Richardson Arms Race Model

Mirha Khan, Faryal Usman and Areeb Khichi

Background

● Lewis Fry Richardson (1881-1953) was a British mathematician and physicist

● He conjectured that arms races were often preparation for war.

● One of the conclusions of his research was a system of differential equations modelling arms races

● The idea behind the model is that the likelihood of two nations engaging in war can be determined by a set of differential equations

Assumptions

▪ By the Richardson’s model, the amount of arms owned by a country at t=k + 1 depends on four principles.

1.The amount of arms the country itself has at time t = k. 2.The amount of arms possessed by neighbouring countries 3. Economic or internal constraints that hinder arms buildup. 4. Exogenous factors such as particular grievances, external pressures or revenge motives.

Two by two general case

❖ This first order, linear system of differential equations can be used to model arms buildups.

❖ This system models only two countries❖ It can be extended to include more countries for example by

adding a dz/dt term. ❖ Each equation corresponds to the rate of change of munition

levels for a particular country. ❖ Meanwhile the solutions x(t) and y(t) represent the total

munitions stockpiles for each country at time t.

How these assumptions apply

For example: ❖ x(t) and y(t) represent the amount of arms possessed by nations x

and y respectively at time t❖ The constants ”a” and ”b” gauge the fear/ reaction to the amount of

arms another country possesses.❖ The constants ”m” and ”n” are known as the ”restraint” or ”fatigue”

factors.

Continued

❖ They represent the desire of a nation to reduce arms stockpiles at a rate directly proportional to what they possess.

❖ The r, s and terms represent factors extraneous to the arms race that have an effect on arms stockpiles such as diplomatic pressure from treaties or other countries, the threat of public opinion etc.

We can write the system of equations involving two countries in matrix notation. Since the system is non-homogeneous, we would have to write the exogenous terms as a separate vector as shown below.

Thus to find the general solution

Finding the general solution

Example

Using data on the military expenditures of NATO and WARSAW in the 1980s, we were able to obtain the following regression equations.

Thus the following will be the parameters for the model

Finding the eigenvalues and vectors of the above matrix, we get

Thus our final general solution will be of the form:

As t → ∞ both exponential terms increase to infinity. Thus it is possible that the 2 countries do not reach any steady state solution and the level of arms blows up over time.

Example from David Bigelow (2003)’s An Analysis of the Richardson Arms Race Model David Bigelow

As mentioned earlier, we can extend this system to model more than 2 countries❖ Consider a system of three countries X, Y and Z. ❖ X is quite aggressive and war-prone. ❖ Y is neutral and rather passive. ❖ Z is a foe of nation X. ❖ Let’s assign variables x, y, and z to them respectively, which indicate

the amount of arms that each nation has.Using the richardson model assumptions we can write this as a system of equations.

In cases involving more than 2 countries

In cases involving more than 2 countries❖ X is war-prone. b1 and c1 will be greater

than one, since nation A overreacts to the arms of nations Y and Z.

❖ Since nation Y is neutral, c3 would have a value of zero and b2 and c2 would also be very small.

❖ Setting b3 to 1 would indicate that nation Z always builds arms if nation X does, alternatively setting it to 1.4 indicates that nation Z always builds 40% more arms than nation X has

❖❖ We might also set the values a1 = 1, a2 =

0, and a3 = 1/3 to express that nation X’s arms budget is not cut, nation Y disarms and that nation Z’s faces a cut

❖ Set g2 = 0, since nation Y is neutral❖ g1 is positive because X is aggressive

and war-pronE❖ g3 is likely to have a zero value to a small

value

Writing the equation

We want to see if the weapons reach a steady state. Since our system is non-homogenous we solve the following set of equations.

Steady State

Depending on the invertibility of , there are three possibilities for the solutions: ❖ No Solutions

❖ A unique solution

❖ A solution with negative components