This article was downloaded by: [Simon Fraser University]On: 16 November 2014, At: 01:22Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

International Interactions: Empiricaland Theoretical Research inInternational RelationsPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/gini20

Disarmament: A theoretical analysisVally Koubi aa Department of Political Science , University of Georgia ,Athens, GA, 30602Published online: 09 Jan 2008.

To cite this article: Vally Koubi (1994) Disarmament: A theoretical analysis, InternationalInteractions: Empirical and Theoretical Research in International Relations, 19:4, 263-281, DOI:10.1080/03050629408434831

To link to this article: http://dx.doi.org/10.1080/03050629408434831

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoeveras to the accuracy, completeness, or suitability for any purpose of the Content. Anyopinions and views expressed in this publication are the opinions and views of theauthors, and are not the views of or endorsed by Taylor & Francis. The accuracy of theContent should not be relied upon and should be independently verified with primarysources of information. Taylor and Francis shall not be liable for any losses, actions,claims, proceedings, demands, costs, expenses, damages, and other liabilitieswhatsoever or howsoever caused arising directly or indirectly in connection with, inrelation to or arising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms& Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

International Interactions, Vol. 19, No. 4, pp. 263-281Reprints available from the publisherPhotocopying permitted by license only© 1994 Gordon and Breach Science Publishers S. A.Printed in the United States of America

DISARMAMENT: A THEORETICAL ANALYSIS

VALLY KOUBI

Department of Political ScienceUniversity of GeorgiaAthens, GA 30602

(Received for Publication November 19, 1993)

The disarmament of rival nations can only take place when all countries involved are made better off and a feasibil-ity criterion is also satisfied. Disarmament is more probable when nations face a similar trade off in the acquisitionof weapons, when developments in military capabilities can be speedily and accurately verified, and when the ex-isting weapon systems are easily reproducible. A successful disarmament also requires that the speed of the dispo-sition of arms be neither too fast nor too slow.

KEYWORDS: disarmament, desirability, feasibility and speed of disarmament

The phenomenon of armament—and in particular its extreme form of the arms race—hasbeen studied extensively in the literature, both theoretically and empirically (Richardson,1960; Brito, 1972; Intriligator, 1975; Intriligator and Brito, 1976; Brams et al, 1979; Brams,1985; Isard, 1988). Its antipodal phenomenon of arms reduction, on the other hand, hasdrawn considerably less attention (Brams and Kilgour, 1988; Downs and Rocke, 1987,1990; Koubi, 1993). The lack of interest is especially astute for a special case of arms reduc-tions which stands on the polar opposite to arms race, that of disarmament. The typical justi-fication for this uneven treatment is that disarmament does not seem to be empiricallyrelevant.

The historical record, however, as manifested in the low frequency of disarmament casesis not on its own sufficient to justify the lack of work on this issue. On the contrary, it is notuncommon for researchers to investigate the existence of conditions that are needed to ruleout the occurrence of certain important events (impossibility theorems abound in all fields ofscientific endeavor). Moreover, there exist good practical reasons for studying disarmamentthat arise from the existence of a strong, positive correlation between arms and conflict. Theexploration of the factors that prevent human societies from disarming themselves mayprove valuable for understanding the process of limited arms reduction. In this sense, amodel of disarmament can serve as a useful benchmark paradigm for arms control. The con-tribution of the present paper lies in its formal attempt to develop insights on what makes

263

Dow

nloa

ded

by [

Sim

on F

rase

r U

nive

rsity

] at

01:

22 1

6 N

ovem

ber

2014

264 VALLY KOUBI

nations want to disarm, and also on what determines whether such desires can be imple-mented.

From a methodological point of view, I find it useful to organize my analysis around theconcepts of desirability and feasibility. Disarmament is said to be desirable when nations arebetter off without any arms rather than with—any positive level of—arms. To determine de-sirability one has to compare the "welfare" level associated with the equilibria with andwithout arms. Feasibility, on the other hand, refers to the existence of a path of arms reduc-tion that connects the two equilibria. A feasible path is characterized by the property that, ateach point, the disarming nations have an incentive to continue with further arms reduc-tions.1 For rival nations to dispose of their weapons it is necessary that disarmament is bothdesirable and feasible. To account for the lack of occurrence of disarmament one then needsto explain why these two properties are not satisfied.

The evaluation of the desirability of disarmament requires the calculation of the net bene-fits (benefits minus costs) associated with alternative arms levels. There are several ways ofdoing this, the most popular involving first the aggregation over the various ingredients ofthe costs and benefits, and secondly, the use of a utility—or objective—function to assignpreference weighs to these two groups (for instance, see Downs and Rocke, 1990, p. 80). Iadopt a standard Richardsonian specification purely for the sake of convenience, but myanalysis is robust to alternative methods of carrying out cost-benefit evaluations.

The main insight obtained is that desirability depends critically on the existence of asym-metries across countries in the ratio of the parameters used to weigh the benefits and costs ofweapons (the utility trade off). It is important to stress that it is not absolute differencesacross countries that matter, but only relative ones. For instance, consider two countries, theUSAF and the USSRF. Let the USSRF place great value to enjoying military superiority (a"hawk") while the USAF is less concerned about possessing a military advantage (a"dove"). If the economic cost of development and production of weapons is significantlyhigher in the USSRF than in the USAF, then enthusiasm in the USSRF for military build upsmay be dampened by the high price it has to pay in order to satisfy its appetite for armament.In these circumstances, disarmament can be desired by both countries.

Now suppose that the two rival nations are sufficiently symmetric so as to satisfy the de-sirability criterion suggested in this paper. Will they be able to dispose of their weapons? Andif yes, how fast? The reason one may offer a negative answer to the former question is that,while disarmament can be mutually beneficial when pursued by both countries, unilateral—or uneven—reductions in arms compromise relative military capabilities and can make acountry worse off. It is the existence of strong temptation to gain a military advantage bycheating that may undermine the existence of a feasible disarmament path.

To address the questions raised above I examine the role of several pertinent factors thathave been previously identified as important by the arms control literature. In the model,verification and arms reproducibility receive special scrutiny.

The inability to observe developments in the military capabilities of other countries is amajor obstacle to undertaking arms reduction activities. No country likes to face big, un-pleasant surprises. A similarly critical role is played by the speed of arms production."Cheating" by any country—say, in the form of hiding rather than destroying weapons—isnot effective in tilting the balance of power when it is economically and technologically fea-sible to replace disposed of weapons almost instantaneously. With immediate reproducibil-ity available, a disarming nation does not expose itself to major risks. It can thus be argued

Dow

nloa

ded

by [

Sim

on F

rase

r U

nive

rsity

] at

01:

22 1

6 N

ovem

ber

2014

DISARMAMENT: A THEORETICAL ANALYSIS 265

that the feasibility of disarmament hinges critically on the nature of weapon technology.While the type of weapon systems has received much attention in the arms control literature,I am not aware of any previous attempts to link these two concepts along the lines proposedhere. Interestingly, this link offers a useful guide for selecting weapons for elimination inarms reduction negotiations. Weapons that can be reproduced easily are more likely to beaxed. An additional empirical implication is that richer and technologically advanced na-tions may be more inclined to engage in arms reduction activities because they have greaterability to substantially reverse the process.

The study of the influence of verification and reproducibility on the prospects for disar-mament generates some intuitively expected results. Better verification and faster repro-ducibility enhance feasibility and also increase the feasible speed of disarmament. Theanalysis, however, goes beyond these standard results by generating a prediction that seemsquite novel and also interesting. I show that the feasible speed of disarmament falls within arange of values, and that neither too fast nor too slow a speed is compatible with feasibility.That too fast a rate may not be feasible is intuitively clear (and also well known) because,under imperfect verification, it can impose excessive risks upon an "honest" nation. That tooslow a rate can undermine feasibility, on the other hand, does not seem that obvious and hasnot appeared in the literature before. I account for this finding by claiming that the futurebenefits associated with a successful but very slow disarmament process are discountedheavily—because they take very long to materialize—while the current benefits from abort-ing disarmament by cheating are low but immediately available.

THE MODEL

Disarmament can take the form of either an explicit formal agreement or can be the result oftacit bargaining (as in Downs and Rocke, 1990). I have chosen to use the former but theanalysis is also applicable to the latter case.

The assumptions employed can be grouped into two categories, behavioral and struc-tural.

A. Behavioral Assumptions

1. The disarmament decision-making is dominated by a single, strong leader. Accordingto Bueno de Mesquita, 1981, in The War Trap, a strong leader can cause a nation to act as if itwere a unitary actor,2 in the sense that although "no leader can completely ignore the desiresand interests of those who follow him...still, it is ultimately the responsibility of a singleleader to decide what to do and how to do it" (Bueno de Mesquita, 1981, p. 28). AlthoughBueno de Mesquita presents historical evidence to support his assertion in the case of warinitiation, one can also find many instances of arms control agreements where the processhas been dominated by a strong executive. In any case, this assumption can help to avoiddealing with internal political conflicts involving competing parties, objectives, etc., thatwould possibly turn armament/disarmament outcomes and preferences into a policy instru-mental in the political scene3 (see Downs and Rocke, 1990, for additional justification forthis assumption).

2. Leaders are rational utility maximizers, in the sense that they never choose an actionthat is expected to produce less utility than some other alternative. Since arms control and

Dow

nloa

ded

by [

Sim

on F

rase

r U

nive

rsity

] at

01:

22 1

6 N

ovem

ber

2014

266 VALLY KOUBI

disarmament strategies involve high security risks (unilateral arms reductions give an ad-vantage to one's adversaries; also inadequate or ineffective verification to a treaty makescheating profitable), a leader would neither adopt a unilateral policy of arms reductions norsign a treaty to reduce weapons or constraint the development of new if the net expectedbenefits from these actions are less than those that would arise if they were not adopted.'*

B. Structural Assumptions5

1. Each nation has complete information about the preferences and capabilities of theother, but it may temporarily have imperfect information regarding the current level of armsin the other country.

2. Verification takes s units of time to be completed. Moreover, there is no gradual accu-mulation of information, that is the results of verification activities become available at theend of the sth period.

Verification has always been the most important issue in arms control because it is theprocess by which one party to an agreement determines the degree to which the other party iscomplying with the provisions of the agreement. The actual process of verification involvestwo distinct phases. The first one is monitoring. Monitoring is a technical exercise aiming atdetecting, identifying, and measuring military developments and activities pertinent to thetreaty partners' behavior regarding the particular agreement (Meyer, 1984). The real verifi-cation of the agreement occurs in the second phase when the collected data is analyzed andinterpreted in order to determine whether the treaty has in fact been violated. Since, both thequality of the monitoring equipment6 and the interpretation7 of the data significantly influ-ence the outcome of the verification process but at different time, I will not consider theirindividual effects on arms control here, but rather their combined final outcome, in order toprevent the analysis from getting too complicated. Finally, the verification time is a constantindependent of the initial stocks of weapons and/or the speed of disarmament.

3. A violation of a disarmament agreement takes s periods to be detected when cheatingtakes the form of non-elimination of the number of weapons provided by the treaty, but it isimmediately detectable when it involves actual military buildup.

4. a) There is a technologically and/or economically maximum feasible rate of rearma-ment, Z.

It is indisputable that the economic, technological, and political constraints imposed onarmament, limit the rate at which a country can rearm itself after the detection of a violationof an arms control agreement. Ikle 1961, for example, argues that if the American and Soviettroops had been withdrawn from Europe during the Cold War years, the United States mighthave found it very difficult to return if the Soviet troops moved back. The difficulty in USredeployment is attributed by Ikle to the possibility that the American troops might havebeen demobilized in the meantime, to high transportation costs, and more importantly, to thelikelihood that the American public might have been unwilling to support such a move if itfelt that the probability of a war had increased.

This assumption reveals the importance of the reproducibility of weapons. Production ofsome weapon systems is much more costly and time consuming than that of others. Coun-tries will be more reluctant to eliminate vital weapons which are not easily reproducible, es-pecially when monitoring and verification are imperfect. Easily produced arms are moreexpendable and hence easier to include in arms control agreements. Technological break-

Dow

nloa

ded

by [

Sim

on F

rase

r U

nive

rsity

] at

01:

22 1

6 N

ovem

ber

2014

DISARMAMENT: A THEORETICAL ANALYSIS 267

throughs that increase reproducibility may improve the prospects for significant arms reduc-tion.

b) The technologically and economically feasible rate of disarmament, is also restrictedto be below a maximum bound. Moreover, it is assumed that up to that level it is costless todestroy weapons.

I postulate that the leader's utility function incorporates two factors. One is the country'srelative military strength, which is assumed to be a linear function of the difference in weap-ons stocks between the two countries. This implies that the leader views favorably militarysuperiority, whether this is for defensive purposes (such as deterrence of the rival) or for of-fensive purposes (expansion). At the same time, armament is a costly process, because of theexistence of a limited quantity of economic and human resources (guns vs. butter). I assume8

that the cost of armament is quadratic.9

These two considerations lead to the following equations10 (assuming two countries, Aand B):

WAt = <xt (ut - vt) - p £ (1)

V 2WBt = a,' (vt - ut) - pt*^- (2)

where WAt(WBt) is the welfare of country A (B) at any time t associated with levels of arms ufor country A and v for country B. at(ut - vt) captures the importance country A attaches to

enjoying a military advantage vis-a-vis country B, and PtU(/2 reflects the perceived cost of

armament (maintenance and production). A similar interpretation applies to at (vt — ut)and

pt*v^/2 of country B.

In order to study disarmament one must explain the initial conditions, that is, how thecountries under consideration found themselves with the existing level of arms. I will nowsketch a simple model of armament.

Suppose that there are two countries only, and that both countries start with zero arms, u =v = 0 (see equations (1) and (2)). I will examine the conditions that will lead a country toinitiate a military buildup. That is, the conditions that will make a country prefer to hold armsindependently of the other country's actions. Assume that changes in armament are detect-able immediately and that there are no time lags in the construction of weapons, so that anydevelopment in one country can be immediately reacted to by the other country. I will relaxthis assumption of perfect monitoring below when I discuss disarmament. To determine theoptimal choice of arms one must compare the welfare functions that result from various al-ternatives. First note that if the optimal u(v) is greater than zero, i.e. there is armament, then itnecessarily takes the value of u= cc/p (v = a*/p*). This is because WA(OC/P) >WA (U), for u > 0. Second, if a country is armed, then the other country will always behavelikewise. Hence both countries will be either armed or unarmed (because WA[U = oc/p, v =<x*/p*] > WA[U = 0, v = a*/p*]). The level of welfare then associated with the two possibili-ties (zero arms or arms) is given by W^ = Wj = 0 under the former and

Dow

nloa

ded

by [

Sim

on F

rase

r U

nive

rsity

] at

01:

22 1

6 N

ovem

ber

2014

268 VALLY KOUBI

"* (" -?• " S O

under the latter possibility.From these two conditions it can be seen that country A will be better off with arms (inde-

pendent of what the other country does) if a/ P > 2 (a* / P*). Similarly, country B will preferarms if a*/p* > 2 (oc/P). If either of these two inequalities holds, then the two countries willend up with the level of arms that corresponds to the equilibrium in the static prisoner's di-lemma game. In other words, an arms build up is inevitable if the two countries are suffi-ciently different (asymmetric11). A country with a relatively higher benefit to cost ratio ofarmament initiates an arms build-up, and forces the other country to respond by building itsmaximum level of weapons. Note that more hawkishness (a > a*) and/or economic superi-ority (P < P*) do not suffice to generate an equilibrium with armament. It has to be the casethat, given similar economic conditions (P = p*), one country is at least twice as hawkish asthe other; or given similar attitudes toward military advantage (a = a*), an economy is atleast twice as efficient as the other in acquiring weapons.

Since hawkishness per se in both countries does not necessarily lead to an arms build-up,an interesting possibility emerges. It may be optimal for a dovish leader whose dovishness isnot publicly known, to behave as a hawkish one (by pretending that it has a large a) as ameans of discouraging an arms race that it does not like. Under certainty about the leadertype, a leader who prefers low levels of arms may embark on a—temporary—massive buildup as a means of convincing the adversary that he attaches great value to military superiority.

If monitoring (verifiability) is imperfect so that it takes s periods to be completed, then therelevant condition for A to initiate an arms race becomes oc/p > 25s (a*/ P*) where 8 is thetime discount factor (0 < 8 < 1). This condition is derived as follows. Suppose that A claimsto be unarmed while in reality it is armed. If it takes country B s periods to detect this cheatingand to react by arming itself, then A's welfare will be given by

The first term gives A's welfare before cheating is detected, and the second gives A's welfareafter B has detected the cheating and has responded to it. Recalling that the welfare levelassociated with zero arms for both countries is zero we have that armament is preferred tozero, arms if the above expression is positive. After some manipulation this is seen to requirethat

Dow

nloa

ded

by [

Sim

on F

rase

r U

nive

rsity

] at

01:

22 1

6 N

ovem

ber

2014

DISARMAMENT: A THEORETICAL ANALYSIS 269

l) - ( f T [(1̂ 6) J > 0

which implies oc/p > 2(a*/p*)8s. The intuition behind this condition is that a country mayfind it optimal to initiate armament solely because of opportunism reasons—the fact that itwill reap a benefit for s periods—even when the symmetry condition under complete in-formation does not favor armament initiation.

Now let us start from a situation with a/ P > 2 (a*/p*) (so that country A prefers and henceinitiates a military build-up), and consider a change in the economic and/or political environ-ment in A (or B), that makes both countries better off in a zero arms equilibrium (that is, achange that produces oc/p < 2 (a*/p*) and cc*/p* < 2(cc/ P). Then disarmament becomes de-sirable. The question is whether it is also feasible.

Consider the following disarmament scenario. Nations A and B agree to disarm at the rateK and M respectively (K and M are the fractions of arms that are removed in each period). Ifthere is a deviation from the agreed upon disarmament path (which is detected s periods lat-er) then there is a punishment. The punishment can take many different forms. A popularone, that has been used extensively in the trigger strategy literature—and is also renegoti-ation proof—involves the reversion to the equilibrium of the one shot game. In my model,such a reversion takes the form of the countries arming themselves at the maximum feasiblerate Z. Note that my mode of analysis can easily accommodate alternative punishment rules.

The following notation is used:

AC BCW t (W t ) denotes the welfare for country A (B) at time t, when both countries con-form to the disarmament agreement.

AN BNW t (W t ) denotes the welfare for country A (B) at time t, when country A (B) iscurrently cheating but it has not been detected yet.

W t (W { ) denotes welfare for country A (B) after detection has been made and as thecountries move towards the single shot equilibrium.

K is the rate of disarmament in country A.

M is the rate of disarmament in country B.

Z is the maximum feasible rate of armament.

Using (1) and (2) we have that if country A honors the disarmament agreement then itenjoys the payoff (given that B conforms to the treaty):

bl W, (3)t = 0

ACwhere 8 is the discount factor and W • is defined as follows:

Dow

nloa

ded

by [

Sim

on F

rase

r U

nive

rsity

] at

01:

22 1

6 N

ovem

ber

2014

270

wfj

w A Cw t + n

VALLY KOUBI

- Kt)ut - (1 - Mt)vt] - ip[(l - Kt)ut]2

: i - K t + 1 ) ( l - K t ) u t - ( l - M t + 1)vt]

i p [ ( l - K t + 1)( l -K t)u t]2

I - Kt + n)...(l - Kt)ut - (1 - Mt + n) .. (1 - Mt)vt]

ip[(l - Kt + n)...(l - Kt)ut]2

(4)

where (1 - Kt)ut is the quantity of arms left over after one period of disarmament in country

A, and so on (the W W . s is similarly defined).If A starts cheating in period n then its welfare is given by:

w A =1-0 i=0 i = 0

i w A

t+n+s+i(5)

The first term in (5) measures welfare before cheating, the second term welfare during cheat-ing (for s periods) and before detection, and the last term welfare after cheating has beendiscovered.

Country A will choose to deviate from the disarmament agreement if (5) exceeds (4) (asimilar argument applies to country B). To make the analysis more transparent I will restrictmy attention to a special case characterized by the condition that the rate of disarmament isthe same in both nations and equal to a constant Mt = Kt = K. Let k = 1 - K and z = 1 + Z. If Adoes not cheat, then (4) reads

w?c = «[k(f - a_

k2[ak u a_

so that (3) can be written as

1 - 8 ka,a 1 k2

2 1 -dk2 (6)

If country A cheats, then the analog to (4) under cheating is

Dow

nloa

ded

by [

Sim

on F

rase

r U

nive

rsity

] at

01:

22 1

6 N

ovem

ber

2014

DISARMAMENT: A THEORETICAL ANALYSIS 271

w

(7)

AN

WA - a ( a _ z r k s o M I B / a ^ (10)

P

/ \ / \ 2

(ii)

According to (7), country A's military superiority increases at the aftermath of an agree-ment to disarm because it cheats while country B is disarming. By period (t+s), country A hasachieved the maximum possible military advantage (8)). In period t+s it gets caught, theagreement collapses, and country B tries to make up by starting to arm itself at the maximumfeasible rate z ((9)). For the sake of simplicity I assume that it takes r periods to rebuild to thelevel corresponding to the equilibrium in the single shot game ((11)).

Disarmament is feasible if (3) is greater than (5). Using (7-11) in (5), and (6) in (3) wehave that disarmament is feasible if g > 0, where g is given by

g(k,z,a,a*,p,P*,S,s,r)=_ k a aa P £ -1 k2 a1

2 1 - <5k2 /?' J

2 1 - 6 p + p [k 1 - <5k +Zdk \-6z + 1 - j(12)

and the country B analog to (12) is satisfied.

Dow

nloa

ded

by [

Sim

on F

rase

r U

nive

rsity

] at

01:

22 1

6 N

ovem

ber

2014

272 VALLY KOUBI

Since condition (12) is too complicated to allow for the analytical treatment of the inter-actions between the various arguments and for the explicit determination of the feasibility ofdisarmament, I have resorted to two alternative strategies. First, in order to get some intu-ition about the role of the various parameters in the determination of the feasibility or thespeed of the disarmament process, I have employed specialized (restrictive) versions of (12)that are susceptible to analytical manipulation. Second, I have used numerical illustrations toexamine how the length of the verification interval, the time it takes to rebuild arms, and themaximum feasible rate of rearmament affect both the likelihood of a successful disarma-ment and its speed. The results are given in graphs 1-7 and will be described shortly.

Let me start with the role of the maximum economically feasible rate of rearmament. Iwill argue that the value of this parameter is critical only for the speed of disarmament. Ifarmament and disarmament were completely symmetric in terms of economic resources (r=1), then it would be optimal for the countries involved to complete disarmament within oneperiod (if, of course, disarmament was feasible at all). This is the case independent of thevalues of s, 8, etc. because those parameters affect the feasibility but not the optimal speed ofdisarmament. To see this, consider the simplest possible case with s = 1. Suppose that, as aresult of asymmetries favoring country A (i.e. (oc/P) / (oc7(P*) > 28), the two nations arepresently armed (i.e., u = a /p , v = a*/p*), and allow for a change in the environment thatmakes disarmament feasible. That is, let

/a /a* . /.tp/pv > 25 (13)

If the two countries completely dismantle their weapons within a single period after theagreement, then WA = WB - 0. If country A cheats, then it enjoys a temporary military ad-vantage until the cheating is detected. Welfare for country A is then given by

AN (a \ i (a\ _ 1 a 2

_ 1 a2

Adding up gives that country A will have an incentive to cheat if

aa*

wAN

1 1 - 8 " «

Simplifying equation (14) gives

2 pl a 2

1-8U p > 0 (14)

s / a ; > 25

which, however, is not possible because the derivation of (14) is based on the assumptionthat equation (13) holds. A similar argument can establish the same result for s > 1.

Dow

nloa

ded

by [

Sim

on F

rase

r U

nive

rsity

] at

01:

22 1

6 N

ovem

ber

2014

DISARMAMENT: A THEORETICAL ANALYSIS 273

One can also use (12) to draw conclusions on the feasibility of disarmament under ex-treme values for 5 and s. It is apparent that if either 5 = 0 or s = °° no arms reduction can evermaterialize. In the former case there is complete disregard of the future, so cheating is inevi-table. In the latter case cheating does not ever get detected, which makes cheating irresistible(recall that, in the model, rational leaders respond to incentives only, not to unspecified mor-als).

It is easy to see that the possibility of initiating a disarmament is decreasing in the lengthof the verification interval. For instance, if detection takes two periods, then disarmamentcan be implemented if (a/ P) / (a*/ p*) < 252. Since 8 < 1, this condition is more stringent (interms of required symmetry) than condition (13), which applies when it only takes one peri-od to verify compliance to an arms control agreement. In other words, a short verificationperiod makes disarmament feasible even when the countries are fairly asymmetric.

NUMERICAL ILLUSTRATIONS

Let me now turn to a numerical example. Expression (12) is not tractable because the time ittakes to rearm following the detection of cheating, r, is a function of z as well as of the timingof cheating. I have chosen to make a simplifying assumption that does not affect qualitative-ly the results. Namely, I have assumed that if the verification process detects a violation, thenthere is no military build-up response for r periods. But once r periods have passed, then the"honest" nation can rearm itself instantaneously. In other words

zs + i = 0,i = l ,2, . . . , r and zs + r + 1 = (1 + Zs + r) = i (15)

Notice that this assumption involves nothing more than a step-wise approximation to thecontinuous response function postulated earlier. With this specification, the term in the se-cond pair of square brackets in (12) takes the very simple form of (12')

and the derivative of g with regard to r is given by

(16)

According to (16) an increase in the length of time that it takes to respond to cheating low-ers the likelihood of disarmament. Also, the smaller the feasible rearmament rate, the lesslikely that disarmament will be feasible.

The results of the numerical exercise are reported in graphs 1—7. Each graph has the feasi-bility of disarmament (the function g) measured along the vertical axis and one or more of thearguments of g measured along the horizontal axis(-es). The values of the variables that are

Dow

nloa

ded

by [

Sim

on F

rase

r U

nive

rsity

] at

01:

22 1

6 N

ovem

ber

2014

274 VALLY KOUBI

not included in the graphs are held constant. Throughout the analysis, I use the followingfixed values12:

a = p = a* = 1, p* = 1.2, 5 = 0.98

Graphs (1) and (2) depict g as a function of verification (s) and the lag of response (r),respectively. As expected, the feasibility of disarmament is a decreasing function of both sand r. An increase in either s or r means that the benefits arising from a surprise military ad-vantage are larger as it takes longer to either detect or counter it.

Graphs (3) and (4) produce a most interesting result. Recall that k = 1 - K, that is k is oneminus the rate of disarmament. According to the graphs, the relationship between the pros-pects for (feasibility of) disarmament and the rate of disarmament is not linear. Feasibilityincreases with a decrease in the rate of disarmament until it reaches a maximum value, andthen it starts decreasing when disarmament proceeds at a low pace (to the right of the maxi-mum value). Consequently, one may argue that disarmament is more likely to succeed whenit involves moderate levels of arms reduction. It may be infeasible when it proceeds at eithertoo fast or too slow a pace. The intuition behind the potential lack of feasibility with too fastrate of disarmament is straightforward. Suppose that two nations agree on a precipitous planof arms reduction. If one of the two countries cheats, then the substantial unilateral arms re-duction results in a significant differential in military advantage favoring the dishonest partyfor several periods (r). The recognition of this incentive makes disarmament infeasible.

The intuition behind the second part of the assertion is not so straightforward. A symmet-ric, slow rate of disarmament means that there are no significant gains from arms reduction inthe short and medium run, as reductions are very small. There are significant benefits in thelong run when both countries have fully disarmed themselves, but those benefits are dis-counted (because 8 < 1). On the other hand, the benefits from cheating at any point in time arealso low (because the "honest" country's unilateral decrease in military strength is very lim-ited due to the slow pace of arms reduction) but those benefits are reaped almost immediate-ly, and hence, discounting plays a smaller role. It is this consideration that makes too slow arate of disarmament not worthwhile to implement.

A comparison of graphs (3) and (4) also reveals that the chances for disarmament de-crease with a longer s and low r. The graph of the g function shifts to the right, which impliesthat disarmament now is harder to occur (recall that g must be positive for disarmament to befeasible; a downward shift of g then makes it more difficult to have a positive g).

Graphs (5), (6), and (7) offer three-dimensional representations of graphs (1), (2), and (3).Graph (5) suggests that s and r play a different role in evaluating the feasibility of disarma-

ment. While both s and r work in the same direction—as both discourage disarmament—thefeasible rate of rearmament seems to play a less important role. In graph (5), g falls fasterwith increases in s rather than with r. A comparison of graphs (1) and (2) reveals a similarpattern. Holding everything else fixed and imposing symmetry (i.e. either r =6 or s = 6), onecan see that the plot of the g function against s lies uniformly below the plot of the g against r.This difference in importance is due to the fact that there is discounting in the model (5 < 1)and that the potential losses due to a long verification interval come earlier from the lossesfrom long delayed rearmament response.

Finally, note that once feasibility has been established (that is g >0), for a certain range ofrates of disarmament, then the optimal rate is identical to the maximum feasible rate. That isdue to the fact that when disarmament is desirable, having arms results in lower welfare than

Dow

nloa

ded

by [

Sim

on F

rase

r U

nive

rsity

] at

01:

22 1

6 N

ovem

ber

2014

DISARMAMENT: A THEORETICAL ANALYSIS 275

GRAPH 1Verification (s) and the Feasibility of Arms Reductionsk = 0.9, r = 3

GRAPH 2Speed of Rearmament (r) and the Feasibility of Arms Reductionsk = 0.9, s = 3

21.4

21.2

21

20.8

20.4

Dow

nloa

ded

by [

Sim

on F

rase

r U

nive

rsity

] at

01:

22 1

6 N

ovem

ber

2014

276 VALLY KOUBI

GRAPH 3Speed of Disarmament (k) and the Feasibility of Arms Reductionss = 2, r = 1

22.4

22.2

22

21.8

21.6

21.4- ^T-" ' ' k0.4 0.6 0.8

21.2

GRAPH 4Speed of Disarmament (k) and the Feasibility of Arms Reductionss = 4, r = 4

20

19

18Dow

nloa

ded

by [

Sim

on F

rase

r U

nive

rsity

] at

01:

22 1

6 N

ovem

ber

2014

DISARMAMENT: A THEORETICAL ANALYSIS 277

GRAPH 5Verification (s), Speed of Rearmament (r) and theFeasibility of Arms Reductionsk = 0.9

GRAPH 6Verification (s), Speed of Disarmament (k) and theFeasibility of Arms Reductionsr = 3

22.5

9 17.5

15

12.50.2 0.4 0.6

Dow

nloa

ded

by [

Sim

on F

rase

r U

nive

rsity

] at

01:

22 1

6 N

ovem

ber

2014

278 VALLY KOUBI

GRAPH 7Speed of Disarmament (k), Speed of Rearmament (r) and theFeasibility of Arms Reductionss = 3

16 •

1 4 •

0.4 0.6 0.8

not having any. A fast rate of disarmament is then the best plan as it moves the two countriesto the preferred position faster and hence, it eliminates faster the costs associated with arma-ment.

MODIFYING THE STRUCTURAL ASSUMPTIONS

Some of the assumptions made may not seem realistic, but I will argue that their replacementwith more general, more realistic ones, will only affect the results in a predictable way.

I have postulated that verification takes exactly s periods with no information being re-vealed until the end of its time interval.13 Alternatively, one could have assumed that there ispartial, gradual accumulation of information which arrives either exogenously, or endoge-nously as a result of costly monitoring activities. In the latter case, one would need to set upproduction functions for the construction of weapons as well as for verification activities.While partial revelation and/or an endogenously determined s add another layer of complex-ity they do not alter the role of the size of s for the feasibility of disarmament. Partial revela-tion makes it harder to cheat and hence improves feasibility. High productivity inverification activities pushes S up and worsens disarmament prospects.

I assumed that if cheating takes the form of not destroying weapons as agreed upon, then ittakes s periods to be detected, while if it takes the form of arms building, then it is immediate-ly detectable. Alternatively, one can postulate that in the later case it takes s\ periods to de-tect, with si = s. If si > 0, then cheating becomes more appealing because it can tilt the

balance of power even more by increasing the level of weapons in the cheating nation beforeit gets detected. Feasibility prospects worsen, and the maximum feasible rate falls.

Finally, I assumed that it is costless to destroy weapons up to a maximum, feasible level.Alternatively, I can make arms destruction economically costly. Such a modification makesdisarmament less feasible because it increases its cost. The incentive to cheat is now strongerbecause a cheating nation avoids bearing the costs associated with arms reduction (whether

Dow

nloa

ded

by [

Sim

on F

rase

r U

nive

rsity

] at

01:

22 1

6 N

ovem

ber

2014

DISARMAMENT: A THEORETICAL ANALYSIS 279

those are indirect, e.g. increased unemployment; or direct, e.g. resources actually used up todestroy existing weapons).

CONCLUSIONS

I have constructed a simple model to study various issues pertaining to disarmament. Whilethe analysis has examined the desirability and feasibility of moving two rival nations from apositive level of arms to a zero level, as well as the properties of the transition path (e.g. thespeed of disarmament) the results are also applicable to partial arms reduction. If the modelhas more than the two equilibria postulated in this paper, and the equilibria are ranked interms of arms levels and welfare, then it is possible to apply the present theoretical frame-work to analyze a move from a high arms to a lower arms equilibrium. In this sense, completedisarmament is just the limiting case of the process of arms reductions. As a move to alowerarms equilibrium may be feasible for some equilibria and infeasible for some other, it wouldbe interesting to investigate the properties of arms reduction in a multiple equilibrium envi-ronment more thoroughly.

Acknowledgments

I am grateful to Bruce Bueno de Mesquita for many valuable suggestions. I have also bene-fited from the comments of two anonymous reviewers. All remaining errors are mine.

NOTES

1. The simple static prisoner's dilemma game (see Brams and Kilgour, 1988) may help illuminate the distinc-tion between these two concepts. If the two countries are identical in every respect, then the equilibriuminvolving positive levels of arms produces a lower level of welfare—or worse payoffs—than the equilibri-um with zero arms (hence, according to the definition of desirability, disarmament is desirable). The fact,however, that both nations would be made better off if they found a way to permanently eliminate theirweapons does not mean that they can actually do it. As, is well known, a zero arms equilibrium cannot beachieved in this static game (disarmament is not feasible).

2. Although the unitary actor assumption has been extensively used in International Relations, especially byscholars in the realist school (Morgenthau, 1973, Waltz, 1979; Bueno de Mesquita, 1981), it has not beenwithout criticism. Those who espouse the view of bureaucratic politics have argued that this characteriza-tion of the state is not accurate since individual bureaucracies, interest groups, and individuals formulate orinfluence foreign policy (Allison, 1971; Bueno de Mesquita and Lalman, 1992).

3. For example, it has been argued that "President Nixon wanted to time the SALT I agreements in a way thatwould help his 1972 political campaign" (Barton and Weiler, 1976, p. 6).

4. The Soviet Union, for example, rejected the Bamch Plan in 1946, on the grounds that the plan gave apermanent nuclear advantage to the US, since the US already had knowledge needed to construct nuclearweapons.

5. At the end of the section I discuss if and how the modification of these assumptions affects the results.6. Wittman (1989) argues that when the quality of the monitoring devices improve, the probability that a

violation will occur decreases.7. Meyer (1984) argues that verification is inherently subjective and judgmental because of differing inter-

pretations and because of its dependence on many other factors like the state of the relations of the partiesinvolved, international and domestic politics, assumptions regarding capabilities, intentions, and motives,etc.

8. A similar quadratic cost variable has been used before in modeling arms races by Downs and Rocke (1991),Gillespieetal. (1977).

Dow

nloa

ded

by [

Sim

on F

rase

r U

nive

rsity

] at

01:

22 1

6 N

ovem

ber

2014

280 VALLY KOUBI

9. It is a standard assumption in economics that cost functions are convex. Also, as more and more resources ofeconomy are devoted to military uses, there is less available for other uses (such as education, privateconsumption, etc.)- This effect is more than proportional to the amount of resources used because of dimin-ishing (social) marginal utility.

10. Note that objective functions similar to ones presented here have widely been used in the arms race litera-ture. Richardson's defense and grievance terms correspond to the relative military strength variable whilehis fatigue term corresponds to the cost of maintenance and production of weapons. McGuire (1965) talksabout the choice of arms as maximizing a national (social) welfare function which depends on nationalsecurity and resources devoted to non-military purposes. Intriligator and Brito (1976) take utility to be afunction of the resources available for civilian goods production and an index of defense. Downs and Rocke(1990) break down defense cost into maintenance and production.

11. Arms race models with other but related types of asymmetries (e.g. objective functions including differentresponse coefficients) have been used in the context of particular, world arms races (Wolfson, 1985; Lam-belet 1975) and also in 3-nation arms races (Hunter, 1980).

12. These particular values were chosen so that α/ß > α*/ß* that is, country A is the initiator, and the two ratiosare close enough to make disarmament desirable. It is straightforward to experiment with alternative val-ues.

13. The first assumption, namely that countries possess complete information regarding the preferences andcapabilities of themselves and their rivals is very difficult to relax within the context of a dynamic modellike that presented in this paper. If uncertainty is not introduced in a trivial way—in the sense that simpleexpectations replace the deterministic values and this is the end of the story—then one needs to worry aboutlearning about the own and rival's characteristics, type signaling and type camouflaging, etc. Althoughthese are very interesting issues they are much harder to treat within a multiperiod model.

REFERENCES

Allison, Graham (1971). The Essence of Decision. Boston: Little Brown.Barton, John H. and Lawrence D. Weiler (1976). International Arms Control: Issues and Agreements. Stanford,

CA: Stanford University Press.Brams, Steven J. and D. Mark Kilgour (1988). Game Theory and National Security. New York: Basil Blackwell.Brams, Steven J. (1985). Superpower Games: Applying Game Theory to Superpower Conflict. New Haven, CT:

Yale University Press.Brams, Steven, J ., Morton D. Davis, and Philip D. Straffin Jr. (1979). "The Geometry of the Arms Race." Interna-

tional Studies Quarterly, Vol. 23, pp. 567-588.Brito, Dagobert L. (1972). "A Dynamic Model of an Armaments Race." International Economic Review, Vol. 13,

pp. 359-372.Bueno de Mesquita, Bruce and David Lalman (1992). War and Reason. New Haven, CT: Yale University Press.Bueno de Mesquita, Bruce (1981). The War Trap. New Haven, CT: Yale University Press.Downs, George W. and David M. Rocke (1990). Tacit Bargaining, Arms Races, and Arms Control. Ann. Arbor:

The University of Michigan Press.Downs, George W. and David M. Rocke (1987). "Tacit Bargaining and Arms Control." World Politics, Vol. 39, pp.

297-325.Gillespie, John U., Dina Zinnes, G. S. Dallin, Philip A. Schrodt, and R. Michael Rubison (1977). "An Optimal

Control Theory of Arms Races." American Political Science Review, Vol. 71, pp. 226-251.Hunter, John E. (1980). "Mathematical Models of a Three-Nation Arms Race." Journal of Conflict Resolution, Vol.

24, pp. 241-252.Ikle, Fred C. (1961). "After Detection-What?" Foreign Affairs, Vol. 39, pp. 208-220.Intriligator, Michael D. and Dagobert L. Brito (1976). "Formal Models of Arms-Races." Journal of Peace Science,

Vol. 2, pp. 77-88.Intriligator, Michael D. (1975). "Strategic Considerations in the Richardson Model of Arms Races." Journal of

Political Economy, Vol. 83, pp. 339-353.Isard, Walter (1988). Arms Races, Arms Control and Conflict Analysis. New York: Cambridge University Press.Koubi, Vally (1993). "International Tensions and Arms Control Agreements." American Journal of Political Sci-

ence, Vol. 37, pp. 148-164.Lambelet, John (1975). "Do Arms Races Lead to War?" Journal of Peace Research, Vol. 12, pp. 123-128.McGuire, Martin C. (1965). Secrecy and Arms Race. Cambridge, MA: Harvard University Press.Meyer, Stephen M. (1984). "Verification and Risk in Arms Control." International Security, Vol. 8, pp. 111-126.

Dow

nloa

ded

by [

Sim

on F

rase

r U

nive

rsity

] at

01:

22 1

6 N

ovem

ber

2014

DISARMAMENT: A THEORETICAL ANALYSIS 281

Morgenthau, Hans J. (1973). Politics Among Nations. New York: Alfred A. Knopf.Richardson, Lewis F. (1960). Anns and Insecurity. Pittsburgh: Boxwood.Waltz, Kenneth N. (1979). Theory of International Politics. Reading, Mass.: Addison-Wesley.Wittman, Donald (1989). "Arms Control Verification and Other Games Involving Imperfect Detection." American

Political Science Review, Vol. 83, pp. 923-945.Wolfson, M. (1985). "Notes on Economic Warfare." Conflict Management and Peace Science, Vol. 8, pp. 1-20.

CONTRIBUTOR

Vally Koubi is an Assistant Professor of Political Science at the University of Georgia. Hercurrent research interests focus on the relationship between military build-ups and arms con-trol agreements; and on the impact of domestic economic performance on international ten-sions.

Dow

nloa

ded

by [

Sim

on F

rase

r U

nive

rsity

] at

01:

22 1

6 N

ovem

ber

2014