In This Article Expand or collapse the "in this article" section Arms Race Modeling

  • Introduction

Political Science Arms Race Modeling
Kanishkan Sathasivam
  • LAST REVIEWED: 12 October 2023
  • LAST MODIFIED: 23 August 2017
  • DOI: 10.1093/obo/9780199756223-0231


Models of the arms race process are among the most extensively tested mathematical models in the social sciences. This is not surprising given their obvious theoretical and mathematical elegance, and the relative ease with which they lend themselves to quantitative analysis. In spite of the vast number of Richardson-type and alternative arms race model specifications proposed by scholars, the results from empirical tests of those models have been decidedly mixed. While some scholars have obtained results demonstrating reasonably good fits of their models to data, others have failed to demonstrate any statistically significant action-reaction relationship between states purportedly engaged in an arms race. This article comprehensively surveys and synthesizes more than fifty years of research on arms races, in an effort to understand what has worked and what has not in specifying and modeling the arms race phenomenon.

The Origins of Mathematically Modeling the Arms Race Phenomenon

The study of arms races as an action-reaction phenomenon essentially began with the pioneering work of Lewis F. Richardson. The classic Richardson arms race model originates from his “linear theory of two nations,” which ascribes certain motives, common to all nations in a very general sense, that drive nations in times of peace to increase or decrease their military preparations for war (Richardson 1960). Some motives such as revenge for perceived historical injustices, unhappiness with the outcomes of treaty negotiations from the past being a good example of this, are independent of the current military capabilities of other nations. These motives tend to be enduring and constant over time. In contrast, the powerful motives of fear and rivalry are generally associated with the actual present-day military capabilities of other nations. The motive of fear is usually based on the overall magnitudes of the military forces of other nations, while the motive of rivalry is based more on the differences between one’s own level of armaments and those of other nations. Finally, the enormous economic costs of adding more armaments to one’s current level of armaments, both in terms of the actual funds expended on those armaments as well as in terms of the resources allocated for military production that are unavailable for civilian use, are viewed as being a restraining motive on ever-increasing military expenditures. These assumptions lead to the very elegant mathematical formulation of the two-nation arms race phenomenon: dX1(t)/dt = k1.X2(t) − a1.X1(t) + g1 dX2(t)/dt = k2.X1(t) − a2.X2(t) + g2, where the two variables X1(t) and X2(t) represent, respectively, the armaments levels of countries 1 and 2 at time t, and dX1(t)/dt and dX2(t)/dt the rates of change over time of X1(t) and X2(t).

  • Richardson, Lewis F. Arms and Insecurity: A Mathematical Study of the Causes and Origins of War. Pittsburgh, PA: Boxwood, 1960.

    The seminal work that originates the modern study of the arms race phenomenon by using mathematical modeling and develops the linear theory of two nations. Reprinted as recently as 2012 (Whitefish, MT: Literary Licensing).

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