Impartial Combinatorial Games and the Nuclear Escalation Ladder

Abstract

We model Herman Kahn's escalation ladder as an impartial combinatorial game. Reindexing each rung by its distance to the nuclear threshold turns the ladder into a subtraction game, the most tractable class in combinatorial game theory, and the doctrinal fact that no side wishes to fire first selects the misere convention. We prove that single-ladder stability is governed by a congruence (Theorem 4.1) and derive a ladder-design corollary that makes the burden of first escalation a function of ladder length and escalation granularity (Corollary 4.2). For simultaneous theaters we show, under normal play, that joint stability is the Nim-sum of the theater-wise escalation distances (Theorem 5.2), a condition that is neither additive nor dominated by the most dangerous theater. We then show the Nim-sum reduction fails under misere play, introduce the misere quotient as its replacement, and prove by exhaustive backward induction that for two-step escalation the quotient is the order-six monoid with generators a, b satisfying a^2=1 and b^3=b, with loss set {a,b^2} (Theorem 6.3). To our knowledge, impartial combinatorial game theory has not previously been applied to nuclear escalation ladders; the existing game-theoretic literature on escalation is classical and payoff-based.