An Optimal Control Model of Arms Races

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Abstract
Lewis Frye Richardson's simple differential equations model of armaments races has been long criticized for its lack of incorporation of the goals of nations. Using the mathematics of optimal control theory, the authors formulate a model which incorporates national goals into an “arms balance” objective function. The goals used are based on the traditional concerns in the balance-of-power literature. From an objective function together with the Richardson model an optimal armaments policy is derived. The United States-Soviet, NATO-WTO, and Arab-Israeli arms races are used as empirical examples, and the parameters in the model are estimated by means of functional minimization techniques. The optimal control model is further examined for its equilibrium and stability properties. The equilibrium and stability conditions are assessed with respect to the empirical examples. The findings are that while the United States and the Soviet Union in direct confrontation pursue strategies that lead to a lack of equilibrium and stability, when taken as part of NATO and WTO, the major powers and their alliance partners do pursue stable and equilibrium strategies. The Israeli policy is found to lead to equilibrium and stability while the Arab policy does not.