Despite intensive investigation, little evidence has been found for a traditional Richardson-style arms race between Greece and Turkey using regression methods. This article uses an alternative model of the arms race, which treats it as a simple repeated two-by-two game such as the Prisoner's Dilemma, in which each country can choose a high or low share of military expenditure. This gives four possible states: both high; Greece high Turkey low; Turkey high Greece low; both low. The strategies of each country, the choice probabilities given the current state, are then estimated using a discrete state regime-switching model, which estimates the transition probabilities between the four states. Various hypotheses about these strategies are tested as restrictions on these transition probabilities. One set of hypotheses is that the countries play 'tit-for-tat', doing what their opponent did in the previous period. This is rejected for both countries. Another hypothesis is that each country plays independently. Each country has its own probabilities of switching between high and low, which do not depend on whether the other country is high or low. This hypothesis is accepted by the data. The estimates of the transition probabilities suggest that the states, high or low shares of military expenditure, are very persistent, with very high probabilities of staying in them. The estimates are not consistent with a traditional 'external' action-reaction explanation of shares of military expenditure, but are more consistent with 'internal' explanations which emphasize bureaucratic and political inertia.