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Subsections


Richardson's Arms Model

Introduction

Lewis Richardson (1881-1953) was a meteorlogist in Britain. A man of wide interests and abilities, he made contributions to science in the areas of meteorology, fluid dynamics, fractals and chaos theory .During World War I, he served for France in their medical corps and saw first hand the horrors of warfare. After the war,being concerned that it would lead to another global conflict, he began to think analytically about the arms buildups going on in Europe. The data he gathered and the mathematical model he developed are the subject of this project.

Background

Suppose for sake of discussion we study the behavior of three nations; A, B, and C. Suppose nation A is quite aggressive and war prone, nation B is a fairly neutral and passive nation (like Switzerland much of this century), and nation C is a reluctant foe of nation A. Suppose we assign variables x, y, and z to them respectively, which indicate the amount of arms that each nation has. A convenient unit of measurement is the money value of the arms that each nation possesses. The arms level that each nation has at time t=k+1, one unit of time from now, may depend on the four general ideas below.
1.
The amount of arms they already had at time t = k.
2.
The amount of arms they might build in response to the other nations arms levels.
3.
The amount of arms they might have gotten rid of due to their internal tendencies. (As we have seen in the US, maintaining armed forces can be expensive and sometimes is the subject of cutbacks in peacetimes due to other priorities or budget deficits.)
4.
If they are particularly warlike or hold grievances against other nations, the amount of arms they would build anyway, even if no other nations presented a threat.
These four factors allow us to consider a the following system of three equations for our three hypothetical nations,

xk+1 = f1 xk + a12 yk + a13 zk + g1

yk+1 = f2 yk + a21 xk + a23 zk + g2

zk+1 = f3 zk + a31 xk + a32 yk + g3

where the fi are the ``fatigue'' coefficients described in item 3 above, the gi are the ``grievances'' described in item 4 above, and the aij represent the response of nation i to the arms level of nation j.

For the three nations we described above, we might set g2 = 0, set a32 = 0 (since B and C are not enemies), give values for a12 and a13 that are greater than one, since nation A overreacts to the arms of nations B and C, and give g1 a positive value. We might also set the values f1 = 1, f2 = 0, and f3 = 1/2 to express that nation A's arms budget is never cut, nation B always disarms unilaterally, and that nation C's arms budget is cut every year. Setting a31 to 1 would indicate that nation C always builds arms if nation A does. On the other hand, setting a31=1.2 indicates that nation C always builds 20% more arms than nation A has.

In matrix form, we could write this model as

\begin{displaymath}
\mathbf{x}_{k+1} = A \mathbf{x}_{k} + \mathbf{g} \end{displaymath}

where the entries in the matrix A for our example would be

\begin{displaymath}
A = \left[ \begin{array}
{ccc}
 f_1 & a_{12} & a_{13} \\  a_...
 ... & f_2 & a_{23} \\  a_{31} & a_{32} & f_3 
 \end{array} \right]\end{displaymath}

and the vector $\mathbf{g}$ would be given by

\begin{displaymath}
\mathbf{g } = \left[ \begin{array}
{c}
 g_1 \\  g_2 \\  g_3 
 \end{array} \right]\end{displaymath}

One important question in such a model is whether there can ever be a steady state, that is, values for the arms levels that don't change. This can only happen if $\mathbf{x}_{k+1} = \mathbf{x}_k$. To determine if this can happen, we denote the steady state vector by $\mathbf{x}_s$ and see if we can solve for it in the equation

\begin{displaymath}
\mathbf{x}_s = A \mathbf{x}_s + \mathbf{g} \end{displaymath}

which we can writen in a more familiar form as

\begin{displaymath}
(I_n = A) \mathbf{x}_s = \mathbf{g} \end{displaymath}

This is a nonhomogeneous system, so there are three possibilities.
1.
There might be no solution.
2.
There might be a unique solution, if (In - A) is invertible.
3.
The solution might exist, but have some negative components, which would not make any physical sense.

As we have done with other models, we might try investigating such a model by computing values of $\mathbf{x}_k$ for $k = 1,
2,3,\ldots$. Doing so produces the following sequence of equations.

\begin{displaymath}
\mathbf{x}_1 = A \mathbf{x}_0 + \mathbf{g} \end{displaymath}

\begin{displaymath}
\mathbf{x}_2 = A \mathbf{x}_1 + \mathbf{g} = A^2 \mathbf{x}_0 + A
\mathbf{g} + \mathbf{g} \end{displaymath}

\begin{displaymath}
\mathbf{x}_3 = A \mathbf{x}_2 + \mathbf{g} = A^3 \mathbf{x}_0 + A^2
\mathbf{g} + A \mathbf{g} + \mathbf{g} \end{displaymath}

or, in general,

\begin{displaymath}
\mathbf{x}_k = A^k \mathbf{x}_0 + A^{k-1}
\mathbf{g} + \ldots + A \mathbf{g} + \mathbf{g} \end{displaymath}

One can imagine results like the following coming out of such simulation.
1.
The magnitude of $\mathbf{x}_k$ might tend to infinity, indicating an unstable arms race.
2.
The vector $\mathbf{x}_k$ might go to the steady state, indicating a stable situation.
3.
The vector $\mathbf{x}_k$ might go to zero, indicating complete disarmament.

Richardson's model of the world in 1935

After World War II, Richardson collected data on ten nations and came up with the following matrix

\begin{displaymath}
A = \left[ \begin{array}
{cccccccccc}
1/2 & 0 & 0 & 1/10 & 0...
 ... & 2/5 & 1/10 & 1/10 & 1/5 & 1/20 & 0 & 1/2 \end{array} \right]\end{displaymath}

and values of $\mathbf{g}$ given by

\begin{displaymath}
\mathbf{g} = \left[ \begin{array}
{c} 
1/20 \\ 1/20 \\ 1/20 ...
 .../20 \\ 1/10 \\ 3/20 \\ 1/20 \\ 1/20 \\ 1/10 \end{array} \right]\end{displaymath}

where the nations, in order, are Czechoslovakia, China, France, Germany, England, Italy, Japan, Poland, the USA, and the USSR.

Exercises

1.
In the basic model, explain why the fatigue coefficients fi should be in the range 0 < fi < 1. How would you explain fi = 0 or fi = 1?
2.
Suppose that the coefficient matrix for a group of four nations is given by the following.

\begin{displaymath}
A = \left[ \begin{array}
{cccc}
 1/5 & 1 1/20 & 1 3/100 & 0 ...
 ...0 \\  2/5 & 0 & 1 & 0 \\  0 & 0 & 0 & 1/4 
 \end{array} \right]\end{displaymath}

(a)
What can you say about the relationships between the four nations?
(b)
Assuming no grievances and initial arms levels of 1 for all four nations, simulate the above arms race and decide if it is stable or not. If it is stable, find the equilibrium.
3.
Suppose we have a situation where all n nations fall into two groups with i of them in the first group and n-i in the second group. Suppose we have organized them so the first group is variables x1 through xi and the others in xi+1 through xn. Further assume that all those in the first group are allied with each other against all those in the second group, and vice versa. What can you say about how the matrix A will look (no actual numbers, just +,- or 0 for individual entries)?
4.
Refer to the matrix that Richardson obtained.
(a)
Which nation is the most aggressive? Explain your choice.
(b)
Which nation is the most disliked or feared? Explain your choice.
5.
If you ignore the diagonal entries, what would row sums represent? Again ignoring the diagonal entries, what would column sums represent?

6.
Again consider Richardson's model. By simulating it, try to determine if it is stable or not. If it is stable, determine the equilibrium. Also determine which country has the most arms in the long run.


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William W. Farr
9/17/1998