Uncertainty and Incentives in Crisis Bargaining: Game-Free ... · In the literature on international conflict, the paper that is closest to ours is Banks (1990). This paper uses a

Uncertainty and Incentives in Crisis Bargaining:Game-Free Analysis of International Conflict

Mark Fey University of RochesterKristopher W. Ramsay Princeton University

We study two different varieties of uncertainty that countries can face in international crises and establish general resultsabout the relationship between these sources of uncertainty and the possibility of peaceful resolution of conflict. Among ourresults, we show that under some weak conditions, there is no equilibrium of any crisis bargaining game that has voluntaryagreements and zero probability of costly war. We also show that while uncertainty about the other side’s cost of war maybe relatively benign in peace negotiations, uncertainty about the other side’s strength in war makes it much more difficultto guarantee peaceful outcomes. Along the way, we are able to assess the degree to which particular modeling assumptionsfound in the existing literature drive the well-known relationship between uncertainty, the incentive to misrepresent, andcostly war. We find that while the theoretical connection between war and uncertainty is quite robust to relaxing manymodeling assumptions, whether uncertainty is about costs or the probability of victory remains important.

Acentral concern in international relations is un-derstanding the obstacles that prevent countriesfrom reaching mutually beneficial settlements in

times of conflict. Why do countries engage in lengthy warswhen the two sides would be better off if they could set-tle their dispute without war? As an answer, the conflictliterature has long held that uncertainty and the resultingincentive to misrepresent private information togetherare a central cause of conflict among states (Blainey 1988;Fearon 1995; Waltz 1979; Wittman 1979). One centralfinding of this theoretical literature is that informationaldifferences regarding aspects of the bargaining processplay a key role in determining bargaining and war behav-ior (Fearon 1995; Powell 1999; Schultz 1998; Slantchev2003b; Smith and Stam 2006; Wagner 2000). The com-mon theme of this work is that, because of private in-formation, the two sides in a conflict may be unable toidentify suitable settlement terms without learning moreabout each other. But because of incentives to misrepre-sent their private information, there is no way to crediblysignal this information short of war.

Mark Fey is Associate Professor of Political Science, 109E Harkness Hall, University of Rochester, Rochester, NY 14627([email protected]). Kristopher W. Ramsay is Assistant Professor of Politics, 033 Corwin Hall, Princeton University, Princeton,NJ 08544 ([email protected]).

We have benefited from comments received at the University of Rochester’s Peter D. Watson Center, Princeton’s International RelationsColloquium, the UCSD IR Seminar, and from participants at various professional conferences. We particularly thank Scott Ashworth, DanBernhardt, Josh Clinton, Tom Christensen, John Duggan, Jay Lyall, Adam Meirowitz, Andy Moravcsik, Tom Palfrey, Tom Romer, CurtSignorino, Randy Stone, and Johannes Urpelainen. We would also like to thank three anonymous reviewers for helpful comments. Anyremaining errors are our own.

One of the limitations of this body of theory, as itexists today, is that it resembles more a collection of the-oretical anecdotes than a systematic body of organizedreasoning linking uncertainty to the risk of costly war.The formal literature on international conflict contains awide variety of modeling approaches. For example, oneclass of models in this literature is characterized by the as-sumption that actors are uncertain about the costs of war,but these models vary widely in their description of a cri-sis. Differences arise over the kinds of strategies availableto decision makers and the timing of interactions. Somescholars model crisis bargaining as a war of attrition incontinuous time (Fearon 1994), others use infinite hori-zon, alternating-offers bargaining as their model (Pow-ell 1996), and still others use discrete choice crisis bar-gaining games with entry (Schultz 1999). Alternatively,sometimes a reader finds consistency in game form, butdifferences in information structures. One such exampleis illustrated by the ultimatum crisis bargaining model,which has been considered with uncertainty about costsby some (Fearon 1995), while the same bargaining model

American Journal of Political Science, Vol. 55, No. 1, January 2011, Pp. 149–169

C©2010, Midwest Political Science Association DOI: 10.1111/j.1540-5907.2010.00486.x

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150 MARK FEY AND KRISTOPHER W. RAMSAY

has been considered with uncertainty regarding the prob-ability of victory in war by others (Reed 2003). Thus,our collective knowledge regarding the relationship be-tween uncertainty, the incentive to misrepresent, and waris entangled with countless other assumptions about thetype of uncertainty, the timing of actions, the bargainingprotocol, and various other assumptions made for eitherpractical or substantive reasons.

While this diversity of models is not necessarily acause for alarm, with some regularity we discover thatcentral conclusions reached from the study of one par-ticular model are overturned when new game forms areconsidered. For example, Leventoglu and Tarar (2008)show that small modifications to the timing structure ofthe alternating-offers bargaining game can generate equi-libria without the “risk-reward trade-off,” Fey and Ram-say (2007) give a formulation of the mutual optimismproblem with private information about the probabilityof winning a war but show there is no war in any equilib-rium, and Slantchev (2003a) points to situations whererational players fight costly wars in a bargaining modelwith no asymmetric information at all. In this article, weaddress this problem by taking a systematic approach toanalyzing the effects of uncertainty on the risk of war incrisis bargaining situations and focus on the two promi-nent forms of uncertainty faced by decision makers ina crisis. We first consider the case where decision mak-ers face uncertainty about the costs of war or the levelof resolve of their opponent (Fearon 1994; Powell 1996;Schultz 1999). Alternatively, decision makers could haveuncertainty and private information about the probabil-ity of victory in war (Smith and Stam 2006; Wagner 2000;Wittman 1979). Our analysis shows how these types ofuncertainty influence the crisis bargaining problem.1

Assessing the effect of uncertainty, even if systemati-cally organized into categories based on its source, is notan easy task. In particular, there is a fundamental prob-lem with using game-theoretic models like those foundin the literature to formulate general claims about therole of uncertainty in international conflict. The root ofthis problem is the fact that the equilibria in any specificgame are typically sensitive to the particular details of thegame form. That is, it is typically not known how far aresult that holds in a specific extensive form generalizesto different extensive forms. This problem is magnifiedin the study of international conflict. Unlike the study ofelections, say, where candidates must first declare theircandidacy, then run their campaigns, which are followed

1A recent paper by Wittman (2009) also makes a distinction re-garding an important difference in the strategic problem whenuncertainty is about the expected outcome of war rather than costs.

by all voters voting simultaneously on election day, thereis no “natural” game form for crisis bargaining. In a cri-sis, who gets to make proposals? Can a state start a wardirectly after its initial proposal is rejected? Can a statestart a war even if its proposal was accepted? Is bargain-ing restricted to being bilateral or can mediators be used?Where would such a mediator fit into the process? Forquestions like these there are no clear answers, and thislack of a “natural” game form limits the applicability ofresults derived from any particular choice of game form.

With this in mind, we employ a methodological ap-proach from economic theory called Bayesian mecha-nism design. This approach makes it possible to analyzethe outcomes of bargaining games even when the preciseprocedures used by the parties are unclear. Indeed, by us-ing a tool known as the revelation principle along with theconstraints implied by a situation where any agreementsmust be voluntary, we find that there are some generalfacts about crisis bargaining that can be characterizedwithout reference to a specific game form. That is, wepresent results that are “game-free” in the sense of Banks(1990) and are not a consequence of particular modelingchoices.

This approach accomplishes three things. First, it al-lows us to cut through the clutter of potentially endlessvariation in the extensive form of crisis bargaining modelsand establish the existence of some truly general resultsfor the crisis bargaining problem. Second, this approachallows us to focus on the fundamental role that privateinformation plays in explaining why crisis bargaining canbreak down even when countries have a common interestin avoiding war. Without such a general analysis, it wouldbe difficult to compare the effects of different types ofinformation structures, as any results would confoundthe effects of uncertainty and the effects of game form.Third, as emphasized by Fearon (1995), just as importantas this uncertainty is the resulting incentive for countriesto misrepresent their private information. In the mech-anism design framework, this concern is captured by an“incentive-compatibility” requirement that turns out toplay a crucial role in our analysis. Indeed, working withthese incentive-compatibility constraints shows that theincentive to misrepresent plays an indispensable role inthe connection between uncertainty and war. This is mostevident in our result that peaceful equilibria can only oc-cur when such incentives are absent.

Our analysis shows that the link between uncertaintyand war in the bargaining problem depends in impor-tant ways on the kind of uncertainty states face. We firstconsider the case in which states are uncertain abouttheir opponent’s costs of fighting, but there is no un-certainty about the probability of success in war. For this

INCENTIVES IN CRISIS BARGAINING 151

commonly assumed type of uncertainty, we show that inany equilibrium of any crisis bargaining game both the ex-pected probability of war and the expected utility of eachside are weakly decreasing in the cost of war. We thenshow that there exist crisis bargaining games in whichwar never occurs and such games have a payoff structurethat is necessarily insensitive to the private informationof each country. The second kind of uncertainty we in-vestigate is one in which the two sides’ values for war areinterdependent. Such interdependence arises naturally insituations where uncertainty is about relative power, butnot costs. We show that when these costs are low, privateinformation and the incentive to misrepresent generatea positive probability of war in every crisis bargaininggame with voluntary agreements. We then describe howexternal subsidies from a third country or an interna-tional organization can be used to peacefully resolve sucha conflict and again offer a characterization of peacefulgame forms. This implies that while uncertainty about theother side’s cost of war may be relatively benign in peacenegotiations, uncertainty about the other side’s strengthin war makes it much more difficult to guarantee peacefuloutcomes.

The approach that we employ to establish these re-sults builds on the mechanism design literature in eco-nomics, particularly on the classic paper of Myerson andSatterthwaite (1983) on bilateral bargaining.2 However,our application to international conflict differs from thestandard economic literature in several important re-spects. First, it is a central facet of international conflictthat bargaining occurs between “sovereign states with nosystem of law enforceable among them, with each statejudging its grievances and ambitions according to the dic-tates of its own reason or desire” (Waltz 2001, 159). Thatis, a country can, at any time, choose to use force andthere is no way a country can commit to not do so. We in-corporate this in our formal analysis by requiring that anyagreements must be voluntary. Thus, in the internationalarena, as no enforcement is possible, the type of bindingcontracts that are implicitly or explicitly assumed in thestandard mechanism design literature do not exist. To beprecise, a standard assumption in the mechanism designliterature is that agents cannot back out of a contract thatthey previously agreed to. Thus, participation decisionsare generally assumed to occur at the interim stage, be-fore the terms of the agreement are finalized. In contrast,because the lack of enforceable contracts is a fundamental

2Several recent surveys of the vast literature on mechanism designin economics include Jackson (2003), Myerson (2008), and Baligaand Sjostrom (2008).

fact in international relations, we suppose that either sidecan veto a potential agreement at any stage.3

A second important element of international con-flict is that countries can learn about their opponentsthrough the process of negotiation (Pillar 1983; Schelling1960; Slantchev 2003b). Specifically, we assume that acountry’s final decision about whether or not to use forceshould incorporate whatever additional information ithas inferred about the opposing country through the ne-gotiation process.4 A third important difference from thestandard mechanism design model of trade in economicsis that war is always costly, and it is therefore commonknowledge that there is some peaceful settlement avail-able that both sides would strictly prefer. This is unlikethe standard model of bilateral trade due to Myerson andSatterthwaite (1983) in which trade is sometimes ineffi-cient.5

In the literature on international conflict, the paperthat is closest to ours is Banks (1990). This paper usesa game-free approach similar to ours but only consid-ers the case of one-sided incomplete information. Banksshows that with this information structure, all equilibriaof all bargaining games are monotonic in the sense thatthe stronger the informed country is, the higher its payofffrom peaceful settlement. We show that these results onlypartially extend to two-sided uncertainty. In addition,because Banks only considers one-sided incomplete in-formation, his analysis cannot evaluate the different con-sequences of uncertainty about costs versus uncertaintyabout relative power.

In the next section, we outline the method we useto generate our results about the relationship betweenuncertainty and war. The third section details our model,defines what is meant by a crisis bargaining game, and ex-plains how our analysis uses the revelation principle. Thefourth section contains our formal results, and the fifthsection provides a discussion of the implications of ourfindings for theories of bargaining, war, and institutionaldesign. The final section concludes.

3However, see Cramton and Palfrey (1995) for a model in whichplayers can opt out of mechanism. This paper differs from ourproject, though, because we permit actors to opt out after thesettlement is generated, instead of only beforehand.

4In the mechanism design literature, similar ideas have been dis-cussed by Matthews and Postlewaite (1989), and Forges (1999) asan ex post participation constraint, and Compte and Jehiel (2009)and Fey and Ramsay (2009) as a veto constraint.

5More specifically, in the standard model of bilateral trade, efficienttrade always occurs if and only if there is common knowledge ofgains from trade.


A Method for Game-Free Analysis

It is perhaps self-evident that in formal models, the re-sults that are obtained depend on the assumptions thatare made in formulating the model. While this fact is wellunderstood, all assumptions are not equal and there islittle discussion by practitioners of formal theory regard-ing how sensitive their results can be to the details of theassumed game form. Often the predictions of our modelsdepend crucially on the precise specification of the gamewe choose. In the game-theoretic literature on bargain-ing, for example, a number of variations of the standardalternating-offers model due to Rubinstein (1982) havebeen studied.6 Taken together, these variations demon-strate that important features of the equilibrium outcomeare often highly sensitive to the exact specification of thebargaining procedure. For example, a seemingly minorchange in who makes the first offer can have a significanteffect on the distributional outcome. Other variations, in-cluding when disagreement leads to a costly inside option,when players can opt out of bargaining, or when playerscannot commit to forego renegotiating their proposal af-ter their opponent accepts, also can have significant effectson equilibrium outcomes.

The bottom line is that consumers of current theo-retical models are necessarily left unsure about how ro-bust existing findings are and how much our theoreticalexpectations depend on the analysis of a specific gameform.7 Indeed, Powell emphasizes the importance of “thepotential sensitivity of informational accounts of war tothe bargaining environment—to the sources of uncer-tainty and the ability to resolve that uncertainty” andcalls for “robustness checks for a particular formalizationof the bargaining environment” (2004). In order to ad-dress these concerns, we make use of a methodologicalapproach from economic theory called Bayesian mech-anism design. This approach enables us to analyze theoutcomes of bargaining games while leaving the preciseprocedures used by the parties unspecified. In particu-lar, we may ask, what possible outcomes could occur forall possible bargaining procedures that could be used? Thisseems, at first glance, to be an intractable question. It isnot even apparent how one might categorize all the dif-ferent kinds of bargaining procedures that could be used.

So how is Bayesian mechanism design able to gener-ate game-free results? The answer is that, through the useof a powerful result known as the revelation principle, we

6For a survey, see the textbook by Muthoo (1999).

7Obviously, there is a trade-off between the ability to make specificpredictions and the generality of results. For a discussion of thistrade-off, see Banks (1990).

are able to reduce the scope of our analysis from the classof all possible Bayesian games to the much smaller classof “incentive-compatible direct mechanisms.” In essence,the revelation principle allows us to include the strategiccalculations and incentives to misrepresent of the bargain-ers as part of the direct mechanism. More specifically, therevelation principle states that the outcome of any equi-librium of any Bayesian game is also the outcome of someequivalent incentive-compatible direct mechanism.8

The important implication of this observation is thatany outcome achievable via any equilibrium, under anybargaining procedure, must be attainable as the equilib-rium outcome of an “information revelation” game inwhich each player finds it optimal to truthfully revealhis information, given the conjecture that all other play-ers will truthfully reveal their information as well. Thisis what is referred to as an incentive-compatible directmechanism. The revelation principle thus implies thatif all incentive-compatible direct mechanisms have someproperty, then every equilibrium of every game form hasthis property. More importantly for our purposes, if noincentive-compatible direct mechanism has some prop-erty, then no equilibrium of any game form has this prop-erty.9 In this way, the revelation principle enables us touse direct mechanisms as a powerful tool for analyzingstrategic behavior in a wide variety of settings.

Formalizing the Approach to CrisisBargaining

As is customary within much of the conflict literature,consider a situation where two states are involved in adispute which may lead to war. We conceptualize theconflict as occurring over a divisible item of unit size,such as an area of territory or an allocation of resources.The expected payoff to war depends on the probabilitythat a country will win, the utility of victory and defeat,and the inefficiencies present in fighting. We normalizethe utility of countries to be 1 for victory in war and 0 fordefeat, and we suppose there is a cost ci ≥ 0 for country ifighting a war. Thus, if pi is the probability that countryi wins the war, the expected utility for country i of goingto war is simply wi = pi − ci .

8A formal description of and intuition for the revelation principleare given below. See Chwe (1999) and Baron (2000) for otherapplications of the revelation principle in political science.

9We will invoke this version of the revelation principle to prove theimpossibility of peaceful resolution of disputes, regardless of thegame form.


At the outset, each state has private information aboutits ability to contest a war. That is, each state has privateinformation regarding its chance of prevailing in a waror the costs of conducting a military campaign. For ex-ample, a state could have unique knowledge about itsrelative value for the issue of dispute (captured by the rel-ative cost of fighting ci ) or the strength and capabilitiesof its military force (reflected in the probability of victorypi ). Formally, we think of each country as having a vari-ety of possible types, where country i’s type ti ∈ Ti ⊆ R

represents its private information. The countries have acommon prior about the joint distribution of types, givenby F(t) for type pair t = (t1, t2) ∈ T = T1 × T2. In gen-eral, we will denote country i’s war payoff for a type profilet by wi (t).10

The two countries can attempt to avoid war by resolv-ing their dispute through some peaceful process, whichmay include direct negotiations, bargaining, threats, me-diation by a third party, or some other interaction. What-ever settlement procedure is available in a given instancecould then, in principle, be described (abstractly) by agame form G which is composed of a set of actions for eachcountry, A1 and A2, and an outcome function g (a1, a2)for a1 ∈ A1 and a2 ∈ A2. It is worth emphasizing thatthis game form can be anything from a simple strategicform game to an arbitrarily complicated extensive form.We denote a pair of actions (a1, a2) by a ∈ A = A1 × A2.Thus, a game form defines the actions available to thecountries (e.g., what negotiation tactic to use, etc.) andhow those actions interact to determine outcomes.

A crisis bargaining game is a game form in whichthe final outcome is either a peaceful settlement or animpasse that leads to war.11 Thus, we can decompose theoutcome function g(a) of a crisis bargaining game intotwo parts: the probability of war �g (a), and, in the caseof a settlement, the value of the settlement to country1, vg (a). We will assume that any potential settlementis efficient and therefore the value of the settlement tocountry 2 is given by 1 − vg (a). We sometimes writev

gi (a) for the value of the settlement to country i. With

this structure, it is easy to see that the payoff to country iof an action profile a is given by

ui (a, t) = �g (a) · wi (t) + (1 − �g (a))vgi (a). (1)

In words, the payoff to country i is the probability of a wartimes the payoff of war plus the probability of a peacefulsettlement times the value of a peaceful settlement for

10For simplicity we assume that F(t) has full support. That is, thesupport of F(t) is equal to T .

11Here a settlement can involve either a peaceful redistribution ofwhatever is being contested or continuation of the status quo.

an action profile a. As in the canonical take-it-or-leave-itbargaining game studied by Fearon (1995), the privateinformation of the two sides only affects payoffs in thecase of war. That is, the terms of a peaceful settlementare determined by the actions of the two sides in thebargaining game and, in the case of peace, it is only theseterms that determine the countries’ payoffs and not theirprivate information about their war-fighting ability.

Of course, a country’s private information can affectits behavior at the bargaining table. Formally, the actionstaken in the game can depend on the countries’ types, andwe reflect this fact by defining a strategy for country i bya function si : Ti → Ai . The set of all possible strategiesfor state i is Si and we let (s1, s2) = s ∈ S = S1 × S2. Theequilibrium concept we employ is Bayesian-Nash equilib-rium. In particular, a strategy profile s ∗ is a Bayesian-Nashequilibrium if each type of each player is playing a bestresponse to the strategies used by the other players. Fora given equilibrium s ∗ of G, define Ui (ti ) to be the ex-pected utility of this equilibrium for a type ti of countryi. Analytically, we let

Ui (ti ) =∫

Tj

ui (s ∗(ti , t j ), t) d F (t j | ti ), (2)

where i �= j and ui (a, t) is given by equation (1).12

In the remainder of this section, we formalize themethod of game-free analysis and discuss how to ap-ply this method to crisis bargaining games. We begin bylinking the game form and the information structure de-scribed above in the following way. Fix an equilibriums ∗ to the overall game. For a type pair t = (t1, t2), thisequilibrium generates an equilibrium probability of war�(t) = �g (s ∗(t)) and an equilibrium value of settlementto country i, vi (t) = v

gi (s ∗(t)). As peaceful settlements

are efficient, the equilibrium value of settlement to coun-try 2 satisfies v2(t) = 1 − v1(t). The functions �(t) andvi (t) form what is called an equivalent direct mechanism,which can be understood as nothing more than a newgame in which each country’s available actions are lim-ited to reporting one of its possible types and outcomesare determined using �(t) and v(t). If it is an equilibriumfor all types to “tell the truth” by sending a report equalto their type in this game, then we say that the directmechanism is incentive-compatible.

With these definitions it is now possible to formallystate the revelation principle.

Result 1 (Myerson, 1979). If s ∗ is a Bayesian-Nash equi-librium of the crisis bargaining game form G, then there

12Here, and throughout the text, we suppose that all the functionswe consider are measurable and thus we take all integrals to beLebesgue integrals.


FIGURE 1 Example of the Revelation Principle: ASimple Crisis Game

exists an incentive-compatible direct mechanism yieldingthe same outcome.

To understand the intuition of the revelation princi-ple, consider the simple extensive form game in Figure 1.This game has two players, country 1 and country 2, andis similar in nature to the crisis subgame in Bueno deMesquita and Lalman (1992). To start, country 1 decideswhether to accept the status quo or make a challenge.If country 1 challenges, then country 2 can capitulate,giving in to country 1’s demand, or resist. If country 2resists, country 1 decides whether to repeal its challengeand back down, or to fight—leading to war. This de-scribes the outcome function g of this game. If a waroccurs, then country 1 wins with probability p1 = pand country 2 wins with probability p2 = 1 − p. Eachcountry also pays a cost of war. In this example, thereis two-sided uncertainty: each country’s cost of war isprivate information and can take on one of two val-ues, cl or ch , with ch > cl > 0. For each country, theprior probability that ci = ch is given by �. For sim-plicity, we focus on the case in which cl is small and ch

is large. Specifically, we assume that cl < min{p, 1 − p}and ch > max{(1 − p(1 − �))/(1 − �), p}.

Now consider the following pair of strategies for thetwo countries. For country 1, both types make a chal-lenge and if country 2 resists, then the high-cost typebacks down and the low-cost type fights. For country 2,after country 1 challenges, the high-cost type capitulatesand the low-cost type resists. Note that as described above,these strategies describe what actions each type of eachcountry takes. It is easy to verify that under the assump-tions on the parameters given above, this pair of strategies

is an equilibrium of this game. Thus we denote this pairof strategies by (s ∗

1 (t1), s ∗2 (t2)).

We now construct the equivalent direct mechanismfor this equilibrium. As each country has two types, ahigh-cost type and a low-cost type, in the direct mecha-nism each country simultaneously chooses to report oneof these two types, which we denote H and L. As thedirect mechanism is just a strategic form game formedby these simultaneous choices, the only other thing weneed to specify is what outcomes are assigned to thesechoices. This is accomplished via an outcome functionfor the direct mechanism equal to g (s ∗(t1, t2)), for allthe possible reported type profiles (t1, t2). In order tosee how this works in our example, consider the profile(H, H) in which both countries report they are the high-cost type. The direct mechanism assigns the outcome“Capitulation” to this combination. Why? Because in theequilibrium of this example the high-cost type of coun-try 1 challenges the status quo and the high-cost type ofcountry 2 capitulates to this challenge. In other words, thedirect mechanism assigns the outcome that would occurif the specified pair of types plays the equilibrium strategyin the original game. In the same way, the direct mech-anism assigns the outcome “War” to the profile (L , L ),“Capitulation” to the profile (L , H) and “Back Down”to the profile (H, L ). Using the payoffs from the gametree, we can complete the direct mechanism as given inFigure 2. In this case, because each side has two possibletypes, the direct mechanism is just a 2 × 2 strategic formBayesian game.

The revelation principle is a statement about equi-libria of a direct mechanism. Specifically, the princi-ple says that in the direct mechanism we construct for


FIGURE 2 Equivalent Direct Mechanism

a given equilibrium of a given game, truth-telling is aBayesian-Nash equilibrium, and it leads to the same out-comes as the original equilibrium in the original game.To see that this is true in our example, consider the directmechanism given in Figure 2. To show that every typereporting truthfully is an equilibrium, consider country1, assuming that country 2 is reporting its type truthfully.Then for the low-cost type of country 1, reporting L givesexpected utility (1 − �)( p − cl ) + �(1), while reportingH gives expected utility (1 − �)(0) + �(1). Thus, report-ing L is optimal if cl < p. Similarly, for the high-cost typeof country 1, reporting H is optimal if ch > p. As bothof these conditions hold given our assumptions on theparameters, it is optimal for country 1 to report its typetruthfully, given that country 2 is doing so. In a similarmanner, it is easy to check that truth-telling is optimal forcountry 2, given this behavior by country 1 in the directmechanism. Finally, it is clear that the outcome in thedirect mechanism is the same as the outcome in the equi-librium of the original game by the way that the directmechanism is constructed.

The construction of the direct mechanism for this ex-ample offers insight into how we use the revelation prin-ciple in this article. As mentioned above, because eachcountry has two possible types, the direct mechanism forthe equilibrium in our example is a 2 × 2 strategic formBayesian game. But as the method of construction makesclear, as long as each country has two possible types,the direct mechanism of any equilibrium of any extensiveform, no matter how complicated, will be a 2 × 2 strategicform Bayesian game. Thus, the revelation principle allowsus to make statements about all equilibria of all extensiveforms, no matter how complicated, by analyzing the set ofpossible direct mechanisms, which in this case is just theset of 2 × 2 strategic form Bayesian games with the givenoutcomes. In this way, we can use the revelation princi-ple to learn about arbitrarily complex games by studyingthe much simpler class of direct mechanisms. Of course,we do not limit ourselves to models with only two types.Also, we will most often make use of the contraposi-tive of the revelation principle which says that if somedistribution of outcomes—say, always peace—is not an

equilibrium to any direct mechanism, then it cannot bean equilibrium to any other extensive form game either.

Thus, by using the revelation principle, we can studyequilibria that satisfy incentive-compatibility constraintsin direct mechanisms and use our findings to establishgeneral results about properties of equilibria in all possi-ble crisis bargaining game forms. Before doing so though,we discuss three important qualitative properties of cri-sis bargaining that play a significant role in our lateranalysis.

First, given the anarchic nature of the internationalsystem, any peaceful agreement reached must be volun-tary. In particular, we suppose that a country always hasthe option of rejecting a proposed settlement vg (a) ifit thinks it will be better off by resolving the crisis byforce. The form of this rejection can literally be the useof military force or it can be an escalation of the crisiswhich either leads to war or to the other side capitulating.Whichever is true, this feature of the bargaining contextensures each side can act in such a way that the outcome iseither war or a voluntary settlement that that side prefersto war. Formally, we say a crisis bargaining game form hasvoluntary agreements if, for i �= j , there exists an actionai ∈ Ai such that either �g (ai , a j ) = 1 for all a j ∈ A j orfor all t ∈ T, v

gi (ai , a j ) ≥ wi (t).13 Put simply, in a crisis

bargaining game form with voluntary agreements, thereis no way to force a country to accept an agreement thatmakes it worse off than it would expect to be by going towar.

A second important observation about crisis bargain-ing is that the process of bargaining has the potentialto reveal, to a greater or lesser extent, the private infor-mation of the bargainers. Of course, a country shouldincorporate this additional information into its decisionwhether to reject a proposed settlement in favor of usingforce. To capture this formally, we let �i (vi , ti ) be coun-try i’s updated belief about the type of country j afterobserving the settlement offer vi .14 As is standard, we as-sume that this belief is formed via Bayes’ Rule, wheneverpossible.

13For an example of such an action, consider the take-it-or-leave-itbargaining game of Fearon (1995). For country 1, the action a1

consists of demanding the whole pie. This demand will either berejected, leading to war, or accepted, which results in a payoff thatis higher than country 1’s war payoff. Likewise, for country 2 inthis game, rejecting any offer leads to war, which thus satisfies thedefinition. So this game has voluntary agreements.

14Although additional information may be revealed by the spe-cific actions taken by countries in a specific game, we focus onthe information revealed by the final settlement offer because thisinformation will be available in all crisis bargaining games. Incor-porating the additional information possible revealed in a specificgame would only strengthen our results.


Combining this updating with our assumption onvoluntary agreements, it is easy to show the followingresult is a consequence of the revelation principle.15

Result 2. Suppose that s ∗ is an equilibrium of a crisisbargaining game form that has voluntary agreements. Thenthere exists an incentive-compatible direct mechanism suchthat vi (t) ≥ E [wi (t) | �i (vi , ti )] for all t ∈ T such that�(t) �= 1.

The intuition for this result is straightforward. Re-ferring back to equation (1), we see that in a game formthat has voluntary agreements, if �g (s ∗(t)) �= 1, then thiscondition simplifies to the requirement that the equi-librium settlement value to country i must be at leastas big as the expected war payoff to country i. That is,when faced with a settlement offer vi , all types of coun-try i have the option of playing ai and receiving a payoffof E [wi (t) | �i (vi , ti )]. Therefore if s ∗ is an equilibriumand �(t) = �g (s ∗(t)) < 1 for some t = (t1, t2), it mustbe that this deviation is not profitable, which is true onlyif vi (t) ≥ E [wi (t) | �i (vi , ti )]. In other words, if agree-ments are voluntary and the unilateral use of force isalways an option, any negotiated settlement must giveeach country a payoff at least as large as the payoff thatit expects to get from settling the dispute by force, givenwhat it has inferred about its opponent as a consequenceof the negotiations. As an example, refer back to Figure 1.This game satisfies our voluntary agreements condition.Therefore, it is easy to verify that, given our assumptionson ch , both players receive at least their war payoff in eachcell of the direct mechanism in Figure 2.

The third and final observation that we incorporateinto our analysis is the simple fact that war is costly. Be-cause of this, we are interested in whether private informa-tion makes war unavoidable or whether there can be casesin which countries always arrive at peaceful settlements.Formally, an equilibrium s ∗ of a crisis bargaining gameform is always peaceful if �g (s ∗(t)) = 0 for all t ∈ T . Inother words, an always peaceful equilibrium of a gameform is one in which no possible pair of types ever endsup abandoning a peaceful resolution of the dispute andresorting to force. Adding this requirement to Result 2gives our final result.

Result 3. Suppose that s ∗ is an always peaceful equilib-rium of a crisis bargaining game form that has volun-tary agreements. Then there exists an incentive-compatibledirect mechanism such that for all t ∈ T, �(t) = 0 andvi (t) ≥ E [wi (t) | �i (vi , ti )].

15This result contains our version of what is usually known as theindividual rationality constraint or the participation constraint.

The importance of this result is that if we canshow that for a given information structure there is noincentive-compatible direct mechanism with these prop-erties, then no always peaceful equilibria exist in any crisisbargaining game form that has voluntary agreements.

We conclude this section by presenting a lemma thatwill be useful in deriving the general results presented inthe next section. This lemma is a direct consequence ofthe revelation principle applied to crisis bargaining gameswith independent types. Formally, we say that types areindependent if there exist distributions F1 and F2 on T1

and T2, respectively, such that F (t) = F1(t1)F2(t2) forevery t ∈ T .

Lemma 1. Suppose types are independent and Ti = [ti , ti ].Suppose also that wi (t) is continuously differentiable in t1

and t2. Let G be any crisis bargaining game form and lets ∗ be any equilibrium of G. Then Ui (ti ) is continuous, isdifferentiable almost everywhere, and can be expressed as

Ui (ti ) = Ui (ti ) +∫ ti

ti

∫Tj

�(s , t j )

× ∂wi

∂ti(s , t j ) dF j (t j ) dFi (s ),

where �(t) = �g (s ∗(t)).

Proof. Suppose that s ∗ is an equilibrium of a crisisbargaining game form. Then by Result 1, there existsan incentive-compatible direct mechanism yielding thesame outcome as s ∗. This direct mechanism is given by�(t) = �g (s ∗(t)) and vi (t) = v

gi (s ∗(t)). Therefore the

expected utility of type ti from equation (2) can be written

Ui (ti ) =∫

Tj

�(ti , t j )wi (ti , t j )

+ (1 − �(ti , t j ))vi (ti , t j ) dF j (t j )

and the expected utility of falsely reporting a type ti is

Ui (ti | ti ) =∫

Tj

�(ti , t j )wi (ti , t j )

+ (1 − �(ti , t j ))vi (ti , t j ) dF j (t j ).

Clearly, Ui (ti ) = Ui (ti | ti ). Moreover, incentive-compatibility requires that for all ti ∈ Ti ,

Ui (ti | ti ) ≥ Ui (ti | ti ),

which is the same as

Ui (ti ) = Ui (ti | ti ) = maxti ∈Ti

Ui (ti | ti ).

Now, viewing Ui (ti | ti ) solely as a function of ti , be-cause wi is differentiable in ti , it follows that Ui (ti | ti )


is differentiable in ti and because wi is continuously dif-ferentiable on a compact set, this derivative is bounded.This implies that Ui (ti | ti ) is absolutely continuous inti . Therefore, the Envelope Theorem (Milgrom and Se-gal 2002, Theorem 2) gives us that Ui (ti ) is absolutelycontinuous and

dUi (ti )

dti= ∂Ui (ti | ti )

∂ti

∣∣∣∣ti =ti

=∫

Tj

�(ti , t j )∂wi

∂ti(ti , t j ) dF j (t j )

∣∣∣∣∣ti =ti

at every point of differentiability. Evaluating this expres-sion at ti = ti yields the desired result. �

Results

In this section we establish several general results thathold for every crisis bargaining game that has voluntaryagreements. Before doing so, though, we must identify theappropriate information structure for our analysis. His-torically, the conflict literature has held that uncertaintyis a central cause of conflict among countries, but hasvaried with respect to the type of uncertainty it views asimportant. Some scholars focus on uncertainty about therelative strength of the countries (Blainey 1988; Organskiand Kugler 1980) while others concentrate on uncertaintyabout the costs of conflict or the resolve of countries (Kydd2003; Morrow 1985; Ramsay 2004; Schultz 1998).

In particular, we focus on two kinds of uncertainty:uncertainty about the costs of war and uncertainty aboutthe distribution of power. These various sources of uncer-tainty correspond to significantly different informationalstructures with important implications for strategic in-teraction in a crisis. Uncertainty about costs implies thatthe values of war to each country are independent, inthe sense that one country’s realized preference for fight-ing does not directly affect another’s utility for fighting.Alternatively, the international system may produce un-certainty about the relative strength and the probability ofvictory. In this situation, the countries’ values for war areinterdependent, although information remains uncorre-lated.

Uncertainty about the Cost of Conflict

We first consider the case in which each country is un-certain about the other’s cost of fighting, which is alsooften interpreted as a country’s level of resolve (Fearon

1994; Schultz 2001) or preference for fighting.16 From atheoretical perspective, this is a situation with uncertaintyabout independent private values for war. That is, the re-alization of one state’s resolve, or relative cost of war, doesnot directly influence the utility of the other side. A classicexample of crisis bargaining with uncertainty regardingresolve occurred between Germany and the United King-dom in the 1930s. While relatively well informed aboutthe distribution of power in Europe and Germany’s mil-itary armaments, the British Foreign Office was unsureabout Hitler’s intentions and how far he would go toachieve his territorial goals (Yarhi-Milo 2009).

To model uncertainty about costs formally, we sup-pose that the probability that country 1 wins a war, p,is common knowledge but there is uncertainty regard-ing each country’s cost for fighting, ci . In this setting,suppose country i’s type, ci ∈ [ci , c i ] = Ci , is its cost ofwar, which is distributed according to a cumulative distri-bution function Fi (ci ), with support Ci .17 War is costly,so we assume that c i > ci ≥ 0. Denote a pair of types(c1, c2) = c ∈ C = C1 × C2.

We begin our analysis of this information structure byidentifying some qualitative features of equilibrium playin crisis bargaining games with uncertain costs. Often,such qualitative features are central components in theanalysis of a particular model. For example, in the classictake-it-or-leave-it bargaining model of conflict (Fearon1995), it is possible to show that the expected probabilityof war is weakly decreasing in the cost of war for bothcountries. The following result demonstrates that thisqualitative feature of this specific model is completelygeneral—it will hold in any equilibrium of any crisis bar-gaining game with uncertainty about the cost of conflict.The result also shows that a similar conclusion can bereached about the overall expected utility of each coun-try. For country i, define the expected probability of war,given strategy profile s, by

�(ci ) = E �g (ci ) =∫

C j

�g (s (ci , c j )) dF j (c j ).

Also, recall that the expected utility of this strategy profileis given by

16It is often the case that uncertainty about costs is said to alsomodel uncertainty about “resolve,” where resolve is loosely definedas how much one side “cares” about the issue or item of dispute.The equivalence comes from the fact that, if costs are known, butthe value of the prize is unknown, we can normalize the value ofthe prize, generating a new “relative cost” c/vi which is a privatelyknown value.

17Because the private information of country i is the cost of war,here we use the notation ci rather than ti for the type of country i.


Ui (ci ) =∫

C j

ui (s (ci , c j ), ci ) dF j (c j ).

We then have the following proposition.

Proposition 1. Suppose costs ci are private information,but each country’s probability of winning a war is commonknowledge. Let G be any crisis bargaining game form andlet s ∗ be any equilibrium of G. Then �(ci ) and Ui (ci )are both weakly decreasing in ci and differentiable almosteverywhere, and Ui (ci ) is continuous in ci .

Proof. Suppose that s ∗ is an equilibrium of a crisisbargaining game form. Then by Result 1, there existsan incentive-compatible direct mechanism yielding thesame outcome as s ∗. This direct mechanism is given by�(c) = �g (s ∗(c)) and vi (c) = v

gi (s ∗(c)). Therefore the

expected utility of type ci from equation (2) can be writ-ten

Ui (ci ) =∫

C j

�(ci , c j )( pi − ci )

+ (1 − �(ci , c j ))vi (ci , c j ) dF j (c j )

and the expected utility of falsely reporting a type c i is

Ui (c i | ci ) =∫

C j

�(c i , c j )( pi − ci )

+ (1 − �(c i , c j ))vi (c i , c j ) dF j (c j ).

Incentive-compatibility requires that for all ci , c i ∈ Ci ,

Ui (ci | ci ) ≥ Ui (c i | ci ) and Ui (c i | c i ) ≥ Ui (ci | c i ).

Adding these two inequalities and simplifying yields∫C j

�(ci , c j )( pi − ci ) −∫

C j

�(ci , c j )( pi − c i )

≥∫

C j

�(c i , c j )( pi − ci ) −∫

C j

�(c i , c j )( pi − c i )

∫C j

�(ci , c j )(c i − ci ) dF j (c j )

≥∫

C j

�(c i , c j )(c i − ci ) dF j (c j ).

From this, it follows that if c i > ci , then �(ci ) ≥ �(c i )and so �(ci ) is weakly decreasing. In addition, as it ismonotonic on a closed interval, it is differentiable almosteverywhere.

We now focus on Ui (ci ). As wi = pi − ci is contin-uously differentiable in ci , it follows from Lemma 1 thatUi (ci ) is absolutely continuous and therefore continuousand differentiable almost everywhere. To show that Ui (ci )is weakly decreasing, take ci < c i and apply Lemma 1 toget

Ui (c i ) − Ui (ci ) =∫ c i

c i

∫C j

(−�(s , c j )) dF j (c j ) ds .

As �(c) is always nonnegative, it follows that Ui (ci ) ≥Ui (c i ). �

Thus, monotonicity of utility and the probability ofwar are general qualitative characteristics of any equilib-rium of any game with uncertainty about the costs ofwar.18 It is then interesting to observe that this mono-tonicity is borne out in a number of well-known, andquite different, models of crisis bargaining. For exam-ple, in the alternating-offers bargaining model of Powell(1996), it is straightforward to deduce from his Lemma 3and Proposition 1 that the probability of war is deceasingin each player’s cost of war. Similarly, the war of attritionmodel in Fearon (1994) has a monotone probability ofwar. From Proposition 2 in Fearon (1994, 584) we can seethat a country with a low cost of war chooses to wait tosome time t and then fight, while a country with a highcost waits for some time and then quits. The probabil-ity of war goes from positive in the first case to zero inthe second. Finally, we can consider the model in Schultz(1999). In this model the probability of war is also a stepfunction, which is positive when a country has a low costof war but is zero when costs are large.

These types of comparative statics represent testablepredictions that are robust to the particular formaliza-tion of crisis bargaining, as advocated by Powell (2004).Moreover, as this result depends only on the incentivecompatibility constraints inherent in the revelation prin-ciple, it holds for all crisis bargaining games, even thosewithout voluntary agreements. In this way, we see that themonotonicities identified in the proposition are purely aconsequence of the incentives to misrepresent private in-formation that are fundamental to the crisis bargainingenvironment.

In addition, this result shows that in any equilibriumof any crisis bargaining game, two types of a country withsimilar costs must receive similar expected utility in equi-librium. Thus a small change in the private informationof the country leads to a small change in its expectedpayoff. This is a feature that is commonly found in “cut-point” equilibria of Bayesian games, but it is present herewith no explicit assumption on the form of equilibriumstrategies.

Of course, in a peaceful equilibrium the probabilityof war is constant at zero, which satisfies Proposition 1

18See Banks (1990) for additional monotonicity results that holdin Bayesian games with one-sided incomplete information. Un-fortunately, these additional results do not extend to the case oftwo-sided incomplete information studied here.


in a trivial way. We now turn to inquiring about thecharacteristics of such equilibria. We begin with a simpleexistence result.

Proposition 2. If costs ci are private information, but eachcountry’s probability of winning a war is common knowl-edge, then there exists a crisis bargaining game form thathas voluntary agreements in which an always peaceful equi-librium exists.

Proof. To prove this result, it is enough to give an exam-ple of a game form that has voluntary agreements andthat has an always peaceful equilibrium. The followingvery simple example shows that this is indeed the case.Consider a game form that has voluntary agreements suchthat g v

i (a) = pi for all a ∈ A and �g (a∗) = 0 for somea∗ ∈ A. In this case, the strategy profile s ∗(c) = a∗ is anequilibrium because if either side deviates and starts awar, then both sides are worse off and if either side de-viates to a different peaceful action, then the settlementamount does not change. Moreover, this equilibrium isalways peaceful by construction. �

This proposition shows that there exists at least onegame form that has voluntary agreements and that caneliminate the possibility of war. The game form describedin the proof has a particularly simple payoff structure inwhich the potential settlement is always equal to a coun-try’s (commonly known) likelihood of success from war.One example of such a game form would be a direct “arbi-tration” game in which an arbitrator would present bothsides with a take-it-or-leave-it offer. Since this agreementwould make both sides better off, regardless of their costsfor fighting, it would provide a rational and preferablealternative to war.19 While a single example is sufficientto prove existence, it would be incorrect to conclude thatthis arbitration game is the only model with uncertaintyabout costs and voluntary agreements that has a peace-ful equilibrium. For example, one kind of equilibriumstudied in the alternating-offers model of Leventoglu andTarar (2008) has uncertainty about costs and zero proba-bility of war. Interestingly, Leventoglu and Tarar observethat a puzzling feature of the standard alternating-offersbargaining model of war is that a country cannot go towar in a period in which it makes a proposal, and whensuch behavior is allowed peaceful equilibria emerge.

At some level, though, the mere existence of gameforms with peaceful equilibria is not a very satisfyingresult. It only tells us that a peaceful equilibrium is theo-

19In this sense, our result lends hope to institutional designers thatthe problem of war is solvable, at least in the case of uncertaintyabout costs or “resolve.”

retically possible; it is not a general result about the pos-sibility of war that applies to all possible crisis bargaininggames. Put another way, Proposition 2 establishes thatthe common understanding in the literature that incom-plete information about costs leads to the risk of war isnot always true; there exist instances where it does nothold. However, without a better characterization of equi-librium incentives, we do not know if this case is just anisolated exception or if peaceful equilibria are the normand the conventional view on the link between incompleteinformation and war must be rethought.

In light of this, we present a general characteriza-tion of peaceful equilibria in all possible crisis bargaininggames that have voluntary agreements. For country i, de-fine the expected settlement value from action ai , givenstrategy s j , by

E vgi (ai | s j ) =

∫C j

vgi (ai , s j (c j )) dF j (c j ).

The following proposition gives a necessary condition forthe existence of an always peaceful equilibrium.

Proposition 3. Suppose costs ci are private information,with Ci = [0, c i ], but each country’s probability of winninga war is common knowledge. Let G be any crisis bargaininggame form that has voluntary agreements. Then an equilib-rium s ∗ of G is always peaceful only if, for i = 1, 2, (1)E v

gi (s ∗

i (ci ) | s ∗j ) = pi for all c i ∈ Ci , (2)E v

gi (ai | s ∗

j ) ≤pi for all ai ∈ Ai and (3) v

gi (s ∗(c)) ≥ pi − ci for all

c ∈ C .

Proof. Suppose that s ∗ is an always peaceful equilib-rium of a crisis bargaining game form that has voluntaryagreements. Then by Result 3, there exists an incentive-compatible direct mechanism such that �(c) = 0 and

vi (c) ≥ E [wi (c) | �i (vi , ci )] = pi − ci

for all ci ∈ Ci and i = 1, 2. Here, E [wi (c) | �i (vi , ci )] =pi − ci because the value of war to country i does notdepend on the value of c j . This direct mechanism is givenby �(c) = �g (s ∗(c)) and vi (c) = v

gi (s ∗(c)).

Because �(c) = 0, applying Lemma 1 yields Ui (ci ) =Ui (0) for all ci and therefore Ui (ci ) = ∫

C jvi (ci , c j )

dF j (c j ) is constant in ci . In addition, because the con-straint vi (c) ≥ pi − ci must hold for ci = 0, we have

∫C j

vi (ci , c j ) dF j (c j ) =∫

C j

vi (0, c j ) dF j (c j ) ≥ pi ,

for all ci ∈ Ci , i = 1, 2. To show that this expression musthold with equality, suppose not. Then


∫C1

∫C2

[v1(c) + v2(c)] dF2(c2) dF1(c1)

=∫

C1

∫C2

v1(c) dF2(c2) dF1(c1)

+∫

C2

∫C1

v2(c) dF1(c1) dF2(c2)

>

∫C2

p dF2(c2) +∫

C1

(1 − p) dF1(c1)

= p + (1 − p) = 1.

But as peaceful settlements are efficient, v1(c) + v2(c) =1 for all c ∈ C and therefore we have a contradiction. Thisproves statement (1) of the proposition. The remainingtwo statements follow directly from the assumption thats ∗ is an equilibrium and thus no profitable deviation ispossible. �

This proposition shows that the simple payoff struc-ture used in the proof of Proposition 2 is, in fact, a gen-eral property of peaceful equilibria. Every peaceful equi-librium must have this same simple payoff structure inwhich the terms that a country accepts are completely in-sensitive to the costs or resolve of the country. High-costand low-cost countries (and all types in between) mustreceive the same expected settlement value. The most nat-ural example of a strategy profile that would generate suchinsensitivity is a (completely) pooling strategy, in whichall types of a country choose the same action.20 A generallesson from this result, then, is that peaceful equilibriamust be “simple” equilibria that do not depend on theprivate information of the countries. That is, an alwayspeaceful agreement necessarily depends only on publiclyobservable information and cannot vary in response tothe privately known costs or resolve of countries.

We can better understand the general idea of thischaracterization of peaceful equilibria by observing that itis the generalized expression of the well-known incentiveto misrepresent private information. It is this incentivethat prevents countries from resolving their uncertaintyby way of “cheap talk” communication prior to bargain-ing (Fearon 1995). Proposition 3 shows that this sameincentive forces peaceful equilibria to be completely in-sensitive to the private information that countries possess.This follows because if a peaceful equilibria did providedifferent expected settlements to different types, the typegetting the worse settlement would have an incentive to

20It would be very useful if this result implied that completely pool-ing equilibria were the only type of peaceful equilibrium possible.However, it can be shown that there also exist nonpooling strate-gies with the property that, in expectation, the payoff to settlementis constant and that therefore satisfy the conditions given in theproposition. Details of this example are available from the authors.

mimic the behavior of the type getting the better settle-ment. Such a deviation is profitable because in a peacefulequilibrium the discipline generated by the risk of wardoes not exist, which means we could not have had anequilibrium that responds to the countries’ private infor-mation.

It is immediately clear from this general lesson thatbecause many examples of crisis bargaining games in theliterature have equilibria that do not conform to this sim-ple payoff structure, these examples support the commonunderstanding that incomplete information about costsleads to war. Indeed, given the severity of the necessarycondition that strong types receive the same expected set-tlement as weak types, it does seem intuitively clear thatmany game forms will fail this necessary condition. Infact, we can strengthen this observation into the follow-ing necessary and sufficient condition:

Corollary 1. Suppose costs ci are private information, buteach country’s probability of winning a war is commonknowledge. Let G be any crisis bargaining game form thathas voluntary agreements and let s ∗ be any equilibrium ofG. Then Ui (ci ) �= Ui (c i ) for some ci , c i ∈ Ci if and only ifthere is a positive probability of war in equilibrium.

Proof. The “if” direction follows directly from the argu-ment in the proof of Proposition 3 that Ui (ci ) is constantin ci if the equilibrium is peaceful. To show the other di-rection, suppose Ui (ci ) = Ui (c i ) for all ci , c i ∈ Ci . Thenby Lemma 1,

Ui (c i ) − Ui (ci ) =∫ c i

c i

∫C j

(−�(s , c j )) dF j (c j ) ds = 0,

for all ci , c i ∈ Ci . This implies that �(c) is zero almosteverywhere. The result follows. �

In other words, this corollary establishes that twotypes can have different expected utilities if and only if waroccurs with positive probability. If we interpret private in-formation about costs as a country’s “resolve,” then thisresult can be stated more clearly in the following form.Crisis bargaining games in which more resolved coun-tries expect to be better off than less resolved countries inequilibrium are those with a positive probability of war.Put another way, this result shows that the only way thatcountries stand to gain from their resolve is through run-ning the risk of war, as in the well-known “risk-reward”trade-off.

As an illustration of Proposition 3 and Corollary 1,we briefly return to the model of Leventoglu and Tarar(2008). As mentioned above, their analysis considers anextension of the alternating-offers bargaining model with


one-sided uncertainty about costs that allows both coun-tries to choose war in any period. In the always peace-ful equilibrium to their game, either both types acceptthe initial offer or both types reject this initial offer andmake the same counteroffer, which is accepted. In eithercase, the payoff of two types is the same, as required byCorollary 1.

This model also allows us to demonstrate howProposition 3 and Corollary 1 can be used to charac-terize equilibria outcomes of games. At the end of theirpaper, Leventoglu and Tarar consider what would hap-pen if their game had two-sided incomplete informationwith a continuum of types. While they do not attemptto solve this version of their model, they conjecture thata peaceful equilibrium would exist and suggest that theagreement would likely be on the division (p, 1 − p), per-haps after some delay. Our results allow us to address theseconjectures directly, without actually solving the model.Suppose that the two countries’ costs are drawn from theinterval [0, c i ] in this model. Then Proposition 3 tells usthat if a peaceful equilibrium like Leventoglu and Tarardescribe exists, the expected value of settlement for everytype pair of the two countries must be (p, 1 − p). More-over, as these expected values sum to one for every typepair and future payoffs are discounted, it must be that inany peaceful equilibrium the game ends with an acceptedoffer in the first round, with probability one. Thus wecan confirm the conjecture that the peaceful settlementwill be ( p, 1 − p) and, additionally, demonstrate that thissettlement will occur without delay in the bargaining.21

In this way, the results in this section can offer significantinsight into equilibrium behavior of complex games withgeneral uncertainty without actually doing the hard workof solving these games.

Uncertainty about Relative Strength

While the previous section dealt with the case of un-certainty about the costs of conflict, in this section wedeal with a second source of uncertainty that has receivedsignificant attention in the literature—uncertainty aboutthe distribution of power and the likelihood of success inwar. In this case, countries are assumed to be informedabout their opponent’s relative cost of fighting, but areuncertain about the likely outcome of conflict. In partic-ular, countries have private information about the quality

21It is worth noting that Leventoglu and Tarar (2008) consider thecase where costs are bounded away from zero, i.e., ci is in theinterval [ci , c i ], ci > 0. Our characterization will hold as long asthere is positive density, although it can be arbitrarily small, oncosts in the interval [0, ci ].

of their military or their combat strategy that leads eachside to hold private beliefs about what will happen as aresult of fighting a war. This is a situation with interde-pendent values. That is, each country’s utility for conflictis not only dependent on its own type but also dependson the type of its opponent. A nice example of this classof problems is the run-up to the 1973 Arab-Israeli War.In this instance, any questions about Israeli resolve, byand large, had been laid to rest in the 1967 war; there waslittle doubt that Israel would defend herself resolutely ifattacked. But even knowing this fact, Egypt and Syria wereuncertain about how effectively the Israeli Defense Forcewould respond to a surprise two-front war. In particular,the Syrian leadership thought their military build up putthem in a strong position to retake the Golan Heights(Yarhi-Milo 2009).

To model this, we assume that the costs of engaging ina war, c1 and c2, are common knowledge, but both coun-tries have private information regarding the probabilityof winning. We implement this in our framework by sup-posing that country i’s type, ti ∈ [ti , ti ] = Ti , is indepen-dently distributed according to a distribution functionFi and the probability that country 1 prevails in a war,p(t1, t2), is a function of both types. We assume that thetypes can be ordered such that the probability of victoryis monotone in the countries’ types. That is, higher typeshave a greater chance of winning, all other things beingequal. Formally, this assumption is that t1 > t ′

1 impliesp(t1, t2) ≥ p(t ′

1, t2), for all t2 ∈ T2. Likewise, we assumethat p is monotonically decreasing in t2. Also, to ensurethere is uncertainty about the distribution of power, weassume that p is not everywhere constant. In this way, thetype ti reflects the “strength” of country i and thus theprobability of victory depends on the relative strength ofthe two combatants.

Our first result deals with how expected outcomesmust vary as the strength of a country varies. Recall fromthe previous section that the expected utility of type ti ofcountry i is denoted Ui (ti ) and the expected probabilityof war is denoted �(ti ).

Proposition 4. Suppose costs are common knowledge buteach country is uncertain about the probability of winninga war. Let G be any crisis bargaining game form and let s ∗

be any equilibrium of G. Then Ui (ti ) is weakly increasingin ti and differentiable almost everywhere. If, in addition,p(t) is continuously differentiable, then Ui (ti ) is continuousin ti .

If p(t1, t2) is strictly increasing in t1 and strictly de-creasing in t2, then there exist t p

i , twi ∈ Ti , with t p

i ≤ twi

such that �(ti ) = 0 for all ti < t pi , 0 < �(ti ) < 1 for all

ti ∈ (t pi , tw

i ), and �(ti ) = 1 for all ti > twi .


Proof. Suppose that s ∗ is an equilibrium of a crisisbargaining game form. Then by Result 1, there existsan incentive-compatible direct mechanism yielding thesame outcome as s ∗. This direct mechanism is given by�(t) = �g (s ∗(t)) and vi (t) = v

gi (s ∗(t)).

As in the proof of Proposition 1, we can write theexpected utility of type ti reporting a type ti as

Ui (ti | ti ) =∫

Tj

�(ti , t j )( pi (ti , t j ) − ci )

+ (1 − �(ti , t j ))vi (�(ti , t j )) dF j (t j ).

Incentive-compatibility requires that for all ti , ti ∈ Ti , Ui

(ti | ti ) ≥ Ui (ti | ti ).To show that Ui (ti ) is weakly increasing, fix ti and ti

such that ti < ti . Then

Ui (ti ) − Ui (ti ) = Ui (ti | ti ) − Ui (ti | ti ) = Ui (ti | ti )

− Ui (ti | ti ) + Ui (ti | ti ) − Ui (ti | ti ).

The above incentive-compatibility condition givesUi (ti | ti ) − Ui (ti | ti ) ≥ 0. In addition, we have

Ui (ti | ti ) − Ui (ti | ti ) =∫

Tj

�(ti , t j )( pi (ti , t j ) − ci )

−�(ti , t j )( pi (ti , t j ) − ci ) dF j (t j )

=∫

Tj

�(ti , t j )( pi (ti , t j ) − pi (ti , t j )) dF j (t j ).

By the monotonicity of pi , the integrand is alwaysnonnegative. Therefore, Ui (ti | ti ) − Ui (ti | ti ) ≥ 0. Thismeans that Ui (ti ) ≥ Ui (ti ), which proves that Ui (ti ) isweakly increasing. It is differentiable almost everywherebecause it is a monotonic function on a closed inter-val. If, in addition, p(t) is continuously differentiable,then wi (t) = pi (t) − ci is continuously differentiable,and Ui (ti ) is continuous by Lemma 1.

We now prove the final part of the proposition. By thesame argument as in the proof of Proposition 1, incentivecompatibility requires that∫

Tj

(�(ti , t j )−�(ti , t j ))( pi (ti , t j )− pi (ti , t j ))dF j (t j ) ≥0.

If we consider ti < ti and assume pi is strictly monotonicin ti , then pi (ti , t j ) − pi (ti , t j ) > 0 for all t j . If �(ti ) = 0,then the above inequality implies �(ti ) = 0. Likewise,if �(ti ) = 1, then the above inequality implies �(ti ) =1. Now define t p

i to be the supremum of the set {ti ∈Ti |�(ti ) = 0} (and t p

i = ti if this set is empty) and twi

to be the infimum of the set {ti ∈ Ti |�(ti ) = 1} (andtwi = ti if this set is empty). The result follows. �

This proposition shows that, regardless of the gameform, a stronger type of a country is never worse off thana weaker type of that country. Thus the monotonicity

result for expected utility given in Proposition 1 with un-certainty about costs continues to hold with uncertaintyabout relative power. With the added assumption that theprobability of winning does not jump discontinuouslywith type, we find that the continuity result of Propo-sition 1 also carries over to this setting. However, theearlier result on the monotonicity of the expected proba-bility of war does not fully carry over. Instead, when theprobability of winning is strictly monotonic in type, theweakest types have zero expected probability of war, mod-erate types have an expected probability of war strictly be-tween zero and one, and the strongest types have war withcertainty.22 Thus, expected probability of war displays alimited kind of monotonicity in this environment.

One important consequence of focusing on uncer-tainty over relative strength is that the process of bargain-ing can reveal important clues as to the likely strength ofthe two countries. In particular, when a country receivesa settlement offer, it can update its prior about the privateinformation of the opposing state by inferring what mustbe true of the other state in order to generate the receivedoffer. Recall that �i (vi , ti ) is country i’s updated beliefabout the type of country j after observing the settlementoffer vi . Let V1(t1, v) = {t2 | v1(t1, t2) = v} be the set ofpossible types of country 2 that a given type t1 of country1 would think are possible after observing a settlementv.23 In this setting, then,

E [w1(t) | �1(v1, t1)] = E [ p(t1, t2) | V1(t1, v1)] − c1,

(3)

and a similar expression holds for country 2. The right-hand side of this inequality is simply the updated expectedutility for war incorporating the inference about the typesof the other country from the observed settlement offer.

For convenience, we use the following notation instating our results:

P1(t1) =∫

T2

p(t1, y)dF2(y) and

P2(t2) =∫

T1

p(x, t2)dF1(x).

In words, P1(t1) is the expected probability of winninga war for type t1 of state 1 and P2(t2) is the expectedprobability of losing a war for type t2 of state 2.

22Note, however, that some of these sets of types could be empty.

23In general, this conditional expectation must be defined ab-stractly. But this abstract definition simplifies in many cases. Forexample, if V1(t1, v1) is an interval, then

E [ p(t1, y) | V1(t1, v1)] =

∫V1(t1,v1)

p(t1, y) dF2(y)

∫V1(t1,v1)

dF2(y).


Let c = P1(t1) − P2(t2). It follows from the mono-tonicity of p that c > 0. Our next result shows that if thecosts of war are less than c , then no matter what bargain-ing procedure is used, there is a positive probability ofwar during a crisis.

Proposition 5. Suppose costs are common knowledge buteach country is uncertain about the probability of winninga war. If c1 + c2 < c , then no always peaceful equilibriumexists in any crisis bargaining game form that has voluntaryagreements.

Proof. The method of proof is by contradiction. Webegin by supposing there is an always peaceful equilib-rium of a crisis bargaining game form that has volun-tary agreements. By Result 3, there exists an incentive-compatible direct mechanism such that �(t) = 0 andvi (t) ≥ E [wi (t) | �i (vi , ti )] for all t and i = 1, 2. Because�(t) = 0, the expected utility of country 1 with true typet1 reporting type t1 is

U1(t1 | t1) =∫

T2

v1(t1, y) dF2(y). (4)

The incentive-compatibility condition is then

U1(t1 | t1) ≥ U1(t1 | t1) for all t1, t1 ∈ T1.

However, from equation (4), it is clear that U1(t1 | t1) doesnot depend on t1. Therefore, the only way the incentive-compatibility condition can be satisfied for all t1 and t1 isif U1(t1 | t1) is a constant, for all t1, and t1.24 We write U1

for this constant.Turning now to the condition that vi (t) ≥ E [wi (t) |

�i (vi , ti )] for all t and i = 1, 2, we use equation (3) eval-uated at the type pair t1 = t1 and t2 = t2 to get

v1(t1, t2) ≥ E [ p(t1, t2) | V1(t1, v1)] − c1.

Taking expectations of both sides, we get

E [v1(t1, t2)] ≥ E [E [ p(t1, t2) | V1(t1, v1)]] − c1.

By the law of iterated expectations, this expression isequivalent to

U1 =∫

T2

v1(t1, t2) dF2(t2) ≥∫

T2

p(t1, t2) dF2(t2) − c1.

(5)

By a similar argument, it follows that

U2 =∫

T1

v2(t1, t2) dF1(t1)

≥∫

T1

[1 − p(t1, t2)] dF1(t1) − c2. (6)

24This result would follow immediately from Lemma 1 if we madethe additional assumption that pi (t) was continuously differen-tiable in ti .

We next show that U1 + U2 = 1. Starting with thefact that v1(t1, t2) + v2(t1, t2) = 1 for all pairs (t1, t2), itfollows that∫

T1

∫T2

[v1(t) + v2(t)] dF2(t2) dF1(t1) = 1

∫T1

∫T2

v1(t) dF2(t2) dF1(t1)

+∫

T2

∫T1

v2(t) dF1(t1) dF2(t2) = 1

∫T1

U1 dF1(t1) +∫

T2

U2 dF2(t2) = 1

U1 + U2 = 1.

Therefore, adding inequalities (5) and (6) yields

1 ≥∫

T2

p(t1, y) dF2(y) − c1 + 1

−∫

T1

p(x, t2) dF1(x) − c2,

from which it follows that

c1 + c2 ≥ P1(t1) − P2(t2) = c .

This contradicts the supposition that c1 + c2 < c andthus proves the proposition. �

This proposition shows that there is a range of costssuch that the countries’ private information about theprobability of winning is always an obstacle to peace inany interaction with voluntary agreements. Phrased asa general result, Proposition 5 shows that small costs ofwar is a sufficient condition for the impossibility of com-pletely peaceful settlements in crisis bargaining gameswith voluntary agreements. We thus can view this resultas extending the claim of Fearon (1995) that when costsare low, private information leads to a positive probabil-ity of war from the case of uncertainty about costs to thecase of uncertainty about relative power. The reason thatthe war problem is particularly hard to solve in gameswith interdependent types is because, in such games, thestrongest type of a given country knows that every typeof their opponent cannot be a stronger adversary. So,from the strongest type’s perspective, war is a relativelyattractive option and therefore in order to persuade thestrongest type to forego this option, such types must getrelatively generous terms from peaceful settlements. Thisin turn creates an incentive for weaker types to pretendthat they are strong in order to secure this high payoff.The only way a peaceful settlement can exist with theseincentives is if the settlement gives every type of a countrythe same payoff as the strongest type of the country. But


when the total costs of war are less than the critical valuec , such settlements are impossible.

As an example of this sufficient condition, considerthe model of Reed (2003). In this model, country 1 makesan offer to country 2, who either accepts the offer or goesto war. This model has one-sided uncertainty: country 2knows the probability of winning the war, while coun-try 1 does not. In fact, in Reed’s model, country 2’s typeis exactly the probability p that 1 wins the war. In ournotation, then, country 2’s type t2 ∈ [a, b] ⊆ [0, 1] and

p(t2) = t2. Therefore, P1 = ∫ ba y dF2(y) and P2(t2) = t2,

because country 2 has no uncertainty. Thus, the suffi-cient condition in Proposition 5 states that if c1 + c2 <

c = ∫ ba y dF2(y) − a , then war must occur in any equi-

librium.25 To understand what this condition means inReed’s model, first note that in order to avoid war, country1 must make an offer that the strongest type of country2 will accept. Thus, country 1 must make an offer thatgives at least 1 − a − c2 in order to ensure peace. Such anoffer gives country 1 a payoff of at most a + c2. However,suppose country 1 makes an offer that all types of coun-try 2 reject. This leads to war with certainty, so country 1

has an expected payoff of∫ b

a y dF2(y) − c1. Thus, peace

is not possible if a + c2 <∫ b

a y dF2(y) − c1, which is ex-actly the sufficient condition in Proposition 5. To be clear,this does not show that such an extreme offer is optimal,only that country 1 will not make an offer that avoids warentirely.

The intuition of Proposition 5 further suggests thatwhen costs are large enough, peaceful settlements may bepossible. This is exactly what we show in the next result.Specifically, we show that if the total costs of war aregreater than the critical value c , it is possible to find somegame form that always avoids war.

Proposition 6. Suppose costs are common knowledge buteach country is uncertain about the probability of winninga war. If c1 + c2 ≥ c , then there exists a crisis bargain-ing game form that has voluntary agreements in which analways peaceful equilibrium exists.

Proof. As with Proposition 2, it is enough to give anexample of a crisis bargaining game form that has vol-untary agreements which has an always peaceful equilib-rium. Consider a crisis bargaining game form that hasvoluntary agreements such that vg (a) = P1(t1) − c1 forall a ∈ A and �g (a∗) = 0 for some a∗ ∈ A. By construc-tion, the strategy profile s ∗(t) = a∗ is always peaceful. Toshow that it is an equilibrium, first note that if either sidedeviates to a different peaceful action, then the settlement

25In this case, t2 = a because this is the strongest type of country 2.

amount does not change. If country 1 takes an action thatleads to war, its expected payoff from war, given its typet1, is∫

T2

p(t1, y) dF2(y) − c1 ≤∫

T2

p(t1, y) dF2(y) − c1

= P1(t1) − c1 = gv(a∗),

where the first inequality follows from the monotonicityof p. Therefore, no type of country 1 will deviate. A similarargument shows that country 2 will not deviate and starta war. Therefore s ∗(t) is an equilibrium and the proof iscomplete. �

The proof of Proposition 6 presents a crisis bargain-ing game form with voluntary agreements that has apeaceful equilibrium when c1 + c2 ≥ c . Thus, the propo-sition demonstrates the existence of a peaceful game form.It should be emphasized that the result does not claim thatall game forms will have such a peaceful equilibrium whenc1 + c2 ≥ c . As an example, consider the model of Reed(2003) described above and assume that F2 is the uni-form distribution on [a, b]. In this case, it is easy to checkthat c = (b − a)/2. However, it is also easy to verify thatthe peaceful equilibrium in which country 1 makes theoffer x = a + c2 that is accepted by all types of country2 exists only when c1 + c2 ≥ b − a . Thus, Reed’s modelis an example in which war still occurs in equilibrium forvalues of c1 + c2 > c .

Another important feature of Proposition 6 is that itimplies that a trivial sufficient condition for the existenceof a peaceful game form is that c1 + c2 ≥ 1. Yet, muchof the time this condition is unlikely to hold, i.e., it israrely the case that a real-world conflict generates relativecosts that are greater than the total value of the object ofdispute. However, if there is a third party that is willing toprovide a sufficiently large subsidy to the peace process,such as an international organization or a superpower, apeaceful settlement is possible. Permitting this possibility,we can say that, if a third party provides a subsidy

� ≥ P1(t1) − P2(t2) − (c1 + c2),

then there exists a crisis bargaining game form that hasvoluntary agreements in which an always peaceful equi-librium exists. In this circumstance, the subsidy amountis the minimum amount that will ensure that there isan agreement that both sides will prefer to unilaterallystarting a war. Thus, in a world with a large powerfulcountry willing to provide sufficient subsidies, the oc-currence of war as a consequence of private informationabout relative power can be avoided.26 Unfortunately for

26This corollary supports the argument that if there is a globalhegemon, then the international system is likely to be more


small (c1 + c2), depending on the technology of war p(t)and the distribution of types, the size of the subsidy canapproach the value of the whole prize.

We conclude our analysis of this case by presenting acharacterization of peaceful equilibrium that is similar tothe results in the previous section.

Proposition 7. Suppose costs are common knowledge buteach country is uncertain about the probability of win-ning a war. Suppose also that p(t) is continuously differen-tiable, strictly increasing in t1, and strictly decreasing in t2.Let G be any crisis bargaining game form that has volun-tary agreements and let s ∗ be any equilibrium of G. ThenUi (ti ) �= Ui (ti ) for some ti , ti ∈ Ti if and only if there is apositive probability of war in equilibrium.

Proof. The proof of Proposition 5 shows that Ui (ti ) isconstant in any peaceful equilibrium. For the other direc-tion, suppose Ui (ti ) = Ui (ti ) for all ti , ti ∈ Ti . Then byLemma 1,

Ui (ti )−Ui (ti )=∫ ti

ti

∫Tj

�(s , t j )∂pi

∂ti(s , t j ) dF j (t j ) ds =0,

for all ti , ti ∈ Ti . As pi is strictly increasing in ti , ∂pi/∂ti >

0, and so �(t) is zero almost everywhere. The result fol-lows. �

This proposition tells us that, as in the previous sec-tion, every peaceful equilibrium of every crisis bargaininggame with uncertainty about relative power must give ev-ery type the same expected settlement. As with the casewith uncertainty about costs, we can use this result toshed new light on strategic problems without having tosolve complicated games. For example, consider the cri-sis bargaining game in Smith (1998). The extensive formof this game is quite involved. To start, a leader from a“defender” country makes a cheap talk announcement oftheir intent to defend a client state if that state is attacked.Next, an aggressor state chooses to attack the client ornot. If the aggressor attacks, then the client can retaliateor capitulate. If the client retaliates, the defender choosesto intervene or not. If the defender intervenes, he changesthe probability that the client wins the war. After the cri-sis is complete, the defender’s citizens decide to keep orreplace their leader, given the cheap talk message, the playof the crisis, and the outcome of the war.

One of the main conclusions of Smith’s paper is thatthe defender’s message “can only be informative if waroccurs with positive probability.” To prove this, Smith

peaceful, given that the hegemon is willing to pay the cost (Gilpin1981; Keohane 1984; Kindleberger 1973).

uses several lemmas,27 but we can derive this result di-rectly from Proposition 7.28 In a peaceful equilibrium, alltypes of the attacker must get the same expected utility.This means that either the client retaliates with positiveprobability when attacked and all types of the attackerstay out, or, if the client never retaliates, all types of theattacker attack and get the capitulation utility. In eithercase, the outcome of the crisis does not vary. Therefore,for the defender country, the only way for its utility tovary is by the probability that the government is replaced.But again by Proposition 7, all types of the defender musthave the same expected utility. Therefore, all types musthave the same probability of replacement. But this impliesthat the cheap talk message must be uninformative, be-cause otherwise differences in the types would be revealedand the less competent types would be removed. There-fore, messages must be uninformative if the probability ofwar is zero. In this way, we can derive useful results fromspecific models using our propositions without actuallysolving for the equilibria in those models.

Discussion

As we have emphasized, one strength of the game-freemethod of analysis presented here is the ability it pro-vides to establish results that hold across a wide varietyof strategic interactions between countries. That is, ourresults apply to any direct bargaining process in whichthe countries can make offers and counteroffers to eachother, with or without communication, as well as anyarbitration mechanism in which the parties communi-cate to a formal institution and this institution decideshow the dispute will be settled. In this way, we have beenable to give a clear picture of the general consequencesof private information and the incentives to misrepresentthis information on the possibility and form of war andpeace. In addition, given our interest in understandinghow different kinds of uncertainty may imply differentgeneral tendencies for strategic behavior, our mechanismdesign approach allows us to clearly and cleanly explorethe effects of various types of uncertainty across manydifferent game forms.

Using our techniques, we have established severalgeneral results. The first group of results concerns themonotonicity properties of equilibria. We show that

27Notably, Smith’s Lemma 1 proves that expected utility is weaklyincreasing in type, which is exactly our Proposition 4.

28Although Smith’s model has three players, it is easy to extend ourresults to this case.


crisis bargaining games with two-sided private informa-tion have some, but not all, of the monotonicity proper-ties described by Banks (1990) for games with one-sidedprivate information. Proposition 1 shows that countries’expected probability of war and expected utility are both(weakly) decreasing in their costs. So the general intu-ition that low-cost types have higher expected utilitiesfrom crisis bargaining, but also face greater risk of war,carries over from the setting with one-sided incompleteinformation. Unlike Banks (1990), however, we are un-able to guarantee monotonicity in expected settlementswith two-sided incomplete information about costs. Sim-ilarly, Proposition 4 shows that with uncertainty aboutstrength, the monotonicity result for utilities also holdsin every crisis bargaining game. For expected probabilityof war, however, we are only able to establish a limitedkind of monotonicity, unlike Banks (1990).

A second interesting general result occurs in the in-terdependent case, where countries are uncertain aboutrelative power. Here we find that the conditions that de-termine whether or not peaceful equilibria can exist neednot depend on any aspect of the game form. As seen inPropositions 5 and 6, the nonexistence of peaceful equi-libria depends only on the costs of fighting and the formof the uncertainty about relative power, and not on thedetails of the bargaining. When the conditions for peace-ful equilibria are met, then a simple mechanism that givesboth sides at least the best they could hope for from war isa simple solution. Otherwise, when such an arrangementis not possible, there is no bargaining process, no matterhow complicated, that ensures peace.

A final interesting result of our analysis is that theeffect of private information and the incentive to mis-represent varies as we vary the structure of the decisionmaker’s uncertainty. This is true even though in eachframework we consider there is common knowledge thata settlement exists that both sides prefer to war. The factthat in each framework the conditions on the game formthat lead to war are different makes clear that the type ofuncertainty that exists in the international environmentcan have important implications for players’ strategiesand the probability of war. At the same time, our focus onthe informational roots of conflict is not meant to suggestthat this is the only potential cause of war. To be clear, thisarticle does not claim to present an all-inclusive modelfor war. Rather, we have attempted to provide game formfree results for an important class of strategic situationsdiscussed in the literature on private information and war.

One basic assumption that we have used to definethis class of games is that any peaceful agreement must beacceptable to both sides of the conflict. This “voluntaryagreements” assumption is intended to be general and

substantively motivated. We require that there be an ac-tion that allows decision makers to reject settlements thatgive them less than they might expect from fighting, butwe do not require that this action leads to a war lotteryor even immediate fighting. This flexibility implies thatour results hold in many different kinds of games. Forexample, the fact that our results hold in game forms thatinclude capitulation as a possible outcome distinct fromwar and peace also aids in clarifying the application of ourmodel to several historical cases. While there are manyclear examples of the direct, unilateral use of force such asthe Japanese attack on Pearl Harbor and the American in-vasion of Iraq in 2003, there are other historical cases thatcould be better categorized as capitulation, such as the in-vasions of Denmark and Luxembourg in 1940. Thus, ourresults apply to both kinds of circumstances, includingthose that involve armed aggression but that ultimatelydo not involve significant armed conflict. Moreover, asnoted above, we can see our results materialize in a widevariety of extensive forms, from alternating-offers bar-gaining and wars of attrition to crisis bargaining modelsand models of extended deterrence (Fearon 1994; Lev-entoglu and Tarar 2008; Powell 1996; Schultz 1999; Smith1998).

While our view is that our “voluntary agreements”assumption is a natural consequence of the anarchic na-ture of the international arena, there may remain somequestion as to how best to describe the process leadingcountries to war. One alternative assumption is that waroccurs only if both countries agree to fight and other-wise the status quo or an imposed settlement prevails.This assumption has been used to consider the idea ofmutual optimism as a cause of war. Central to this litera-ture (Blainey 1988; Wittman 1979) is the claim that bothcountries must want to fight for a war to occur, hence thename mutual optimism. Not surprisingly, this alternativeassumption yields different conclusions about the possi-bility of war. Thus our results for the case of voluntaryagreements highlight the role of private information andthe incentives to misrepresent and can be interpreted asthe complement to the mutual optimism argument.29

The other basic assumption we use to define the classof crisis bargaining games is that whatever form the in-teraction of the two sides takes, the end result is alwayseither a negotiated settlement or war. Dividing outcomesinto these two categories seems to us to be quite natural.At the same time, because we make the common assump-tion that the war outcome is a game-ending costly lottery,we should consider how our results speak to the recent

29For more on the issue of mutual optimism and assumptions abouthow countries end up in war, see Fey and Ramsay (2007, 2008).


formal conflict literature on “war as a bargaining pro-cess” (Filson and Werner 2002; Powell 2004; Slantchev2003b). In a manner similar to Powell (2004), we can re-lax the assumption that fighting is a game-ending moveby supposing that fighting creates a chance of a militarilydecisive outcome but can also be indecisive and lead tofurther negotiations or further fighting. We can fit sucha model into our theoretical framework by identifying allterminal nodes with a decisive military outcome as “war”outcomes and all other terminal nodes as “peaceful set-tlement” outcomes. However, as a number of rounds offighting can precede any such outcome, we must makechanges to two of the assumptions we make about ourclass of crisis bargaining games. First, we must modifythe assumption that peaceful settlements are efficient be-cause if the countries fight for some time before reachinga peaceful settlement, the payoffs to the two countries forsuch a strategy profile will sum to less than one. Second,for a similar reason we must modify our assumption thata country’s war payoff does not depend on the actionstaken in the game. The longer the countries choose tofight before a decisive military outcome results, the loweris the expected value of this outcome.

How do our results fare if we make these two changesto our framework in order to accommodate models ofwar as a bargaining process? Reassuringly, it is possibleto show that all of our main results continue to hold.In the case of the first change, we show in a companionpaper (Fey and Ramsay 2009) that permitting settlementoutcomes to be inefficient does not change any of thequalitative features of our results. This is also true withthe second change, permitting the war payoff to vary withthe actions taken in the game. Rather than provide formalproofs here, instead we briefly explain why our resultscontinue to hold.30 Our reasoning rests on the fact thateven if a country’s war payoff depends on its actions, asits equilibrium actions depend on its type, we can use therevelation principle to find a direct mechanism in whicha country’s war payoff depends directly on its type. In thisway, Lemma 1 continues to hold. As this lemma is a keyingredient in our proofs, this result, coupled with the factthat the actions of a country can only change its payoff byincurring the costs of multiple rounds of fighting, yieldsthe same characterizations as found in our propositions.

Conclusion

This article has set out to establish general results aboutthe fundamental incentives inherent in crisis bargaining.

30Details are presented in a supplemental appendix available online.

Rather than fix a particular model and derive results thatare limited to this single extensive form, we develop amethod to identify properties shared by all equilibria ofa large class of crisis bargaining models. The value of thisapproach is twofold. First, such an approach allows us tocut through the clutter of the endless variety of modelingassumptions, in particular assumptions about sequencingof moves, and to show how uncertainty and the incentiveto misrepresent affect the probability of war in a variety ofcrisis bargaining settings. Second, our “game form free”approach allows us to compare the effect that differencesin the kinds of private information possessed by actorshave on strategic choices of the actors. We have developedthese results across a range of information environmentsin order to identify conditions under which the positiveprobability of war is an unavoidable consequence of pri-vate information and conditions under which peacefulresolution of conflicts is possible. We have also charac-terized the nature of such peaceful resolutions. In short,we have found that while it is possible to give general re-sults that support the rationalist explanation of war as aconsequence of private information and the incentive tomisrepresent this information, the link between privateinformation and war depends in important ways on thetypes of uncertainty that states face.

Throughout this article, we have emphasized the gen-erality of our approach and the necessity of utilizing such ageneral approach to justify broad claims about the causesof war. Although we are able to give results that iden-tify specific relationships between information and out-comes, there is an obvious trade-off between general re-sults and specific predictions. The relative value of the twomust depend on the question being asked. In particular,questions regarding particular behaviors in a specific in-stitutional setting call for an approach very different fromthe one we take here. On the other hand, as a theoreti-cal matter, we may be interested in results that are notinstitution specific, or that apply to a wide class of differ-ent institutions and strategic settings. To the extent thatanswers to general questions about the relationship be-tween uncertainty and war, independent of institutionalspecifics, are the object of interest, no iteration of theprocess of analyzing specific protocols will reveal thesegeneral equilibrium phenomena. For these questions, thegame-free approach is a useful tool. As a guiding prin-ciple, we endorse the “two-step” approach outlined inBanks (1990) in which general results are established firstusing the approach developed here and then additionalbehavioral predictions are generated from a particulargame form that captures specific institutional features ofinterest.

In the end, while this article has laid out a frame-work for analyzing some general questions about war


and peace, several additional aspects of this approachremain to be explored. First, while we have focused onhow various theoretical sources of uncertainty may in-fluence the possibility for “well-designed” institutions toeliminate unwanted, inefficient conflict, it is also true thatinstitutions do not just arrive from nowhere; they are en-dogenous to the negotiation process between states. It isnatural, then, to consider the process by which countriesbargain over possible game forms that then govern theirinteraction. It should be possible to use our approach todeal with such a situation by viewing the institution selec-tion process as itself part of a more broadly defined gameand applying the revelation principle.31 Second, althougheach of the two kinds of uncertainty that we have consid-ered in this article have involved one-dimensional types,it should also be possible to generalize our approach toallow for multiple-dimensional types. One example ofthis kind of uncertainty would be a case in which coun-tries possessed private information about both their owncost of conflict as well as the relative balance of power.We leave the analysis of this case as a topic for furtherresearch.

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Uncertainty and Incentives in Crisis Bargaining: Game-Free ... · In the literature on international conflict, the paper that is closest to ours is Banks (1990). This paper uses a