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, he began to analytically think about the arms buildups going on in Europe, being concerned that it would lead to another global conflict. The data he gathered and the mathematical model he developed are the subject of this project.

Background

Familiarity with solutions of systems and matrix algebra; iterating functions.

The Model

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 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 money.

The arms level that each nation has at time t=k+1, one unit of time from now; may depend on four general things:

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, the amount of arms they would build anyway, even if no other nations presented a threat

These four factors allow us to consider a system of three equations for our three hypothetical nations:

where, in general, the f_i are "fatigue" coefficients described in (3), above, the g_i are "grievances" described in (4), and the a_ij would represent the response of nation i to the arms level of nation j.

These equations might be for any three nations; for our three hypothetical nations, one might assume that g₂ = 0, g₁ is positive, a₃₂=0 (since B and C are not enemies), and perhaps that a₁₂ and a₁₃ are greater than 1 (indicating that A overcompensates for everything the other nations have). Possibly f₁=1, f₂ might be zero, and f₃<1 indicating that in the absence of other armed nations, A keeps all its arms anyway, B gets rid of all of its arms, and C keeps some of its arms. Possibly a₃₁ might be 1 indicating that for every arm that A has, C will build one. On the other hand, if a₁₃ = 1.2, this would indicate that for every arm that C has, A will build 20% more.

In matrix form, we have

where in the above example,

In general, notice that matrix A has response terms in the off-diagonal entries and fatigue factors on the diagonal.

Steady State

It is possible to have an equilibrium situation. This would be where x(k+1) = x(k) or that

which is algebraically equivalent to (I_n - A )x_s = g .

Several considerations are in order here. This is a nonhomogeneous system so

• there might be no solution if the rank of I - A is less than n
• there might be a unique solution
• the solution might have some negative components, which would physically make no sense

If a nonnegative steady state occurs, it might suggest that an "uneasy peace" existed.

A Solution to the System

We may iteratively solve this

x(1) = Ax(0) + g

x(2) = Ax(1) + g = A( Ax(0) +g) + g = A²x(0) + Ag + g

x(3) = Ax(2) + g = A(A²x(0) + Ag + g) + g = A³x(0) + A²g + Ag + g

.

.

.

and apparently in general, that

x(k) = A^kx(0) +( A^k-1 + A^k-2 + ...+A + I_n) g

(note that the term in front of g in parentheses is a partial geometric series in A). The solution, as k goes to infinity, may either

• tend to infinity (unstable)
• tend to the steady state (stable)
• go to zero

At this point, we only have enough tools to determine this by direct simulation.

Richardson's Model of the World in 1935

Richardson spent considerable time and effort after WWI gathering data to describe 10 nations and their arms dynamics. He published the 1935 version which includes the following values for the matrix A:

The Report

Introduction and Summary : This section should give a very brief account of the subject matter discussed in your report.

Summary of Ideas: Provide a 1-2 page summary of how one uses linear algebra techniques to model the buildup of arms in a given collection of countries. You can just summarize what I've written above if you like, but put it in your own words (do not simply cut and paste my words into your report!)

Exercises:

1. In the basic model, argue why the fatigue coefficients, f_i, must be in the range

   0 < f_i < 1

   what would the f_i  =1  mean? f_i = 0?

2. 1. Suppose the coefficient matrix for a group of 4 nations is

      [.2	1.05	1.03	0]
      |.5	.4	0	0|
      |.4	0	1	0|
      [0	0	0	.22]

      what can you say about the relationships of the 4 nations?

   2. Assuming no grievances and equal initial arms levels (1 1 1 1)^t, simulate the above arms race and decide if it is stable or not. If it is stable, what is the equilibrium? If it is not stable, find the factor by which it is eventually geometrically growing.

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 x_i and the others in x_i+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. In terms of the entries in the matrix A, how would you go about spotting nations that are only defensive in their posture? How would a neutral nation like Switzerland appear?

5. In Richardson's 1935 matrix above,

   1. what nation is the most aggressive? Why/how did you decide?
   2. what nation is most disliked or feared? Why/how did you decide?

6. In general, neglecting diagonal entries, what would

   1. row sums represent?
   2. column sums represent?

7. Consider Richardson's 1935 model of the world.

   1. By simulating it, decide if it is stable or not. If it is stable, what is the steady state? If it is unstable, what is the factor by which it geometrically grows? (1 decimal place in either) What nation has the most arms in the long run?
   2. What is the steady state for this system?

8. Extra Credit (up to 5 points):

   1. There have been many attempts in this century in the area of "arms controls", especially after the development and escalation of nuclear weapons following World War II and more recently, the existence of biological and chemical weapons. Suppose you work for the United Nations. Describe in words a scheme you might find realistic for arms control. Once you have described it, interpret it mathematically and modifiy Richardson's model accordingly.
   2. Apply your arms control scheme to the problem in 2a and show that it does indeed do what you have intended it to.
   3. Is your scheme "fair"? Will some nations feel discriminated against?

Points

• Mathematical Methodology: Your calculations, approach and procedure in your solution of the exercises should be clear, complete and correct. -- 12 points
  
  
• Clarity of your explanations / summary: It is very important to be able to communicate your solution clearly to non-experts -- 7 points
  
  
• Good Grammar: Grammar, spelling, general professionality of completed document (the spell check is just the beginning....) -- 3 points
  
  
• Mathematical Notation: Use correct mathematical notation in the correct context -- 3 points
  
  
• Extra Credit: Problem 8 above -- up to 5 points
  

References

Richardson, Lewis F. Arms and Insecurity 1953 Pittsburgh, Boxwood Press.

Richardson,Lewis F. Statistics of Deadly Quarrels. 1960 Pittsburgh, Boxwood Press.

Richardson's Arms Model and Arms Control" J. Goulet. SIAM Conference on Discrete Math and Its Application. MIT, June 1983.

Luenberger,D.G. Dynamical Systems New York. John Wiley and Sons 1979.