cient differential equations such as the Lanchester-square-law. 1 Lanchester equations will probably remain quite useful for making particular points in the classroom (e.g., illustrating the power of concentration or the value of “crossing the T” in classic naval engagements) or theoretical papers, but to argue about their more general validity is to chase red herrings. It is the simulations, not the Lanchester differential equations, that should be examined.

Today's higher-level combat simulations (e.g., those at division, corps, and theater levels) are best seen as implementing aggregate state-space models (something much broader than Lanchester models). The basic notion is that the “state” of the system (the two opposed forces, their strategies, and the environment in which they fight) can be represented by a collection of variables such as counts of personnel and vehicles in an area, and terrain factors characterizing that area, rather than the locations and current behaviors of all the individual entities such as individual soldiers and tanks. Usually, the simulation then generates the predicted future state as a function of the current (aggregate) state. In more general formulations, there can be “memory effects” of previous states as well. Again, the variables affecting this prediction are not just the sides' strengths (much less their scalar strengths, as in the simpler Lanchester equations). Instead, the predicted change of state depends on many other factors such as terrain, defender preparations, flank exposure, strategy, and tactics. One important change of state, typically made at the end of time periods or when some significant event occurs, is a change of strategy or tactics (e.g., a decision to attack or withdraw, or to maneuver reinforcements to a trouble area). It is then true that the close-combat ground-force attrition in a given time step is sometimes approximated by a local use of some Lanchester equation, but the “coefficients” used can be highly situation dependent, that is, dependent on many other state variables that change over time (Allen, 1992, 1995). Thus, the simulation does not (or at least is not intended to) behave like a constant-coefficient Lanchester equation. 2

Breakdown of Aggregate State-Space Models

It has long been a reasonable hypothesis—but only that—that a relatively aggregated close battle in a particular area will have attrition that can be reason-

1  

The principal reference for discussion of Lanchester equations is Taylor (1983b), which also covers many generalizations of the original work (Lanchester, 1916), including generalizations such as Bonder-Farrell theory (Bonder and Farrell, 1970) used in simulations. See also the recent collection of papers in Bracken et al. (1995), which includes historical analysis, a translation by Helmbold and Rehm of work by Osipov, and considerable thoughtful discussion. Wise (1991) explains some of the fundamental ambiguities in using and calibrating Lanchester laws. Hughes (1986) and Deitchman (1962) discuss applications of Lanchester models to naval and guerrilla warfare, respectively. Dupuy (1987) includes discussion of how his extensive history-based work on combat modeling relates (and does not relate) to Lanchester theory.

2  

In fact, simulations do sometimes generate behaviors that look remarkably like what could be generated by such an equation, but that is an artifact of the particular application.



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