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When Is an Arms Rivalry a Prisoner's Dilemma? Richardson's Models and 2 2 GamesAuthor(s): Mark Irving LichbachSource: The Journal of Conflict Resolution, Vol. 34, No. 1 (Mar., 1990), pp. 29-56Published by: Sage Publications, Inc.Stable URL: http://www.jstor.org/stable/174133Accessed: 26/11/2010 06:14

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30 JOURNALOF CONFLICTRESOLUTION

GAME-THEORY MODELS OF ARMS RIVALRIESOne important pproach o answering he questionof the outcomeof anarmsrivalry s to focus on the equilibrium utcome of the 2 x 2 armsrivalrygame thatthe nations might be thoughtof as playing.A 2 x 2 game-theoryarms rivalry(GT-AR)model for example, one of Brams'(1985) formula-tions that used the Prisoner'sDilemma (PD)-is a model of two or morenations'weapons acquisitionprocesses in which there is an interdependentchoice of the levels of militaryexpendituresby the two nations.'Consideranytwo levels of armsexpenditures f nationX, suchasX1and

X2,andof nationY.such as Y1 andY2. If X1< X2 andY1< Y2, then we mayrefer to X1 and Y1 as "cooperation"n low armament xpendituresandX2andY2 as "defection" o high armament xpenditures.Thepayoffsto X andY from the four possible outcomes may be represented n the following 2 x2 matrix(Exhibit 1) thatrepresentsa GT-ARmodel:Exhibit 1: A GT-ARModel

yY1 C) Y2(D)

Xi (C) P. (C,C), Py(C,C) Px(C,D), Py(C,D)X2 (D) |Px (D,C) , Py (D,C) | Px (D,D) , Py (D9D)|

NOTE: X, and Y1 representcooperation (C) or "low" armamentexpenditures;X2 and Y2representdefection (D) or "high"armament xpenditures; ndpayoffsaredenoted, orexample,as Px(CD) [Py(D,D) ], thepayoffto nationX[Y] from X's choice of C andY's choice of D.

The GT-ARapproachproducesan answerabout heequilibrium utcomeof anarmsrivalry:Theoutcome is a resultof thegamethatnationsplay.Forexample,if an armsrivalry eadsto aPD, then theequilibrium utcomewillbe an"arms ace"; f an armsrivalry eadsto Stag Hunt,thenanequilbiriumoutcomewill be "arms ontrol"; nd f an armsrivalry eads to Chicken, henan equilibrium utcomewill be "armsdomination."This answerto thequestionof the outcome of anarmsrivalry s by itselfincompletebecauseit simply raises anotherquestion:If armsoutcomesare1. There have been three types of game-theoretic models of arms rivalries. One usescontinuousarmsrivalries n the static case and emphasizesvarious ypes of reactioncurves(seeBoulding, 1962; Case, 1979, p. 22; Friedman,1986, p. 180; Intriligator, 964). Anotherusescontinuousarmsrivalries n the dynamiccase andemphasizesdifferential ames (see the reviewin Isardand Anderton,1985). The focus of this article s the third ype, which uses 2x2 games.

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LichbachIARMSRIVALRYAND RISONER'SDILEMMA 31

the result of arms games, then which 2 x 2 game best representsan armsrivalry? Knowing which game characterizesa particulararms rivalry iscrucialto the GT-ARapproach.Unfortunately,he answer o this question sin dispute. Arms rivalries take place in many differentdecision-makingenvironments.The payoff structure, he extent of mutualand conflictingpreferences,and the types of interactionsor "games" hatlead nations intoan arms rivalry are thus open questions. There are, moreover, variousgame-theoreticmodels for the two actor-twochoice case. RapoportandGuyer (1966) showed that if preferencesare strictlyordered, hereare 78conceptuallydifferent2 x 2 games. If preferencesare weakly ordered tiesin preferencesover outcomesare permitted), hereare morethan500 differ-ent typesof games.Using thebasicformulation f Exhibit1, analystshavethuscharacterizedarmsrivalriesby manyof the 78 different2 x 2 games structures.Most haveargued hatparticipantsn an armsrivalryare nvolved ina PD game(Brams,1985, chap. 3; Rapoport,1960; Russett, 1983a; Schelling, 1960; Snyder,1971; andSnyder and Diesing, 1977). But many have suggestedthat othergames mayrepresentan armsrivalry:Chicken,Deadlock,and Stag Huntarecommonlyused. Moregenerally,analystshave argued hat it is possible torepresentanarmsrivalryby several different2 x 2 games. Schelling (1984:244-247) analyzed16 different2 x 2 armsgames thatheconsidered hemostlikely to occur.Downs et al. (1985) offered 12 different2 x 2 games thatpotentiallygeneratearmsrivalriesand even argued hat"under ufficientlyunluckycircumstances lmost any game canresult n an armsrace"(p. 133,emphasisin original).Hardin 1983) suggestedseven different2 x 2 gamesthatmay generate henucleararmsrivalrybetweentheUnitedStatesandtheSoviet Union.Evidently,as Downs andRocke(1987:302) put it, "nosinglegame characterizes ll arms races."One consequenceof this uncertaintys thatthe connection between anarms rivalry and a PD has been nebulous and confused. Questions, forexample,havebeen raisedabout he conditionsunderwhich an armsrivalryis really a PD. Manycontendthata particular rmsrivalrydoes not fit thePD payoff structure,or that a particularPD armsrivalryhas changed itspayoff structureand hence is no longer a PD. No one, however,has offeredprecise guidelinesfor determining hese conditions;only informalspecula-tions or suggestionshave been made about the relevanceof internal actors(for example,bureaucratic olitics or the military-industrialomplex, as inRussett, 1983b) or the technology of war (for example, its offensive ordefensivenature,as inJervis, 1978).Although everyonerecognizesthatnotall armsrivalriesfit the PD model, no one has outlinedan approach hatindicates which ones do. Gowa (1986: 172), for example,has pointedout

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thatAxelrod argued hat the PD fits armsrivalries as well as other nterna-tionalphenomena)but "doesnot offer any clues to his rationale or distin-guishingbetweenproblems n international oliticsthatare only 'related' o(p. 4) PD games and hose that takethe form'(p. 190) of PD games."Anotherconsequenceof the uncertaintys that he connectionbetweenan armsrivalryand Chicken has also been nebulous and confused. While some (Hardin,1983) have modeled an armsrivalryusing Chicken,Downs et al. (1985: 121)argued hat "Chicken s highly unstable,however, and not a realistic modelfor an arms race as continual defection."The linkage between arms rivalries and GT-AR models has also beenconfused because the same games, such as PD, Chicken,Deadlock and StagHunt, that have been used to representan arms rivalryhave also been usedto representseveral other phenomena in international elations:war, (nu-clear) deterrence,bargaining, rises, alliance formation, radewars, and thegeneral problem of international ooperationunder anarchy.Dacey (1987:165) thus refersto the "arms ace/deterrence ame,"whereas Snyder(1971)argues hatdeterrence s best modeled as Chickenandan armsrivalryas PD.Such confusionhas not gone unnoticed. Gowa (1986: 172) pointedoutthat "there is a growing literaturefocused precisely on the issue of theapplicability of PDs and other game theoretic concepts to the study ofinternational elations."We evidently need a close examinationof the appli-cability of the game-theory ramework o armsrivalries.

RICHARDSON MODELS OF ARMS RIVALRIESThedisputeoverthemostappropriateGT-ARmodelhasoccurredbecauseGT-ARmodelers have not inquired nto theprocess thatgeneratesprefer-ences over outcomes.No one has asked,Whatmechanisms n armsrivalriesgenerate the particular2 x 2 arms rivalry games that, in turn, produceequilibriumarmsoutcomes?For example,whatproducesa PD armsrivalryrather han a Deadlock arms rivalry?The typical procedure n the GT-ARliteraturewas shown above:Posit nations'utilitiesfor possible outcomesandlabelpayoffs by single parameters epresenting rdinalpreferenceorderings(Brams,1985, p. 90; Dacey, 1987, p. 165;Downsetal., 1985, p. 121; Russett,1983a, p. 101). No one, however, has inquired into the origins of theserankings n an armsrivalry.2This gap severely limits game-theoreticmodels2. 1now speak of the 2 x 2 game formulationsof arms rivalries,not of the dynamicgameformulationsas in Isardand Anderton 1985). The "graduated" ame models of Snidal(1981,1985) andMcGinnis(1988) come closest to examininghow continuousRichardson-type tilityfunctionsyield 2x2 games.

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of arms rivalriesbecause such models aremore realisticif the process bywhichpayoffsorutilitiesaregenerated s specified.As Morrow 1986: 1131)argued, "rational choice models are most compelling when the actor'sutilitiesfor outcomes canbe tied to primitivequantities."Simply assumingthe existence of particular referenceorderings s inherentlyunsatisfying.The classic statementof the process behind an armsrivalrywas made byRichardson 1960). A Richardsonarmsrivalry(R-AR) model is one of twoor morenations'military xpenditure rocesses nwhich each nation'schoiceof the level of its militaryexpenditures s partiallydependentupon what theothernation(s) does.3All such models assume that the change in a nation'sarms expenditures is a function of the external threat created by theopponent'sarmsand the internal atigue created by one's own arms. Thevariations n the models come in how one's own and one's opponentsarmsentera nation's expenditure unction.The R-ARapproachalsoproducesan answer o the equilibrium utcomeof an arms rivalry:The outcome is a result of the grievances nations have,thefatigue comingfrom a nation'sownexpenditures narms,andthethreatscoming from the other nation's expenditureson arms. Richardson howed,for example, thatgiven low grievances, low threats,and high fatigue, anequilibriumoutcomemay exist and be stable at a low level of armaments(Zinnes,1976:339-354).Because the process behind armsrivalriesare explored n R-AR modelsbut not GT-ARmodels, perhaps he conditionsthatproducea particular x2 armsrivalrygame may be sought in Richardson'sapproach o explainingthe equilibriumoutcomes of armsrivalries.This article tries to accountforthe equilibriumoutcomesof armsrivalries n a moresatisfactoryway thando existing GT-ARmodelsby usingR-AR models to helpformulateGT-ARmodels. In otherwords, I do more thansimply posit thatboth nations in aGT-ARreceivepayoffs.I relatea nation'spayofforutility unction n a 2 x 2gameto aRichardson-type rms-acquisition rocess.Hence the specific question that guides this article:Whatare the condi-tions, if any,under which the utility unctions of a particularR-AR modelhelp determinethe equilibriumoutcomesof a particular GT-ARmodel? Iinvestigate this question by using Richardson-typeassumptions about anation'sutilityfunction o constructparticular x 2 armsrivalrygames, oneswith particular referenceorderingsover armsrivalryoutcomesandpartic-ularequilibrium utcomes.This article ills animportant ap.Althoughthereis muchinformal peculationabout hesourcesof various2 x 2 armsrivalry

3. For an overview of the literatureon Richardson-type rmsrivalries,see Zinnes(1976),Intriligator 1982), Anderton (1985a, b), Isardand Anderton (1985), and Leidy and Staiger(1985).

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games, there are no formal demonstrations f the conditions(if any)underwhich an armsrivalrymodeledby Richardson-type tilityfunctions mplies,for example, thepreferenceorderingand equilibriumoutcome of a PD.My results arehighly general. I first establish the most general possiblerestrictions nRichardson-type tility unctions hatdetermine our common2 x 2 arms rivalrygames: a PD Arms Rivalry(PD-AR), a Deadlock armsrivalry(Deadlock-AR), a Chicken arms rivalry(Chicken-AR), and a StagHuntarmsrivalry Stag Hunt-AR). then establish hegames thatmay occurundertwo commonlyused but morespecific Richardson-type tility func-tions. Twocounterintuitive esultsemerge: 1) It is impossiblefor one of thecommonlyused Richardson-type tilityfunctionsto everleadto a Chicken-or Stag Hunt-AR;and(2) it is impossiblefor either of the commonlyusedRichardson-type tility functionsever to lead to a StagHunt-AR.

THE ASSUMPTIONSThere are threesets of assumptionsmade in the models developedhere:

general assumptionsaboutAR models, assumptionsabout the utilityfunc-tions of R-AR models, and assumptionsaboutthe preferenceorderingsofGT-ARmodels.GENERALARASSUMPTIONS

I begin with six assumptionsaboutthe natureof thearmsrivalry hatwillbe explored.Due to space constraints, canoffernojustificationshere(seeLichbach,1989).ASSUMPTIONAl: Two nations,X andY,are involvedin the armsrivalry.ASSUMPTIONA2: The only strategiesavailableto nationsX and Y arelevels of defenseexpenditures.ASSUMPTIONA3: Eachnation s a unitaryactor.ASSUMPTIONA4: Eachnation s a utilitymaximizer.ASSUMPTIONA5: The nationsplay a noncooperativegame of completeinformation.4

4. These rulesof the game lead me to analyzeNash pure-strategy quilibria. donot considermixed-strategyequilibriabecause it appears arfetched o conceptualizenations as "spinning

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LichbachIARMSRIVALRYANDRISONER'SDILEMMA 35

ASSUMPTIONA6: Nations makea one-shot binarychoice of armamentsfromamong a continuumof possible armament evels.5R-AR ASSUMPTIONS ABOUT UTILITYFUNCTIONS

What are the sources of a state's preferencesabout the possession ofarmaments?What are its intentionsandgoals that areaffectedby its arma-mentprocurement?As indicatedpreviously,Richardsondevelopedseveralvariationsof his basicmodel while subsequentanalystsproposedadditionalones. In this section, I propose threedifferentvariantsof Richardson-typeutility functions.I first consider the most general possible Richardson-typeutilityfunction hatdetermines ourcommon2 x 2 armsrivalrygames.I thenexamine two commonly used, but more specific, Richardson-typeutilityfunctions.A General R-AR Model

The most general assumptionabouta R-AR utility functionis that eachnation'sutilityis a functionof its own armsandthe othernation's arms:ASSUMPTIONSB1: The two nationshave the following utilityfunctions:

UX= UX(X,Y) [2.1]UY= Uy (X,Y) [2.2]

WhereUxandUyare heutilityevelsofX andY;andX andY are hearms xpendituresfX andY.spinnersorrolling dice" (Ordeshook,1986: 181) to choose armamentevels, and becausemixedstrategiesproduceparadoxesninterpretationAumannandMaschler,1972). Hence,the conceptof mixed strategies"does not offer a useful model of choice" (Ordeshook,1986: 137) in thecontextof armsrivalries.For anattempt o justifymixed-strategy quilibria, ee Harsanyi 1973).5. The first five assumptionsare commonto both modeling traditions.The final assumptionis not, for while GT-AR models adopt this assumption,R-AR models typically assume thatplayers choose from among a continuumof armamentevels over time. How do the two armsexpendituresat one time in the 2x2 models correspond o the continuumof arms expendituresover timein theRichardsonmodel? One possibility s that f the Richardsonmodel had multipleequilibria, he 2x2 game moveswould correspond o choosing amongthose stable levels. But Ido not wish to pushthis equivalence oo far:Binary one-shot choices areclearlydifferent hancontinuouschoices made over time. I need not establish the exact equivalence between theapproaches ecausethepurpose f this article snot to compare he conclusionsaboutequilibriumfrom the two models it is to explore how the assumptionsaboutthe utility functionsbehindR-AR modelshelp determineGT-ARmodels.

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Ifone assumes hatnationsmay choosefrom amonga continuumof possiblearmament evels, then one may use resultsfromFriedman 1986: 29-46) toimpose restrictionson (2.1) and (2.2) to establish the existence andunique-ness of an equilibriumpoint in thecontinuousarmsrivalrybetween nationsX andY I am concernedwith the equilibriumoutcomes of the 2 x 2 armsgames often discussedin the literature.Hence, I assume(A6) that nationsmake binarychoices and impose restrictionson (2.1) and(2.2) to establishthe existence anduniqueness of the equilibriumoutcome of the 2 x 2 ARgamethenationsmaybe thoughtof as playing.To produce restrictions on the preference orderings and equilibriumoutcomesof the2 x 2 gamesconsideredhere, t turnsout thatonlythe mildestof restrictions n (2.1) and(2.2) arerequired.Two sets of restrictionsare onthe first-orderpartialderivatives,or how X and Y separatelyaffect eachnation'sutility.A thirdset of restrictionss on the totaldifferentials,or howX and Y togetheraffecteach nation'sutility.The first set of possible assumptions elateto how increases n a nation'sown arms affect the nation'sutility. One assumption s thatincreases in anation'sown armsalways increasethe nation'sutility:ASSUMPTIONB1.1: Over the intervalsX1to X2and Y1to Y2,aUX/aX> 0andaluyiY > 0.

Two otherassumptionsare that,depending upon the level of the othernation's armaments,ncreasesin a nation's own arms sometimes increaseandsometimes decreasethe nation'sutility:ASSUMPTIONB1.1': Over the intervalsX1 to X2 andY1to Y2,aUx/aX> 0if YsY1 andaUx/aX< 0 if Y > Y1; andaUY/aY> 0 if X 5 X1andauy / aY X1.ASSUMPTIONB1.1": Over the intervalsX1to X2 and Y1 to Y2,auiaX 0 if Y > Y1;andauy/aY < 0 if X 5 X1andaUy/aY>OifX>X1.The second set of possible assumptionsrelate to how increasesin the othernation'sarmsaffect a nation'sutility.The only assumptionrequiredhere isthat ncreases n the othernation'sarmsalwaysdecrease thenation'sutility:ASSUMPTIONB1.2: Overthe intervalsX1to X2 andY1to Y2,aUxlaY< 0andaUy/aX< 0.

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The thirdset of possibleassumptions ombine the first two and state howincreases in a nation's own and the other nation's arms affect the nation'sutility. One assumption s thatthe positive impact of increases n a nation'sown armson its utility (in B1.1, B1.1', or B1.1 ") is outweighed by thenegative impact of increasesin the other nation's armson its utility(131.2);thus, the total impact of increases n the arms of both nations is to decreasethe nation'sutility:ASSUMPTIONB1.3: Overthe intervalsX1 to X2 and Y1 to Y2,

dUx = (aUx/3X)dX + (aUx/aY)dY < 0dUy = (3Uy/IY) dY + (aUy/aX) dX < 0

The alternative assumptionis that the positive impact of increases in anation's own arms on its utility (B1.1) outweighs the negative impact ofincreases in the othernation's arms on its utility (B1.2), and thus the totalimpactof increases n the armsof bothnations s to increase henation'sutility:6ASSUMPTIONB1.3': Overthe intervalsXjto X2 and Y1 to Y2,

dUx = (aUx/aX)dX + (aUx/aY)dY > 0dUy= (3Uy/3Y) dY + (aUy/aX) dY > 0

A Specific R-AR ModelA morespecific R-AR assumptionabout utility functions, employed by

Downs and Rocke (1987: 309) and Friedman 1986: 180), is the following:ASSUMPTIONB2: Thetwo nationshave the following utility functions:

Ux = kx (X-Y) - jX X2 [2.3]Uy = ky (Y-X) - lyy2 [2.4]

Wherekx andky are the threatparameters f X andY;and1x and1 are thefatigue parameters f X andY.

6. Note thatbecause dX, dY > 0, the following are true:assumptionB1.1, B1.1', or BL1"can be combinedwith assumptionB1.2 to produceassumptionB1.3; assumptionB1.1 can becombined with assumptionB1.2 to produce assumptionB1.3'; however,neitherassumptionB1.1' norB1.1" can be combinedwith assumptionB1.2 to produceassumptionB1.3'.

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tion adoptedby Gillespie et al. (1977), McGuire (1982), and McGinnis(1985):ASSUMPTIONB3: The two nationshavethe following utilityfunctions:

U = kx,X_py)2 X1xX2 [2.9]UY= ky (Y- qX)2- lyY2 [2.10]

Wherep(q)is the minimum ecurity evel of Y's (X's) armaments cceptabletoX(Y),p,q>0.Importantpropertiesof this R-AR model are also revealed by takingderivatives.Consider hepartialderivativesof Uxwith respectto Y:

ay = 2pkx(pY- X) [2.11]a2Ux 2 [2.12]

2= 2p kx

Note thatthepartialderivativeof Uxwith respecttoY,as shown in (2.11), isnegativewhen X > pY andpositive when X < pY Moreover,as shown in(2.12), as its rival's militaryexpenditures ncrease, a state loses utilityat adecreasingrate or gains utilityat an increasingrate. Hence, Y's threat o Xnow varies with levels of Y: When X > pY, very high levels of Y aremarginally ess threateningo X.Now considerthe partialderivativesof Uxwith respectto X:au, 2.3ax = 2kx(X - pY) - 1xX [2.13]

ax2 = 2 (kx - 1) [2.14]Note thatthe marginalutilityof Ux with respect to X, as shown in (2.13),given thatX > pY, is positive if the firsttermon the RHSof (2.13) exceedsthesecond term.Equation 2.14) indicates hatthis marginalutility increases(decreases)at an increasingrate if kx > 1, and increases(decreases)at adecreasingrate f kx< l, Thisrepresentshe fatiguedueto opportunity ostsasX approaches he state'sbudget constraint.Note thatthis fatigue factorisdependenton Y. As Y increases, X's gain from moreX always decreases.Thispropertywill also be shown to havedramatic onsequences or thetypeof GT-ARallowed.

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Lichbach IARMSRIVALRYAND RISONER'SDILEMMA 41

in Stag Hunt.Hence, in these four games, nations do not always prefertospend moreon arms.Thesecond setof utilityfunctionassumptions,how X'sutility is affected by changes in its opponent'sarms,relates to horizontalmoves in the game, or moves from mutual cooperation to Y's unilateraldefection and from X's unilateralcooperationto mutual defection. Suchmoves are harmful o X in all fourgames, and hence in these gamesnationsalways prefer hattheiropponents pend ess on arms.Thethirdsetof utilityfunctionassumptions,how X's utility is affected by changesin its own andits opponent'sarms,relates to diagonal moves in the game, or moves frommutualcooperation o mutualdefection.Suchmoves are harmful o X in PD,Chicken,andStag Hunt, but beneficial to X in Deadlock. In these games,nationsmaypreferanarmsrace o armscontrol, r armscontrol o anarms ace.

THE GENERAL R-AR MODELUsing assumptionsA1-A6, theB1-ARmaybe represented s the follow-ing game (Exhibit3):

Exhibit 3: The Bl-AR Gamey

Y. (C) Y2 D)X1 ( U, (Xi, Y0) UY (X1, Y1) Uy (X1, Y2), UY (X1, Y2)

X .X2 (D) Ux NX, Y1), UY (X2, Y1) U, (X2, YA) UY NX, Y2)

Threehighly generalsets of restrictions nUxandUYweresuggestedearlier.Which are the sufficientconditions hat allow this mostgeneralRichardson-type armsrivalry o producea PD, Chicken,Deadlock,or StagHuntgame?This section assesses the ability of the highly general restrictionson theB 1-AR modeltoyieldtheequilibrium utcomesof C1-ARtoC4-ARmodels.PD

Thefollowingpropositionestablishesthe restrictions n utilityfunctions(2.1) and (2.2) thataresufficient for a Bl-AR to be a Cl-AR (PD).

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PROPOSITION .1: GivenA1-A6, if B1.1 andB1.3 hold,then an armsraceis theNash equilibriumoutcome of Exhibit3.PROOF:B1.1 impliesPx(D,C) > Px (C,C) andPx (D,D) > Px(C,D). B1.3implies Px(C,C)> Px(D,D). Combiningyields a preferenceorderingas inPD. A similarresultholdsforY,andthus mutualnoncooperations theNashequilibrium.QEDHence, the resultsproducea classic summaryof a PD-ARin Richardsonianterms:If, overthe intervalsX1to X2 andY1 to Y2,X's (Y's) gains fromX's(Y's) armsare alwayspositive(B1.1), but X's (Y's) gainsfrom X's andY'sarmsarealways negative(B1i.3), hen bothnationsarein a PD.CHICKEN

Thefollowingpropositionestablishes he restrictions n utilityfunctions(2.1) and(2.2) that aresufficient for a B1-AR to be a C2-AR(Chicken).PROPOSITION .2: Given A1-A6, if B1.1' and B1.2 hold, then the armsdominationoutcomes aretheNashequilibriaof Exhibit3.PROOF:B1.1' implies Px(D,C)> Px(C,C)and PX(C,D)> PX(D,D).B1.2implies Px(C,C)> Px(CD) [as well as Px(D,C)> PX(D,D),which is redun-dant].Combiningyields apreferenceorderingasinChicken.A similarresultholdsforY.and thus the unilateraldefections are the Nashequilibria.QEDHence, the results produce a classic summary of a Chicken-AR inRichardsonian erms:If, overthe intervalsX1to X2and Y1 to Y2,X's (Y's)gains fromX's (Y's) armsarepositive ata low Y (X) butnegativeat a highY (X) (B1.1'),andX's (Y's) gains fromX's andY's armsarealways negative(B1.2), then both nationsare in a Chickengame.DEADLOCK

Thefollowing propositionestablishes he restrictions n utility functions(2.1) and(2.2) thataresufficientfor a B1-ARto be a C3-AR(Deadlock).PROPOSITION .3:GivenA1-A6, if B1.2 andB1.3'hold, then anarmsraceis the Nashequilibriumoutcome of Exhibit3.PROOF:B1.2 implies Px(CC) > Px(CD) and Px(DC) > PX(DD). B1.3'implies Px (D,D) > Px(CC). Combiningyields a preferenceorderingas inDeadlock.A similarresultholds forY.andthusmutualnoncooperations theNashequilibrium.QED

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Lichbach ARMS RIVALRYAND RISONER'SDILEMMA 43

Hence, the results produce a classic summary of a Deadlock-AR inRichardsonian erms:If, over the intervalsX1 to X2 andY1 to Y2,X's (Y's)gains from Y's (X's) arms are always negative (B11.2)butX's (Y's) gainsfrom X's and Y's arms are always positive (B1.3'), thenboth nationsareina Deadlock game.STAG HUNT

The following propositionestablishes he restrictions n utility functions(2.1) and (2.2) that are sufficient for a B1-ARto be a C4-AR(Stag Hunt).PROPOSITION .4: GivenA1-A6, if B1.1"andB1.2 hold, thenarmscontrolis a Nash equilibriumoutcome of Exhibit3.PROOF:B1.1" implies Px(C,C)> PX(DC)andPx(D,D) > Px(C,D). B1.2implies PX(D,C)> Px (D,D) [as well as Px(C,C)> Px (C,D), which isredundant].Combining yields a preference ordering as in Stag Hunt. AsimilarresultholdsforY,andthusmutualcooperations a Nashequilibrium.QEDHence, the results produce a classic summary of a Stag Hunt-AR inRichardsonian erms:If, over the intervalsXI to X2 andY, to Y2,X's (Y's)gains fromX's (Y's) armsarenegativeat a low Y (X) butpositiveat a highY (X) (B1.1"),and X's (Y's) gainsfrom Y's (X's) armsarealways negative(B1.2), then both nationsare in a Stag Huntgame.In Summary,PD, Chicken,Deadlock,andStagHuntarms rivalriesmayexist from a Richardsonian erspective.Quite generalrestrictionson Rich-ardson-typeutilityfunctionsmay produce hepreferenceorderingsandNashequilibriaof the four most commonlydiscussed2 x 2 armsrivalrygames.Giventhattherestrictions re all individuallyreasonabledescriptionsof realworldarmsrivalries, uch realworldarmsrivalriesmayindeedapproximatecertain 2 x 2 games.These resultsaboutthe use of Richardson-type tilityfunctionsin 2 x 2 GT-AR models aregratifyingfor two reasons:We nowknow,in one important ense, that the two traditionsareconsistentand that2 x 2 GT-ARmodelsarerealistic.

But just because highly general Richardson-typeutility functions canproducecertain2 x 2 armsrivalrygames does not mean that all conceivableRichardson-type tilityfunctionscanproduceall conceivable2 x 2 games.Specific Richardson-type tilityfunctionsembody specific restrictions hatmay make certain 2 x 2 arms rivalry games impossible. This potentialsituation s investigated n the next two sections.

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THE FIRST SPECIFIC R-AR MODELUsing assumptionsA1-A6, the B2-ARmaybe represented s thegameinExhibit4.

Exhibit4: The B2-AR GameY

Y1 (C) Y2 (D)X1 (C) k. (XI-YI) - 1X12, ky(Y1- X1) - 1yY12 k. (X1-Y2) - 1X12, ky (Y2- X1) - ly Y2x X2 (D) k. (X2-Yl) - 1xX2, k (Y1- X2) - l2 k. (X2-Y2) - 1,X22, ky 2-y2 - ly Y2

Note that he armsrivalryoriginsof armsrivalrypayoffsis specifiedmoreexactlyinExhibit4 thanwas possiblein Exhibits1-3. But arethe restrictionsinvolved in utility functions (2.3) and (2.4) sufficient to produce a PD,Chicken,Deadlock,or StagHuntgame?Thissection assesses the abilityofthe highly specific restrictions n theB2-AR model to yield the equilibriumoutcomesof Cl-AR to C4-AR models.THE PD-AR

The following propositionshows what inequality is needed for thisB2-AR to be a PD.PROPOSITIONS .1: GivenassumptionsA1-A6, if

(X1+ X2) + lx(X - j) 'l > (X1 + X2) [4.1]lxX2 -X1) lxthen anarmsrace is theNashequilibriumoutcomeof Exhibit4.PROOF:Cl lists threeconditions, foundin the appendix, or a PD to existgiven utility functions(2.3) and (2.4). First,Px(DC) > Px(C,C).Second,Px(D,D)> PX(CD). Note that these two conditionsare equivalent. Third,Px(CC) > PX(D,D).Combiningandrearranginghese conditionsyields thegeneralconditiononpreferenceorderings orB2-AR to be aPD-AR in (4.1).Hence, if (4.1) holds, Exhibit 4 is a PD, andmutualnoncooperations theNash equilibrium.QED

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Anotherconditionsometimes imposed on the PD is2R>T+S [4.2]

This eliminates hepossibilityof an alternatingolution:X receives T andYreceives S on one round, hen X receives S andY receivesT on the next.B2impliesthis PD condition f2[kx(Xi-Yi)-1XXI2I[kx(X2-Y1)-1XX22][kx(Xi-Y2)-1jXi2] [4.3]

This reduces to the simplification of Px(CC) > PX(DD) shown in theappendix,and adds nothingnew to (4.1).What does Proposition4.1 say? First, considerwhat each of the threecomponentssay.Px(DC) > Px(CC) implies thatthe gains to X frombeingtheexploiterexceed itsgainsfrommutualcooperation.Thesimplificationofthis expression,listed in the appendix,says that(a) if the "threat/fatigue"ratio(kx/1x) s high, then thearmsrace is a PD; but(b) if thesumof the twopossible levels of militaryexpenditures X1+X2) s high, thenthe armsraceis not a PD. In otherwords, at high levels of militaryexpenditures, atigueoverwhelms hreatand the state no longer prefers o spendmore on its arms.Px(D,D)> PX(C,D) mpliesthatthegainsto X frommutualdefection exceedits gainsfrom being exploited.Giventhatthe sameanalysisholdshereas inPX(D,C)> PX(C,C), f the threat/fatigue atio is low or the total level ofmilitaryexpenditures s high,Cwill alwaysbepreferredoD and theB2-ARgame becomes a simple game of coordination.PX(C,C)> PX(D,D) mpliesthat X's gainsfrom mutualcooperationexceed its gainsfrom mutualdefec-tion.The simplificationof this expression, isted in the appendix,says thatin orderfor X to prefer mutualcooperation o mutualdefection the threat/fatigueratio,kx/lx, mustbe less thanthe sum of thetwo possible armamentlevels, X1+ X2, plusthethreat/fatigueatiotimesthe ratioof the increase nY's armaments o the increase in X's armaments,kx(Y2-Y1)/1X(X2-X1).Note that forPx(D,C)> Px(C,C) o hold,kx/1x> (X1+X2).The first termontheRUS of thePx(CC) > PX(D,D) ondition,X1+ X2,is, by itself, less thankx/1x.Thusthe last term on the RUS, kx(Y2-Y1)/1x(X2-X1),must be largeenough to offset the difference.This is less likely if the changes in Y aresmaller han hechanges nX. Forsymmetricgames,the ratiobecomes 1andtheconditionalways holds.Thetermgets at the relative size of thechangesin armament evels: The changesin Y must be as threateningas those in Xareattractive.Now, put all threecomponentstogetheras in (4.1). Proposition4.1 thussays thatfor aPD to exist,X's losses frommutualdefection must exceedX's

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gains from its unilateraldefection, which must exceed X's costs from itsunilateraldefection. The dilemma s as follows: The inducement o cooper-ation, (X1+X2)+ kx(Y2-Y1) / lx(X2-Xl) > kx/lx,and the inticement todefection,kx/1x> (X1+X2),are linkedtogether.Thatis, the condition for aPD armsrivalryis that the rewardsfrom mutualcooperationexceed therewards rom one's own unilateralnoncooperation thus,cooperate),but therewards rom one's own unilateralnoncooperationxceed the cost of one'sown unilateral ooperation thus,defect).Whatdo these results imply about the equilibriumof the PD-AR? Thecondition n theappendix orPX(D,D)> PX(C,D)mustholdforbothplayersin order for (D,D) to be a Nash equilibrium.Thus, a Nash equilibriumat(D,D) occurs ntheB2-ARif, foreach nation, tsthreat/fatigueatio s greaterthanthesum of its two possible levels of militaryexpenditures.CHICKEN

The next propositionshows that while a B2-AR may, under certainconditions,become a PD game, there areno conditionsunderwhich it maybecome a Chickengame.PROPOSITION .2:GivenassumptionsA1-A6, B2 andC2 are nconsistent.PROOF:C2 lists the requirements or a Chicken game to exist, Px(DC) >Px(CC) > PX(C,D)> PX(D,D).Hence, Px(DC) > Px(CC) and Px(CD) >Px(D,D).Theappendix howsthat,exceptin the trivialcase where allpayoffsare thesame,these two requirements reinconsistent.QEDDEADLOCK

The following proposition hows what inequality s neededfor a B2-ARto be a Deadlockgame.PROPOSITION .3: GivenassumptionsA1-A6, if

(+X)+kx(Y2 - Y) >kx [4.4]+x(X2 -x1) 1xthen an armsraceis the Nashequilibriumoutcomeof Exhibit4.PROOF:C3 lists threeconditions,found in the appendix,for a Deadlockgametoexistgiven utilityfunctions 2.3)and(2.4).First,Px(DC) > PX(DD).Second,Px(C,C)> PX(CD).These conditionsarealwaysfulfilledbecauseitis assumedthatY2> Y1. Third,PX(DD)> Px(CC), or (4.4). Hence, if (4.4)

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holds, Exhibit 4 is a Deadlock game, and mutual noncooperation is the Nashequilibrium. QED

STAG HUNTThe next proposition shows that while a B2-AR may, under certain

conditions, become a PD or Deadlock game, there are no conditions underwhich it may become a Stag Hunt game.PROPOSITION 4.4: Given assumptions A1-A6, B2 and C4 are inconsistent.PROOF: C4 lists the requirements for a Stag Hunt game to exist, Px(CC) >Px(DC) > Px(DD) > Px(CD). Hence, Px(CC) > PX(DC) and Px(DD) >PX(CD). The appendix shows that, except in the trivial case where all payoffsare the same, these two requirement are inconsistent. QED

In sum, commonly used Richardson-type utility functions (2.3 and 2.4)may, under certain conditions, be a PD or Deadlock game, but there are noconditions under which there may be a Chicken or Stag Hunt game. If onebelieves that utility functions (2.3) and (2.4) are a realistic representation ofan arms rivalry, then one must also believe that a Chicken-AR and a StagHunt-AR are unrealistic representations of that arms rivalry.

THE SECOND SPECIFIC R-AR MODELUsing assumptions A1-A6, the B3-AR may be represented as the game in

Exhibit 5.Exhibit 5: The B3-AR Game

yY. Y2

|X1 kx (Xl-pY1) - 1,Xx2 Iy (Y1- qX) - _ Y12 kX(Xl-pY2)2 - lXX12, ky (Y2- qX)2 - ly Y2XI

X2 kx (X2-pYl)2 - 1XX2, ky (Yj- qX2) - ly Y1 kx (X2-pYj)2 - 1XX22,ky (Y2- qX2)2 - ly y22

PDThe following proposition shows what inequality is needed for this B3-AR

to be a PD.

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PROPOSITION .1: GivenAssumptionsA1-A6, if

[ kx] [X2 ]>X2 [5.1]1X(X22-Xi2)>kx(X2-py2)2 kx(XlpYl)2 [5.2]

thenanarms raceis the Nash equilibriumoutcomeof Exhibit5.PROOF:Cl lists three conditions,foundin the appendix, or a PD to existgiven utility functions (2.9) and (2.10). First,Px(D,C)> Px(C,C).Second,Px(D,D)> Px(C,D).Note thatthese two conditionsareno longerequivalent.However,Y2> Y1,and the LHS of both expressionsarethe same.Only (5.1)holds as a PD requirement ere.Third,Px(C,C)> PX(D,D),or (5.2). Hence,if (5.1) and (5.2) hold,Exhibit5 is a PD, andmutualnoncooperations theNash equilibrium.QEDThe crucialdifferencebetweenthe two specificR-AR models discussedhereis thattwo of the conditions for a PD, Px(D,C)> Px(C,C)and Px(D,D) >PX(C,D), are equivalent for a B2-AR and different for a B3-AR. Thisdifference does not affect the possibilityof the two specific R-AR modelsbecoming PD-ARs,butwill have crucialimplications or theR-AR modelsbecomingChicken-ARs.CHICKEN

The following propositionshows what inequality s neededfor a B3-ARto be a Chickengame.PROPOSITION .2: GivenAssumptionsA1-A6, if

PY2 > [k X ][ +X2 ] PY1 [5.3]then the two arms-dominationoutcomes are the Nash equilibria of Ex-hibit 5.PROOF:C2lists threeconditions, ound n the appendix,or aChickengameto exist. First, PX(D,C)> Px (C,C). Second, Px(C,D)> Px(D,D).These twoconditionsmaybe combined into (5.3). Third,Px(C,C)> PX(C,D),which isalways fulfilled because it is assumed that Y2 > Y1. Hence, if (5.3) holds,Exhibit 5 is a Chicken game and the unilateral defections are the Nashequilibria.QED

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Lichbach IARMSRIVALRYAND RISONER'SDILEMMA 49

In sum, a B3-AR may,undercertainconditions,become a Chicken-AR,buta B2-AR may never,underany conditions,become a Chicken-AR.It isimportant o realize how the utility functionsdefinedin B2 andB3 producethese differentresults. Equation 2.7) shows that values of Y do not affectX's gains from greater militaryexpenditures n a B2-AR. Thus, if a statederives increasedbenefit fromincreasedexpenditures it prefersD to C orX2 to X1), then it does so for allpossible valuesof Y. It thereforecannotbe,as in Chicken, that D is preferred o C for low Y andC preferred o D forhigh Y. On the otherhand,(2.13) shows thatvalues of Y do affectX's gainsfrom greatermilitaryexpenditures n a B3-AR. High levels of Y hurt X'sgains frommoreX. Hence, it canbe, as in Chicken,thatat low Y,X prefersD to C [Px(DC) > Px(C,C)],while at highY,X prefersC to D [Px(C, D) >Px(DD].

DEADLOCKThe following proposition hows whatinequality s neededfor a B3-ARto be a Deadlockgame.

PROPOSITION .3: GivenassumptionsA1-A6, ifkx(X2_py2)2- kx(Xl-pyl)2 > x(X22-X1 [5.4]

then anarmsrace is theNash equilibriumoutcome of Exhibit5.PROOF: C3 lists threeconditions, found in the appendix,for a Deadlockgame to exist given utility functions (2.9) and (2.10). First, Px(DC) >PX(D,D).Second, Px(CC) > Px(CD). These two conditions are alwaysfulfilled because it is assumed thatY2 > Y,. Third,PX(DD) > Px(CC) or(5.4). Hence, if (5.4) holds, Exhibit 5 is a Deadlock game, and mutualnoncooperation s the Nash equilibrium.QEDSTAG HUNT

The next propositionshows that while a B3-AR may, under certainconditions,become a PD, Chicken,or Deadlockgame, thereare no condi-tionsunderwhich it maybecome a StagHuntgame.PROPOSITION .4: GivenassumptionsA1-A6, B3 andC4 areinconsistent.PROOF:C4 lists threeconditions,foundin the appendix,for a Stag Huntgame to exist. First,Px(CC) > Px(DC). Second,PX(D,D)> Px(C,D).Thesetwo conditionsmaybe combined nto

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pY, > [1 lx 2 >l+X py2>But (5.5) cannotholdbecause Y2> Y1, andhencepY1< PY2. QED

In sum, neither a B2-AR nor a B3-AR may ever, underany conditions,become a Stag Hunt-AR.The informalanalysesof GT-ARshas againgoneawry.It is importanto realize how the utility functionsdefined in B2 andB3 producethese similarresults.Equation 2.7) shows thatvalues of Y donot affectX's gains fromgreatermilitaryexpendituresn a B2-AR. Thus, ifa statederives increasedbenefit fromincreasedexpenditures it prefersD toC orX2toX1), thenit does so for all possiblevalues of Y. It therefore annotbe, as in Stag Hunt,that C is preferred o D for low Y andD preferredo Cfor high Y. On the otherhand, (2.13) shows that values of Y do affect X'sgains from greatermilitaryexpenditures n a B3-AR. But high levels of YhurtX's gainsfrom more X. Hence, it cannotbe, as in StagHunt,thatatlowY, X prefersC to D [Px(C,C)>Px(D,C)]ndyet at high Y, X prefersD to C[Px(DD)>PX(C,D)].Thepattern an only be asinChicken,whereY's greaterexpenditures orces X to prefermore armament xpendituresat low Y, andless armament xpendituresat highY

SOME CONCLUSIONS ABOUT GT-AR MODELSPD

Examples of PD-ARsabound.Downs andRocke (1987: 302) cited "theAnglo-Germannaval race in its later stages, the French-German acejustbeforetheoutbreakof WorldWar , theJapanese,U.S. andBritishnaval racethat was temporarilyhaltedby the WashingtonTreatyof 1923, and theU.S.-Sovietraceprior o SALT ."As aconsequence, heanalystscitedearlierconsidered the PD as a reasonablerepresentation f an armsrivalry.Myresultsshow thatall threeR-ARmodels will generatePD-ARmodelswhen(a) the marginalutilityof increasesin one's own armsarepositive, (b) themarginalutility of increasesin one's opponentsarmsare negative,and (c)the total differentialof the increases of both are negative, or when (b)overwhelms(a). All threeassumptionsare quite reasonable.The conven-tional wisdom aboutthe usefulness of the PD as a model of anarmsrivalryis confirmed.

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DEADLOCKExamples of Deadlock-ARsalso abound.Downs andRocke (1987: 302)cited several examples of participants n arms rivalriesthat appear to bemotivatedby Deadlock:Germanyduring he1930s,theSoviet Union duringthe 1950s,Chinaduring he 1960s, Iranduring he 1980s, Britainduring he1870s, and the United States during he Reaganyears.PreviousresearchhasconsideredDeadlock as a reasonablerepresentation f an arms rivalry.Myresults show thatall threeR-AR models will generateDeadlock-ARmodelswhen (a) the marginalutility of increases n one's own armare positive, (b)

the marginalutility of increases in one's opponentsarmsarenegative,and(c) the total differentialof the increases of both arepositive, or when (a)overwhelms (b). All three assumptionsare quite reasonable.The conven-tionalwisdom about the usefulness of Deadlock as a model of armsrivalryis also confirmed.CHICKEN

Many have speculatedaboutthe possibilityof a Chicken-AR.The exam-ple usually given is a nucleararmsrivalry nwhich it is reasonable o expectthatthe marginalutility of increases n X's armaments hould be positive atlow levels of Y's armamentsbecausesome deterrence reventsnuclearwar)but negativeat high levels (because too much deterrence s destabilizing).Although,mymostgeneralR-ARmodel,theB1-AR,shows that tispossibleto make this assumption,the fairly common B2-AR does not have thisproperty.A Chicken-AR s more unusual han expectedby theconventionalwisdom.STAG HUNT

And finally,many have speculatedabout the possibilityof a StagHunt-AR. Jervis (1978) was first. Hardin 1983: 247) referred o a StagHunt-ARas a CoordinationGame,and Stein (1982: 303) as an AssuranceGame.Onescenario offeredfor this model is that a state may not be very competitiveand not very interested n "beating ts rival" throughunilateraldefection.Thus,a StagHunt armsrace"describes, or example,theclassic situation nwhich a stateacquiresa weaponitwould nothave builton its own initiativebecause it believes that the rival is buildingit" (Downs et al., 1985: 135).Another scenario offered for this model is that the costs of unilaterallybuildingsome armamentmightbe very high.A StagHunt-AR s consistent

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with "generalresults of the earlyRichardsonmodeling literature .. thatastableequilibriumwill exist wheneverthe economic constraintsassociatedwithlargemissile stocks captured y the fatigueterm outweighthe desireto acquire new missiles in responseto growing adversarymissile stocks"(Leidy and Staiger, 1985: 515). Such reasons are not compelling becausethey do not explainwhy the marginalutilityof increases n X's armamentsshouldbe negativeatlow levels of Y's armaments ut positiveathighlevels.Although the most general R-AR models, the Bl-AR, shows that this ispossible, neitherof the specific R-AR models, the B2-AR or B3-AR, havethis property.The Stag-Hunt AR is more unusualthan expected by theconventionalwisdom.

CONCLUSIONS ABOUT THECOMPATIBILITYOF R-AR AND GT-ARMODELSIs thereanyrelationshipbetweenthe two explanationsof theequilibriumoutcome of an arms rivalry? Given the considerableoverlap, it is quite

reasonable to attempt to link the two formal traditionsof arms-rivalrymodeling. The lack of overlapwas, in fact, one of the earlycriticismsmadeby game theorists Rapoport,1957;Harsanyi,1962)of Richardson'smodelsof armsrivalries;critics focused on Richardson's 1960: 12) "whatpeoplewould do if they did not stop to think"assumptionandurgedmoregame-theoreticassumptions.Aset of questionsarises:Are there ormalconnectionsbetween the two traditions or modelingarmsrivalries?When is borrowingappropriate?May one be used to provide insightandunderstanding boutthe other?This articleaddresseda specific version of thesequestions:Given partic-ular R-AR assumptions,which GT-ARequilibriumoutcomes can exist?Toanswer hisquestion, twas necessary o formallyspecify the set of assump-tions behinda particularR-ARmodel, and then determine f and when it ispossible for these assumptions o generatethe set of assumptionsbehind aparticularGT-ARmodel.I showed, forexample, f andwhencertainassump-tions behind the R-AR model in (2.1) and (2.2) formally imply the set ofassumptionsbehind a PD gamein Exhibit 1.Whatsorts of assumptionswere involved?R-AR models makeassump-tions abouta nation'sutility functiondefinedover its goals. GT-ARmodelsmake assumptionsabouta nation'spreferenceorderingsdefined over thepossibleoutcomes.Theabilityof Richardson's ssumptionso determine heequilibrium utcomes of 2 x 2 gamesboils down tothecompatibilityof theR-ARmodel'sassumptionsaboututilityovergoalswith theGT-ARmodel's

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assumptionsabout preferenceover outcomes. Finding the compatibilitybetween the two differentsets of assumptions s the same as discoveringwhether a particularRichardson-type tility functioncan producethe pref-erence orderings(and the Nash equilibria)of a particular2 x 2 game. Thedebate over the propergame representation f an armsrivalryis really adebate over the properutility functions and preferenceorderingsof thenations nvolved. Nationshave many differentmotives,andthereforeutilityand preferences,for the possession and nonpossessionof arms.Differentutilityfunctionsproducedifferentgamesthathave different mplications orthe outcomesof armsrivalries.

Twotypesof resultsoccurred.First,one set of R-ARassumptionsmpliedanother et of GT-ARassumptions. twas possible, forexample, to derive aPD preferenceordering and hence an equilibriumoutcomethat is an armsrace)from a particularRichardsonmodel. Inthis case, I developedproposi-tions of the form: GivenassumptionsA and conditionsB, if R-ARmodel Cholds, then GT-AR model D holds. The R-AR model, in other words, issufficientfor the GT-ARmodel, andthe GT-ARmodelis necessaryfortheR-AR model.Second,a relationship etweenthetwomodelingtraditionsdidnot hold:The two sets of assumptionswereinconsistent. f aparticularR-ARrepresentation f an arms rivalry s not consistentwith a particularGT-ARrepresentation, hen making one set of R-AR assumptionswill precludeanother et of GT-ARassumptions. n thiscase, I developedpropositionsoftheform:Given AssumptionsA, if R-AR modelCholds,thenGT-ARmodelD cannothold.Theutilityfunctionof aparticularR-AR modelmightbe inconsistentwiththe preferenceorderingsof a particularGT-ARmodel. For example, therequirementsora particularR-AR may,undercertainconditions, mplytherequirements or a GT-AR to be a PD game, but be inconsistentwith therequirementsora GT-AR o be a Chickengame.A R-AR modelmay implyone GT-ARmodel andbe inconsistentwithanotherbecause theset of utilityfunctions for a R-AR may notbe able to yield both a single equilibrium,asin PD, andtwoequilibria,as in Chicken.Similarly,a set of Richardson-typeutility functionsmay not be able to yield botha pareto-dominated quilib-rium,as in PD, and a pareto-optimalquilibrium,as in StagHunt.This analysis has revealed the importanceof explicating the preciseRichardson-typeutility functionsin a GT-AR,rather hansimply positingpreferenceorderings. Several informalconjecturesaboutthe relationshipbetween armsrivalriesand2 x 2 gameswereshownto be incorrect.Themoreexactinganalysisadoptedherehighlightedhe substantivemeaningof GT-ARs.As a resultof the analyses,externalthreat and internal atigue, previouslyconsideredonly in R-AR formulations,are now also examinedin GT-AR

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formulations Lichbach,1989). These resultsaremoreprecise andrelevantto the armsoutcome question than those offeredby the previousinformaltreatments f theconnectionbetweenarmsrivalriesandPDs.Showingif andwhen a R-ARbecomes a GT-ARgreatlyenhancesthe substantiventerpre-tation and theoreticalrelevance of the latter.Unless we explore the utilityfunctions behindthe payoff functions of the nationsinvolved in a GT-AR,these issues are hidden.

APPENDIXThefollowingR-ARrequirementsorGT-ARconditions o hold areusedinproofsof the propositionsin the sections discussing the two specific R-AR models andconclusionsaboutGT-ARmodels.

R-ARGT-AR B2-AR B3-AR

Px (D,C) > Px (CC) -> (X1 + X2) |1l| [ X1+ X2] PYkx kXY2Y1 222

Px (C,C) > Px (D,D) - > (X1 + X2) + ( ) 1X(X2 - X1 ) > kX(X2 - PY2)- kX(Xl - pYj)ix iX(X2- Xi)

Px (D,D) > Px (C,D) > (X1 + X2) [its] [X1 +X2} [ 2

Px (C,C) > Px (C,D) Y2 > Y1 Y2 > Y1

Px (D,C) > Px (D,D) Y2 > Y1 Y2 > Y1

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