Major Powers and Weak Allies: Stability and Structure in Arms Race Models

Two general problems of formal models of arms races are examined. First, we must expect that the true processes (and thus correct models of them) will be nonlinear, and nonlinear equation systems are considerably more difficult to solve analytically than linear systems. Second, even the linear models are difficult to solve in many-nation systems without unrealistic assumptions of behavioral symmetries between governments. So long as the system remains near equilibrium, however, linear models will usually be good approximations of more complicated nonlinear models. And by examining a general linear reaction process model under the constraints of alternative structures of influence and reaction, we can draw a number of conclusions without asserting behavioral (coefficient) equalities between governments. Such different reaction structures may be used to model power differences, alliance ties, and military aid flows. The mathematical solutions are frequently straightforward, and the conditions for stability are sometimes simple. While the causes of arms races are complex, reaction to the acts of others has been cited as a key factor in explaining both contemporary and historical arms races (c.f. McNamara 1967). Such a reaction process, in which armaments are purchased at least partly because potential opponents purchase arms, is postulated in the well-known family of models of arms expenditures developed by Richardson (1960). Although the Richardson models are deterministic, and although most can be reduced to a very general linear form, applications of these models typically have been limited to two nation-states or two groups of states, o r to other dyadic reaction processes such as perception and expression of hostility. This limitation reflects the difficulty of obtaining more than very general solutions when several nations are modeled.