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 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
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.
These four factors allow us to consider a the following system of
for our three hypothetical nations,
- The amount of arms they already had
at time t = k.
- The amount of arms they might build in
response to the other nations arms levels.
- 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.)
- 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
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
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 over-reacts 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
In matrix form, we could write this model as
where the entries in the matrix A for our example would be
and the vector would be given by
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 . To
determine if this can happen, we denote the steady state vector by
and see if we can solve for it in the equation
which we can write in a more familiar form as
This is a nonhomogeneous system, so there are three possibilities.
- There might be no solution.
- There might be a unique solution, if (In - A) is invertible.
- The solution might exist, but have some negative components,
which would not make any physical sense.
One of the main techniques for investigating models like this is
simulation. That is, we choose an initial vector and then compute
values of for . Doing so produces the following sequence of equations.
or, in general,
One can imagine results like the following coming out of such
After World War I, Richardson collected data on ten nations and came
up with the following matrix
- The magnitude of might tend to infinity,
indicating an unstable arms race.
- The vector might go to the steady state,
indicating a stable situation.
- The vector might go to zero, indicating complete
and values of given by
where the nations, in order, are Czechoslovakia, China, France,
Germany, England, Italy, Japan, Poland, the USA, and the USSR.
- In the basic model, explain why the fatigue coefficients fi
should be in the range . How would you explain fi = 0
or fi = 1?
- Suppose that the coefficient matrix for a group of four nations
is given by the following.
- What can you say about the relationships between the four
- 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.
- Refer to the matrix that Richardson obtained.
- Which nation is the most aggressive? Explain your choice.
- Which nation is the most disliked or feared? Explain your
- If you ignore the diagonal entries, what would row sums
represent? Again ignoring the diagonal entries, what would column sums
- Again consider Richardson's model. By simulation, try to
determine if it is stable or not. You should find that it is
unstable. Can you change the outcome by changing the initial arms
level? Try at least two different initial arms levels besides the one
supplied in the Getting started worksheet.
- In 1935, Germany and Italy were both in the midst of a massive arms
build-ups. What evidence for this can you see in the matrix and the
grievance vector? Can you
change the behavior of the model by reducing the fatigue coefficients
for Germany and Italy to 1/5? That is, can you make the
arms race a stable one just by reducing these coefficients? The answer
to this should be yes. Find the equilibrium.
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