Attrition games

Abstract

This paper is a summary report of investigations of C. Tompkins, the authors, and others concerning r a t h e r general mathematical models of military situations. The basic assumptions on the mechanism of attrition a r e given by certain differential equations modified and generalized f r o m the classical Lanchester equations by R. N. Snow, C. Tompkins and the authors. P a r t I deals with the attrition equations and is thus a study of a family of stochastic processes. It is also the study of combat wherein strategy in a game thcoretic sense does not enter; the combat is described by physical constants expressed as probabilities--no provision is made f o r rational choices by the players. P a r t s I1 and I11 deal with games, i.e. minimax problems, which isolate certain aspects of the problem of military (strategic, tactical, o r logistical) decision. In P a r t I1 the decision means the commitment of a p a r t of the available f o r c e to battle, retaining the remainder as a reserve. In P a r t I11 the f o r c e s a r e in the field, and the decision means a distribution of the f i r e of each type of unit among the s e v e r a l possible types of targets. P a r t IV is devoted to the problem of estimating intrinsic values of weapons. Results f r o m P a r t s I and 11, along with related work by R. H. Brown, a r e employed to motivate a linear estimate f o r weapons of one type against one enemy type. This estimate is then generalized f o r the case of several weapon types.More precise details are given in the introductions to the individual parts of the paper. It will be c l e a r to the r e a d e r that this paper, while contributing somewhat to the study of attrition and military decision, presents several unsolved problems. Of particular note a r e difficult computational problems on a theoretical level. PART I -ATTRITION PROCESSESIn this Part we are concerned with bivariate stochastic processes based upon probabilities of survival of individual units. We summarize work of various authors [9, 11, 121 with some added results and comments. The central problem is to determine P (x, E ; r, p ; t) for all arguments non-negative and x, E , r, p integral; P is the probability that at time t, (x, E ) units survive of (r, p ) which began at t = 0. With the exception of 1.4, weconsider homogeneous attrition: individual units available to a player are identical but not necessarily like units available to his opponent. In 1.1 we consider a probabilistic case: P is positive for t > 0, 0 5 x 2 r and 0 $ E 5 p . We give a partial solution in closed form for a special case (1.1.6) and indicate that numerical computation of the general case appears promising. We next introduce a truncated process, Section 1.2, in which we ignore the behavior of the process in time in favor of the direction of travel of the system. This process yields to analysis and is justified somewhat by -lResearch supported ( i n p a r t ) by the Office of Naval Research. This a r t i c l e was presented to the American Mathematical

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