Diplomatic Calculus in Anarchy: How Communication Mattersan essential aspect of interstate diplomacy. The anar-chic strategic context can provide less resolved states with a disincentive

American Political Science Review Vol. 104, No. 2 May 2010

doi:10.1017/S0003055410000158

Diplomatic Calculus in Anarchy: How Communication MattersROBERT F. TRAGER University of California at Los Angeles

When states come to believe that other states are hostile to their interests, they often reorient theirforeign policies by realigning alliance commitments, building arms, striking first, mobilizingtroops, or adopting policies to drain the resources of states that menace them. This article

presents a crisis bargaining model that allows threatened states a wider array of responses than the choiceto back down or not. Two implications are that (1) “cheap talk” diplomatic statements by adversaries canaffect perceptions of intentions, and (2) war can occur because resolved states decline to communicatetheir intentions, even though they could, and even though doing so would avoid a war. The model relatesthe content and quality of diplomatic signals to the context of prior beliefs about intentions and strategicoptions. In simulations, this form of diplomatic communication reduces the likelihood of conflict.

In the fall of 1876, Russia queried Germany as towhat its position would be if Russia were to goto war with Austria-Hungary. Germany replied, as

delicately as it possibly could, that it could not guar-antee its neutrality in such an eventuality, and that“a lasting weakening of Austria would be contrary to[German] interests.”1 The German chancellor, Ottovon Bismarck, thought that this created a “new situ-ation” in the German-Russian relationship.2 He meantthat Germany had increased Russian expectationsabout Germany’s willingness to go to war on behalf ofAustria, and perhaps decreased Russia’s appraisal ofthe extent to which Germany would support Russianaims generally.

According to most theories of information transmis-sion between states, this should have been impossible.Germany communicated its position privately throughdiplomatic channels. Thus, German elites did not staketheir reputation before a domestic audience, nor didtheir actions carry explicit costs, two commonly recog-nized mechanisms of information transmission.3 Fur-ther, Germany’s noncommittal statements can hardlybe considered to have so engaged its reputation beforethe Russian and interstate audiences that anyone couldhave believed Germany significantly less willing toback down for fear of the repercussions of having beencaught in a bluff. Theories of information transmis-sion that rely on the staking of bargaining reputations,

Robert F. Trager is Assistant Professor, Department of Political Sci-ence, University of California at Los Angeles, 4289 Bunche Hall,Box 951472, Los Angeles, CA 90095-1472 ([email protected]).

I am grateful to Michael Chwe, Maria Fanis, Erik Gartzke, MikeHorowitz, Robert Jervis, Andy Kydd, Helen Milner, Barry O’Neill,Kris Ramsay, Sebastian Rosato, Ken Schultz, Art Stein, DessieZagorcheva and seminar participants at the Olin Institute for usefulcomments and suggestions, as well as to Bob and Judith Terry andOriel College, Oxford, for a conducive environment in which to write.I bear responsibility for remaining errors and omissions.1 Schweinitz (1927, 360). The Russians did not explicitly threatenAustria, but it was at this time that the saying, “the road to Con-stantinople leads through Vienna” (Rupp 1941, 232) gained popu-larity in Russian diplomatic and military circles and many Russiansand Germans thought that the tense, ongoing Austro-Russian ne-gotiations could lead to war. See Ignatyev (1931, 391), Rupp (1941,297), Saburov (1929, 82), Schweinitz (1927, 359), and Grosse Politik,II, 54–66, 74–9.2 Bismarck 1915, 286.3 See, for instance, Fearon (1994) and Kydd (1997).

therefore, also fail to explain the case.4 Yet Bismarck’sappraisal was correct: thereafter, Russia reckoned onGerman support for Austria-Hungary.5

In this article, I describe a bargaining model in whichstates have a richer array of options than the previousliterature has allowed. This results in a mechanism ofinformation transmission that provides a convincingexplanation for this case, and, I shall argue, representsan essential aspect of interstate diplomacy. The anar-chic strategic context can provide less resolved stateswith a disincentive to misrepresenting their intentions.Very often, when states negotiate over important se-curity concerns, there is a danger that a threat (orother noncooperative behavior) will result in a breachin relations, not merely with respect to the issue athand, but also with respect to other aspects of therelationship. In such cases, states can send “costless”signals about their intentions. Information is conveyedby threats because states understand the dangers ofaltering other states’ perceptions of their intentions,and yet choose to threaten anyway when they are suffi-ciently resolved. Alternatively, if foreign policy choicesare responsive to perceptions of other states’ intentionsat all, then explicit threats from one state to another,whether in public or private, can convey information.Thus, this article contributes to the literature on theconditions under which costless diplomatic signals areinformative.6

A second result of the model is a path by whichincomplete information can lead to conflict that is dif-ferent from the mechanism in other models in theliterature. War can occur in some equilibria becauseresolved states decline to communicate their resolve

4 See, for instance, Sartori (2005). See Fearon (1995) for an analysisof the efficacy of costless diplomatic signals.5 As the German Ambassador wrote, “nothing was more natural”(Schweinitz 1927, 350–1) than for the Tsar to expect a guarantee ofGerman neutrality, and he remained openly disturbed by the Ger-man response for days (Rupp 1941, 204; Schweinitz 1927, 360–364).According to the Russian diplomat Kartsov, the idea of declaring waron Austria “was abandoned” because “Prince Bismarck forewarnedthat Austria was necessary for Germany for reasons of politicalbalance of power and that Germany would therefore not permitdelivering Austria a death-blow” (Rupp 1941, 297). See Trager (2007,chap. 3).6 Other works on this subject, in addition to those cited above, in-clude Guisinger and Smith (2002), Jervis (1970), Kurizaki (2004),Kydd (2003), Ramsay (2004), and Schelling (1966; 1980).

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even though they could, and even though this wouldprevent the war. This can occur when one state is highlyresolved, and believes that another is also, but believesthat the other state does not believe that the first ishighly resolved. In such cases, if the first state wereto signal its resolve, there would be a high probabil-ity that the other state would prepare for conflict. Toavoid this likelihood, the first state declines to con-vey its resolve to the other. Sometimes, however, thesecond state would have been willing to comply withthe first state’s demand in order to avoid war. Thus,in this situation, resolved states mimic the behaviorof unresolved states in order to catch the other sideunprepared. This contrasts with other signaling modelsin which it is generally unresolved states that would liketo imitate the signals sent by resolved states.

This second point is similar to results derived fromother recent models that have begun to expand therange of state options beyond those conceived of intraditional bargaining models. Ritter (2004, chap. 2) ar-gues that states sometimes make their alliances secret,in spite of the drawbacks from the point of view of de-terrence, to prevent potential adversaries from takingcountermeasures. Slantchev (n.d.) presents an ultima-tum game in which rejected offers lead to conflicts inwhich each side must choose a level of effort in thefighting. Because the optimal effort of states dependson each side’s perception of the strength of the otherside, states sometimes have an incentive to make offersin the preconflict stage that hide their true strength. Themodel below reveals a similar dynamic in the contextof costless diplomatic encounters across a wide rangeof seemingly disparate contexts, from alliance politicsto nuclear brinkmanship.

The article has four sections. In the first, I locatethe argument in the literature on communication ininternational relations. In the second, I argue that thestrategic options available to threatened states go wellbeyond the binary choice to comply or not in tradi-tional models of coercion. The third section presents amodel that allows states that are threatened to preparefor conflicts and to choose themselves to engage inconflict. When these preparations are effective, infor-mative costless signaling occurs in plausible equilibria.This section discusses the conditions under which sig-naling can be effective, and under which wars occurthat available but unused signals could prevent. A finalsection discusses implications of the analysis.

COMMUNICATION, CONSTRUCTION, ANDDETERRENCE

Social aspects of the human world exist in language,through which they are created, altered, and reified.Within the social realm, some complex constructs, forinstance, those that would be identified as sources ofcultural difference, are the products of many agentsinteracting over long periods of time and consequentlychange only very gradually in general. But other as-pects of the social world, such as the “situation” de-scribed by Bismarck, change more quickly, and such

change is more often intentional. To bring about thissort of change, attempts at communication must suc-ceed in conveying information. That is, they must bebelieved, at least partially. This article will not addresshow symbols acquire their meanings or how individualsare initiated into language groups.7 Rather, I assumefor analytical purposes that a shared and unchanginglanguage exists and ask this second question, namely,how it is that attempts at communication by adversariescan convey information.

In particular, I look at one of the most adversarialaspects of relations between states, when one is at-tempting to coerce another through a threat to useforce.8 Here, more than elsewhere, we expect commu-nication to be difficult because the state making thethreat has an incentive to misrepresent its preferencesin order to make the threat appear more credible. Insuch situations, as Jervis (1970) recognized, analystsdistinguish potential sources of information that can bemanipulated from those that cannot. The latter can bebelieved, whereas the former only convey informationin certain contexts.

Within the category of information sources that canbe manipulated, some messages are credible, even be-tween adversaries, because senders would not want tomisrepresent their preferences although they could.When the message causes the sender, in some con-tingencies, to incur costs as a direct result of sending themessage, such messages are called “costly signals.” Ingame-theoretic terms, the decision to send the messageor not has a direct effect on the payoff function ofthe sender. Perhaps the clearest example from inter-national relations is the decision to build arms. Thisaffects the balance of capabilities directly, but if thearms are costly enough,9 it may also send a signal toan adversary about the state’s intentions.10 The adver-sary may conclude that a less resolved state would beunwilling to pay those costs.11

7 The literature that addresses these questions is too large to cite evena representative sample. Among philosophers, the work of LudwigWittgenstein is the most widely appreciated. Within international re-lations, see for instance, Finnemore and Sikkink (1998), Jervis (1970,139–224), O’Neill (1999), and Wendt (1999, 313–69).8 See Schelling (1966, 70–1) for the definition of coercion as subsum-ing “deterrence” and “compellence.” But see also Morgan (2003, 3)for a discussion of the intimate relationship of these two from ananalytical point of view.9 If spending on arms does not reach a threshold related to thepossible values states could place on the issue in question, it willgenerally be be uninformative (at least as a costly signal).10 For a definition of intentions that is similar to the game-theoreticconcept of a strategy, see Jervis (1976, 48–57). For another definitionof intention, see Bratman (1999).11 It is not argued here that states must draw this conclusion whenthey observe other states choosing to build arms, but merely thatsuch a dynamic is plausible and easily grasped. There is a largeliterature that discusses costly signaling, especially as it relates tocrisis bargaining. See, for instance, Fearon (1995; 1997), Jervis (1970,28–9), Kydd (1997), Morrow (1989), and Powell (1990). It also maybe worth noting that, as Sartori (2005, 58) points out, although “audi-ence costs” have been modeled as costly signals, this is probably bestthought of as a modeling shorthand (when voters are not modeledexplicitly) for what is really a costless process. See also Ramsay(2004) and Smith (1998).

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But messages need not be associated with direct costsfor there to be disincentives to lie, and thus for themessage to enable to the receiver to learn somethingfrom it. Whether such disincentives exist depends onthe strategic context. Suppose, for instance, that anemployer wants to give a high-paying position to anexperienced applicant and a low-paying position toan inexperienced one. If there is no way to verifywhether a candidate is telling the truth, the candidatemay have a hard time convincing the employer that heor she deserves the high-paying position. On the otherhand, if the high-paying position is for a pilot and thelow-paying one for a flight steward, the candidate’sstatements about his or her hours of flight experiencemay well be at least partially informative. Misrepre-senting qualifications in this case may result in his orher flying planes, which for the inexperienced may re-sult in a crash that has negative consequences for bothemployer and employee. In this employment example,therefore, the two sides have interests in common; com-munication of private information may be possible. Inthe game-theoretic literature, such signals are calledcostless because the actions available to agents andthe payoffs associated with material outcomes are allunchanged no matter which message is sent.12

Models in the economics literature show that themaximal degree of precision of credible costless mes-sages increases as the interests of sender and receivergrow more aligned.13 In two-player games, when re-lationships are zero-sum, no “cheap talk” communi-cation is possible at all. Understanding the degree towhich interests are aligned, however, turns out to befar more difficult than it appears at first glance.14

Many scholars of international relations have con-cluded that relations between states are too adversar-

12 The term “costly signal” is sometimes used differently from itsdefinition in the game-theoretic literature to refer to communicationmechanisms that rely on any disincentives associated with sendingthe signal, whether direct or arising out of the strategic interaction ofagents. Because any signal of resolve that conveys information mustresult from a disincentive to unresolved types to send the signal, allinformative signals must be “costly” in this sense. The staking ofprivate reputations, which convey information because of the “cost”of not being believed in the future (Sartori 2005), and every othermechanism of information transmission are costly signals on this un-derstanding, but not according to the way these terms are understoodin the game-theoretic literature. Recent work in political science hasused the game-theoretic terminology. In the international relationsliterature, see Kydd (2003), Ramsay (2004), and Sartori (2005, 51).In other literature see, for instance, Gilligan and Krehbiel (1987).13 See Crawford and Sobel (1982), Farrell and Gibbons (1989), andfor a nontechnical discussion, Farrell and Rabin (1996). To betterunderstand the distinction between costly and costless signals, it maybe useful to consider the following. If sending the signal costs thesender $100 and the receiver of the signal infers something fromthe sender’s willingness to pay $100, this constitutes a costly signal.Suppose the signal costs nothing to send, however, but conveys in-formation that causes the receiver to take $100 from the sender. Thesignal may only have affected the receiver’s beliefs because of theanticipated reaction of the receiver and the sender’s willingness tosuffer the receiver’s reaction. Nevertheless, this scenario representsa costless signal because the disincentive to send a misleading signalarises from the strategic interaction of the two agents rather thanfrom a direct cost of taking the action itself. This is so even thoughthe ultimate result—the loss or not of $100—is the same.14 See Axelrod (1970).

ial for costless communication to occur. For instance,Schelling noted that “words are cheap, not inherentlycredible when they emanate from an adversary. . . .”15

Fearon, careful not to overstate the case, argued that“it remains unclear whether cheap talk is important ininternational disputes. . . .” His work as a whole sug-gests, however, that, “costless signals. . . have no ef-fect” on international outcomes.16 Many other schol-ars have followed their lead. I show below that thepreferences of state adversaries are in fact sometimessufficiently aligned to allow costless communication tooccur.

This distinction between costly and costless signalsmay seem merely technical, with little substantive im-portance. This is not the case. On it hinges the questionof whether diplomacy, particularly that practiced byleaders behind closed doors, can play a role in de-termining the course of events in a rationalist frame-work. Macro-level theories of the international systemhave generally claimed to explain international eventswithout reference to diplomatic communication.17 Thisorientation has received support from micro-level the-orizing of conflict processes, which has suggested (ifnot stated outright) that only directly costly activitiessuch as mobilizing troops, building arms, and creating“backdown costs” can convey information. As a result,international relations scholarship has neglected thestudy of the role of private diplomatic communications,both theoretically and empirically.

There are important exceptions to this characteriza-tion of the literature, however, and recent scholarshiphas shown a renewed interest in the mechanisms ofprivate diplomacy. Sartori (2002; 2005) has demon-strated that private diplomatic communications canbe made credible by the desire of states to maintaina bargaining reputation, which makes them hesitantto send misleading signals for fear of being caughtin a bluff.18 Guisinger and Smith (2002) combine twostrains of the literature to show that in the presence ofa reputational mechanism along the lines of Sartori’s,democratic selection of leaders results in an endoge-nous additional disincentive against bluffing. Kurizaki(2004) shows that if a publicly threatened state will loseface by backing down, then it is sometimes optimalfor states to make private threats. Such threats do notincrease the probability that the threatened state as-signs to the threatener following through, but neitherdo these private threats cause that probability to goto zero.19 Kydd (2003) presents a cheap-talk model ofthird-party mediation.20

15 Schelling (1980, 150). Elsewhere, Schelling argues that words usedto frame an issue affect perceptions of resolve. See Schelling (1966,35–91).16 Fearon (1997, 69).17 For instance, Mearsheimer (2001); Schweller (1998); Walt (1987);Waltz (1979).18 Jervis (1970, 78–83), Schelling (1966; 1980), and others have alsoargued in favor of a reputational signaling mechanism in privatediplomacy.19 See also Fearon (1997, 84).20 Ramsay (2004) describes a cheap-talk model of public diplomacy.

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Earlier work describes several other mechanisms ofprivate diplomacy.21 Schelling (1966, 1980) argued thatthreats sometimes risk an undesired event that nei-ther side directly controls, which causes the threats toconvey information. The essence of brinkmanship isthat when one climber who is roped to another movescloser to the edge, he or she may slip. By demonstratinga willingness to slip, the climber conveys informationabout his or her resolve in the issue being negotiated.22

Schelling’s analysis focuses on how engaging in limitedforms of conflict can constitute an implicit threat be-cause it demonstrates a willingness to risk even morecostly conflict. Private threats may have a similar ef-fect, however, in that they delay resolution of the issueand create a crisis atmosphere in which conflict maybe more likely. Schelling’s work bears important sim-ilarities to the analysis to follow. Here, however, theemphasis will be on the intentional action of actorsrather than the partly exogenous danger of sliding offa cliff and on the implications of including the option toprepare in models of the conflict process. The analysisto follow also makes it clear that some of the dynamicsof engaging in limited conflict are equally germane tocostless signals and thus to private diplomacy.23

Of these mechanisms for private information trans-mission, the staking of bargaining reputations has re-ceived the most attention. Despite some scholarshipthat argues to the contrary, I believe the diplomatic–historical record supports the proposition that themaintenance of a reputation for being true to one’scommitments has been an important force in inter-national affairs.24 As in the case of Germany’s tacitthreat to Russia in 1876, mentioned above, however,important information is often conveyed by costlessdiplomacy that cannot be explained by reputationaldynamics. In this case and in many others, in makinga threat, policy-makers on both sides were not primar-ily concerned with bargaining reputation. Rather, Ger-many worried about consequences that would followif, as it expected, its private threat were found credible.

21 Jervis (1970) proposes three additional mechanisms through whichprivate threats could convey information. First, if leaders are reticentto lie for moral reasons, credibility may attach to their statements.Second, if a country has a stake in the current functioning of theinternational system, a reticence to lie may derive from a desireto ensure that states do not too often deceive each other becausea baseline of honest communication may be required to maintainthe overall systemic equilibrium. Third, lies may result in unwanted“changes in the international environment.” Sometimes, if a state-ment is believed, other actors may act in such a way that the actormaking the statement has an additional incentive to follow throughon the statement. For instance, if one state professes hostility towardanother, the reaction of the second state may make it in the firststate’s interest to take hostile actions it had not originally plannedon taking.22 Powell (1988) formalizes this idea in the context of inadvertentwar.23 The constructivist literature has emphasized the role of behaviorin creating norms and shaping expectations, interests, and identi-ties, but has not focused on the communication of intention amongadversaries. See, however, Der Derian (1987).24 For criticisms of a reputational signaling mechanism, see Jervis(1984), Mercer (1996), Morrow (1994), and Press (2005) .

Further, in many cases where reputations may appearto have been staked, the mechanism I propose hererepresents an alternative and sometimes more plausi-ble explanation of communication.

The model in this article contributes to this growingliterature by providing a new mechanism of informa-tion transmission between states that applies to certainsystemic contexts and appeals neither to the costlinessof the signal, nor to bargaining reputation. The theoryapplies equally to messages conveyed in public and inprivate. It also demonstrates that a similar signalingmechanism is at work in seemingly diverse strategiccontexts, such as nuclear brinkmanship and alliancebargaining.

THE STRATEGIC CONTEXT OF ANARCHY

In prominent models in the international relations lit-erature, the strategic options of the target of a threatare limited: it can stand firm or concede. This conceptu-alization does not correspond to the options and incen-tives of states in the anarchic international context. If astate comes to believe it cannot achieve its key strategicaims through its current relationship with another state,it may choose to alter that relationship. In particular,rather than merely deciding whether or not to backdown, threatened states must decide how to preparefor conflict if they believe a breach with a threateningstate is imminent. In such cases, states often reorienttheir security policies in order to drain resources fromthe threatening state, and they also tend to form newalliances that are contrary to the threatening state’s se-curity interests. In addition, when the target of a threatbelieves future conflict is more likely, it will sometimeschoose to increase arms production, mobilize troops,or strike first. These decisions are often made beforethe threatening state chooses to back down or followthrough on its threat and have consequences whetheror not the states involved ultimately go to war.25

To illustrate the strategic choices available to Targets,consider the responses of U.S. President Kennedy andSoviet Premier Khrushchev to each other’s threats intheir June 1961 meeting. Put simply, each threatened toescalate to war if the other did not accept a settlementof the Berlin question favorable to his side. Followingthe meeting, Khrushchev responded by adopting a newset of policies designed to drain the resources of theUnited States. On August 1, he “approved most of aKGB plan to create ‘a situation in various areas ofthe world that would favor the dispersion of attentionand resources by the United States and their satellites,and would tie them down during the settlement of aGerman peace treaty and West Berlin.”’26 For his part,

25 Works that argue that states react to perceptions of intentions inways described here include Jervis (1976, chap. 3), Schelling (1966),Schultz (2001), Schweller (1994), and Walt (1987). One work thatargues against state responsiveness to the intentions of other statesis Mearsheimer (2001). Waltz (1979) does not argue against a causalrole for perceptions of intentions. Rather, he argues only for a sep-arate and independent effect of the distribution of capabilities. SeeWaltz (2003, 53).26 Fursenko and Naftali (1999, 138).

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hard as it is to imagine today, Kennedy began to takethe idea of a nuclear first strike more seriously. Hetold the Joint Chiefs of Staff that “Berlin developmentsmay confront us with a situation where we may desireto take the initiative in the escalation of conflict fromthe local to the general [nuclear] war level.”27 Thus,because both leaders were unwilling to back down,they considered or adopted policies to prepare for aconflict they thought might be imminent.

Alternatively, consider the Japanese strategic cal-culus in 1941. In November of that year, the UnitedStates demanded that Japan withdraw from China. Thisamounted to “surrendering her position as a power inthe Far East.”28 Japan was unwilling to accede to U.S.demands, and, precisely because it found U.S. threatscredible, decided to take radical action to prepare forthe coming conflict. In the hope of demonstrating itsresolve to resist, and of engaging in only a limited warwith the United States, Japan opted to destroy the of-fensive capability of the U.S. fleet at Pearl Harbor.29

The history of international relations is full of exam-ples of states responding to diplomatic pressure withactions that go well beyond a simple refusal to com-ply with a demand. In fact, where important questionsof security are concerned, simply declining to complywith demands is likely the exception rather than therule. The Japanese response to U.S., British, and Dutchpolicy in 1941 was exceptional only in scale and deci-siveness.

In response to Austrian threats during the CrimeanWar, Russia took actions it would not have takenotherwise. These included colluding with France andSardinia to strip Austria of northern Italy, tipping thebalance in Germany in favor of Prussia, permittingthe revolution in Hungary that resulted in the Austro-Hungarian Ausgleich (instead of assisting in suppress-ing the Hungarians as Russia had before the war), anddeclining to renew generous offers of Russo-Austriancooperation in the Balkans, leading to drastically in-creased security competition in the region between thetwo countries.30 More speculatively, but with some jus-tice, the historian Norman Rich argues that the Aus-trian threats during the Crimean War resulted in “abitter hostility that was to culminate in war in 1914, thedestruction of both imperial houses, and the liquidationof the Habsburg Empire.”31

A particularly common response to dissatisfactionwith another country’s conduct of foreign policy andits perceived hostile intentions is the realigning of al-liance commitments. In 1864, for instance, NapoleonIII wished to use a European conference to revisethe post–Napoleonic Wars settlement of 1815. WhenBritain, with which France was closely aligned, refusedto support a conference, Napoleon was explicit: “So itseems we shall have no Congress. Well! I shall have to

27 Cited in Press (2005, 5); italics added.28 Feis (1950, 327).29 See George (1991, 19) and Russett (1967).30 For a detailed comparison and analysis of Russian foreign policybefore and after the Austrian threats, see Trager (2007, chap. 2).31 Rich (1965, 123).

change my alliances.” With that, the alignment betweenthe two countries ended.32

The danger of a great power realignment resultingfrom a general breach in Russo-German relations isalso the explanation for the information conveyed byGermany’s statements in 1876 discussed above. Bothsides understood that even a tacit threat from Germanymight lead Russia to form an alliance with France,Germany’s arch rival since the Franco-Prussian warsix years before. The fact that Germany understoodthe danger, and yet chose to tacitly threaten anyway,meant that Germany had effectively communicated itsresolve to defend Austria-Hungary.33

To fix ideas, and to see the relationship betweenpreparation for conflict and communication, considerthe following stylized examples of international con-texts in which costless communication is possible. Themodel presented in the next section is designed to rep-resent these situations in a stylized way. As these nar-ratives make clear, signaling dynamics can be similarin seemingly diverse strategic contexts.

Example 1 (External Balancing). A conflict arisesbetween two states, a Deterrer and a Target, over aspecific issue. In order to get its way, the Deterrerwould like to signal its willingness to go to war overthe issue to the Target. The Deterrer knows, however,that if the issue is particularly important to the Target,the latter may form an alliance with a third country inorder to get its way or prepare for a possible conflict.The Target would prefer not to make the concessionsto the third state required to get an agreement, andthe new alliance is also likely to have a negative ef-fect on the Deterrer’s security position, especially ifthe third country is already hostile to the Deterrer.Threat-making therefore has both an advantage and adrawback. The advantage is the increased likelihoodthat the Target state will concede the issue to the De-terrer; the drawback is the possibility that the Targetwill “balance” against the Deterrer by forming a hostilealliance. As has been mentioned, this was the princi-pal concern in German relations with Russia after theFranco-Prussian war. Deterrers for whom the issue isnot sufficiently important are unwilling to incur the riskof such a breach in relations by making a threat. Whenthe Target state observes a threat, therefore, it learnsthe issue is relatively important to the threatening state.

Example 2 (Internal Balancing). As the name im-plies, this scenario is similar to the last, except that theprinciple strategic option of the Target (the threateningstate’s principal concern) is to transfer resources to itsmilitary sector in order, one day, to resist the demandsof the threatening state. The timing of China’s decisionin the 1950s to devote enormous diplomatic and ma-terial resources to the pursuit of nuclear weapons, forinstance, may have been partly a result of U.S. threatsin the first and second Taiwan Strait Crises. Because

32 Mosse (1958, 142).33 For an analysis of this case, see Trager (2007, chap. 3). For a relatedargument on the follow-on effects of alliance realignments, see Healyand Stein (1973).

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such arms production alters the future bargaining re-lationship between the nations, internal balancing willoften constitute a significant long-term drawback tothreat-making.

Example 3 (First Strike). If a Deterrer threatens aTarget, the military and strategic context may be suchthat if the Target is unwilling to back down and believesthe Deterring state is also sufficiently unlikely to backdown, the Target’s best option is to strike first. This wasthe situation for Japan in 1941. It was also a worry forU.S. President Kennedy during the Cuban missile crisis.He recognized that a U.S. threat to destroy the missilesin Cuba in four days could result in a Soviet threat totake action in three days, as well as further escalationsthat could result in nuclear war in that time frame.34

In such military-strategic contexts, therefore, threat-making once again involves a trade-off similar to thatin Examples 1 and 2. Threats increase the chance that astate gets what it wants with respect to the issue at hand,but can also create a danger that the threatened sidewill begin an unwanted military conflict. Once again,the willingness of the Deterrer to incur such a risk cancause the threat to convey information to the Target.

Example 4 (Too Costly Deterrence). Faced with athreat, the Target might consider adopting policies thatwould deter the threatening state from attacking. Suchactivities, for instance mobilizing forces on a border,may be too costly to sustain for long. Rather than main-tain a high level of preparation, therefore, the Targetstate might prefer to go to war. Powell (1993) analyzes astrategic context that leads to a similar dynamic. Fromthe perspective of diplomatic signaling, if we think ofthe Target’s decision as preceding the Deterrer’s deci-sion to go to war or back down, the danger of such anoutcome plays a similar role to the risk of a first strikein Example 3.

Example 5 (Resource Drain). Other reactions thatstates may have to threats also have long-term conse-quences for the threatening state. As the Soviet Uniondid following the Vienna meeting, for instance, a statemay choose to drain the resources of another state inorder to get its way on a particular issue. As in Ex-amples 1–4, the risk that a threatened state will adoptsuch a course provides a disincentive to less resolvedstates to signal their willingness to engage in conflictover such a contentious issue, making communicationpossible.

Example 6 (Mobilization). When a threatened statedeclines to back down and believes conflict likely, itmay elect to mobilize its troops. With the cost of the mo-bilization paid, the choice to go to war looks more at-tractive to the mobilized state than it had previously.35

If we allow for the possibility that the mobilized statewill go to war, we once again have a tradeoff for statesthat consider threat-making, and this again results inthe possibility of informative signals.

34 May and Zelikow (2002, 43–4).35 Slanchev (2005).

In each of these narratives, the decisions the Target ofa threat takes when it means to resist the deterring (orcompelling) state’s demands have a negative impacton the Deterrer’s utility. If the Target chooses eitherinternal or external balancing, the Deterrer’s utilitywill be negatively affected even if it chooses not to goto war. This is true when the increased capabilities ofthe Target increase the likelihood that the Target willdecide to go to war itself, and may also be true when theTarget will not contemplate war in the near term. In thecontext of Russo–German relations in 1876, Bismarckwas explicit on this point. Even a tacit German threat toRussia, he argued in one foreign policy circular, “couldinduce [the Tsar] to conclude flawed resolutions andalliances that would be very disadvantageous for bothsides.”36 A Franco-Russian alliance would have hadconsequences for Germany whether or not the Ger-mans backed down in 1876.37

Internal and external balancing on the part of a Tar-get negatively affects the Deterrer’s expected utilityfrom peace because it changes the balance of powerbetween them. If the two countries are involved in an-other crisis in the future, the weaker relative positionof the Deterrer will usually mean that it is less likelyto get its way and more likely to fare badly if conflictshould actually break out. Thus, if future crises areof the sort described here or, for example, in Fearon(1994, 1997, 1998), Schultz (1998, 2001) or Slanchev(2005), the Deterrer’s expected future utility decreasesin the capabilities of its adversary.38 Although theremay be some contexts in which states are indifferentto the increasing capabilities of an adversary, they willmore often view such developments with understand-able concern.

In the model described below, we allow for the pos-sibility that the Target of a threat, having made costlypreparations for war, might choose to begin one. Thisresults in a concrete risk to the coercing state of ap-pearing to menace the other state. Even if the coercingstate backs down, the other state may make prepara-tions for war and choose to fight one. More generally,however, we might think of the preparations of theTarget negatively affecting the coercing state’s utilitybecause of the effect of preparations on the balance ofpower, even when the coercing state backs down andthe sides are not in immediate conflict.

36 Grosse Politik, v. II, p. 37.37 Note that Russia elected not to attack or explicitly threatenAustria-Hungary and that Russia also did not pursue an alliancewith France at this time. The Franco-Russian alliance came about15 years later, after much intervening history, including Russia’sdeclared frustration with German policy in the Congress of Berlin,the Austro-German alliance of 1879 (directed partly, and only partlysecretly, against Russia), and the German failure to renew the Rein-surance Treaty with Russia in 1890.38 Kydd (2005) is a partial exception because increased relative ca-pabilities of an adversary may on occasion make it more trusting andthereby result in a net improvement of a threatening state’s expectedutility. We could also consider multistate interactions in which theincreased capabilities of an adversary are beneficial, but the reverseis of course the more usual case.

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COMMUNICATION IN ANARCHY

We now turn to a formal model of coercion in whichwe allow for the possibility that threatened states maydecide to take actions that go beyond standing firm orbacking down. In the model, there are two players, aDeterrer or threatening state and a Target or threat-ened state. The players are indexed by i ∈ I ≡ {d, t}.

The model below is different from previous modelsin the international relations literature in two impor-tant respects. First, the Target can chose to preparefor conflict when it believes it is sufficiently likely, orreorient its foreign policy in other ways that are hostileto the interests of the threatening state. If the Deterrertakes a threatening posture, and if the Target’s costsof war are sufficiently low relative to the importanceof the issue in question, the Target may be convincedto take these sorts of measures. Second, in addition tothe Deterrer having the option to attack the Target,the Target has the opportunity to attack the Deterrer.Having prepared for war, the Target may opt for iteven if the Deterrer does not. Because the Target’sactions may lead to an outcome that is even worsefor an unresolved Deterrer than the peaceful outcomewhere the Target is undeterred, a resolved Deterrer hasan incentive to make threats vis-a-vis a particular issuethat less resolved states would be unwilling to make. AsI show below, this dynamic causes threats to be mean-ingful even though no direct cost is associated withmaking them. One general implication of the model istherefore that if states respond at all to perceptions ofother states’ intentions in formulating foreign policy,then a verbal threat made by one state against another,whether in public or private, can convey information.

Following other models in the international relationsliterature, we shall suppose there is a bargaining spaceX ≡ [0, 1] such that the Deterrer prefers outcomescloser to 1 and the Target outcomes closer to 0. Thestatus quo at the beginning of the game is s ∈ X. Playershave von Neumann–Morgenstern utility functions de-fined over outcomes in the bargaining space, x ∈ X, andwhatever costs of fighting and preparation for conflictthe players pay, ci. Specifically;

ud(x) = x − cd

and

ut(x) = 1 − x − ct.

Thus, players have risk-neutral preferences over out-comes in the bargaining space. We assume this becauseit simplifies the exposition without substantively alter-ing the key points of the analysis. The costs playerspay in the game, cd and ct, will be defined below asfunctions of other variables to reflect the outcomes ofplayer actions during the game, so that, for instance,the players do not pay a cost of conflict when no waris fought. (They may still pay a cost of preparing forconflict, however.)

Figure 1 depicts the stages of the game. It beginswith the Deterrer’s attempt to influence the Target bysending a costless signal. Whatever message the Deter-rer sends, the game that follows is precisely the same;

FIGURE 1. Stages of the Game

First stage

Second stage

Third stage

Makes a costless threat or acquiesces

Decides (1) whether to take the action in question and (2) whether to prepare for conflict

Deterrer

Target

Deterrer and TargetEach decides whether or not to go to war

action sets, the sequence of moves, and player utilitiesare all unchanged. If the initial communication stageis important at all, therefore, it is only because theDeterrer’s message affects the Target’s beliefs aboutthe Deterrer’s intentions. This is possible in equilib-rium when the Deterrer is aware of the conclusions theTarget will draw from a particular messaging strategyof the Deterrer (and thus also aware of how theseconclusions will likely affect its actions), and yet theDeterrer’s best option is nevertheless to use that verymessaging strategy. We restrict attention to the Deter-rer’s decision to threaten or not, which we represent bythe message m ∈ M ≡ {0, 1}. We shall interpret m = 1as a promise to take violent action if the Target takesa particular action, as well as a promise not to takeviolent action in the event the Target cooperates.

In the second stage, the Target has two decisionsto make: whether to take the action in question or not(choosing a1 ∈ A1 ≡ {0, 1}), and whether to prepare forconflict (choosing a2 ∈ A2 ≡ {0, 1}). If the Target takesthe action, setting a1 = 1, it unilaterally moves the sta-tus quo to s − ε, toward its ideal point, where ε ∈ (0, s].If the sides remain at peace, this will then be the bar-gaining outcome. Then, in the third stage, the Deterrerand Target decide whether or not to go to war by choos-ing ri ∈ Ri ≡ {0, 1}, where 1 represents conflict initia-tion. First, the Deterrer chooses rd. If rd = 1, conflictoccurs; if not, the Target chooses rt. War occurs if eitherside opts to begin one; both sides must choose peaceto obtain that outcome. We let r represent an indicatorvariable that equals 1 when the sides go to war and 0otherwise. Thus, r = 1 if and only if ri = 1 for some i.

If the sides should fight a war, the Deterrer will winthe war with common knowledge probability p(a2),and is then able to choose its ideal outcome in thebargaining range X. Similarly, if the Target should winthe war, it may choose its ideal bargaining outcome.We assume that one of the two sides will win the war.When the Target prepares, the chances that it wins awar may increase: 1 − p(1) ≥ 1 − p(0) ⇔ p(0) ≥ p(1).

We shall model the Target’s costs of war and prepa-ration as consisting of several components. If the sidesgo to war, the Target’s war costs are ηt ∈ [η

t, ηt] ≡ �t,

where ηt> 0 and ηt is private information of the Target.

If the Target chooses to prepare, it incurs some prepa-ration costs, kt ≥ 0, whether or not a war is fought,

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but reduces its costs of conflict by βt ∈ [0, ηt). Thus,

preparations imply an increase in the sunk costs thatthe Target pays whether or not it goes to war and a de-crease in the variable costs associated with the conflictitself. (We make no assumption about the net effect ofpreparations on the overall cost of conflict.) Thus,

ct(a2, r, ηt) = kta2 + r(ηt − βta2).

We take a similar approach to modeling the Deter-rer’s costs, but we suppose the Deterrer has alreadymade any relevant preparations for conflict and do notmodel the Deterrer’s choice of preparations explicitly.For simplicity, we suppose the Target’s preparationsdo not affect the Deterrer’s costs of conflict—only itsprobability of victory. Thus,

cd(r, ηd) = rηd,

where ηd ∈ [ηd, ηd] ≡ �d, η

d> 0, and ηd is private in-

formation of the Deterrer. Both sources of private in-formation are independently distributed according tothe continuous, strictly increasing, common knowledgedistribution functions �ηi .

Thus, the Deterrer’s utility depends on the bargain-ing outcome, whether the players go to war, and theDeterrer’s type, whereas the Target’s utility dependson those same factors and also on whether or not theTarget chooses to prepare. Therefore, we shall write theplayers’ utility functions as ud(x, r, ηd) : X × R × �d →R and ut(x, r, ηt, a2) : X × R × �t × A2 → R.39 Substi-tuting the cost functions into the player utility functionsyields the following utilities for peace:

ud(s, 0, ηd) = s − εa1

ut(s, 0, ηt, a2) = 1 − s + εa1 − kta2.

Similarly, the players’ expected utilities for war are40

Eud(r = 1 | a2, ηd) = p(a2) − cd(1, ηd) = p(a2) − ηd

Eut(r = 1 | a2, ηt) = 1 − p(a2) − ct(a2, 1, ηt)

= 1 − p(a2) − ηt + βta2 − kta2.

The formal structure of the game is shown in Fig-ure 2. Whichever choice the Deterrer makes in the firststage, the Target has the four Stage 2 options shownin the figure. Whichever of these the Target opts for,the same Stage 3 structure shown follows. As the figureshows, the players’ utilities over the Stage 3 terminalnode outcomes depend on the Target’s Stage 2 choice.The full game tree, therefore, has 24 terminal nodes (2

39 Note that although the Target’s choice of preparation does notaffect the Deterrer’s utility directly, it does affect the likelihood ofoutcomes (victory and defeat) over which the Deterrer has differentpreferences.40 So long as player utility functions are bounded, we can de-rive these expected utilities without assuming risk neutrality overbargaining outcomes by setting ud(1, 0, ηd) = ut(0, 0, ηt, 0) = 1 andud(0, 0, ηd) = ut(1, 0, ηt, 0) = 0 without loss of generality. Becausethe sides’ war utilities do not depend on which side chooses war inthe third stage, the order of player choices in this stage will have nosubstantive impact on the analysis.

Deterrer options in the first stage × 4 Target optionsin the second stage × 3 terminal nodes in each Stage 3branch).

To make the signaling problem interesting, we shallmake several assumptions about payoffs. First, we as-sume the Deterrer prefers not to go to war when theTarget complies with its wishes by not taking the action,and that there is a possibility the Deterrer would bewilling to fight if the Target did not prepare and took theaction in question and a possibility the Deterrer wouldnot be willing to fight in this case. This is equivalent to

s > p(0) − cd(1, ηd) > s − ε > p(0) − cd(1, ηd). (1)

Second, we assume the Target prefers peace when ittakes the action in question and does not prepare toits most preferred war outcome and that there is apossibility the Target would prefer a prepared war toaccepting the initial status quo and a possibility theTarget prefers the initial status quo to a prepared war.This is equivalent to

1 − s + ε > 1 − p(1) − ct(1, 1, ηt) > 1 − s > 1 − p(1)

− ct(1, 1, ηt). (2)

Similarly to other models of coercion, these assump-tions imply that the Deterrer’s most preferred outcomeoccurs when the Target complies with the Deterrer’swishes by not taking the action in question (and also notattacking the Deterrer). The Target’s most preferredoutcome occurs when it takes the action in question,does not prepare for conflict, and the Deterrer doesnot attack. Players are willing to go to war when theyconsider the costs of war to be low relative to theirevaluation of the issues at stake and their chances ofvictory.41 An example of parameters satisfying theseassumptions is shown in Figure 3. The figure also repre-sents the model in terms similar to those used in Fearon(1995) and can usefully be compared to Figure 1 of thatpaper.

The preparations of the Target may or may not havean effect on the Target’s costs of conflict and the play-ers’ probability of victory. We shall say that prepara-tions are effective if and only if kt > 0, p(0) > p(1),and [(1 − p(1)) − (1 − p(0))] > kt − βt. The last con-dition states that the Target prefers a prepared to anunprepared war, or in other words, that the overall in-crease in the Target’s costs as a result of preparation (ifpreparing does involve a net increase in costs) must beoutweighed by the benefit of the increased likelihoodof victory. We shall say preparations are ineffective ifkt = βt = 0 and p(0) = p(1).42 Note that in Figure 3,preparations are effective.

41 Here, players are uncertain about each other’s costs of conflict.We might also consider a model in which players are uncertain aboutthe importance of the issue in question to the other side. A model ofthis form, which generates results similar to those presented below,can be found in Trager (2007, chap. 3).42 The effective/ineffective distinction does not exhaust all regions ofthe parameter space. We shall not analyze cases where preparationsimpact the probability of victory but are not costly or are costly butdo not affect the chances of victory.

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FIGURE 2. Formal Structure

1st Stage 2nd Stage

3rd Stage

Deterrer Target

(Threaten)

Deterrer

(Don’tThreaten)

Target

(Attack)

(Attack)(Don’tAttack)

(Don’tAttack)

FIGURE 3. Player Utilities Satisfying Model Assumptions

Deterrer’s peace utility if the Target takes the action

D t ’ tilit

Target’s peace utility if it takes the action and does not prepare

High cost Target’s expected utility from a

prepared conflict

T t’ tilit

Low cost Deterrer’s expected utility from conflict when the Target does not prepare

High cost Deterrer’s expected utility from conflict when the

Target does not prepareLow cost Target’s expected

utility from a prepared conflict

Deterrer’s peace utility at the status quo

Target’s peace utility at the status quo

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Equilibria with Ineffective Preparations

We can now ask whether, in such a strategic context,the Target can learn from the costless signals sent by theDeterrer. Commonly, in cheap talk games, signalers canconvey information that does not influence the courseof events. If communicated information will not influ-ence actions, it makes little difference to the receiverof the information whether or not he or she believes itto be true. In these equilibria, signals are informativebut not influential. We shall focus on whether equilib-ria with influential signals can exist. In such equilib-ria, there are signals that increase the probability thatthe Target both does not take the action in question(a1 = 0) and does not attack the Deterrer (rt = 0).Let qt(m) = Pr(a1 = 0, rt = 0 | m) be the probabilitythe Target’s strategy satisfies a1 = 0, rt = 0 given signalm induced by the players’ strategies and beliefs in aparticular equilibrium.

Definition. An equilibrium is influential if there existsignals m′ �= m′′ played with positive probability in theequilibrium such that qt(m′) > qt(m′′).

If the Target’s preparations do not affect its probabil-ity of victory or the players’ costs of conflict, the Deter-rer’s costless signals cannot convey information to theTarget in a way that influences the Target’s actions; noinfluential equilibrium exists. If preparations have noeffect on player utilities over outcomes, then this modelis very similar to others in the international relationsliterature. In such cases, the reason informative signal-ing is impossible is well understood: there is a benefitto being thought willing to fight and no drawback.43

Therefore, the Deterrer always tries to send the signalthat will most convince the Target of its resolve, butbecause that signal is sent in every case regardless ofwhether it corresponds to the truth of the matter, theTarget learns nothing from it. Proposition 1 expressesthis formally.

Proposition 1. If preparations are ineffective, no in-fluential perfect Bayesian equilibrium exists (in purestrategies).

Influential Equilibria

When preparations have an effect, in some cases in-fluential equilibria exist in the model. To understandhow these equilibria function intuitively, suppose theDeterrer makes a threat and the Target learns fromthis that the Deterrer is more likely willing to go to war

43 The logic here is only slightly more complicated because, unlikethe situation in most models in the literature, the Target also decideswhether or not to go to war later on. The proof of Proposition 1 mustalso demonstrate, therefore, that no influential equilibrium existsin which resolved, threatened Targets decline to take the action inquestion, thereby committing themselves to fighting a war. If sucha dynamic were possible, less resolved Deterrers might decline tothreaten, resulting in influential signaling. As the proof of Proposi-tion 1 in the Appendix demonstrates, however, this cannot occur.

over the issue in question than the Target had previ-ously thought. (We shall see that, in this equilibrium,this supposition is correct.) Realizing that the Deterreris more likely to follow through, the Target is morelikely to decide that the issue is not worth risking afight over and to back down. If the Target’s costs ofwar are sufficiently low relative to the value it placeson the issue, however, the Target may decide not toback down and to make costly preparations for war.The Target’s decision may be a difficult one because onthe one hand, preparations such as alliances, buildingarms, or striking first may provide additional security,but on the other hand, such actions carry their own ad-ditional costs. Because the Target’s preparations affectits own calculations about the relative benefits of warand peace, having prepared, it may decide to attack theDeterrer.

In the first stage, in deciding whether or not tothreaten, the Deterrer understands these dynamics. Ifthe Deterrer’s costs of war are high relative to the valueit places on the issue in question, then even though bythreatening it stands a better chance of getting its way,it will not be willing to make a threat. This benefit tothreatening does not outweigh the increased risk of abreach in relations with the Target and the attendantincreased risk of a costly conflict. Thus, the Deterrerwill only be willing to threaten if it is relatively highlyresolved. This, in turn, implies that our supposition thatthe Target learns from the threat is correct.44

A principal obstacle to influential cheap talk signalsis the incentive that unresolved types have to mimicthe signals sent by resolved types. In other models inthe literature, unresolved types have every incentiveto mimic because they can always back down and en-sure themselves of an outcome they like just as wellas the one in which they make no threat. In the modeldescribed here, however, there may be a danger thatthe Target will respond to a threat by preparing forwar and, having done so, launch a strike of its own.When this is possible, low-resolve types face a risk inmisrepresenting their levels of resolve by mimickingthe behavior of high-resolve types.45

44 As described, the logic may appear circular. But note that it iscircular only in the way that the logic of any truly strategic Nashequilibrium (where players’ optimal actions are dependent upon theactions of other players) must be circular.45 In the model, the Target can take an action that shifts the bar-gaining outcome in its favor by ε or not take that action. As a result,the Target sometimes prefers to go to war because the shift in thebargaining outcome is not sufficient to cause a fully prepared Targetto prefer peace. If it were feasible for the Target to take an actionunilaterally that moved the bargaining outcome sufficiently furthertoward the Target’s ideal point following preparations for conflict,however, and if we allowed for this possibility in the model, thenthe Target would never choose war in equilibrium and influentialsignaling would be impossible. The assumption of a fixed action ne-gotiated over by the states is probably reasonable in some cases andnot in others. Even if we were to change the model to allow for anincrease in ε following preparations, however, influential signalingwould still be possible if we relaxed the assumption of risk neutralityover bargaining outcomes. More generally, for alternate bargainingprotocols, influential signaling would be possible as long as Targettypes with higher expected values for war take actions that imply astrictly higher likelihood of conflict. This is an assumption or a result

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For this logic to operate, several conditions must bemet. On the one hand, Deterrers cannot have so muchto gain from threatening that even the least resolvedDeterrers would be willing to do it. If the Target isthought too likely to make concessions, for instance,the Deterrer will always find it preferable to try its luckwith a threat. Similarly, and as we saw above, if thepreparations of the Target are not effective enough,there will be no disincentive to making threats all thetime.

On the other hand, resolved Deterrers must haveenough to gain from threatening. If the Target isthought too likely to make preparations, if those prepa-rations have too large an effect on the balance of power,or if there is not a high enough likelihood that the Tar-get will concede the issue, then the Deterrer will preferto mislead the Target about its true level of resolve.States willing to go to war will see it in their interestto convince Targets that they are not, in order to catchthe latter unprepared. This too can make informativesignaling impossible.

Proposition 2 establishes one set of sufficient condi-tions for influential signaling. So long as preparationsare minimally effective and both sides are not too cer-tain about the resolve or irresolve of the other side, aninfluential perfect Bayesian equilibrium exists whenthe sunk costs of preparation, kt, are at least as greatas the value of the issue in contention, ε. The signifi-cance of this last condition is that it ensures that whenthe Target finds war an unattractive option, it prefersto comply with the Deterrer’s demand rather than tomake preparations in order to deter an attack from theDeterrer while declining to comply. If Targets preferrednoncompliance with preparations to compliance and ifthe impact of Target preparations on the probability ofvictory were sufficient to prevent Deterrers from everattacking, then Deterrers would have no incentive tosignal resolve. They would have nothing to gain fromdoing so, and an influential equilibrium would not exist.

Proposition 2. When preparations are effective, forsome set of beliefs �ηi , an influential perfect Bayesianequilibrium exists if kt ≥ ε.

As the discussion above makes clear, the sufficientconditions provided in Proposition 2 are not necessaryfor the existence of influential equilibria. In particular,if there is a sufficiently high likelihood that the Deter-rer would be willing to fight even against a preparedTarget, influential equilibria exist in some cases whenkt < ε. The reason is that the willingness of the Deterrerto fight ensures that the Target might sometimes beunwilling to risk war and thus would prefer to complywith the demand rather than preparing and declining tocomply. Thus, by demonstrating its resolve, the Deter-rer would have something to gain, enabling influentialsignaling.

of nearly all crisis bargaining models. See Banks (1990) in particular,which demonstrates in a fairly general setting that the probability ofwar is at least weakly increasing in the expected conflict utility of aninformed player.

To better understand how signaling operates in themodel, we now consider a particular parameterization.Let us suppose that if the sides fight a war, there isa p(0) = 50% chance that each side wins when theTarget has not made preparations. The status quo inthe issue space, s, favors the Deterrer at 0.6, but if theTarget takes the action in question, it unilaterally shiftsthe status quo by ε = 0.2 to 0.4. If the Target signs analliance with a third country, mobilizes its troops, andinitiates a dramatic armaments program, the probabil-ity that the Deterrer is victorious in war decreases top(1) = 30%. The Deterrer’s costs of war range fromsmall to quite large: ηd is uniformly distributed over[0.15, 0.75]. We shall suppose that the Target’s costs ofwar are very high when it does not prepare, but similaralthough somewhat less than the Deterrer’s costs whenthe Target does prepare: βt = 0.585, ηt is distributeduniformly over [0.6, 0.85], and the Target’s fixed costsof making preparations are kt = 0.2. Thus, if the Targetprepares, the additional costs it pays as a result of theconflict (ηt − βt) range from 0.015 to 0.265.

In such a world, there is an equilibrium in which theDeterrer sends influential, costless signals of its resolve.Before the Target observes the Deterrer’s choice tothreaten or not, it believes there is only a 14% chancethat the Deterrer would be willing to fight over theissue. But the Target also knows that if it observes athreat, the Deterrer is willing to take a risk of war.This is because the Target knows that the Deterrer be-lieves there is a 34% chance that the Target’s war costsonce it prepares (ηt − βt) are in the range [0.015, 0.1).If the Target’s costs of war are in this lower range,it prefers a prepared conflict to accepting the statusquo. Its optimal response to a threat from the Deterreris to make maximal preparations for war. There arefixed costs of kt = 0.2 associated with these prepara-tions that the Target pays whether or not the coun-tries go to war. Having paid this cost, the Target’sexpected utility from conflict is 1 − p(1) − kt + βt −ηt = 1 − 0.3 − 0.2 + 0.585 − ηt = 1.085 − ηt. For low-cost Targets (e.g., η

t= 0.6), this is greater than its

utility when conflict is avoided (even when it gets itsway with respect to the issue in question), which is1 − s + ε − kt = 0.4. Thus, if the Target is threatenedand if it is highly resolved (has relatively low costs ofwar), the Target will make preparations and go to warwith the Deterrer.

Optimal behavior by the Target therefore impliesthat threat-making by the Deterrer entails a risk ofconflict. For this reason, the Deterrer would only bewilling to make a threat when its privately known costsof conflict (ηd) were less than 0.42. Thus, the Deterrerthreatens only 46% of the time. Because the Deter-rer is willing to go to war against an undeterred andunprepared Target only when ηd < 0.235, the Targetknows for sure when the Deterrer declines to threatenthat it will also decline to initiate conflict if the Tar-get takes the action in question and does not prepare.On the other hand, if the Deterrer does make a threat,the Target will believe there is a 31% chance that theDeterrer would initiate conflict if the Target did notback down and did not prepare. This, in turn, causes the

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equilibrium to be influential: the probability that theTarget backs down is 0% when the Deterrer declinesto threaten, and 66% when the Deterrer does.

The reason the equilibrium is influential is that theTarget learns from the Deterrer’s costless statements.The Deterrer’s initial belief is that there is a 14%chance the Deterrer would be willing to go to war overthe issue. When the Deterrer threatens, that probabilitymore than doubles to 31%. When the Deterrer declinesto threaten, that probability falls to 0%.

The difference from other deterrence models wherethe Deterrer’s statements could not convey informa-tion is the possibility that the Target will take an actionthat negatively affects the Deterrer’s utility whether ornot the Deterrer later opts for war itself. In decidingwhether or not to threaten, therefore, the Deterrer con-siders the tradeoff described in the stylized examplesof the previous section. If it declines to threaten, itreveals itself as a low type, ensuring that the Target willnot concede the issue in question. Less resolved De-terrers, those that have a high relative cost of conflict,are nevertheless willing to make this choice becauseof the risk of unwanted conflict that threat-makingentails. On the other hand, the possibility the Targetwill back down without the need for conflict makes itworth it for resolved Deterrers to apprise Targets oftheir intentions by making a threat.

Noninfluential Equilibria

Even when preparations are effective, influential equi-libria may not exist. When Deterrers risk too littleby making threats, they will not be able to influenceTargets. When they risk too much, they will not bewilling to make threats at all. In either case, influentialsignaling is impossible.

Consider, for instance, Deterrers that are known tohave relatively low costs of conflict, when there is a lowupper bound on Deterrer cost types (ηd). In such cases,Deterrers risk relatively little by threatening. So longas there is a high enough likelihood that the Targethas relatively high costs of war and therefore will bewilling to back down, the Deterrer will always preferto make a threat. Thus, in such cases, signaling will beimpossible for the traditional reason: low-resolve types(high–cost of war types) prefer to mimic high-resolve(low-cost) types.

Conversely, if the Target is thought very likely to be arelatively low-cost type and thus likely to prefer war tocomplying with the Deterrer’s demand, signaling willagain be impossible, but for a different reason. Insteadof an incentive to misrepresent itself as resolved evenwhen it is not, the Deterrer would have an incentiveto misrepresent itself as unresolved even when it is.Rather than threaten in the hope the Target wouldimprobably be willing to comply, resolved Deterrerswould prefer to pretend to be unresolved in hopes ofcatching the Target unprepared. If the incentive of re-solved states to misrepresent themselves as unresolvedis strong enough, influential signaling will again be im-possible.

When resolved states have such incentives to mimicunresolved states, wars will occur that both sides wouldhave preferred to avoid—and that could have beenavoided if the Deterrer had sent a different signal. Weshall refer to these as “diplomatically avoidable wars.”The intuition for why such wars occur is simple. Con-sider a Deterrer willing to fight an unprepared Targetunless the Target makes a concession. If the Deterrermakes a threat, there is a chance that the Target willgive in, but there is also a chance that the Target willmake preparations, which means either that the De-terrer will itself be deterred from attacking or that theDeterrer will have to fight a prepared Target. There-fore, if the probability of Target resolve is too high,the Deterrer will decide not to warn the Target aboutits intentions. To illustrate these dynamics, we mightthink of the Israeli attack on the Osirak reactor in Iraqin 1981. If Israel had made a direct threat beforehandto bomb the reactor unless Iraq halted the reactor’sconstruction, Iraq would most likely have mobilized itsair defenses rather than comply. Israel would then havehad to acquiesce to the Iraqi nuclear program or fighta prepared opponent, in which case Israel would havebeen better off hiding its intentions in order to catchthe Iraqis unprepared.

Proposition 3 shows that in any material context,there always exist sets of beliefs of the two sides thatimply that diplomatically avoidable wars occur. As theproof in the Appendix makes clear, such wars occurwhen the Deterrer believes the Target very likely tobe highly resolved, whereas the Target believes theDeterrer relatively unlikely to be willing to fight. Insuch cases, the incentive of resolved Deterrers to mimicunresolved Deterrers completely eliminates the possi-bility of influential signaling. As we shall see below,however, surprise attacks and their correlate, diplo-matically avoidable wars, also occur in some influentialequilibria.

Definition. A diplomatically avoidable war is anequilibrium outcome in which (1) the players go to war(r = 1), (2) the Deterrer sends some signal m′, and (3)the players’ strategies imply that had the Deterrer senta signal m′′ �= m′, no war would have occurred (r = 0).

Proposition 3. When preparations are effective, forsome set of beliefs �ηi , a noninfluential perfect Bayesianequilibrium exists in which diplomatically avoidablewars occur.

Another noninfluential equilibrium also exists. Be-cause this is a cheap talk model, an uninformative,“babbling” equilibrium exists in all regions of the pa-rameter space. In this equilibrium, the Target does notinterpret the Deterrer’s signals as conveying informa-tion. Because all Deterrer types are therefore indif-ferent between signals, the condition that each typeis optimizing does not prevent it from sending signalsthat are uncorrelated with its costs of conflict. In sucha case, the Target cannot use Bayes’s rule to learn fromthe Deterrer’s signal.

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FIGURE 4. Equilibrium Signaling Properties

Det

erre

rco

sts

of c

onfl

ict

uppe

r bo

und

Target costs of conflict upper bound

0.7 0.75 0.8 0.85 0.9

0.25

0.35

0.45

0.55

0.65

0.75

0.85

0.95

Pooling:

No influential signals

Diplomatically avoidable wars

occur

Deterrer never threatens

Pooling:


Deterrer always threatens

Semiseparating:

Influential signals

Diplomatically avoidable wars

occur

Surprise attacks occur

Pooling:


Semiseparating:

Influential signals

Diplomatically avoidable wars do not occur

Surprise attacks do not occur

When influential equilibria exist, empirical analysisof cases stands the best chance of judging whether theinfluential or babbling equilibrium most closely tracksinternational reality.46 There are reasons to supposethe influential equilibrium is a more sensible socialequilibrium, however. When the parameters of themodel are such that an equilibrium exists in which allDeterrer types willing to go to war would be willingto threaten, for instance, then all Deterrer types thatwould be willing to go to war (when the Target did notprepare) would prefer to be in the informative ratherthan the uninformative equilibrium. If we thereforeassume such types will attempt to communicate theirresolve, then the uninformative equilibrium requiresthat the very least resolved types (those unwilling tothreaten in the informative equilibrium) choose to im-itate the behavior of the resolved types by threateningthemselves. This, in turn, requires that we presume that

46 It may be that process tracing of individual cases stands the besthope of evaluating whether influential or noninfluential equilibriaare better descriptions of state behavior. Documentary evidence willsometimes show very clearly, for instance, whether decision maker’sbeliefs changed after a threat was made.

these least resolved types attempt to send a signal thatthey would prefer not to have sent if the signal werebelieved. If this is unlikely, the informative equilibriummay be more reasonable than the uninformative one.47

Discussion of Equilibrium Dynamics

We can understand comparative statics in the model byconsidering the example of an influential equilibriumdiscussed above. The effect on signaling dynamics ofthe players’ beliefs about each other’s willingness tofight is shown in Figure 4. This figure and others thatfollow were generated through numerical simulationsof the model. Recall that in our example, the players’costs of conflict are drawn from uniform distributionsover [η

i, ηi]. We can model shifts in the players’ beliefs

about each other’s resolve by shifts in ηd and ηt. InFigure 4, ηd is on the vertical axis and ηt is on the hori-zontal axis. Thus, as we move rightward in the Figure,

47 For other arguments for the reasonableness of informative overuninformative equilibria, see Crawford and Sobel (1982, 1443) andChen, Kartik, and Sobel (2008).

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FIGURE 5. Effectiveness of the Signal and Likelihood of Surprise

0%

10%

20%

30%

40%

50%

0.7 0.74 0.78 0.82 0.86 0.9


Target’s evaluation of the likelihood of

deterrer resolve after threat

Target’s evaluation of the likelihood of a surprise attack

Target’s evaluation of the likelihood of deterrer resolve

before and after signalLik

elih

ood

Range where influential equilibria exist

the Deterrer is less convinced of the Target’s resolve,and as we move upward, the Target is less convincedof the Deterrer’s resolve (before a signal is sent).

When the Deterrer is too much or too little con-vinced of the Target’s resolve (very low or very high ηt),or when the Target is too convinced of the Deterrer’sresolve (low ηd), influential signaling is impossible. Inthe middle of the figure are the two ranges where in-fluential signaling is possible, but there are interestingdifferences in the dynamics of signaling and conflictbetween these two ranges. In both, the threats of theDeterrer convince the Target that the Deterrer is morelikely to be willing to fight over the issue in questionthan the Target had previously thought. In the right-most semiseparating equilibrium, however, the Targetknows for sure when the Deterrer declines to threatenthat the Deterrer will not fight over the issue. In theleftmost semiseparating equilibrium, this is not thecase. There, the Deterrer sometimes, but not always,misrepresents itself as less resolved than it actually is inorder to catch the Target unprepared.

Figure 5 provides us with a closer look at the signal-ing properties when ηd is held fixed at 0.75, as in theexample of the previous section. ηt is on the horizontalaxis, as in Figure 4. Within the middle range whereinfluential signaling is possible, the more convinced

the Deterrer is of the Target’s resolve (the lower ηt),the more informative is the Deterrer’s signal. Whenηt increases, the probability that the Target complieswith the Deterrer’s demand increases, which causes theprobability that the Deterrer is willing to threaten toincrease as well, and the signaling value of a threat todecline. Thus, when ηt = 0.775 a threat causes the Tar-get’s evaluation of the likelihood the Deterrer wouldfight to jump from 14% to 49%. When ηt = 0.875, incontrast, a threat causes the Target’s evaluation of De-terrer resolve to change only from 14% to 26%.

When ηt increases enough, the change in the Target’sbeliefs as a result of the Deterrer’s signal suddenly fallsdiscontinuously to zero as signaling becomes impossi-ble. This can be seen on the right-hand side of Figure 5.The reason is complex and relates to the interactionof a number of factors. As η̄t increases and the Targetcomes to believe that the Deterrer is less likely to followthrough on a threat (because the Target knows that theDeterrer knows that the Target is itself less likely tobe willing to fight over the issue and thus the Deterrerhas less to lose from making a threat), Target typesin the middle of the range of cost types have a newoptimal strategy. Even if the Target did find the De-terrer’s signal to be somewhat informative, rather thanlow Target cost types preparing and then fighting and

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high Target cost types backing down, Target cost typesin the middle of the ηt range would prefer to take theaction in question without preparing at all. They findthe chance that the Deterrer is willing to fight to be toolow to justify the expense of preparation, but neitherare they deterred from taking the action in question.As a result of the change in the Target’s strategy, theDeterrer would have both less to fear from threateningand less to gain. The former effect dominates the latter,causing the Deterrer to be even more willing to makethreats, which in turn would cause even more Targettypes to prefer taking the action in question withoutpreparation, and so on, with the result that influentialsignaling is impossible.

When the Deterrer believes it relatively likely thatthe Target will not back down over the issue, so that ηtis low, but still in the range where influential signalingpossible, surprise attacks (i.e., attacks not preceded bya threat) occur in equilibrium. When the Deterrer’scosts of conflict are very low, it prefers not to threatenthe Target at all in order to be able to attack theTarget when the Target is unprepared. In such cases,as we saw in our analysis of noninfluential equilibriaand diplomatically avoidable wars, conflict can occurin equilibrium because the Deterrer is unwilling torisk communicating its resolve. A signal exists that theDeterrer could send that would result in complianceand avoid the need for either side to go to war, butbecause the Deterrer does not know that the Targetwould comply with a demand if the Target knew thethreat were credible, the Deterrer declines to send thesignal.

In traditional crisis bargaining models, war can oftenoccur because the Deterrer does not have the meansavailable to communicate its resolve.48 As we sawabove, in diplomatically avoidable wars, the Deterrerhas the ability to communicate but chooses instead tocatch the Target unprepared.49 Instead of the weakDeterrer types mimicking the strong, in such cases, thestrong pretend to be weak because they believe theTarget is unlikely to back down even if it knew theDeterrer were in earnest.

Unlike the situation in the noninfluential equilib-rium of Proposition 3, however, in this example, diplo-matically avoidable wars and influential signalingcoexist in a single equilibrium. Deterrer’s adopt fourdifferent approaches, depending on their level of re-solve. The very most resolved Deterrers are unwillingto risk Target preparations and choose to use a surpriseattack. They will attack an unprepared Target withoutfirst attempting to coerce through a threat. The nextmost resolved Deterrers make threats that they willfollow through on if the Target does not acquiesce. De-terrer types that are slightly less resolved than these willmake threats they would later be unwilling to prosecute

48 See, in particular, Fearon (1995).49 Fearon (1995, 395–6) also mentions this incentive to conceal in-formation: “States can also have an incentive to conceal their capa-bilities or resolve if they are concerned that revelation would makethem militarily (and hence politically) vulnerable or would reducethe chances for a successful first strike.”

and, finally, the least resolved Deterrer types make nothreat at all and later choose not to attack.

Because the likelihood of a surprise attack increasesas ηt decreases, the lower ηt, the less the Target can learnfrom the Deterrer’s decision to decline to threaten.When ηt = 0.775, for instance, the Target’s evaluationof Deterrer resolve declines only slightly when the De-terrer declines to threaten: from 14% to 13%. Whenηt = 0.875, the Target’s evaluation of the likelihood ofDeterrer resolve falls from 14% to 0%. Thus, whenthere is a high probability that the Target is highlyresolved, the Target learns the most from a threat;but when there is a high probability that the Targetis unresolved, it learns the most when the Deterrerdeclines to threaten.

As can be seen on the left-hand side of Figure 5,at the extreme, when the Deterrer believes the Targetvery unlikely to back down in response to a threat,influential signaling is impossible because the incen-tive for a resolved Deterrer to attempt to surprise theTarget is too great. When the Deterrer is highly re-solved, it always prefers that the Target not know this.Thus, in this region, the Deterrer never threatens, in thesense that low-cost Deterrer types have an incentive toimitate high-cost types.50 Once again, diplomaticallyavoidable wars can occur in equilibrium.

The existence of a communication mechanism low-ers the probability of war in general. In the parameteri-zation discussed here, when none exists, the probabilityof war is 14% for all values of ηt shown in Figure 6.The reason is that when the Deterrer cannot affect theTarget’s beliefs, the Target’s optimal strategy for allvalues of ηt is to take the action in question withoutpreparing. Thus, war occurs if and only if the Deterrerprefers to fight one, given that the Target has taken theaction and not prepared.

When a communication mechanism exists, the prob-ability of war is reduced over parameter ranges whereinfluential signaling is possible, as Figure 6 illustrates.As we approach the left border of the range whereinfluential equilibria exist from the right, the proba-bility of war is arbitrarily close to the probability ofwar when no communication mechanism exists. Thisis so for two reasons. First, even though threats arevery informative, as shown on the left-hand side ofFigure 5, they are rarely used, as can be seen from theupward-sloping line in Figure 6. When the Deterreris resolved, it often prefers the gamble of a surpriseattack to the gamble of diplomatic signaling. Second,when ηt is relatively low, there is a low probability thatthe Target is willing to back down. As ηt increases, sothat the Target is more likely to be a high-cost type,even though threats are less informative, they becomemore used, surprise attacks decline, and the Target ismore likely to be willing to back down, with the re-sult that the probability of conflict decreases. When

50 In a cheap talk model, messages have no inherent meaning, so ifthere is an equilibrium in which all types coordinate on m = 0, thereis also an equilibrium in which all types send m = 1. Nevertheless,we say that the Deterrer “never threatens” in the left-hand rangeof Figures 4 and 5 because the resolved types have an incentive tomimic the unresolved types.

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FIGURE 6. Probability of War

0%

10%

20%

30%

40%

0.7 0.74 0.78 0.82 0.86 0.9


Probability of war without cheap talk

Probability of war with cheap talk

Lik

elih

ood

Probability that the deterrer makes

a threat

Range where influential equilibria exist

ηt = 0.875, for instance, the probability of war is 4%,less than one-third of what it would be in the absenceof a communication mechanism.

When ηt increases enough so that signaling becomesimpossible, however, the probability of war jumps dis-continuously back from 4% to 14% for the reason givenabove. If we were to consider parameterizations whereηt was much higher, increases in ηt would again causethe probability of war to decline because Targets forwhom war was extremely costly would either declineto take the action in question or make preparationssuch that the Deterrer would itself be deterred frominitiating a conflict. Thus, communication results ina nonmonotonic relationship between the probabilitythat the sides are willing to fight and the probability ofwar.

CONCLUSION

In international politics, the anarchic strategic contextis such that states must develop their foreign policiesunder conditions of uncertainty about the foreign poli-cies of others. The structure of the system leaves awide scope for agency in some cases, if not in all, anddifferent leaders react differently to the same foreignpolicy questions. States also tend to adopt more ad-

versarial policies against states they believe are morethreatening to their interests. States have a wide rangeof options in this regard, such as increasing arms pro-duction, forming hostile alliances, first strikes, drainingresources, and joining the opposing side in conflicts.Taken together, these aspects of the system imply thatcostless signals of resolve sent by adversaries can con-vey information. Diplomatic communication shapesstates’ perceptions of the threat they pose to each other.A second result of conceiving of the Target of a threat ashaving a wider array of responses is that conflict can oc-cur because resolved states do not communicate theirprivate information even when they could. In contrastto most other crisis bargaining models, the resolvedsometimes have incentive to imitate the behavior ofthe unresolved.

For the sake of clarity, the model presented in theprevious section considered a particular crisis in isola-tion, but some state decisions have far-reaching effects,and this has implications for the dynamics describedhere. A key aspect of the model is the possibility thatthreat-making can result in outcomes that are worse,from the point of view of the threatening state, thanallowing the Target to take the action in question.In the model of a particular crisis, this results fromthe possibility that a Target might elect to prepare for

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war, and having prepared, to fight one. More generally,however, the reaction of a Target may have a negativeeffect on the Deterrer’s utility even if no war is foughtor contemplated. If the Target builds arms or forms analliance, its increased capabilities may tip the balanceof power and result in diminished bargaining outcomesfor the Deterrer down the road. Thus, even in the eventof a peaceful outcome in the current crisis, the Target’sresponse to a threat may not be welcomed by the De-terrer, because the latter feels its security decreasedas the Target forms hostile alliances, mobilizes, buildsarms, or initiates other policies to drain the Deterrer’sresources.

Similarly, the incentive of a threatened Target to de-cline to make preparations when it declines to resistthe Deterrer in a particular crisis may not hold in amore general context. If the Target learns from a threatthat the Deterrer defines its interests as more opposedto the Target’s than the Target had previously thought,the Target may elect to form an alliance with a thirdstate that would be detrimental to the interests of theDeterrer, even when the Target has no plans to con-test the particular issue of the current crisis. In theseinternational contexts, the dynamics of signaling arelikely to be similar to those in our model of an isolatedinternational crisis; the same essential logic of costlessdiplomacy will apply.

Whether information can be conveyed in particularcases depends on the context of beliefs and strategicoptions in which costless statements are made. For in-stance, there must be a significant risk, but not a nearcertainty, of a breach in relations. Put differently, theremust be a possibility that a threatened state will reactagainst the threatener, not merely by refusing to coop-erate on the issue at hand, but with respect to otheraspects of the relationship. Up to a point, the greaterthe likelihood of a breach, and the more serious itsconsequences for the Deterring state, the greater thechange in the Target’s perceptions of the Deterrer’sintentions as a result of a threat. If the likelihood ofa breach is too great or its consequences too severe,however, highly resolved Deterrers may have incen-tive to deceive Targets by refraining from threateningbehavior while planning to attack, thereby catching theTarget unprepared. Because Targets are aware of De-terrers’ incentives, this dynamic can impair the abilityof the Deterrer to credibly signal peaceful intentions.

These considerations have several implications forthe study of interstate coercion. First, we should notalways expect that the quality of the Target’s options toprepare for and respond to hostilities will be inverselyrelated to the credibility of the Deterrer’s threats. Be-cause, in some contexts, the increased efficacy of Targetoptions will make states more reticent to threaten inthe first place, such signals will carry more weight whenthey are actually used. Second, as in models of publicsignaling and signaling based on reputation, beliefs areintersubjective. The statements and actions of otherstates are interpreted in light of what is believed aboutthese states, and also what these states are known tobelieve about one’s own state. If both sides understandthat the Deterrer believes the Target fairly likely to

back down, for instance, statements by the Deterrerwill have little marginal effect on the Target’s beliefsabout the Deterrer’s intentions. Third, the dynamics ofcommunication introduce unexpected nonmonotonici-ties into the relationship between the factors that affectthe likelihood that states are willing to fight over anissue and the probability of war. A decrease in theprobability that the Target of a threat is willing to fightcan increase the probability of war because of the effectof the Target’s perceived likely resolve on the possi-bility of communication. If the decrease in perceivedTarget resolve makes influential signaling impossible,as is sometimes the case, this will increase the likeli-hood of war. Fourth, sometimes resolved states have anincentive to hide their resolve, which results in surpriseattacks and wars that could have been avoided if stateshad just informed each other of their willingness tocontest the issues of the day.

The mechanism described here fits many cases. Inunderstanding the dynamics of threat-making, it is in-structive to study the fears of decision-makers contem-plating such action. Very often, they worry not aboutthe consequences for their reputations of being caughtin a bluff, but about the effect on their security positionsif their threats are believed. In a multipolar context,Bismarck’s tacit threat to Russia in 1876 during theGreat Eastern Crisis conveyed information because allsides understood the danger that a frustrated Russiawould form an alliance with France. In a bipolar con-text, Kennedy and Khrushchev’s threats to each otherover the status of Berlin in June of 1961 conveyed infor-mation because the two sides understood the dangerthat each would increase its efforts to harm the interestsof the other side, exacerbating the security dilemmadynamic between the two countries.51

The 1876 and 1961 cases are examples of privatediplomacy, but we should expect to find the mechanismdiscussed here operating alongside other mechanismsof information transmission in cases of public diplo-macy as well. When U.S. President G. W. Bush saysthat Iran is part of an “axis of evil,” for instance, thismay constitute a tacit threat to take actions against thecountry’s leadership.52 If Iran is unwilling to complywith U.S. demands, it may adopt policies designed todrain U.S. capabilities, for instance, by frustrating U.S.objectives in Iraq. By demonstrating a willingness torisk such a response, the President’s statement mayconvey information about U.S. resolve to force changesin Iranian policy. A similar dynamic is at work whenU.S. Defense Secretary Robert Gates says that unlessRussia acquiesces to Western demands in Georgia, “theU.S.–Russian relationship could be adversely affectedfor years to come.” If Russia does not comply andtherefore believes the U.S. to be more hostile to itsinterests, Russia will reorient its security posture inways that have negative consequences for both sides.Though such statements must be interpreted in the light

51 See Foreign Relations of the United States, 1961–1963 (1993), andJervis (2001).52 White House Press Release, January 29, 2002, “President DeliversState of the Union Address.”

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of a multitude of factors, including subsequent interac-tions between the two states, by increasing the risk ofthis negative outcome, these statements convey infor-mation. In both public and private contexts, therefore,the existence of costless communication mechanismsis likely to have important effects on conflict processesand the likelihoods of war and peace.

Capabilities alone do not determine outcomes, com-munication between states is not purely epiphenome-nal on clashes of power and interest, and it is not thecase, as even the diplomatic historian A. J. P. Taylorhas written, that, “wars make the decisions; diplomacymerely records them.”53 There is wide scope for hu-man agency and diplomats in particular in shapingthe form that clashing interests will take. Diplomaticconversations, even those between adversaries, are anintegral part of the processes that construct perceptionsof intention and determine the course of events in theinternational system.

APPENDIX

A strategy for the Deterrer is a pair (m(ηd), rd(ηd, h)), whereh ∈ H ≡ M × A1 × A2. A strategy for the Target is a triple(a1(ηt, m), a2(ηt, m), rt(ηt, h)). Let the updated beliefs of theTarget following the Deterrer’s Stage 1 signal be µηd(ηd | m).Let p(0) = p(1) = p if p(0) = p(1). For some particular PBE,let qd(a2, m) = Pr(rd = 0 | a1 = 1, a2, m), which is induced by�ηd and the Deterrer’s strategy, and let qt(m) be the probabil-ity that the Target’s strategy satisfies a1 = 0, rt = 0 for somesignal m (which can be written Pr(a1 = 0 | m)Pr(rt = 0 | a1 =0, m)) induced by �ηt and the Target’s strategy.

The following two lemmas are used in the proof of Proposi-tion 1. Without loss of generality, we suppress the a2 notationin the proofs of the following lemmas and Proposition 1 sothat the Target’s strategy is a pair, (a1(ηt, m), rt(ηt, h)), andqd(a2, m) is instead written qd(m).

LEMMA 1. In an influential PBE with ineffectivepreparations, if qt(m′) > qt(m′′), m(ηd) = m′ ∀ηd suchthat p − ηd > s − ε.

Proof. Suppose p − ηd > s − ε. The Deterrer can set rd = 1and obtain its highest expected utility in any subgame follow-ing a1 = 1: p − ηd. Therefore, any equilibrium strategies mustresult in an outcome with this Deterrer utility in any subgamefollowing a1 = 1. By assumption, s > p − ηd ∀ηd, so optimal-ity of the Deterrer’s Stage 3 choice implies rd(ηd, h) = 0 forall h such that a1 = 0 when qt(m) > 0 (which implies Pr(rt =0 | a1 = 0, m) > 0). Therefore, ∀ηd < p − s + ε, we can writethe Deterrer’s expected utility in an equilibrium as a functionof its signal: Eud(m) = qt(m)s + (1 − qt(m))(p − ηd), which isincreasing in qt(m). �LEMMA 2. In any influential PBE with ineffective prepara-tions, qd(m) > 0 ∀m.

Proof. We shall use a proof by contradiction: supposeqd(m̃) = 0 for m̃ ∈ M. Note that in an influential equilibrium,both signals m = 0 and m = 1 must be sent with positiveprobability.

First, observe that

qt(m̃) = 1 − �ηt (s − p) > 0. (3)

53 Taylor (1954, 246).

This follows because in any PBE, the expected Targetutilities in a subgame following m̃ are as follows:Eut(a1 = 1, rt = 0 | m̃) = Eut(a1 = 1, rt = 1 | m̃) = Eut(a1 =0, rt = 1 | m̃) = 1 − p − ηt, Eut(a1 = 0, rt = 0 | m̃) = 1 − s.(The last expected utility uses that, in an influ-ential PBE, rd(ηd, h) = 0 ∀h such that a1 = 0. IfPr(rt = 0 | a1 = 0, m = m̃) > 0, this follows directly. IfPr(rt = 0 | a1 = 0, m = m̃) = 0, then qt(m̃) = 0. For theequilibrium to be influential, we need qt(m̌) > 0 for m̌ �= m̃.But then m(ηd) = m̌ ∀ηd < p − s + ε by Lemma 1, whichimplies either that qd(m̃) �= 0, which contradicts our originalassumption, or that m̃ is not sent with positive probabilityin which case the equilibrium is not influential.) qt(m̃) istherefore the probability that 1 − s > 1 − p − ηt.

Second, observe that in equilibrium,

qd(m̌) = 1 for m̌ �= m̃ (4)

and

qt(m̌) = 1 − �ηt

(s − p + ε

qd(m̌)1 − qd(m̌)

)< qt(m̃). (5)

To see this, note that in equilibrium,

rt(ηt, h) = 0 ∀ηt, h such that a1 = 1. (6)

Then, let Pr(m̃) be the probability the Deterrer sendssignal m̃ induced by the Deterrer’s strategy, so that, ina PBE, qd(m̌) = 1−�ηd (p−s+ε)

1−Pr(m̃) > 0. The Target’s expectedutilities over strategies in the subgame following m̌are the same as its expected utilities following m̃, ex-cept that Eut(a1 = 1, rt = 0 | m̌) = qd(m̌)(1 − s + ε) + (1 −qd(m̌))(1 − p − ηt) > Eut(a1 = 0, rt = 1 | m̌). Thus, in an in-fluential PBE, the Target’s strategy in the subgame fol-lowing signal m̌ must satisfy the following three condi-tions: (1) a1(ηt, m̌) = 1 ∀ηt < s − p + ε

qd(m̌)1−qd(m̌) , (2) a1(ηt, m̌) =

0 ∀ηt > s − p + εqd(m̌)

1−qd(m̌) , and (3) rt(ηt, h) = 0 ∀ηt > s − p +ε

qd(m̌)1−qd(m̌) , h such that m = m̌. This implies that qt(m̌) = 1 −

�ηt (s − p + εqd(m̌)

1−qd(m̌) ) < qt(m̃). Using Lemma 1, this impliesthat

m(ηd) = m̃ ∀ηd such that p − ηd > s − ε. (7)

Using Bayes’ rule, this in turn implies (4).Third, I show that

m(ηd) = m̃ ∀ηd < p − s + ε

1 − qt(m̃). (8)

Given (4), the Target’s optimal strategy in the Stage 2 sub-game must satisfy a1(ηt, m̌) = 1 ∀ηt. Using (6), this impliesthat for ηd such that p − ηd < s − ε, Eud(m̌ | ηd) = s − ε andEud(m̃ | ηd) ≥ qt(m̃)s + (1 − qt(m̃))(p − ηd). Using (7), thisimplies (8).

Finally, observe that because qt(m̃) > 0, for h such thatm = m̃,

rd(ηd, h) = 0 ∀ηd > p − s + ε. (9)

Thus, since �ηd is strictly increasing and qt(m̃) > 0, in a PBE,Bayes’ rule, (8), and (9) do not imply qd(m̃) = 0, which con-tradicts our original assumption. �Proof of Proposition 1. Suppose qt(m′) > qt(m′′).By Lemma 1, in an influential equilibrium, m(ηd) =m′ ∀ηd < p − s + ε. We have already observed that ina PBE rt(ηt, h) = 0 ∀ηt, h such that a1 = 1. Then, usingLemma 2, the Target’s expected utilities in the Stage2 subgame for strategies and beliefs consistent with aPBE are Eut(a1 = 0, rt = 1 | m) = 1 − p − ηt < Eut(a1 = 1,

364


rt = 0 | m) = qd(m)(1 − s + ε) + (1 − qd(m))(1 − p − ηt). Thus,in any PBE, if rt(ηt, h) = 1 for h such that a1 = 0, thena1(ηt, m) = 1 ∀ηt, m. This implies that in a PBE, for ηd suchthat p − ηd ≤ s − ε,

Eud(m) = qt(m)s + (1 − qt(m))(s − ε).

Because this is increasing in qt(m), in any PBE, m(ηd) =m′ ∀ηd such that p − ηd ≤ s − ε. Since m′′ is not sent withpositive probability in equilibrium, no PBE can beinfluential. �Proof of Proposition 2. Let the Deterrer’s strategy bem(ηd) = 1 ∀ηd < η̂d and m(ηd) = 0 ∀ηd ≥ η̂d; rd(ηd, h) = 1 iffηd < p(a2) − s + ε & h such that a1 = 1. Let the Stage 3component of the Target’s strategy be as follows: forh such that a1 = 1 & a2 = 1, rt(ηt, h) = 1 iff ηt < s − p(1) −ε + βt ≡ η̌t; for h such that a1 = 1 & a2 = 0, rt(ηt, h) = 0 ∀ηt;for h such that a1 = 0, rt(ηt, h) = 1 iff 1 − p(a2) − ηt + βta2 >1 − s. Let the Stage 2 component of the Target’s strat-egy be a1(ηt, 0) = 1, a2(ηt, 0) = 0 ∀ηt; a1(ηt, 1) = 1, a2(ηt, 1) =1 ∀ηt < η̂t; a1(ηt, 1) = 0, a2(ηt, 1) = 0 ∀ηt ≥ η̂t where η̂t ≡ s −p(1) − kt + βt. Let the Target’s updated beliefs be µηd(ηd |1) = �ηd (ηd)

�ηd (η̂d) ∀ηd ∈ [ηd, η̂d) and 0 otherwise, and µηd(ηd | 0) =

�ηd (ηd)−�ηd (η̂d)1−�ηd (η̂d) ∀ηd ∈ [η̂d, ηd] and 0 otherwise.An equilibrium with these strategies and Target beliefs is

influential when η̂i ∈ (ηi, ηi) ∀i. Note that the updated beliefs

of the Deterrer must also follow from Bayes’s rule in theequilibrium, but that the Deterrer’s optimal actions in thethird stage do not depend on these beliefs. The Target’s beliefsabove follow directly from Bayes’ rule, given the Deterrer’sstrategy. I now show that for some �ηd and �ηt , if kt ≥ ε, theabove strategies are optimal and that η̂i ∈ (η

i, ηi) ∀i.

The optimality of the Stage 3 component of the Target’sstrategy follows directly from the Target’s preferences overoutcomes. Given the Target’s strategy, the optimality of theStage 3 component of the Deterrer’s strategy follows by back-wards induction.

We turn now to the optimality of the Deterrer’s Stage 1choice. Given the Stage 2 and 3 components of the players’strategies, because η̂t ≤ η̌t because kt ≥ ε, the Deterrer’s ex-pected utility from threatening is

Eud(m = 1 | ηd) = �ηt (η̂t)(p(1) − ηd) + (1 − �ηt (η̂t))s, (10)

which is decreasing in ηd because our assumptions implythat η̂t ∈ (η

t, ηt). Given the Stage 2 and 3 components of the

players’ strategies, the Deterrer’s expected utility from notthreatening is

Eud(m = 0 | ηd) ={

p(0) − ηd ηd < η̃d

s − ε ηd ≥ η̃d,

where η̃d = p(0) − s + ε > ηd.

Type ηd = ηd

prefers to threaten when

�ηt (η̂t) ≤s − p(0) + η

d

s − p(1) + ηd

≡ �t ∈ (0, 1). (11)

We can choose �ηt such that condition (11) holds. For in-stance, let �ηt be such that �ηt (η̂t) = �t. Then

Eud(m = 0 | ηd = ηd) = Eud(m = 1 | ηd = η

d). (12)

Note also that

∂Eud(m = 0 | ηd < η̃d)∂ηd

<∂Eud(m = 1 | ηd < η̃d)

∂ηd. (13)

By (12) and (13), note that if Eud(m = 0 | η̂d) = Eud(m = 1 |η̂d) for η̂d �= η

d, then η̂d > η̃d, and in particular,

η̂d = p(1) − s + ε/�ηt (η̂t).

Below, we shall choose ηd such that ηd > η̂d. With this speci-fication of η̂d and �ηt , and the expected utilities given above,we can easily check that the Deterrer’s Stage 1 strategy isoptimal given the players’ beliefs and the Stage 2 and 3 com-ponents of the players’ strategies. Using (12) and (13) again,we see that Eud(m = 1 | ηd) ≥ Eud(m = 0 | ηd) ∀ηd < η̂d. Be-cause ∂Eud(m= 0|ηd ≥ η̃d)

∂ηd= 0 and �ηt (η̂t) > 0, ∂Eud(m= 1|ηd)

∂ηd< 0 and

Eud(m = 0 | ηd) ≥ Eud(m = 1 | ηd) ∀ηd > η̂d. Thus, using theone-stage deviation property, the Deterrer’s strategy is opti-mal given the Target’s strategy and the players’ beliefs.

We turn now to the optimality of the Target’s strategyin Stage 2 subgames. Given the Deterrer’s strategy and ourdefinition of η̂t, the Target’s equilibrium strategy in Stage 3,and the Target’s beliefs, the Stage 2 component of the Target’sstrategy at the m = 0 node (a1(ηt, 0) = 1, a2(ηt, 0) = 0) ∀ηt,gives the Target its best outcome (1 − s + ε) with certainty,so the Target cannot gain by deviating.

To complete the proof, it remains only to show that the Tar-get’s strategy is optimal in m = 1 subgames. Recall that η̌t ≡s − p(1) + βt − ε, and let η̇t ≡ s − p(1) + βt and η̈t ≡ s − p(0).Then, given the Stage 3 component of the players’ strategiesand the Target’s updated beliefs at m = 1 nodes, we can writethe Target’s expected utilities over its four actions at Stage 2nodes as follows:

Eut(a1 = 1, a2 = 1 | m = 1, ηt)

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

1 − p(1) − ηt − kt + βt ηt < η̌t

�ηd (p(1) + ε − s)�ηd (η̂d) (1 − p(1) − ηt − kt + βt)

+(1 − �ηd (p(1) + ε − s)�ηd (η̂d) )(1 − s + ε − kt) ηt ≥ η̌t

(note that p(1) + ε − s ≤ η̂d);

Eut(a1 = 1, a2 = 0 | m = 1, ηt)

= min(

1,�ηd(p(0) + ε − s)

�ηd(η̂d)

)(1 − p(0) − ηt)

+(

1 − min(

1,�ηd(p(0) + ε − s)

�ηd(η̂d)

))(1 − s + ε)

(note that�ηd (p(0) + ε − s)

�ηd (η̂d) > 0 because s − ε < p(0) − ηd

by as-

sumption);

Eut(a1 = 0, a2 = 1 | m = 1, ηt)

={

1 − p(1) − ηt − kt + βt ηt < η̇t

1 − s − kt ηt ≥ η̇t

;

Eut(a1 = 0, a2 = 0 | m = 1, ηt)

={

1 − p(0) − ηt ηt < η̈t

1 − s ηt ≥ η̈t

.

Because kt ≥ ε and [(1 − p(1)) − (1 − p(0))] > kt − βt,η̇t > η̌t ≥ η̂t > η̈t. We first show that Target types ηt < η̂t

have no incentive deviate from the proposed equilibrium.For ηt < η̂t, the Target has no incentive to deviate to(a1(ηt, 1) = 0, a2(ηt, 1) = 1) because Eut(a1 = 0, a2 = 1 |m = 1, ηt) = Eut(a1 = 1, a2 = 1 | m = 1, ηt) ∀ηt < η̂t. Since

365


Eut(a1 = 1, a2 = 1 | m = 1, ηt) > Eut(a1 = 0, a2 = 0 | m =1, ηt) ∀ηt < η̈t and Eut(a1 = 1, a2 = 1 | m = 1, ηt) ≥ Eut(a1 =0, a2 = 0 | m = 1, ηt) ∀ηt ∈ [η̈t, η̂t], Target types ηt < η̂t haveno incentive to deviate to (a1(ηt, 1) = 0, a2(ηt, 1) = 0).Further, Eut(a1 = 1, a2 = 1 | m = 1, ηt) ≥ Eut(a1 = 1, a2 =0 | m = 1, ηt) ∀ηt ∈ [η

t, η̂t) when

�ηd(p(0) + ε − s)�ηd(η̂d)

≥ −s + p(1) + ε + ηt + kt − βt

−s + p(0) + ε + ηt.

Because the RHS is increasing in ηt, let

�d ≡ −s + p(1) + ε + ηt + kt − βt

−s + p(0) + ε + ηt,

so that, taking the other components of the players’ strategiesas given, Eut(a1 = 1, a2 = 1 | m = 1, ηt) > Eut(a1 = 0, a2 =1 | m = 1, ηt) ∀ηt ∈ (η

t, η̂t) if

�ηd (p(0)+ε−s)�ηd (η̂d) > �d. Below, we shall

choose �ηd so that this is true. (Note that our assumptions,in particular (2) and (1 − p(1)) − (1 − p(0)) > kt − βt, implythat �d ∈ (0, 1).)

We now show that all Target types ηt ≥ η̂t preferthe action prescribed by the Target’s equilibriumstrategy in a Stage 2 subgame following m = 1 (againtaking other elements of the equilibrium as given).Eut(a1 = 0, a2 = 0 | m = 1, ηt = η̂t) ≥ Eut(a1 = 1, a2 = 0 |m= 1, ηt = η̂t) ⇔ �ηd (p(0)+ε−s)

�ηd (η̂d) ≥ ε

p(0)−p(1)+βt−kt+ε≡ �̈d ∈ (0, 1).

Because ∂Eut(a1 = 1,a2 = 0|m= 1,ηt)∂ηt

<∂Eut(a1 = 0,a2 = 0|m= 1,ηt)

∂ηt∀ηt > η̂t,

if�ηd (p(0)+ε−s)

�ηd (η̂d) ≥ �̈d, Eut(a1 = 0, a2 = 0 | m= 1, ηt) ≥ EUt(a1 =1, a2 = 0 | m= 1, ηt) ∀ηt ≥ η̂t. Let �̂d = max(�d, �̈d).

We can now choose �ηd such that ηd > η̂d and�ηd (p(0) + ε − s)

�ηd (η̂d) = �̂d.54 Given such a �ηd , Eut(a1 = 1, a2 = 1 |m = 1, ηt) is at least as high as the expected utility ofany other Stage 2 action ∀ηt < η̂t. Further, Eut(a1 = 0, a2 =0 | m = 1, ηt) ≥ Eut(a1 = 1, a2 = 0 | m = 1, ηt) ∀ηt > η̂t since�ηd (p(0)+ε−s)

�ηd (η̂d) ≥ �̈d.Because η̈t < η̂t ≤ η̌t < η̇t, and using the definition of η̂t and

the fact that Eut(a1 = 1, a2 = 1 | m = 1, ηt) is decreasing inηt, Eut(a1 = 0, a2 = 0 | m = 1, ηt) ≥ Eut(a1 = 0, a2 = 1 | m =1, ηt) ∀ηt ≥ η̂t and Eut(a1 = 0, a2 = 0 | m = 1, ηt) ≥ Eut(a1 =1, a2 = 1 | m = 1, ηt) ∀ηt ≥ η̂t. Thus, no Target type can gainby deviating from its equilibrium strategy only in Stage 2.Again using the one stage deviation property, this ensuresthat the Target’s equilibrium strategy is optimal in the Stage2 subgame, given our specification of �ηd . �Proof of Proposition 3. We shall first show that thefollowing strategies and beliefs constitute a PBE. Letthe Deterrer’s strategy be m(ηd) = 0 ∀ηd, rd(ηd, h) = 1 iffηd < p(a2) − s + ε & h such that a1 = 1, and let the Stage3 component of the Target’s strategy be as follows:for h such that a1 = 1, rt(ηt, h) = 0 iff 1 − s + ε − kta2 ≥ 1 −p(a2) − ηt − kta2 + βta2; for h such that a1 = 0, rt(ηt, h) = 1iff 1 − p(a2) − ηt + βta2 > 1 − s. The Stage 3 components ofthe players’ strategies follow directly from backward in-duction. Let the Stage 2 component of the Target’s strat-egy be a1(ηt, 0) = 1, a2(ηt, 0) = 0 ∀ηt; a1(ηt, 1) = 1, a2(ηt, 1) =

54 Note that such a choice of distribution is sufficient but not nec-essary for the proposed equilibrium to exist. We need only that (1)there is some possibility that the Deterrer has high enough costs sothat it would not be willing to make a threat in a particular context,and (2) this probability that the Deterrer has high costs is not sogreat that the Target never takes the Deterrer’s threats seriously andthus never makes preparations for conflict.

1 ∀ηt ≤ η̃t; a1(ηt, 1) = 0, a2(ηt, 1) = 0 ∀ηt > η̃t. Because theDeterrer pools on m = 0, the Target does not update itsbeliefs following this signal. Off the equilibrium path, let theTarget’s beliefs be µ(ηd | 1) = 1 (and note that these beliefsare unconstrained by Bayes’s rule in a PBE).

We now verify that the Stage 2 component of the Tar-get’s strategy is optimal. Given the other components of theplayers’ strategies and beliefs, the Target’s expected utilitiesfrom actions in the Stage 2 subgame following m = 1 are asfollows:

Eut(a1 = 1, a2 = 0 | m = 1, ηt) = 1 − p(0) − ηt,

Eut(a1 = 0, a2 = 1 | m = 1, ηt)

={

1 − s − kt ηt ≥ s − p(1) + βt

1 − p(1) − ηt − kt + βt ηt < s − p(1) + βt,

Eut(a1 = 0, a2 = 0 | m = 1, ηt)

={

1 − s ηt ≥ s − p(0)

1 − p(0) − ηt ηt < s − p(0),

and

Eut(a1 = 1, a2 = 1 | m = 1, ηt)

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

1 − p(1) − ηt − kt + βt ηt < s − p(1) − ε + βt

1 − p(1) − ηt − kt + βt ηt ≥ s − p(1) − ε + βt

& p(1) − ηd

> s − ε

1 − s + ε − kt ηt ≥ s − p(1) − ε + βt

& p(1) − ηd

≤ s − ε.

(14)

Let case (a) be where kt > ε or p(1) − ηd

> s − ε, and letcase (b) be where kt ≤ ε and p(1) − η

d≤ s − ε. Let η̃t = s −

p(1) − kt + βt in case (a), and η̃t = ηt in case (b). By comparingthe expected utilities for each range of Target types, we canthen easily verify that the Stage 2 component of the Target’sstrategy in the subgame following m = 1 is optimal given theother components of the equilibrium.

Given the players’ strategies and the Target’s beliefs, theTarget’s expected utilities from actions in the Stage 2 sub-game following m = 0 are as follows:

Eut(a1 = 1, a2 = 0 | m = 0, ηt) = qd(0, 0)(1 − s + ε)

+ (1 − qd(0, 0))(1 − p(0) − ηt),

Eut(a1 = 0, a2 = 1 | m = 0, ηt)

={

1 − p(1) − ηt − kt + βt ηt < s − p(1) + βt

1 − s − kt ηt ≥ s − p(1) + βt,

Eut(a1 = 0, a2 = 0 | m = 0, ηt)

={

1 − p(0) − ηt ηt < s − p(0)

1 − s ηt ≥ s − p(0),

366


and

Eut(a1 = 1, a2 = 1 | m = 0, ηt)

=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

1 − p(1) − ηt − kt + βt ηt < s − p(1) − ε + βt

qd(1, 0)(1 − s + ε − kt)

+ (1 − qd(1, 0))(1 − p(1)

− ηt − kt + βt) ηt ≥ s − p(1) − ε + βt.

(15)

Because 1 − s + ε is the Target’s most preferred out-come, we can choose �ηd so that qd(0, 0) is high, whichimplies that Eut(a1 = 1, a2 = 0 | m = 0, ηd) is higher thanthe expected utility of any other Stage 2 option for all ηd

in the subgame following m = 0. To see this explicitly, leta = max(1 − p(1) − η

t− kt + βt, 1 − s + ε − kt), which is at

least as high as the highest utility that any Target type canachieve from a Stage 2 deviation. We can then calculate theqd(0, 0) required to make Eut(a1 = 1, a2 = 0 | m = 0, ηt) ≥ afor all ηt:

qd(0, 0) ≥ a − 1 + p(0) + ηt

−s + ε + p(0) + ηt≡ q̂d(0, 0) < 1. (16)

We can see that q̂d(0, 0) < 1 by comparing the numeratorand denominator of q̂d(0, 0) for all possible values of a. Thus,we can let �ηd be such that �ηd(p(0) − s + ε) = 1 − q̂d(0, 0),which implies that the Target has no incentive to deviate inthe m = 0 subgame given the Target does not deviate fromits equilibrium strategy in its Stage 3 subgame. Thus, by theone-shot deviation property, the Target’s strategy is optimal.

We now turn to the optimality of the Deterrer’s strategyin the Stage 1 subgame. The Target’s strategy implies thatfor h such that (a1 = 0, a2 = 0), we must have rt(ηt, h) = 0∀ηt ≥ s − p(0). Let η′

d ≡ p(1) − s + ε and η′′d ≡ p(0) − s + ε.

Then, because η̃t > s − p(0), given the other components ofthe players’ strategies, we have

Eud(m = 1 | ηd)

=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

(1 − �ηt (η̃t))s + �ηt (η̃t)(p(1) − ηd) kt > ε

(1 − �ηt (η̃t))s + �ηt (η̃t)(p(1) − ηd) kt ≤ ε and ηd < η′d

(1 − �ηt (η̃t))s + (�ηt (η̃t)− �ηt (s − p(1) − ε + βt))(s − ε)+ �ηt (s − p(1) − ε + βt)(p(1) − ηd) kt ≤ ε and ηd ≥ η′

d,

(17)

and

Eud(m = 0 | ηd) ={

s − ε ηd ≥ η′′d

p(0) − ηd ηd < η′′d.

(18)

Because η′′d > η′

d, Eud(m = 0 | ηd < η′d) ≥ Eud(m = 1 | ηd <

η′d) when

�ηt (η̃t) ≥ s − p(0) + ηd

s − p(1) + ηd.

Because the RHS is increasing in ηd, let γ ≡ s − p(0) + ηds − p(1) + ηd

so that ∀ηd < η′d Eud(m = 0 | ηd) ≥ Eud(m = 1 | ηd)

if �ηt (η̃t) ≥ γ. Eud(m = 1 | η′d ≤ ηd) is at most

(1 − �ηt (η̃t))s + (�ηt (η̃t) − �ηt (s − p(1) − ε + βt))(s − ε) +�ηt (s − p(1) − ε + βt)(p(1) − ηd) < (1 − �ηt (s − p(1) − ε +βt))s + �ηt (s − p(1) − ε + βt)(p(1) − ηd). Setting this lessthan Eud(m = 0 | η′

d ≤ ηd < η′′d) = p(0) − ηd yields

�ηt (s − p(1) − ε + βt) ≥ s − p(0) + ηd

s − p(1) + ηd,

which must hold when �ηt (s − p(1) − ε + βt) ≥ γ. By sim-ilar reasoning, Eud(m = 1 | ηd ≥ η′′

d) ≤ Eud(m = 0 | ηd ≥ η′′d)

when�ηt (s − p(1) − ε + βt) ≥ ε

s − p(1) + ηd.

Because the RHS is decreasing in ηd, let γ′ ≡ ε

s−p(1)+η′′d

∈ (0, 1).

Thus, we can choose �ηt to be such that �ηt (min(η̃t, s − p(1) −ε + βt)) ≥ max(γ, γ′), which implies that, fixing the othercomponents of the equilibrium, the Deterrer has no incentiveto deviate from its equilibrium strategy in Stage 1. Thus, thestrategies and beliefs given above constitute a PBE for the�ηi specified.

Finally, note that (1) this equilibrium is not influen-tial because only one signal is sent with positive proba-bility, and (2) there is a strictly positive probability thatηd ∈ [ηd, p(0) − s + ε) and that in case (a) ηt ∈ (η̃t, ηt] or in case(b) ηt ∈ [η

t, ηt]. In either case (a) or case (b), if ηd and ηt are

in the ranges specified here, r = 1, m = 0, and if the Deterrerwere to deviate to m = 1, no war would occur. �

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Documents

Diplomatic Calculus in Anarchy: How Communication Mattersan essential aspect of interstate diplomacy. The anar-chic strategic context can provide less resolved states with a disincentive