \right)\right)x_i\left(\theta \right).\end{array}$

Let $U_i\left({\stackrel{\sim }{\theta }}_i|{\theta }_i\right)$ be a type- ${\theta }_i$ leader’s expected payoff if she reports ${\stackrel{\sim }{\theta }}_i$ in the mechanism. Then,

$U_i\left({\stackrel{\sim }{\theta }}_i|{\theta }_i\right)={\int }_{{\underset{¯}{\theta }}_j}^{{\overline{\theta }}_j}{u}_i\left(\stackrel{\sim }{{\theta }_i},{\theta }_j\right)d{F}_j\left({\theta }_j\right)={\int }_{{\underset{¯}{\theta }}_j}^{{\overline{\theta }}_j}\left[\pi \left({\stackrel{\sim }{\theta }}_i,\theta \right)w_i^l\left({\theta }_i,{\theta }_j\right)+\left(1-\pi \left(\stackrel{\sim }{{\theta }_i},{\theta }_j\right)\right)v_i\left({\stackrel{\sim }{\theta }}_i,{\theta }_j\right)\right]d{F}_j\left({\theta }_j\right).$

By truthful reporting, she receives a payoff of $U_i\left({\theta }_i\right|{\theta }_i)$ , which is denoted $U_i\left({\theta }_i\right)$ for simplicity.

Budget balancedness

I rule out the uninteresting possibility that a third-party may subsidize the peace process. ³ Accordingly, if peace has to be financed by two states, the resource being divided must be large enough to appease them across all types. Such a requirement corresponds to the notion of “budget-balancedness” in the mechanism design literature.

I assume that for each realization of type $\theta$ within a direct mechanism $\left(\pi \right(\theta ),x(\theta \left)\right)$ , state i receives an aggregated war payoff of ${\gamma }_i{w}_i^l\left(\theta \right)+\left(1-{\gamma }_i\right)w_i^c\left(\theta \right)$ and an aggregated peace payoff of ${\gamma }_i{v}_i\left(\theta \right)+(1-{\gamma }_i)x_i\left(\theta \right)$ . That is, state i’s aggregated payoff is the average of its leader’s and citizen’s payoff. The weights can literally represent the population share of grassroots and elites within a state; in this way, a state’s aggregated payoff simply takes its face value. Generally, the weights measure whether a state’s overall preference for war and peace is closer to its citizen’s or leader’s preference. When the weights are degenerate (i.e., models of unitary states), one player’s preference for war and peace is elevated into the state’s preference, which often governs the state’s decision to go to war.

Comments

The model setup is flexible enough to accommodate situations in which the citizen of a state is excluded from crisis bargaining. Whenever state i fits this description, the game-free analysis can proceed by setting the leader’s weight ${\gamma }_i$ to 1. In this way, state i’s incentive to go to war is determined by its leader’s incentive; such a state can be viewed as a unitary state in crisis bargaining games.

Analysis

I sketch out the steps of the game-free analysis. I start by characterizing the set of incentive compatible direct mechanism $\left(\pi \right(\theta ),x(\theta \left)\right)$ ; focusing on these mechanisms entails no loss of generality. Then, I provide conditions that guarantee incentive compatible direct mechanisms to be peaceful and respect players’ voluntary participation constraints, where “peaceful” is modeled as $\pi \left(\theta \right)\equiv 0$ for all $\theta \in \Theta$ (up to a measure-zero set). I call such mechanisms “ICDPP” mechanisms for short. Finally, I check whether the ICDPP mechanisms also satisfy budget-balancedness (i.e., whether they are feasible given the resource constraint faced by two states). If so, I can invoke Proposition 6 of Fey and Ramsay (2011, 164) to construct a game form that supports a peaceful bargaining outcome. Otherwise, by the contrapositive of the revelation principle, “always peace” cannot be an equilibrium of any crisis bargaining game.

Characterizing ICDPP mechanisms

Incentive compatibility

Lemma 1

For any direct mechanism $\left(\pi \right(\theta ),x(\theta \left)\right)$ ,

1. it is incentive compatible if and only if

$\begin{array}{c}{U}_i\left({\theta }_i\right)-U_i\left({\underset{¯}{\theta }}_i\right)={\int }_{\underset{¯}{{\theta }_i}}^{{\theta }_i}{\int }_{{\underset{¯}{\theta }}_j}^{{\overline{\theta }}_j}\pi \left(t,{\theta }_j\right)\frac{\partial w_i\left(t,{\theta }_j\right)}{\partial t}d{F}_j\left({\theta }_j\right)dt,\end{array}$

(3)

$\begin{array}{c}{\int }_{{\underset{¯}{\theta }}_j}^{{\overline{\theta }}_j}\pi \left(t,{\theta }_j\right)\frac{\partial w_i\left(t,{\theta }_j\right)}{\partial t}d{F}_j\left({\theta }_j\right)\text{is} \text{weakly} \text{increasing} \text{in}t.\end{array}$

(4)

• 2. leaders’ equilibrium war probability $\overline{\pi }\left({\theta }_i\right)$ is increasing in ${\theta }_i$ if $w_i({\theta }_i,{\theta }_j)$ can be written as $w_i({\theta }_i,{\theta }_j)=h_i\left({\theta }_i\right)-g_i\left({\theta }_j\right)$ where $h_i,g_i$ are strictly increasing functions.

Proof. In Online Appendix 1

Lemma 1 fully characterizes incentive compatible mechanisms (Part 1) and derives monotonicity results about leaders’ bargaining strategy (Part 2). Condition (3) implies that higher types obtain higher equilibrium payoffs because the integrand of RHS is weakly positive. Substantively, higher types can always pretend to be lower types during crisis bargaining, which brings them exactly the same peacetime payoffs but strictly higher wartime payoffs compared with lower types. Condition (4) and Part 2 provide sufficient conditions under which higher types are more likely to initiate war than lower types; its rationale lies in higher types’ comparative advantage in resolving crises with war.

Participation of peaceful mechanisms

I highlight a particular feature of the incentive compatible direct peaceful mechanisms: across all types, the leader of a given state must receive the same payoff. This can be seen mathematically by setting $\pi \left(\theta \right)=0$ to Conditions (3): a type- ${\theta }_i$ leader receives an expected (peace) payoff of $U_i\left({\theta }_i\right)=U_i\left({\underset{¯}{\theta }}_i\right)$ , which is a constant. As this observation holds regardless of the audience cost, I conclude:

Result 1

Fix any incentive compatible direct peaceful mechanism. For any given audience cost, leaders’ payoffs remain constant across types.

Proof. The result follows from the arguments in the text.

Intuitively, for a mechanism to qualify as “peaceful,” leaders’ misrepresentation of types cannot be penalized by an increased probability of war. Thus, within peaceful mechanisms, leaders are no longer subject to the commonplace “risk–return tradeoff” in crisis bargaining. Instead, they tend to pool on the strategy that brings them the highest peacetime payoff net of audience cost, which makes it difficult for students of war to infer leaders’ true types.

Various mechanism designs paired with audience costs can hold a leader’s peacetime payoff constant. If the audience cost is zero, then a mechanism can assign constant settlements across all reported types. If the audience cost goes “you (leaders) surrender all gains from a settlement to your motherland,” then leaders’ peacetime payoffs are constant zero from any mechanism.

Now, I rewrite the participation constraints. For a leader of type ${\theta }_i$ to participate, the settlement net of audience cost must be preferred to war, that is, $W_i^l\left({\theta }_i\right)\le X_i\left({\theta }_i\right)-A_i\left({\theta }_i\right)$ for i = 1, 2. By Result 1, the RHS is a constant, which I denote by ${\overline{V}}_i$ . To preclude war, this condition must hold for all types. Thus, the highest types’ participation constraints are relevant, which can be written as $W_i^l\left({\overline{\theta }}_i\right)\le {\overline{V}}_i$ for i = 1, 2 . Citizens’ participation can be written as $W_i^c\left({\overline{\theta }}_i\right)\le X_i\left({\theta }_i\right)$ for i = 1, 2 and for each ${\theta }_i$ .

Existence

Budget-balancedness

Fix an ICDPP mechanism, which is uniquely identified by its message-contingent settlement $x\left(\theta \right)$ . From this mechanism, state i of type ${\theta }_i$ is expected to bring back a payoff of ${\overline{V}}_i$ for its leader and $X_i\left({\theta }_i\right)$ for its citizen. State i’s total payoff is ${\gamma }_i{\overline{V}}_i+\left(1-{\gamma }_i\right)X_i\left({\theta }_i\right)$ , which can be viewed as its “price of peace.” Budget balancedness requires ${\sum }_i\left({\gamma }_i{\overline{V}}_i+(1-{\gamma }_i)X_i\left({\theta }_i\right)\right)\le 1$ , which means that peace must be financed by the resource being divided.

Existence condition

Result 2.

1. There always exists a pair of audience costs $\left(a_1^w,a_2^w\right)$ that prevents peaceful resolution.

2. An ICDPP mechanism exists only if the existence condition holds: ${\displaystyle \sum }_i({\gamma }_i{W}_i^l\left({\overline{\theta }}_i\right)+(1-{\gamma }_i)W_i^c\left({\overline{\theta }}_i\right))\le 1$ . In this case, there exists a crisis bargaining game form and a pair of audience costs $\left(a_1^p,a_2^p\right)$ in which an always peaceful equilibrium exists.

Proof. In Online Appendix 2

Result 2 describes the limits of peaceful resolution. The “impossibility” result (Part 1) is straightforward: extremely hawkish citizens can maneuver crisis bargaining by committing to ostracize leaders who strike a peaceful settlement. The existence condition (Part 2) extends the two-player bargaining results to the agency models of conflicts: with properly specified audience costs, peace is possible when the total size of resource being divided is large enough to appease two states. In line with Propositions 3, 5, and 6 of Fey and Ramsay (2011: 159,163–164), the existence condition is more likely to hold when the information uncertainty is about the cost of war than relative strength. Furthermore, political bias directly affects the existence condition, because it alters each state’s expected war payoff.

Now I elaborate on the design of audience costs supporting a peaceful outcome. Generally, if war is inefficient (as illustrated by the existence condition), the audience costs that support peaceful outcomes are guaranteed to exist. The logic is best seen by applying the famous Coase theorem ⁴ twice. Internationally, whenever two states jointly find the war inefficient, they may delegate leaders to sign a mutually beneficial peaceful settlement. Domestically, as the peaceful settlement reached between leaders becomes the efficient outcome in lieu of war, the citizen may agree to ratify the settlement, after properly transferring resources within the state. Within-state transfers are conditional on the settlement, so they can be viewed as a version of audience costs/benefits.

The relevance of audience costs in peaceful resolution is affected by the weights assigned to domestic players. Notably, a player weighted zero can be easily bought off by the other player. For example, a state composed of predominantly dovish citizens (leaders’ weight ${\gamma }_i\approx 0$ ) can dissuade its hawkish leaders from going to war by paying each of them a unit out of pocket, whose total value adds up to virtually zero. Thus, if an agency model assigns degenerate weights for domestic players, then for any specification of audience costs (i.e., the peacetime preference divergence between leaders and citizens), the possibility of peaceful resolution hinges on the player assigned the entire weight. ⁵

When the existence condition fails, for any settlement there exists a discontented player who sometimes refuses to settle down; consequently, war becomes inevitable. Although the strategic interactions between states may vary with model specifications, ⁶ they share common patterns summarized in the “sharpened” monotonicity results below. Let ${\mathrm{\Theta }}^w:={\mathrm{\Theta }}_1^w\times {\mathrm{\Theta }}_2^w$ be the set of types who initiate war with strictly positive probability. Then,

Corollary

Suppose ${\sum }_i({\gamma }_i{W}_i^l\left({\overline{\theta }}_i\right)+(1-{\gamma }_i)W_i^c\left({\overline{\theta }}_i\right))>1$ . Then in any equilibrium of the crisis bargaining game, leaders’ equilibrium payoffs are strictly increasing in ${\theta }_i$ for all ${\theta }_i\in {\mathrm{\Theta }}_i^w$ , where ${\mathrm{\Theta }}_i^w$ is nondegenerate. Furthermore, if the functional form assumption on $w_i({\theta }_i,{\theta }_j)$ in Lemma 1 holds, then leaders’ equilibrium war probability is strictly increasing in ${\theta }_i$ for all ${\theta }_i\in {\mathrm{\Theta }}_i^w$ .

Proof. In Online Appendix 3

Thus, if the existence condition fails, the monotonicity results must be strict over some range of types.

Conclusion

This paper analyzes a standard agency model of crisis bargaining where two states have private information about war payoffs. Rather than fully specifying the game form and domestic constraints, I derive conclusions with the game-free approach. To achieve peaceful outcomes, a necessary condition is that the resource being divided can appease both states. From this, I conclude that political bias matters for peaceful resolution because it alters states’ war payoffs. Moreover, if the necessary condition holds, I show that properly specified audience costs can guarantee peaceful bargaining outcomes. Thus, the necessary condition about resource can be viewed as prescribing the limits of peaceful resolution in the face of information asymmetry and agency issues.

By examining incentive compatibility constraints, I also derive robust results about political leaders’ equilibrium behaviors and payoffs in crisis bargaining games. Typically, higher types are more likely to initiate a war and obtain higher equilibrium payoffs. In any equilibrium that has zero probability of costly war, a leader’s payoff cannot vary with her private information. As these results hold regardless of the game form and domestic constraints, the value of a fully specified agency model lies in other empirically relevant predictions.

Supplemental Material

Supplemental Material - Domestic constraints in crisis bargaining

Supplemental Material for Domestic constraints in crisis bargaining by Liqun Liu in Research & Politics