¹ Richardson, Lewis Frye, Arms and Insecurity (Pittsburgh: Boxwood Press, 1960), p. 12 Google Scholar.

² Alker, Hayward R. Jr., “The Structure of Social Action in an Arms Race” (paper prepared for delivery at the 1968 Conference of the North American Peace Research Society (International), Cambridge, Massachusetts, November, 1968)Google Scholar; Rapoport, Anatol A., “Lewis F. Richardson's Mathematical Theory of War,” Journal of Conflict Resolution, 1 (September, 1957), 249–299 CrossRefGoogle Scholar, and Fights, Games and Debates (Ann Arbor: University of Michigan Press, 1960), pp. 15–30 Google Scholar.

³ Numerous scholars have attempted to examine the degree of empirical fit between the Richardson model or modified Richardson models and armaments races or have suggested reformulations of Richardson's basic model. See, for example: Alcock, N. Z. and Lowe, Keith, ‘The Vietnam War as a Richardson Process,” Joumal of Peace Research, 6 (April, 1969), 105–112 CrossRefGoogle Scholar; Blumberg, Avron A., “Model for a Two-Adversary Arms Race,” Nature, 234 (November 19, 1971),158 CrossRefGoogle Scholar; Caspary, William R., “Richardson's Model of Arms Races: Description, Critique, and an Alternative Model,” International Studies Quarterly, 11 (March, 1967), 63–88 CrossRefGoogle Scholar; Chase, P. E., ‘The Relevance of Arms Race Theory to Arms Control,” General Systems Yearbook, 13 (1968), 91–98 Google Scholar, and “Feedback Control Theory and Arms Races,” General Systems Yearbook, 14 (1969), 137–149 Google Scholar; Chatterji, Manas, “A Model of Resolution of Conflict between India and Pakistan,” Peace Research Society Papers, 12 (1969), 87–102 Google Scholar; Ferejohn, John, “On the Effects of Aid to Nations in Arms Races” (Paper delivered at the Annual Meeting of the American Statistical Association, December, 1970)Google Scholar; Friberg, Mats and Jonsson, Dan, “A Simple War and Armament Game,” Journal of Peace Research, 5 (July, 1968), 233–247 CrossRefGoogle Scholar; Intriligator, Michael D., “Some Simple Models of Arms Races,” General Systems Yearbook, 9 (1964),143–147 Google Scholar; Ivanilov, Yu. P., “A Model of Competition between Two Countries,” Engineering Cybernetics, 1 (Jan.-Feb., 1969),19–30 Google Scholar; Kupperman, Robert H. and Smith, Harvey A., “Strategies of Mutual Deterrence,” Science, 176 (April 7, 1972), 18–23 CrossRefGoogle ScholarPubMed; Lambelet, John C, “A Dynamic Model of the Arms Race in the Middle East, 1953–1965,” General Systems Yearbook, 16 (1971), 145–167 Google Scholar, and ‘Towards a Dynamic Theater Model of the East-West Arms Race,” Journal of Peace Science, 1 (Autumn, 1973), 1–38 CrossRefGoogle Scholar; Midgaard, Knut, “Arms Races, Arms Control, and Disarmament,” Cooperation and Conflict, 5 (February, 1970), 20–51 CrossRefGoogle Scholar; Moll, Kendall D., The Influence of History Upon Seapower, 1865–1914 (Menlo Park, Cal.: Stanford Research Institute, September, 1968)Google Scholar; O'Neill, Barry, ‘The Pattern of Insta-bihty among Nations: A Test of Richardson's Theory,” General Systems Yearbook, 15 (1970), 175–181 Google Scholar; Rathjens, George W., ‘The Dynamics of the Arms Race,” Scientific American, 220, no. 4 (April, 1969), 15–25 CrossRefGoogle Scholar; Smoker, Paul, “A Mathematical Study of the Present Arms Race,” General Systems Yearbook, 8 (1963), 61–76 Google Scholar, “Fear in the Arms Race: A Mathematical Study,” Joumal of Peace Research, 1 (February, 1964), 55–64 CrossRefGoogle Scholar, “Trade, Defense and the Richardson Theory of Arms Races: A Seven Nation Study,” Journal of Peace Research, 2 (April, 1965), 161–176 CrossRefGoogle Scholar, “Nation State Escabtion and international Integration,” Journal of Peace Research, 4 (February, 1967), 61–75 CrossRefGoogle Scholar. ‘The Arms Race As an Open and Closed System,” Peace Research Society Papers, 7 (1967), 41–42 Google Scholar; Wagner, D. L., Perkins, R. T. and Taagepera, R., “Fitting the Soviet-U.S. Arms Race Data to the Complete Solution of Richardson's Equations” (paper presented at the 1974 Meetings of the Western Regional Peace Science Society)Google Scholar; Zinnes, Dina A. and Gillespie, John V., “Analysis of Arms Race Models: USA vs. USSR and NATO vs. WTO,” Modeling and Simulation, vol. 4, ed. Vogt, William G. and Mickle, Marlin H. (Pittsburgh: Instrument Society of America, 1973), pp. 145–148 Google Scholar; Wotfson, Murray, “A Mathematical Model of the Cold War,” Peace Research Society Papers, 9 (1968), 107–124 Google Scholar. Some schokrs have suggested goals for nations involved in various specific aspects of armaments races. See McGuire, Martin C, Secrecy and the Arms Race (Cambridge, Mass.: Harvard University Press, 1965)Google Scholar, and Saaty, Thomas L., Mathematical Models of Arms Control and Disarmament New York: John Wilay and Sons, 1968)Google Scholar.

⁴ Brito, Dagobert L., “A Dynamic Model of an Armaments Race,” International Economic Review, 13 (June, 1972)CrossRefGoogle Scholar; Brito, Dagobert L. and Intriligator, Michael D., “Some Applications of the Maximum Principle to the Problem of an Armaments Race,” in Modeling and Stimulation, vol. 4, pp. 140–144 Google Scholar, and “An NOountry Model of an Armaments Race” (mimeoed, 1974)Google Scholar.

⁵ Intriligator, Michael D., Strategy in a Missile War: Targets and Rates of Fire (Los Angeles: University of California, Los Angeles, Security Studies Project, Paper #10, 1967)Google Scholar, and “The Debate over Missile Strategy: Targets and Rates of Fire,” Orbis, 11 (Winter, 1968), 1138–1159 Google Scholar.

⁶ Simaan, Marwan and Cruz, José B. Jr., “A Differential Game Formulation of Arms Race and Control and Its Relationship to Richardson's Model” (mimeoed, 1973)Google Scholar, and “A Multistage Game Formulation of Arms Race and Control and Its Relationship to Richardson's Model,” in Modeling and Simulation, vol. 4, 149–153 Google Scholar.

⁷ Other objections have been raised about the Richardson model. See Alker, “Structure of Social Action”; Rapoport, “Richardson's Mathematical Theory”; and Caspary, “Richardson's Model.”

⁸ The most useful explanations of optimal control theory are Athans, Michael and Falb, Peter L., Optimal Control (New York: McGraw-Hill, 1966)Google Scholar; Bryson, Arthur E. Jr. and Ho, Yu-Chi, Applied Optimal Control (Valtham, Mass.: Ginn and Co., 1969)Google Scholar; Citron, Stephen J., Elements of Optimal Control (New York: Holt, Rinehart and Winston, 1969)Google Scholar; Cruz, Jose B. Jr., ed., Feedback Systems (New York: McGraw-Hill, 1972)Google Scholar; Dyer, Peter and McReynolds, Stephen R., The Computation and Theory of Optimal Control (New York: Academic Press, 1970)Google Scholar; Hsu, Jay C and Meyer, Andrew U., Modern Control Principles and Applications New York: McGraw-Hill, 1968)Google Scholar; Ogata, Katsuhiko, Modern Control Engineering (Englewood Cliffs, N.J.: Prentice-Hall, 1970)Google Scholar; Perkins, William R. and Cruz, Jose B. Jr., Engineering of Dynamic Systems (New York: John Wiley and Sons, 1969)Google Scholar; Varaiya, Pravin P., Notes on Optimization (New York: Van Nostrand Reinhold, 1972)Google Scholar. See also: Arrow, Kenneth, Applications of Control Theory to Economic Growth (Stanford, Cal.: fastitute for Mathematical Studies in Social Science, Technical Report #2, Stanford University, July, 1968)Google Scholar; Dorfman, Robert, “An Economic Interpretation of Optimal Control Theory,” American Economic Review, 10 (December, 1969), 817–831 Google Scholar; Intriligator, Michael D., Mathematical Optimization and Economic Theory (Englewood Cliffs, N.J.: Prentice-Hall, 1971)Google Scholar; Gillespie, John. V., “Optimal Control Theory and Comparative Foreign Policy: A Promising Approach to Future Research,” in In Search of Global Patterns, ed. Rosenau, James N. (New York: Free Press, forthcoming)Google Scholar and “Some Observations on Dynamic Modeling,” in Measurement and Research Design in Political Science, ed. Leege, David (Chicago: Aldine, forthcoming)Google Scholar.

⁹ Not all models in which manipulable variables appear are controllable. m general, if the desired state is specified for all time, the requirements for the existence of a control input that will generate the desired x(t) are very stringent. A less ambitious but more realistic goal is to require only a partial specification of the state variables. One such partial specification is forcing the state of the system to attain a specified value at some finite time in the future. That is, given an initial time, t ₀, an initial state x(t₀)=x₀ , and a final state x_f, t₀ ≤ t ≤ t + T, for some finite T, such that s(t₀ + T) = xt, there may or may not be any control, u(t), which can force the system to attain the state x_f . By a suitable choice of error coordinates such as state variables, the problem of reaching and maintaining a specified state is the problem of matching a desired dynamic response. A system is said to be completely state-controllable if, for any initial x₀ , for any initial time t₀ , it is possible to generate an unconstrained control vector, u(t), that will take any given initial state x(t₀) to any final state x(t_f) in a finite time interval t₀ ≤ t ≤ t_f For a linear system, the final state may be taken to the origin without any loss of generality. The model we have developed here is completely state-controllable.

¹⁰ Bellman, Richard E., Dynamic Programming (Princeton, N.J.: Princeton University Press, 1957)Google ScholarPubMed; Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidge, R. V. and Mishchenko, E. F., The Mathematical Theory of Optimal Processes, trans. Trirogoff, K. M. (New York: Wiley Interscience, 1962)Google Scholar.

¹¹ By obtaining an optimal trajectory in this way we are incorporating feedback into the model. Feedback in our case will allow countries to respond to the foreign policy decisions of other countries. A solution which incorporates feedback is a closed-loop solution. It is closed-loop in the sense that the output from the model is an input at a later time in the dynamic process.

¹² Richardson, p. 16.

¹³ See, for example: Zinnes, Dina A., “An Analytical Study of the Balance of Power Theories,” Journal of Peace Research, 4 (July, 1967),270–288 CrossRefGoogle Scholar, and “Coalition Theories and the Balance of Power,” in The Study of Coalition Behavior, ed. Groennmgs, Sven, Kelley, E. W. and Leiserson, Michael (New York: Holt, Rinehart and Winston, 1970),351–36Google Scholar & See also: Qaude, Inis L. Jr., Power and International Rektions (New York: Random House, 1962),11–93 Google Scholar; Pollard, A. F., ‘The Balance of Power,” Journal of the British Institute of International Affairs, 2 (March, 1923), 53–64 CrossRefGoogle Scholar; Haas, Ernst B., ‘The Balance of Power: Prescription, Concept, or Propaganda?” World Politics, 5 (July, 1953), 442–477 CrossRefGoogle Scholar.

¹⁴ Although the variable u(t) is manipulable by the decision maker, if and only if the model is completely state-controllable is it the case that a manipulable variable becomes a control variable. For our model the condition of controllability is satisfied. We are assuming that the variable u(t) is completely manipulable, i.e., in principle, any values of u(t) can be chosen.

¹⁵ For block diagramming and network methods, see Perkins and Cruz, pp. 11–94.

¹⁶ Our analysis makes use of the Ricatti equations whereby the equations allow decomposition because certain terms must equal zero. The general form of our model is the “regulator problem with a Unear system and quadratic cost.” See Varaiya, pp. 176–184.

¹⁸ Since we have not specified any “scrap value,” i.e., value that must be shown by x(t) when t = T, the adjoint variables at time p(T) must be equal to zero. The adjoint variables are composed only of the state equation and contain no additional constraints. This property holds for all models of the general form of the linear-quadratic reguktor problem.

¹⁹ Because in equation (26), k₁ is dependent on x(T) and in equation (24) k₁(t) contains no k₂ terms, it is the case that k₁(t) is a zero function. In equation (25) k₂(t) is dependent on k₁(t), butinequation (26) it must be the case that k₂(T) is equal to 0.

²⁰ The assumption that all parameters are positive is a very strong assumption. It can be shown that Richardson could have obtained his results with a less strict assumption. The quantity (b – la) is a very important quantity for this model as will be demonstrated below.

²¹ The assumption that a nation has a given objective function should not be thought to be unnecessarily strict because the objective function used here is sufficiently general to incorporate many differing instances. It is important to note that our derivation is both necessary and sufficient.

²² The method of analysis here can be described as: Find a G(λ) such that F[α, β, γ, G(*#x03BB;)] meets some criterion.

²³ The associated measures of significance in ordinary least-squares analysis do not apply in our case. This is because we do not have, strictly speaking, a Unear estimation equation. We have “Unearized” equation (36) by using pseudoparameters, one of which is not chosen strictly by least-squares analysis.

²⁴ In any minimization one may encounter local minima (shallow valleys in a hypersurface separated by hills from deeper valleys) which may mask themselves as global minima (the deepest valley in the hypersurface). To assure that we have obtained a global minimum we have begun our analysis from several randomly selected initial values and have required convergence of final estimates from these initial values.

²⁵ See Brent, Richard P., Aleorithms for Minimization with Derivatives (Englewood Cliffs, N.J.: Prentice-Hall, 1973).Google Scholar

²⁶ Powell, M. J. D. “An Efficient Method for Finding the Minimum of a Function of Several Variables without Calculating Derivatives,” Computer Journal, 7 (July, 1964), 155–162.Google Scholar

²⁷ Davidon, William C., Variable Metric Method of Minimization, rev. ed. (Washington, D.C.: Atomic Energy Commission Research and Development Report, ANL-5990,1959)Google Scholar.

²⁸ Our criterion was convergence. If for severalruns the analysis produced the same minimization, we assumed that a global minimum was found. The Davidon-Powell method is not constrained by the range selected for initial values. Values outside of the range are eligible as minima.

²⁹ International Institute for Peace and Conflict Research, SIPRI Yearbook of World Armaments and Disarmaments (Stockhohn: Stockholm International Peace Research mstitute, 1968 and forward)Google Scholar. Our data sets run from 1947-1972 for the United States and Soviet Union; 1950–1972 for NATO, WTO, Israeland Arab states.

³⁰ See Becker, Abraham S., Soviet Military Outkys Since 1955 (Rand Corporation Memorandum RM-3886 PR) (Santa Monica: The Rand Corporation, 1964)Google Scholar. To adjust for the Soviet accounting change, the mean value of the four years after 1960 was compared to the mean value of the four years immediately prior to 1960. The prior-1960 mean value was subtracted from the post-1960 mean value, and this mean difference was then subtracted from all post-1960 total defense expenditures. This adjusted Soviet expenditure was used in calcuhting the WTO aggregate defense expenditure.

³¹ The Arab nations were defined as Egypt (UAR), Syria, and Jordan. The data are expressed in terms of noninflationary dollars using mean yearly market exchange rates for determining conversion into American dollars. The rates of inflation were obtained from the Stockhomi International Peace Research Institute.

³² Autocorrelation is not the only statistical problem in time-series analysis. Multicollinearity is an additional problem. By the very nature of our model, one might expect multicollinearity. Our analyses show that there is some multicollinearity in our data but that it is not particukrly disturbing. There are no agreed-upon methods for treating multicollinearity in time-series analysis. Some may challenge the need for treating autocorrektion in a model with a lagged endogenous variable. Once again there is no agreement on whether adjustments for autocorrelation are meaningful in the kind of model we have posited. See Schrodt, Philip A., The Rektionship Between Arms Races and the Preservation of Peace (Boomington: Indiana University, Ph.D. dissertation, 1976), chapter 7Google Scholar; and Nerlove, Marc and Wallis, Kenneth F., “Use of the Durbin-Watson Statistic in blappropriate Situations,” Econometrica, 34 (March, 1966), 235–238 CrossRefGoogle Scholar. The conclusions from our analyses are the same, whether or not corrections for autocorrelation are made. More conservative findings are obtained when the correction is made. However, by no means do we fully endorse use of the Durbin-Watson statistic or the Geary sign test for systems with togged endogenous variables. We have used these analyses because they do not alter the conclusions, and because they provide more conservative estimates of the parameters.

³³ See Johnston, John, Econometric Methods, 2nd ed. (New York: McGraw-Hill, 1972), pp. 246–266 and 312–313 Google Scholar; Goldberger, Arthur S., Econometric Theory (New York: John Wiley and Sons, 1964), pp. 153–155, 267–268 and 270–272 Google Scholar.

³⁴ Durbin, J. and Watson, G. S., ‘Testing for Serial Correkttion in Least Squares Regression, I,” Bio-metrika, 37 (June, 1950), 409–428 Google Scholar, and “Testing for Serial Correiation in Least Squares Regression, II,” Biometrika, 38 (June, 1951), 159–178 CrossRefGoogle Scholar; Habibagahi, Hamid and Pratschke, John L., “A Comparison of the Power of the Von Neumann Ratio, Durbin-Watson and Geary Tests,” Review of Economics and Statistics, 54 (May, 1972), 179–185 CrossRefGoogle Scholar; Durbin, J., “Estimation of Parameters in Time-Series Regression Modek,” Journal of the Royal Statistical Society, Series B, 22 (January, 1960), 139–153 Google Scholar; Cochrane, D. and Orcutt, G. H., “Application of Least-Squares Regression to Rektion-ships Containing Autocorrekited Error Terms,” Journal of the American Statistical Assocktion, 44 (March, 1949), 32–61 Google Scholar.

³⁵ Kadiyala, Koteswara Rao, “A Transformation Used to Circumvent the Problem of Autocorrektion,” Econometrica, 36 (January, 1968), 93–96 CrossRefGoogle Scholar.

³⁶ In obtaining the parameter from this analysis, γ has been standardized by dividing by the autocorrektion parameter, The reader should note that, whereas GLS provides more conservative findings, the conclusions are the same whether the analysis is completed under GLS or OLS.

³⁷ Care must be taken in interpreting these results, given the measures of fit in Tables 1 and 2.

³⁸ Richardson and his followers have accepted the idea that all parameters must be positive. See Richardson and, for example, D. L. Wagner et al.

³⁹ See for example Smoker's work, and Caspary.

⁴⁰ It must be remembered that the parameter b is multiplied by a negative unity in the state equation.

⁴¹ For NATO all the signs ofthe coefficients are in the direction Richardson stiputated. See Richardson.

⁴² For readable summaries of equillbrium analysis, see Intrifigator, Mathematical Optimization, pp. 220–257 Google Scholar; Perkins and Cruz, pp. 72–78.

⁴³ Since our solution is for a steady-state equilibrium, one must be careful not to interpret equilibrium as total disarmament. Equilibrium for our analyses simply means that there the rate of change in armaments is zero, not that armaments expenditure becomes zero.

⁴⁴ In equilibrium, if ẋ equals zero, it is not the case that u is equal to zero. The nation U is the controller in our model and may increase or decrease its armaments expenditure while nation X is in equilibrium.

⁴⁵ This method has been shown to be entirely tegitimate. See, Weiss, Leonard and Inifante, E. F., “On the Stability of Systems Defined over a Finite Time Interval,” Proceedings of the National Academy of Sdences, 54 (July 15, 1965), 44–48 CrossRefGoogle Scholar. Normally one assumes T to be infinity, but such an assumption is not appropriate for our analysis. The actual value of T is irrelevant.

⁴⁶ For the optimal control reguktor problem of the form here, the assumption that equillbrium and stabllity exist in an infinite time horizon is incompatible with the boundary conditions used to solve for u*(t). Hence the weaker, but still legitimate, assumption of setting T equal to a krge number is employed.

⁴⁷ For this analysis, estimates of the constant h needed to be derived. Consistent with our analysis is the use of the mean difference in expenditures. For our analysis h is equal to the following values for the six cases.

⁴⁸ This numerical evaluation uses the values of h in footnote 47 and sets (T - t⁰ ) equal to numbers ranging from 50 to 1.

⁴⁹ The actual values for the equliibrium are mean-ingless in that we have set the initial conditions for all nations equal to a small positive number. The numbers are interpretable in terms of whether or not the values yielded using series of (T - t⁰) numbers are about the same.

⁵⁰ Liapunov, Aleksandr M., Stability of Motion (New York: Academic Press, 1966)Google Scholar; Liephote, Horst, Stability Theory (New York: Academic Press, 1970)Google Scholar.

⁵¹ For equation (56) from (55):

⁵² An examination of equation (55) will show that when (la - b) = -∞ then x(t) is equal only to the forcing function.