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)
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 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
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:
(5)
Let A(n) be the vector representing 
and A(n-1) be the
vector representing 
. The model can now be written, more easily, as
. (6)
Finding and using eignevalues and eignevectors, we obtain
the following solution to the homogeneous part of the system:
(7)
We use the formula (or conjecture) D=(I-R)-1B to
find the nonhomogeneous part of the solution.
D =
. (8)
The final general solution is
. (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) 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.