# Lanchester Equations and Scoring Systems

### Appendix

### General Discussion

Lanchester equations are differential equations describing the time dependence of attacker and defender strengths*A*and

*D*as a function of time, with the function depending only on

*A*and

*D*.[12] One partly generalized version of the Lanchester equations has the following form

(A.1)

in which the attrition rates and exponents are time-independent parameters. Sometimes the equations are extended to include constant reinforcement-rate terms.

Most authors doing analytical work (as distinct from computer simulations) have focused on one of two special cases: the "square law" corresponds to *s*=*u*=1 and *r*=*t*=0; the "linear law" corresponds to *r*=*s*=*t*=*u*=1.

(a.2) square law

(A.3) linear law

It is usually said that the square law applies to "aimed fire" (e.g., tank versus tank) and the linear law to "unaimed fire" (e.g., artillery barraging an area without precise knowledge of target locations). Alternatively, it is sometimes said that the key feature of the square law is that it describes concentration of fire.

Although the simple Lanchester equations with constant coefficients remain useful for demonstrating some features of combat (e.g., the value of concentrating effort and the associated penalty for breaking up one's forces), especially when it is desirable to do so analytically, they are a poor basis for describing most combat situations. Computer simulations may use Lanchester expressions "locally" (i.e., for attrition estimates within a given time interval), but the coefficients of those equations change from time step to time step as conditions of terrain, defender preparations, and many other factors change. Good computer simulations recognize that the losing side may choose to break off battle rather than be annihilated. Some use equations in which the exponents are much smaller than called for in the square law and in which there are some differences in exponent between attacker and defender (e.g., to reflect the different mix of aimed and unaimed fire that might result from the defender having better cover and the attacker relying more heavily on artillery preparation).[13] Most computer simulations deal separately with different classes of weapon-on-weapon interactions and treat maneuver as fundamental, not an annoying complication. Unfortunately, such computer simulations are then more complicated to understand and discuss. Hence, Lanchester equations continue to have a place in explaining simple points.

For readers interested in understanding the relationship between Lanchester equations and "physics-level calculations," a recent study may be illuminating (Hillestad, Owen, and Blumenthal, 1993). It illustrates how a Lanchester square law can--in simple cases--be a reasonable approximation of events when the opponents approach each other frontally. The authors began with item-level simulations with individual shooters (e.g., tanks) and kill-per-shot probabilities dependent on range. They assumed flat, featureless, terrain. Even in this case, moving to and understanding the Lanchester representation was nontrivial and, in practice, was informed by theory and experimentation with the higher-resolution simulations.

### Estimating the Strengths or Scores Used Within Lanchester Formulations

Lanchester equations assume that the sides' strengths can be characterized by scalar quantities that are usually called scores. In practice, estimating appropriate scores can be very troublesome, especially when the sides each have a mix of equipment and especially when the opponents have different equipment, organization, and doctrine. The most important considerations are accounting for the number of items of relevant equipment and gross features of context (type terrain, type battle, and whether there is a serious mismatch of capabilities). Early scoring methods, known as static methods, did not reflect context, but a newer situational scoring method does so, albeit in a way dependent on expert judgment for correction factors (Allen, 1992). The situational scoring method is used by RAND in the RSAS and JICM theater-level models. It has been used in Germany for NATO-sponsored work on multipolar stability concepts (Huber and Helling, 1995). Another subtle problem in using scores involves the treatment of qualitative factors (e.g., the effects of terrain or the differences in competence between equally sized and equipped forces of different nations). Lanchester intended that *A* and *D* measure numbers of entities (e.g., people or tanks). Applying Lanchester laws to force *strength* (i.e., scores reflecting both numbers and qualitative features of combatant entities) requires great care to avoid logical inconsistencies (Lepingwell, 1987; Homer-Dixon, 1987).[14] It is mathematically cleaner to treat qualitative effects by modifying the attrition coefficients rather than the scores. In this report I assume that appropriate scores can be constructed.

### The 3:1 Rule

The 3:1 rule in ground combat has been discussed for centuries, but it is difficult to find authoritative sources justifying it in any detail. For discussion and some citations, see Mearsheimer (1989). For rejoinders see Epstein (1989) and Dupuy (1989). My own view (consistent, I believe, with Mearsheimer's intended message) is that for modern mechanized combat the 3:1 rule applies approximately, when applied to scores such as WEI/WUV or equivalent-division scores determined largely by the number of pieces of major equipment such as tanks. It applies only to equally competent opponents when one of them is fighting from prepared positions in good defensive terrain and the other is conducting a frontal attack. In other situations, the defender advantage is normally less. Dupuy (1987) deals with this by assessing a "combat power," which is something like a WEI/WUV score modified by a series of correction factors for terrain, defensive preparations, surprise effects, and so on. After making such corrections, Dupuy treats break-even as a ratio of 1:1--in "combat power." Models like RAND's RSAS or JICM treat the same effects in somewhat different ways.

[12] For extensive discussion of Lanchester equations see the treatises by Taylor (1980, 1983) and more recent work by Lepingwell (1987), Homer-Dixon (1987) and Epstein (1990), all of which describe the shortcomings of Lanchester theory. See also Wise (1991), which discusses effects of maneuver and command-control, and Helmbold (1993, 1994), which discuss alternative formulations useful for examining empirical data and appreciating some of the more subtle implications of the formulation. The Lanchester equations were discovered simultaneously and independently by the Russian scientist Osipov.

[13] See, e.g., Allen (1992, p. 41) for the expressions used in RAND's RSAS and JICM models. (*dA*/*dt*)/*A* goes as 1/*F ^{.93}*; (

*dD*/

*dt*)/

*D*goes as

*F*; the ratio of fractional loss rates goes as 1/

^{.64}*F*. These were based on loose fits to historical data as well as approximate theoretical arguments.

^{1.6}[14] For example, models such as the RSAS and JICM that use qualitatively adjusted scores to compute attrition must calibrate the scores so that the results are the same as if the situation-dependent effects had been included in the attrition coefficients. Further, they must keep separate track of the unadjusted and adjusted force levels, because the ratios of loss rates are different for these quantities. See Allen (1992, p. 41).

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