Arms race modeling
(Adapted from various sources. For general worldwide military expenditure and arms race issues see http://www.sipri.org)Background
Worldwide military expenditures have grown sharply during the past 15 years, even after adjusting for inflation -- more than a 60% increase between 2001 and 2010 alone. One of the major factors that causes escalation in weapons spending across the world is arms races. Irrationality abounds in the competition between groups of nations to outdo one another in weapons acquisitions and strengthening their military power. The undisputed leader of the pack, of course, remains the U.S., with a military budget nearly 6 times larger than its nearest competitor, China, and with 43% of the world's total share of military spending.
One of the earliest and best known mathematical models for arms competitions is the Richardson Arms Race Model, named after a Quaker mathematician who developed the model in the 1930s. Richardson believed that as nations accumulated larger and larger stockpiles of weapons, their willingness to use them also increases, eventually leading to war. In his view, the arms buildup process itself is the precursor to, and key predictor of, impending war.
I. A simple 2-nation arms race model
The scene: Two countries named "Green" and "Purple" are engaged in a competition to buildup their military strength and weapons stockpiles. We will model this competition via functions that measure their respective annual military expenditures as a function of time (in some common units of currency). Here is our modeling strategy:
- We will model the rate of change in their military expenditures, with respect to time.
- Let G(t) = military expenditure of country Green at time t.
- Let P(t) = military expenditure of country Purple at time t.
- Let t be in units of years. Assume G(t), P(t) are always positive.
- The schematic diagram helps us see the trend/shape of vs and vs , if we assume and are positive. Sketch graphs showing those qualitative behaviors.
- From the general trend in your graphs, what can you say about the sign of and ? Notice how the schematic diagram also tells us exactly that, even if we ignore our graphs.
- We want to capture this behavior in a model that looks like: , where is a function of and/or and/or . Conjecture a form for the function . For example, you could say: , or , or , or, ... Given the assumptions in this scenario, what would be a good choice for ?
- Model in a similar way.
- Consider the task of modeling , or . The simplest way to account for the two factors that influence is to set: term1 +term2 What sign should these terms have? Why?
- We already have a reasonable form for term1 from Model 0. Conjecture a similar reasonable form for term2.
- Put everything together and write the complete model for and .
- There are 4 parameters in this model, say a, b, c, and d. Their numerical values (assumed positive) have a significant effect on the long-term predictions from the model. What range of values might be reasonable for these parameters? Why?
- Consider country , and suppose is positive. What effect does it have on 's military spending? Does your conclusion depend in any way on the other country 's military spending?
- Likewise, if is negative, it has a certain effect on , independent of other factors.
- Clearly, the parameter is trying to capture some behavior pattern of that persists independent of other things. A common interpretation is that it accounts for historical grievances (if is positive), or good will (if is negative) that country feels towards country . A similar interpretation applies to parameter .
- Note that the range of typical numerical values for and is very different from those of the other parameters. Why? Hint: Look closely at the terms in the equations; think about comparing their magnitudes.
To explore some of these possibilities, let's try simulating a few different parameter values and initial conditions using the interactive segment below. Discuss what you find, e.g., do you get runaway arms buildups? do you get complete mutual disarmament? is it possible to get a fixed point (i.e., a constant value of and that doesn't change as increases)? etc.?