Imperfect Collective Security and Arms Race Dynamics: Why a Little Cooperation Can Make a Big Difference

This paper uses simple mathematical models to explore the relationship between security regimes and arms race dynamics. The main focus is a regime known as collective security, which is receiving wide attention. Little of the attention is mathematical, however, and, to my knowledge, none of it involves dynamical systems. One aim of this paper, then, is to formalize collective security in a dynamical systems context, which will allow us to extract some unexpected results. This formalization, of course, requires a rigorous definition of collective security. To wit: Imagine three countries $x$, $y$, and $z$. “Perfect” collective security would then operate as follows: If $x$ attacks $y$, $z$ allocates all force to $y$; if $y$ attacks $z$, $x$ allocates all force to $z$; and so on. The general rule is simply that “the odd man out instantly allocates all force to the attacked party.” In more biological--or sociobiological--terms, perfect collective security is a form of reciprocal altruism.