Arms Races and Price Wars

Two rivals each build up in response to the other, hold back because building costs money, and carry old grudges. Whether they settle at a balance or spiral into an arms race depends on one number: reaction versus restraint.

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How it works

Modeling with systems

In one sentence

A linear system $\mathbf x' = A\mathbf x + \mathbf b$ with $\det A \ne 0$ has one equilibrium $\mathbf x^* = -A^{-1}\mathbf b$ ; its type comes from the eigenvalues of $A$ , and $\det A < 0$ means a saddle.

General formula

{\htmlData{sym=0}{x}}' = {\htmlData{sym=2}{a}}\,{\htmlData{sym=1}{y}} - {\htmlData{sym=4}{m}}\,{\htmlData{sym=0}{x}} + {\htmlData{sym=6}{g}}, \qquad {\htmlData{sym=1}{y}}' = {\htmlData{sym=3}{b}}\,{\htmlData{sym=0}{x}} - {\htmlData{sym=5}{n}}\,{\htmlData{sym=1}{y}} + {\htmlData{sym=7}{h}}

Each equation reads "my change = my reaction to you − my restraint + my grievance". Strong restraint pulls both back to a balance; strong reaction pushes both away from it.

What the symbols mean

$x$
X's arms
How much country X spends on weapons.
$y$
Y's arms
How much country Y spends.
$a$
X's reaction
How strongly X responds to Y's arms.
$b$
Y's reaction
How strongly Y responds to X's arms.
$m$
X's restraint
How much the cost holds X back.
$n$
Y's restraint
How much the cost holds Y back.
$g$
X's grievance
Arming X does regardless of Y.
$h$
Y's grievance
Arming Y does regardless of X.

Each side arms in response to the other ( $ay$ , $bx$ ), restrains itself because arms cost money ( $-mx$ , $-ny$ ), and has standing grievances ( $g$ , $h$ ).
The equilibrium solves two linear equations; it exists and is stable when restraint wins, $mn > ab$ .
If the sides react more than they restrain, $ab > mn$ , the equilibrium becomes a saddle and almost every path runs away.
The sign of $\det A = mn - ab$ decides between balance and escalation.

Learn the math

Modeling with SystemsDiff Eq · Lesson 36, Systems of Differential Equations