ArmsControlandWarfareWilliamP.FoxIntroductionWhat causes nations to wage war?Historyshowsthattheexistenceofweaponslargemilitaryarsenals-increasesthelikelihoodofviolent conflict.Without destructive weapons,perhapsnations sometimes would settledisputesbyothermeans.Itwasthisassumptionthatled LewisFryRichardsontobeginhisstudyandanalysisofarmsraces.RichardsonwasaQuakerandwastroubled bybothwwiandWWil.His scientific training in physics led himto believethat wars were aphenomena that could be studied and mathematically modeled.Richardsonconjecturedthatarmsraceswereoftenpreludestowar.Ifnationswere increasingtheirexpenditures ondefensebudgetsthena small spark couldstarta majorconflagration.if twonations weredecreasingtheirdefensebudgets, then a small incident might not trigger a war.Ultimately,Richardsonwantedtobuildamodel toexaminecertainconditions inordertopredictwhetheranarmsracewas"stable"or“unstable”TheArmsRaceModelWeexaminetheRichardson'sArmsRaceModel initiallyasa systemof lineardifferenceequationa systemofdiscretedynamicalsystems.Welet,X(n)=the armament of Nation X at time t=n.Thechange inarmament levelfromt=n-1tot=n isrepresentedby:(1)△X(n) = X(n)-X(n-1)Simarly this model is also true for nation Y:Y(n)=thearmamentof NationYattimet=n.Thechange inarmament levelfromt=n-1tot=n isrepresentedby(2)△Y(n) = Y(n)-Y(n-1)Richardsonenvisionedtheeffectsoneachnation'sarmamentontheothernation.Headdedterms consideringdefensecoefficientsorhoweachnationiseffectedbythe strength of the other nation(1a)AX(n)= 8,Y(n-1)(2a)△Y(n)= 82X(n-1)103
103 Arms Control and Warfare William P. Fox Introduction What causes nations to wage war? History shows that the existence of weapons— large military arsenals— increases the likelihood of violent conflict. Without destructive weapons, perhaps nations sometimes would settle disputes by other means. It was this assumption that led Lewis Fry Richardson to begin his study and analysis of arms races. Richardson was a Quaker and was troubled by both WWI and WWII. His scientific training in physics led him to believe that wars were a phenomena that could be studied and mathematically modeled. Richardson conjectured that arms races were often preludes to war. If nations were increasing their expenditures on defense budgets then a small spark could start a major conflagration. If two nations were decreasing their defense budgets, then a small incident might not trigger a war. Ultimately, Richardson wanted to build a model to examine certain conditions in order to predict whether an arms race was “stable” or “unstable”. The Arms Race Model We examine the Richardson’s Arms Race Model initially as a system of linear difference equation— a system of discrete dynamical systems. We let, X(n) = the armament of Nation X at time t=n. The change in armament level from t=n-1 to t=n is represented by: DX(n) = X(n)-X(n-1) (1) Simarly this model is also true for nation Y: Y(n) = the armament of Nation Y at time t=n. The change in armament level from t=n-1 to t=n is represented by: DY(n) = Y(n)-Y(n-1) (2) Richardson envisioned the effects on each nation’s armament on the other nation. He added terms considering defense coefficients or how each nation is effected by the strength of the other nation DX(n)= d1Y(n-1) (1a) DY(n)= d2X(n-1) (2a)
Thenhe consideredfatigueand expensecoefficients ofkeepingupanarmsrace.(1b)AX(n)= 8,Y(n-1) - αX(n-1)(2b)△Y(n)= 82X(n-1) - α2Y(n-1)Finally,grievancesorambitionsareaddedtothemodelas constants.(1c)4X(n)= S,Y(n-1) - αiX(n-1) + g(2c)4Y(n)= 82X(n-1) - α2Y(n-1) + hWecallthesefinaltwoequations (1c)and (2c),asystemofdiscretedynamicalsystems.EstimatesoftheModel'sParametersConsider thedata in Table 1 for the arms build up in Iraq and Iran before their1975war.Thedatacollected istheexpendituresforarmsbythetwocountriesfrom1954to1974.Let'suseourmodel toanalyzewhat occurred to causethiswartotakeplace.YearIranIrag19547875671071955941956126151102195724311019581959129271196029214519613201852061962345271196338719644253594021965435196646045047348019671968498513196953454961272319701971732781840921197219739801292197413081632TABLE1.DefenseExpendituresforIranandIrag(1954-1974)104
104 Then he considered fatigue and expense coefficients of keeping up an arms race. DX(n)= d1Y(n-1) - a1X(n-1) (1b) DY(n)= d2X(n-1) - a2Y(n-1) (2b) Finally, grievances or ambitions are added to the model as constants. DX(n)= d1Y(n-1) - a1X(n-1) + g (1c) DY(n)= d2X(n-1) - a2Y(n-1) + h (2c) We call these final two equations (1c) and (2c), a system of discrete dynamical systems. Estimates of the Model’s Parameters Consider the data in Table 1 for the arms build up in Iraq and Iran before their 1975 war. The data collected is the expenditures for arms by the two countries from 1954 to 1974. Let’s use our model to analyze what occurred to cause this war to take place. Year Iran Iraq 1954 78 75 1955 107 67 1956 126 94 1957 151 102 1958 243 110 1959 271 129 1960 292 145 1961 320 185 1962 345 206 1963 387 271 1964 425 359 1965 435 402 1966 460 450 1967 473 480 1968 498 513 1969 534 549 1970 612 723 1971 732 781 1972 840 921 1973 980 1292 1974 1308 1632 TABLE 1. Defense Expenditures for Iran and Iraq (1954-1974)
We usemultiple linearregressiontoestimatetheparametersof ourmodel.WeletX(n)stand for thedefense expendituresfor Iran intimeperiod n.Similary,weletY(n)standforthedefenseexpendituresforIraqintimeperiodn.WeregressX(n)—theresponsevariableonthepredictors—X(n-1)and Y(n-1).Wealsoregress Y(n) on its two predictors-—Y(n-1) and X(n-1).Using MINITAB to performthemultiple linear regression models, we achievethefollowingresults (MINITABprintout):Worksheet size:1o0o0o cellsMTB > Regress c5 2 c2 c3;SUBC>Constant.Regression AnalysisThe regression equation isX(n) =37.1+0.651 x(n-1)+0.432Y(n-1)2o cases used 1 cases contain missing valuesTPPredictorCoefStDev37.0626.351.410.178Constant0.65080.16513.940.001x(n-1)0.43173.590.002Y(n-1)0.1204S=38.91R-Sq = 98.5%R-Sq(adj)= 98.3%Analysis of VarianceFDFPssMSSource21689279844639557.840.000Regression17257401514Error191715019TotalDFseq ssSource1x(n-1)16698161Y(n-1)19463UnusualObservationsObsFitx(n-1)X(n)StDev FitResidualSt Resid209801308.001232.5628.5675.442.85RX21?*13081592.7934.58* XR denotes an observation with a large standardized residualX denotes an observation whose x value gives it large influence.MTB>Regressc62c2c3;SUBC>Constant.Regression AnalysisThe regression equation isY(n) =-52.9+0.195X(n-1))+1.13Y(n-1)20 cases used 1 cases contain missing valuesTPCoefStDevPredictor-52.9140.06-1.320.204Constant105
105 We use multiple linear regression to estimate the parameters of our model. We let X(n) stand for the defense expenditures for Iran in time period n. Similary, we let Y(n) stand for the defense expenditures for Iraq in time period n. We regress X(n)— the response variable on the predictors— X(n-1) and Y(n-1). We also regress Y(n) on its two predictors— Y(n-1) and X(n-1). Using MINITAB to perform the multiple linear regression models, we achieve the following results (MINITAB printout): Worksheet size: 100000 cells MTB > Regress c5 2 c2 c3; SUBC> Constant. Regression Analysis The regression equation is X(n) = 37.1 + 0.651 X(n-1) + 0.432 Y(n-1) 20 cases used 1 cases contain missing values Predictor Coef StDev T P Constant 37.06 26.35 1.41 0.178 X(n-1) 0.6508 0.1651 3.94 0.001 Y(n-1) 0.4317 0.1204 3.59 0.002 S = 38.91 R-Sq = 98.5% R-Sq(adj) = 98.3% Analysis of Variance Source DF SS MS F P Regression 2 1689279 844639 557.84 0.000 Error 17 25740 1514 Total 19 1715019 Source DF Seq SS X(n-1) 1 1669816 Y(n-1) 1 19463 Unusual Observations Obs X(n-1) X(n) Fit StDev Fit Residual St Resid 20 980 1308.00 1232.56 28.56 75.44 2.85RX 21 1308 * 1592.79 34.58 * * X R denotes an observation with a large standardized residual X denotes an observation whose X value gives it large influence. MTB > Regress c6 2 c2 c3; SUBC> Constant. Regression Analysis The regression equation is Y(n) = - 52.9 + 0.195 X(n-1) + 1.13 Y(n-1) 20 cases used 1 cases contain missing values Predictor Coef StDev T P Constant -52.91 40.06 -1.32 0.204
0.19490.25100.78x(n-1)0.448Y(n-1)1.12680.18306.160.000S =59.15R-Sq = 98.2%R-sq(adj)= 98.0%Analysis of VarianceDFFQsourcessMS233373411668671476.900.000Regression17Error59484349919Total3396825DFSourceSeq ss113204724x(n-1)132617Y(n-1)ObservationsUnusualObsx(n-1)Y(n)FitstDev FitResidualst Resid198401292.01148.628.2143.42.76R209801632.01593.943.438.10.95X21?1308*2041.052.6* XR denotes an observation with a large standardized residualX denotes an observation whose x value gives it large influence.ExtractingtheregressionequationsfromtheMINITABoutput,wegetthecouplednonhomogeneoussystemmodelas:(3)X(n) = 37.1 + 0.651 X(n-1) + 0.432 Y(n-1)Y(n) = -52.9 +0.195 X(n-1) +1.13 Y(n-1)(4)ModelSolutionandSystemLongTermBehavior(StabilityAnalysis)Using linearalgebra,wecansolveforthestabilityofthesystem.Themodel(inmatrix form with Iran(n) = X(n) and Iraq(n) = Y(n) is:[.651.432Tran(n-1)],Iran(n)37.1(5).1951.13Irag(n-1)-52.9Irag(n)[Iran(n)Let A(n) be the vector representingandA(n-1)bethevectorIrag(n)[Iran(n -1)representingThemodel can nowbewritten,moreeasily,asIraq(n-1).43237.1.651(6)A(n)A(n-1)+.1951.13-52.9106
106 X(n-1) 0.1949 0.2510 0.78 0.448 Y(n-1) 1.1268 0.1830 6.16 0.000 S = 59.15 R-Sq = 98.2% R-Sq(adj) = 98.0% Analysis of Variance Source DF SS MS F P Regression 2 3337341 1668671 476.90 0.000 Error 17 59484 3499 Total 19 3396825 Source DF Seq SS X(n-1) 1 3204724 Y(n-1) 1 132617 Unusual Observations Obs X(n-1) Y(n) Fit StDev Fit Residual St Resid 19 840 1292.0 1148.6 28.2 143.4 2.76R 20 980 1632.0 1593.9 43.4 38.1 0.95 X 21 1308 * 2041.0 52.6 * * X R denotes an observation with a large standardized residual X denotes an observation whose X value gives it large influence. Extracting the regression equations from the MINITAB output, we get the coupled nonhomogeneous system model as: X(n) = 37.1 + 0.651 X(n-1) + 0.432 Y(n-1) (3) Y(n) = - 52.9 + 0.195 X(n-1) + 1.13 Y(n-1) (4) Model Solution and System Long Term Behavior (Stability Analysis) Using linear algebra, we can solve for the stability of the system. The model (in matrix form with Iran(n) = X(n) and Iraq(n) = Y(n)) is: ú û ù ê ë é - + ú û ù ê ë é - - ú û ù ê ë é =ú û ù ê ë é 52.9 37.1 ( 1) ( 1) .195 1.13 .651 .432 ( ) ( ) Iraq n Iran n Iraq n Iran n (5) Let A(n) be the vector representing ú û ù ê ë é ( ) ( ) Iraq n Iran n and A(n-1) be the vector representing ú û ù ê ë é - - ( 1) ( 1) Iraq n Iran n . The model can now be written, more easily, as ú û ù ê ë é - - + ú û ù ê ë é = 52.9 37.1 ( 1) .195 1.13 .651 .432 A(n) A n . (6)
Findingand usingeignevaluesand eignevectors,we obtainthefollowingsolutiontothehomogeneouspartof thesystem:.70152A(n)=c,(1.266798)+c,(.5142019)-.31662(7)We use the formula (or conjecture) D=(l-R)'B to find the nonhomogeneous partofthesolution..349-.43237.1213.531D元(8)-.195-.13-52.986.626The final general solution is.70152213.531+C(.5142019)A(n)=c,(1.266798)(9).3166286.626As k → co, the term (1.266798)* grows without bound. This system is not stable.Thus,thisis an unstablesystemand is conducivetowar.A stable arms race would indicate that there is at leastoneequilibriumpointwherebothnationsaresatisfied.There isnoneed to escalatearmamentbuild upbeyondthisequilibriumpoint.Theequilibriumpointinanarmsracerepresentsthelevelofarmssuchthatthedynamicsof thearms race ceases.It is thevalueof thearmaments thatresults in noneedto change the armaments ofthe two nations.An unstable arms race indicatesthatnoeguilibriumpointexists.Theexpenditurescontinuetoescalateas doesthe buildup ofdestructiveweapons.Thedynamics of thearms race continuesasanypositivechange in armaments in onenation results inapositive changein armaments to the other nation.Perhaps a small spark or act can triggeraconflictinthisunstablecaseExercises1.Findtheparticularsolutiontothelran-lragarmsracemodeliftheinitialconditions are Iran(0)=78 and Irag(0)=752. Write a shortessay on the nature of war and how eigenvalues can help todeterminethestabilityofthearmsrace.3.GiventhefollowingdataforthearmsracebetweentheWarsawPactforcesandtheNATOforces,usetheRichardson'sArmsRacemodelto107
107 Finding and using eignevalues and eignevectors, we obtain the following solution to the homogeneous part of the system: ú û ù ê ë é - + ú û ù ê ë é = .31662 1 (.5142019) 1 .70152 ( ) (1.266798) 1 2 k k A n c c (7) We use the formula (or conjecture) D=(I-R)-1 B to find the nonhomogeneous part of the solution. D = ú û ù ê ë é = ú û ù ê ë é - ú û ù ê ë é - - - - 86.626 213.531 52.9 37.1 .195 .13 .349 .432 1 . (8) The final general solution is ú û ù ê ë é + ú û ù ê ë é - + ú û ù ê ë é = 86.626 213.531 .31662 1 (.5142019) 1 .70152 ( ) (1.266798) 1 2 k k A n c c . (9) As k ® ¥, the term (1.266798)k grows without bound. This system is not stable. Thus, this is an unstable system and is conducive to war. A stable arms race would indicate that there is at least one equilibrium point where both nations are satisfied. There is no need to escalate armament build up beyond this equilibrium point. The equilibrium point in an arms race represents the level of arms such that the dynamics of the arms race ceases. It is the value of the armaments that results in no need to change the armaments of the two nations. An unstable arms race indicates that no equilibrium point exists. The expenditures continue to escalate as does the build up of destructive weapons. The dynamics of the arms race continues as any positive change in armaments in one nation results in a positive change in armaments to the other nation. Perhaps a small spark or act can trigger a conflict in this unstable case. Exercises 1. Find the particular solution to the Iran-Iraq arms race model if the initial conditions are Iran(0)=78 and Iraq(0)=75. 2. Write a short essay on the nature of war and how eigenvalues can help to determine the stability of the arms race. 3. Given the following data for the arms race between the Warsaw Pact forces and the NATO forces, use the Richardson’s Arms Race model to
(a)EstimatetheparametersfortheNATO-WarsawForces(b) Solvethe model(c)Determinethestabilityofthearmsrace(d)Writea shortessayconcerningyourmodelingresultandtherealityofthe1980-90'sscenarioinEasternEurope.YearNATOWTO1971206.1166.6173.91972209.61973205.6180.91974208.6188.51975206.1195.31976202.8203.81977209.9206.91978212.7210.11979218.8212.61980218.9229.8References1.Fox,WilliamP.,FrankGiordano,and MauryWeir.AFirstCourseinMathematicalModeling,Brooks/ColePublishing.PacificGrove,CA.19972.Schrodt,Phillip.Richardson'sArmsRaceModel.NationalCollegiateSoftwareClearinghouse.Raleigh,NC.1987.3.Zinnes,DinaA.John Gillespie,andG.S.Tahim.TheRichardson'sArmsRaceModel,UMAPModule308.COMAP.Boston,MA.1990108
108 (a) Estimate the parameters for the NATO-Warsaw Forces (b) Solve the model (c) Determine the stability of the arms race (d) Write a short essay concerning your modeling result and the reality of the 1980-90’s scenario in Eastern Europe. Year NATO WTO 1971 206.1 166.6 1972 209.6 173.9 1973 205.6 180.9 1974 208.6 188.5 1975 206.1 195.3 1976 202.8 203.8 1977 209.9 206.9 1978 212.7 210.1 1979 218.8 212.6 1980 229.8 218.9 References 1. Fox, William P., Frank Giordano, and Maury Weir. A First Course in Mathematical Modeling, Brooks/Cole Publishing. Pacific Grove, CA. 1997. 2. Schrodt, Phillip. Richardson’s Arms Race Model. National Collegiate Software Clearinghouse. Raleigh, NC. 1987. 3. Zinnes, Dina A. John Gillespie, and G.S. Tahim. The Richardson’s Arms Race Model, UMAP Module 308. COMAP. Boston, MA. 1990