- 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.

*x*_{k+1} = *f _{1}*

*y*_{k+1} = *f _{2}*

*z*_{k+1} = *f _{3}*

For the three nations we described above, we might set *g _{2}* = 0, set

In matrix form, we could write this model as

where the entries in the matrixOne 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.- 1.
- There might be no solution.
- 2.
- There might be a unique solution, if (
*I*_{n}-*A*) is invertible. - 3.
- 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 simulation.- 1.
- The magnitude of might tend to infinity, indicating an unstable arms race.
- 2.
- The vector might go to the steady state, indicating a stable situation.
- 3.
- The vector might go to zero, indicating complete disarmament.

- 1.
- In the basic model, explain why the fatigue coefficients
*f*_{i}should be in the range . How would you explain*f*_{i}= 0 or*f*_{i}= 1? - 2.
- Suppose that the coefficient matrix for a group of four nations
is given by the following.
- (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.
- 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.

- 4.
- If you ignore the diagonal entries, what would row sums represent? Again ignoring the diagonal entries, what would column sums represent?
- 5.
- 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. - 6.
- 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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