Richardson's Arms Race Model

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Sep 30, 2005 - war,being concerned that the arms buildups going on in Europe would lead .... After World War I, Richards
Outline Richardson’s Arms Race Model

Richardson’s Arms Race Model MA 2071 A ’05 Bill Farr

September 30, 2005

MA 2071 A ’05 Bill Farr

Richardson’s Arms Race Model

Outline Richardson’s Arms Race Model

1

Richardson’s Arms Race Model Introduction A Simple Model Richardson’s Model

MA 2071 A ’05 Bill Farr

Richardson’s Arms Race Model

Outline Richardson’s Arms Race Model

Introduction A Simple Model Richardson’s Model

Introduction

Lewis Richardson (1881-1953) was a meteorologist 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 the arms buildups going on in Europe would lead to another global conflict, he began to think analytically about modeling the process. The data he gathered and the mathematical model he developed are the subject of this presentation.

MA 2071 A ’05 Bill Farr

Richardson’s Arms Race Model

Outline Richardson’s Arms Race Model

Introduction A Simple Model Richardson’s Model

A Simple Example

We start by considering a system of only three nations, denoted A, B, and C. A is quite aggressive and war-prone. B is neutral and rather passive. 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.

MA 2071 A ’05 Bill Farr

Richardson’s Arms Race Model

Outline Richardson’s Arms Race Model

Introduction A Simple Model Richardson’s Model

Richardson’s Four Principles In Richardson’s model, the amount of arms possessed by a nation at time t = k + 1 depends on the four principles 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. MA 2071 A ’05 Bill Farr

Richardson’s Arms Race Model

Outline Richardson’s Arms Race Model

Introduction A Simple Model Richardson’s Model

A Linear Model These four principles allow us to generate 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.

MA 2071 A ’05 Bill Farr

Richardson’s Arms Race Model

Outline Richardson’s Arms Race Model

Introduction A Simple Model Richardson’s Model

Setting Coefficient Values For the three nations we described above, we might set the following values. Set g2 = 0, since nation B is neutral. 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. MA 2071 A ’05 Bill Farr

Richardson’s Arms Race Model

Outline Richardson’s Arms Race Model

Introduction A Simple Model Richardson’s Model

Matrix Form In matrix form, we could write this model as xk+1 = Axk + g where the entries in the matrix A  f1 A =  a21 a31

for our example would be  a12 a13 f2 a23  a32 f3

and the vector g would be given by   g1 g =  g2  g3 MA 2071 A ’05 Bill Farr

Richardson’s Arms Race Model

Outline Richardson’s Arms Race Model

Introduction A Simple Model Richardson’s Model

Steady States 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 xk+1 = xk . To determine if this can happen, we denote the steady state vector by xs and see if we can solve for it in the equation xs = Axs + g which we can writen in a more familiar form as (In − A)xs = g This is a nonhomogeneous system, so there are three possibilities. 1 There might be no solution. 2 There might be a unique solution, if (I − A) is invertible. n 3 The solution might exist, but have some negative components, which would not make any physical sense. MA 2071 A ’05 Bill Farr

Richardson’s Arms Race Model

Outline Richardson’s Arms Race Model

Introduction A Simple Model Richardson’s Model

Simulation Another approach to investigating such a model is by computing values of xk for k = 1, 2, 3, . . .. Doing so produces the following sequence of equations. x1 = Ax0 + g x2 = Ax1 + g = A2 x0 + Ag + g x3 = Ax2 + g = A3 x0 + A2 g + Ag + g or, in general, xk = Ak x0 + Ak−1 g + . . . + Ag + g

MA 2071 A ’05 Bill Farr

Richardson’s Arms Race Model

Outline Richardson’s Arms Race Model

Introduction A Simple Model Richardson’s Model

Possible Results of Simulation

One can imagine results like the following coming out of such simulation. 1

The magnitude of xk might tend to infinity, indicating an unstable arms race.

2

The vector xk might go to the steady state, indicating a stable situation.

3

The vector xk might go to zero, indicating complete disarmament.

MA 2071 A ’05 Bill Farr

Richardson’s Arms Race Model

Outline Richardson’s Arms Race Model

Introduction A Simple Model Richardson’s Model

Richardson’s Model of the World in 1935 After World War I, Richardson collected data on ten nations and came up with the following matrix A                

 0.5 0 0 0.1 0 0 0 0.05 0 0 0 0.05 0 0 0 0 0.2 0 0 0.01   0 0 0 0.1 0.2 0 0.2 0 0 0   0.2 0 0.2 5 0.1 0 0 0.05 0 0.04   0 0 0 0.2 0.25 0.3 0.1 0 0 0   0 0 0.1 0 0.2 0.75 0 0 0 0.1   0 0.2 0 0 0 0 0.5 0 0.2 0.2   0.05 0 0 0.05 0 0 0 0.5 0 0.05   0 0 0 0.1 0.1 0.1 0.2 0 0.65 0.1  0 0.1 0 0.4 0.1 0.1 0.2 0.05 0 0.5

MA 2071 A ’05 Bill Farr

Richardson’s Arms Race Model

Outline Richardson’s Arms Race Model

Introduction A Simple Model Richardson’s Model

Richardson’s Model, cont. and values of g given by         g=       

1/20 1/20 1/20 3/20 1/20 1/10 3/20 1/20 1/20 1/10

               

where the nations, in order, are Czechoslovakia, China, France, Germany, England, Italy, Japan, Poland, the USA, and the USSR. MA 2071 A ’05 Bill Farr

Richardson’s Arms Race Model

Outline Richardson’s Arms Race Model

Introduction A Simple Model Richardson’s Model

Exploring Richardson’s Model To do numerical computations, use the Maple worksheet. To test your understanding of the modeling process, here are some questions to think about. 1 In Richardson’s model, explain why the fatigue coefficients f i 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.   1/5 11/20 13/100 0  1/2 2/5 0 0   A=  2/5 0 1 0  0 0 0 1/4 What can you say about the relationships between the four nations? MA 2071 A ’05 Bill Farr

Richardson’s Arms Race Model