Stochastic Models for the Duration and Magnitude of a “Deadly Quarrel”

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Abstract
Once a war has been initiated, its continuation or termination may be considered to be determined by two transition probabilities—one describing the probability that in a small interval of time an additional death occurs, and the other describing the probability that in the same small interval of time the war is terminated. Given forms for these transition probabilities, one may obtain distribution functions for the duration and magnitude of a war, given that it has begun. The inverse problem of determining the form of the transition probabilities, given the distribution of war durations and magnitudes, is more difficult. However, functions are developed in this paper that will generate distributions corresponding to those presented in the tabulation of 315 wars prepared by L. F. Richardson. The expressions for the transition probabilities are derived based on the assumption that they depend at each instant only as the cumulative number of deaths to that point and the time over which the deaths have been incurred. Some of the implications of the expressions and the possibility of their use to categorize classes of wars are discussed.

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