approximations and empirics for stochastic war equations

Approximations and Empirics for Stochastic War Equations

Kjell Hausken,1 John F. Moxnes2

1 Faculty of Social Sciences, University of Stavanger, N-4068 Stavanger, Norway

2 Department for Protection, Norwegian Defence Research Establishment, P.O. Box 25, 2007 Kjeller, Norway

Received February 2002; revised January 2005; accepted 12 July 2005DOI 10.1002/nav.20105

Published online 25 August 2005 in Wiley InterScience (www.interscience.wiley.com).

Abstract: The article develops a theorem which shows that the Lanchester linear war equations are not in general equal to theKolmogorov linear war equations. The latter are time-consuming to solve, and speed is important when a large number ofsimulations must be run to examine a large parameter space. Run times are provided, where time is a scarce factor in warfare. Fourtime efficient approximations are presented in the form of ordinary differential equations for the expected sizes and variances ofeach group, and the covariance, accounting for reinforcement and withdrawal of forces. The approximations are compared with“exact” Monte Carlo simulations and empirics from the WWII Ardennes campaign. The band spanned out by plus versus minusthe incremented standard deviations captures some of the scatter in the empirics, but not all. With stochastically varying combateffectiveness coefficients, a substantial part of the scatter in the empirics is contained. The model is used to forecast possiblefutures. The implications of increasing the combat effectiveness coefficient governing the size of the Allied force, and injectingreinforcement to the German force during the Campaign, are evaluated, with variance assessments. © 2005 Wiley Periodicals, Inc.Naval Research Logistics 52: 682–700, 2005.

Keywords: war; battle; Kolmogorov; Markov process; Lanchester linear warfare; differential equations; Ardennes campaign

1. INTRODUCTION

After World War II stochastic analysis of warfareemerged, for example, by Morse and Kimball [12], to sup-plement earlier deterministic analysis, most notably byLanchester [11]. Stochastic analysis is time consuming tocarry out even on today’s computers. Modern warfare, withincreased capability of surveillance and intelligence gather-ing of fluctuating characteristics and parameters throughtechnological and human means, makes time-efficient cal-culations of alternative war scenarios crucial. This suggeststhe need for approximations to stochastic analysis in theform of ordinary differential equations. This article providessuch equations for the expected value, variance, and covari-ance, which we consider to be superior to the classicalLanchester approximation.

A deficiency of the Lanchester equations was first pointedout by Morse and Kimball [12, p. 70] and Ventisel [18, pp.183–184]. To wit, that the Lanchester equations appear toperform unsatisfactorily for small groups and that a variance

equation is missing. To remedy the deficiency, this articledevelops a foundational model at the individual level forwarfare based on probabilistic reasoning. This gives theKolmogorov linear war equations which are correct forarbitrary group sizes. The equations can be solved directly,or Monte Carlo simulations can be utilized. Both thesemethods are time-consuming. Various closure approxima-tions are proposed in order to construct ordinary differentialequations which allow for time-efficient computation. Theobjective of approximations is to reduce the computationtime by solving a limited set of ordinary differential equa-tions for the expectation, variance, and covariance. Just asthe Lanchester equations are extendable to account forreinforcement and withdrawal, we show what roles rein-forcement and withdrawal play in the expected value, vari-ance, and covariance of the Kolmogorov equations.

Section 2 develops the Kolmogorov linear war equations.Section 3 shows how four approximations differ from theKolmogorov linear war equations and from classicalLanchester linear war. Section 4 simulates an example oflinear warfare, comparing with Monte Carlo simulations.Section 5 evaluates the approximations against empiricsfrom the WWII Ardennes campaign. Section 6 forecasts

Correspondence to: K. Hausken ([email protected]); J.F.Moxnes ([email protected])

© 2005 Wiley Periodicals, Inc.

possible futures. Section 7 discusses run times and furtherwork. Section 8 concludes.

2. THE KOLMOGOROV EQUATIONS OF WAR

Lanchester’s [11] classical and well-known suggestionfor linear warfare, generalized to account for reinforcementand withdrawal, is

ni�t� � ��i�t�n1�t�n2�t� � �i�t�wi�n1, n2, t�

� �i�t�ri�n1, n2, t�, i � 1, 2, (1)

where ni(t) is the size of group i at time t, �i(t), �i(t), �i(t)are combat, reinforcement, withdrawal effectiveness coef-ficients, ri(n1, n2, t) is reinforcement, and wi(n1, n2, t) iswithdrawal. Smaller values of �i(t) correspond to increasedcombat effectiveness for group i. Conversely, larger valuesof �i(t) and �i(t) correspond to increased reinforcement andwithdrawal effectiveness, respectively, for group i. �i(t)depends on the offensive and defensive capabilities ofweaponry and materiel, where the fire power, shot rate, andaiming capabilities are crucial, and on the training andcapabilities of personnel. �i(t) and �i(t) are similar to �i(t)for reinforced and withdrawn weaponry, materiel, and per-sonnel, and also accounts for group i’s willingness to rein-force/withdraw and capabilities of deploying/withdrawingeffectively in combat.

We refer to hi(n1, n2, t) � n1(t)n2(t) � n1n2 as thecombat equation, where we hereafter suppress the argumentt on n1 and n2. There are many reasons why we analyzen1n2. First, the nature of approximations is such that thegeneral equation hi(n1, n2, t) cannot be analyzed.1 Analyz-ing approximations for all possible forms that hi(n1, n2, t)can take is an insurmountable task. Some specific choice hasto be made. Second, this article makes a first step for linearwarfare, one of the most common forms of warfare in theliterature.2 Third, analyzing n1n2 provides a nice illustrationand is suggestive of analyses of alternative combat equa-tions. Fourth, demonstrating closure of approximated solutionsis more straightforward for linear warfare than for other formsof warfare.3 Fifth, although the issue is not settled, linearwarfare appears often to be descriptive of real warfare.

In Lanchester linear warfare agents in one group shootinto the other group without knowing the location of theopposing agents and thus cannot know when a hit has been

made. The loss rate for group 1 is proportional to n2, sincea larger group 2 is likely to produce a larger number ofn1-casualties, and proportional to n1, since the larger is n1,the larger is the probability that a shot from group 2 will hit.Define Nit (stochastic process) as the number of agents ingroup i at time t. We let the Nit � ni individuals in eachgroup be independent, and define pi

h(n1, n2, h) as theprobability that a specific agent in group i is hit during a timeinterval h, given Nit agents in group i. The joint binomial pointprobability for the number Ni,t�h of agents in group i at timet � h, given Nit agents in group i at time t, is then

p�N1,t�h � n1 � x, N2,t�h � n2 � y�N1t � n1, N2t � n2�

�mod� n1

x �p1h�n1, n2, h�x�1 � p1

h�n1, n2, h��n1�x� n2

y �� p2

h�n1, n2, h�y�1 � p2h�n1, n2, h��n2�y, (2)

where mod means that this is a model assumption. ThePoisson point process for the probability of zero hits duringh is Exp[��]. We model the Poisson parameter as � � nj

�tt�h �i(u) du, that is, proportional to the time interval h,

�i(t), and the size nj of the opposing group, since the largeris any of these three factors, the larger is the probability thatthe agent in group i is hit. The probability of being hit is

pih �

def

pih�n1, n2, h� �

mod

1 � Exp��nj �t

t�h

�i(u) du�

� �i�t�hnj � O�h2�, i � j, i, j � 1, 2, (3)

where last equation follows from including terms up to andincluding order O(h) in the Taylor series. Inserting (3) into(2) for combat gives


� �O�h2� when x � �1 and y � 0,O�h2� when x � 0 and y � �1,

1 � h �i�1

2

�i�t�n1n2 � O�h2�, x � 0 and y � 0,

�1�t�hn1n2 � O�h2� when x � 1 and y � 0,�2�t�hn1n2 � O�h2� when x � 0 and y � 1,O�h2� when x2 � y2 1.

(4)

Analogously, pir � �i(t)hri(n1, n2, t)/ni � O(h2) and pi

w

� �i(t)hwi(n1, n2, t)/ni � O(h2) are the probabilities thata specific agent is reinforced and withdrawn, respectively,for group i during h. When one of the groups goes extinct,that is N1t � 0 or N2t � 0, we assume that the war ends and

1 See Hausken and Moxnes [8, 9] for a stochastic analysis of thegeneral war equation hi(n1, n2, t).2 See Chen and Chu [3] for a recent example.3 The most plausible alternative to linear warfare is square warfarewith combat equations for example h1(n1, n2, t) � n2 and h2(n1,n2, t) � n1, where the analysis of approximations is morecomplex.

683Hausken and Moxnes: Approximations and Empirics for Stochastic War Equations

ri(N1t, 0, t) � 0 and ri(0, N2t, t) � 0.4 The war may startup anew with positive group sizes. Consistently with thesigns in (1), we express the stochastic variables as

Ni,t�h � Ni,t � R�Bin�ni, �i�t�hnj � O�h2��

� R�Bin�ni, �i�t�hwi�n1, n2, t�/ni � O�h2��

� R�Bin�ni, �i�t�hri�n1, n2, t�/ni � O�h2�� O�h2�,

where R[ � ] means random draw and Bin[ � ] meansbinomial distribution. The withdrawal term has the samenegative sign as the combat term, while the reinforce-ment term is positive. This means that withdrawal andcombat operate when x or y equals 1, while reinforcementoperates when x or y equals �1. More specifically, ac-counting for reinforcement and withdrawal, (4) general-izes to


�mod�

�1�t�hr1�n1, n2, t� � O�h2� when x � �1 and y � 0,�2�t�hr2�n1, n2, t� � O�h2� when x � 0 and y � �1,

1 � h �i�1

2

��i�t�n1n2 � �i�t�wi�n1, n2, t� � �i�t�ri�n1, n2, t�� O�h2�, x � 0 and y � 0,

�1�t�hn1n2 � �1�t�hw1�n1, n2, t� � O�h2� when x � 1 and y � 0,�2�t�hn1n2 � �2�t�hw2�n1, n2, t� � O�h2� when x � 0 and y � 1,O�h2� when x2 � y2 1. (5)

When the interval h 0 from t to t � h is sufficientlysmall, at most one agent in group i gets hit or withdrawnreducing ni to ni � 1, and at most one agent in group i getsreinforced increasing ni to ni � 1, where x, y � {�1, 0,1} and xy � 0. Applying the law of total probability gives

p�N1,t�h � n1, N2,t�h � n2� � �i1�0

�i2�0

p�N1,t�h � n1, N2,t�h

� n2�N1t � i1, N2t � i2�p�N1t � i1, N2t � i2� � p�N1t

� n1, N2t � n2��1 � �i�1

2

�i(t)hn1n2 � �i�1

2

�i(t)h

� wi�n1, n2, t) � �i�1

2

�i(t)hri(n1, n2, t)� � p�N1t � n1 � 1, N2t

� n2��1�t�h�n1 � 1�n2 � �1�t�hw1�n1 � 1, n2, t�� p�N1t

� n1, N2t � n2 � 1��2�t�hn1�n2 � 1� � �2�t�hw2�n1, n2

� 1, t�� p�N1t � n1 � 1, N2t � n2��1�t�hr1�n1 � 1, n2, t�

� p�N1t � n1, N2t � n2 � 1��2�t�hr2�n1, n2 � 1, t� � O�h2�,

(6)

where rearranging and taking the limit as h tends to zerogives the following equation, which is of the same class or

type of stochastic differential equations as those presentedby Kolmogorov (Cox and Miller [5], p. 181):

p�N1t � n1, N2t � n2� � �p�N1t � n1, N2t � n2�

� �i�1

2

��i�t�n1n2 � �i�t�wi�n1, n2, t� � �i�t�ri�n1, n2, t��

� p�N1t � n1 � 1, N2t � n2��1�t��n1 � 1�n2 � �1�t�w1�n1

� 1, n2, t�� p�N1t � n1, N2t � n2 � 1��2�t�n1�n2 � 1�

� �2�t�w2�n1, n2 � 1, t�� p�N1t � n1 � 1, N2t

� n2��1�t�r1�n1 � 1, n2, t� � p�N1t � n1, N2t � n2

� 1��2�t�r2�n1, n2 � 1, t�, (7)

where a dot above a variable means time derivation andwhere we assume that p(N1t � n1, N2t � n2) is differen-tiable which ensures continuity of p(N1t � n1, N2t � n2)and all derived expressions. After some lengthy calculationsprovided in Appendix A, (8) gives what we hereafter referto as the Kolmogorov linear war equations

E�N1t� � ��1�t��E�N1t�E�N2t� � Cov�N1t, N2t��

� �1�t�E�w1�N1t, N2t, t�� 1�t�E�r1�N1t, N2t, t��,

Var�N1t� � �1�t��E�N1t�E�N2t� � Cov�N1t, N2t�

� 2Cov�N1t, N1tN2t�� 1�t��E�w1�N1t, N2t, t��

� 2Cov�N1t, w1�N1t, N2t, t�� 1�t��E�r1�N1t, N2t, t��

� 2Cov�N1t, r1�N1t, N2t, t��,

4 Trivially, when group 1 is extinct (n1 � 0) at time t, it cannotsuffer combat loss up to time t � h; hence the joint probability in(3) equals zero when x � 1 and y � 0.

684 Naval Research Logistics, Vol. 52 (2005)

Cov�N1t, N2t� � ��1�t�Cov�N2t, N1tN2t�

� �2�t�Cov�N1t, N1tN2t� � �1�t�Cov�N2t, w1�N1t, N2t, t��

� �2�t�Cov�N1t, w2�N1t, N2t, t�� 1�t�Cov�N2t,

� r1�N1t, N2t, t�� 2�t�Cov�N1t, r2�N1t, N2t, t��. (8)

Equation (8) embodies five nice features. The first four ofthese are best illustrated without reinforcement/withdrawal:�i(t) � �i(t) � 0. First, if a stationary situation ariseswhere E(N1t) � 0, for example, after sufficiently large timet, positive expected group sizes E(N1t) 0 and E(N2t) 0 inserted into the first equation implies Cov(N1t, N2t) �0. If initially Cov(N1t, N2t) � 0, ensuring E(N1t) � 0requires a time period where Cov(N1t, N2t) � 0 whichcauses Cov(N1t, N2t) to be negative. According to the lastequation, this requires a time period when either Cov(N1t,N1tN2t) 0 or Cov(N2t, N1tN2t) 0. One of the lattertwo events is bound to happen; otherwise Cov(N1t, N2t)would increase monotonically, which would cause E(N1t)to decrease monotonically, which is impossible sinceE(N1t) cannot be negative.

Second, assume that all covariances are zero. The term�1(t) E(N1t) E(N2t) is negatively present on the RHS ofE(N1t), and positively on the RHS of Var(N1t). Hence theexpected value E(Nit) decreases with time at the same rateas the variance Var(Nit) increases with time: Var(Nit) ��E(Nit). During a time interval h, Var(Nit) �Var(Nit)h � �E(Nit)h � � E(Nit), where denotes in-cremental change. An example provides intuition. A sim-plification of the joint point probability in (3) gives

p�N1,t�h � n1 � i�N1t � n1�

� � 1 � p � 1 � �1hn1n2 � O�h2� when i � 0,p � �1hn1n2 � O�h2� when i � 1,

E�N1,t�h � N1,t�N1t � n1� � ��1��1hn1n2 � �p,

Var�N1,t�h � N1,t�N1t � n1� � E��N1,t�h � N1,t�2�N1t � n1�

� E�N1,t�h � N1,t�N1t � n1�2 � p � p2 � p � �E�N1,t�h

� N1,t�N1t � n1� when h is sufficiently small, (9)

which specifies the point probability for group 1 whengroup 2 has fixed size n2. When p is sufficiently small, dueto , the expected value and the variance have approxi-mately equal absolute values but opposite signs. Such aresult is quite common. Although this example is differentfrom warfare, it has in common with warfare that one agentis hit with a probability that is constant for this alternativeexample, and depends on both groups in warfare. TheMarkov process laying the foundation for (3) ultimately

generating (8) causes E(Nit) and Var(Nit) to decrease andincrease at the same rate. That the signs are opposite isintuitive since combat over time causes smaller group sizescombined with increased variation or uncertainty in theassessment of these group sizes through time.

Third, the covariance Cov(N1t, N2t) does not affect thatE(Nit) decreases with time at a rate that is 2�i(t)Cov(Nit,N1tN2t) larger than the rate by which Var(Nit) increaseswith time. The negative Cov(N1t, N2t) � 0 (see Section 4)reduces this rate of change, preserving the status quo, ame-liorating the adverse effects of war on both groups.

Fourth, the covariance Cov(Nit, N1tN2t) does not affectE(Nit) directly, but affects both Var(Nit) and Cov(N1t,N2t) with a negative sign. That is, when Cov(Nit,N1tN2t) 0 (�0), E(Nit) decreases with time at a larger(slower) rate than Var(Nit) increases with time.

Fifth, and naturally, positive reinforcement ri(N1t, N2t,t) 0 causes the expected value E(Nit) to decrease lesswith time, which is the intention of positive reinforcement.The variance Var(Nit) increases more with time whenE(ri(N1t, N2t, t)) � 2Cov(Nit, ri(N1t, N2t, t)) 0. Withconstant positive reinforcement ri(N1t, N2t, t) � c 0, thelatter assumption reduces to c 0 and is thus valid. Thisconfirms the intuition that combat loss combined with con-stant positive reinforcement generates more variation anduncertainty in the estimation of group sizes than combat lossalone. For example, reinforcing to counteract all combatloss, ri(N1t, N2t, t) � N1tN2t, �i(t) � �i(t), �i(t) � 0 givesE(Nit) � E(Nit0

), Var(Nit) � 2�i(t)[E(N1t)E(N2t) � Cov(N1t,N2t)], Cov(N1t, N2t) � Cov(N1t0

, N2t0), where t0 is the initial

time.Reinforcement and withdrawal influence E(Nit), Var-

(Nit), and Cov(N1t, N2t) as specified in (8), which furtherinfluence the loss of agents in group i. Hence we present theKolmogorov linear war loss equations, which account fortotal combat loss, that is,

EL�N1t� � ��1�t��E�N1t�E�N2t� � Cov�N1t, N2t��,

VarL�N1t� � �1�t��E�N1t�E�N2t� � Cov�N1t, N2t�

� 2Cov�N1t, N1tN2t��,

CovL�N1t, N2t� � ��1�t�Cov�N2t, N1tN2t�

� �2�t�Cov�N1t, N1tN2t�,

IEL�Nit� �def

EL�Nit� h, IVarL�Nit� �def

VarL�Nit�h,

ICovL�N1t, N2t� �def

CovL�N1t, N2t�h. (10)

Although �i(t) and �i(t) play no role in (10), this does notmean that �i(t) � �i(t) � 0. Quite the contrary. Reinforce-


ment and withdrawal influence (8) directly, and influencewar loss in (10) indirectly. Analogous expressions for group2 are found by permuting the indices 1 and 2 in (8) and (10),though not permuting within ri(N1t, N2t, t) and wi(N1t,N2t, t), which are not necessarily symmetric in their firsttwo arguments. There are thus five linear war equationsE(N1t), E(N2t), Var(N1t), Var(N2t), Cov(N1t, N2t) andfive linear war loss equations EL(N1t), EL(N2t), VarL(N1t),VarL(N2t), CovL(N1t, N2t). Together these constitute a setof ten coupled differential equations. Equation (10) pro-vides the expected value, variance, and covariance of theaccumulated war loss over time, from t0 to t. The incre-mental war loss expressions IEL(Nit), IVarL(Nit),ICovL(N1t, N2t) are the increments in the expected value,variance, and covariance for one time period of length h,where h is sufficiently small, forward in time t.

If the deterministic group sizes in (1) are replaced withstochastic group sizes, the first equation of (8) is obtained bytaking expectations of both sides, and allowing the inter-change of differentiation and expectation on the LHS. Thatis, the stochastic version of (1) is, in expectation, the firstequation of (8). The second and third equations of (8)constrain (1) with additional differential equations that in-volve higher-order moments of the stochastic process in (5).Assume that the deterministic variables in the Lanchesterlinear war equations in (1) can be interpreted as expectedvalues. The Lanchester linear war equations in (1) aremathematically equivalent with the Kolmogorov linear warequations [the first equation in (8)] when Cov(N1t, N2t) �0, E(wi(N1t, N2t, t)) � wi(E(N1t), E(N2t), t), andE(ri(N1t, N2t, t)) � ri(E(N1t), E(N2t), t). We thus pro-pose the following theorem.

THEOREM: The Lanchester linear war equations are notin general equal to the Kolmogorov linear war equations.

PROOF: It suffices to prove the Theorem without rein-forcement and withdrawal. Hence when Cov(N1t, N2t) � 0and �i(t) � �i(t) � 0, the first equation in (8) equals (1).We prove that there exists a time t0 � h, where t0 is theinitial time and h is suitably chosen, where Cov(N1,t0�h,N2,t0�h) � 0. Initial independence between N1t and N2t attime t � t0 implies

p�N1t0 � n1, N2t0 � n2� �mod

p1�N1t0 � n1�p2�N2t0 � n2�, (11)

where p1(N1t0� n1) and p2(N2t0

� n2) are the marginalprobabilities. This implies

Cov�N1t0, N2t0� � 0,

Cov�Nit0, Nit0Njt0� � E�Njt0�Var�Nit0�. (12)

When the expectations and variances are initially chosensuch that

��1�t0�Cov�N2t0, N1t0N2t0� � �2�t0�Cov�N1t0, N1t0N2t0� � 0,

(13)

Eq. (8) implies Cov(N1t0, N2t0

) � 0. After (7), we assumedcontinuity of p(N1t � n1, N2t � n2) which ensures con-tinuity of E(Nit), Var(Nit), Cov(N1t, N2t). When thederivative of a function differs from zero at t � t0, itfollows from continuity of the function at t � t0 that thereexists a positive h 0 such that the function itself differsfrom zero at t0 � h, that is

Cov�N1t0, N2t0� � 0 f Cov�N1,t0�h, N2,t0�h� � 0. �

(14)

3. APPROXIMATIONS

Var(N1t) and Cov(N1t, N2t) in (8) are determined fromE(N1t), E(N1t

2 ), and E(N1tN2t) as shown in Appendix A.The RHSs of E(N1t), E(N1t

2 ), E(N1tN2t) contain the ex-pressions E(N1t

2 N2t), E(N1tN2t2 ), which are not present as

derivatives on the LHSs. The derivatives E(N1t2 N2t) and

E(N1tN2t2 ) are straightforward to set up as two new equa-

tions, analogous to the development in Appendix A, butthese will contain further expressions on the RHSs whichare not present as derivatives on the LHSs. This process ofadding new equations to account for ever more complexexpressions on the RHSs of the earlier equations continuesad infinitum, giving an infinite number of equations, that is,an infinite set of differential equations. The system of equa-tions (8) is thus not closed, and approximations are neces-sary to close the system so that the equation system can besolved. The closure consists in reducing an infinite set ofordinary differential equations, which is countably infinitelymany ordinary differential equations, to a set of three ordi-nary differential equations for the expectation, variance, andcovariance. It is of interest to assess several approximationsup against each other, and up against the “exact” MonteCarlo simulation.

The simplest approximation to the Kolmogorov (linearwar) equations in (8) is the Lanchester (linear war) equa-tions in (1), where Lanchester’s deterministic variables areinterpreted as expected values. The first equation in (8) ismathematically equivalent to (1) when Cov(N1t, N2t) � 0,E(wi(N1t, N2t, t)) � wi(E(N1t), E(N2t), t), and E(ri(N1t,N2t, t)) � ri(E(N1t), E(N2t), t). Independence betweenN1t and N2t implies Cov(N1t, N2t) � 0, but Cov(N1t,N2t) � 0 does not imply independence between N1t andN2t. Hence mathematical equivalence between the firstequation in (8) and (1) does not require independence be-


tween groups 1 and 2. If N1t and N2t are independent, itfollows from the definition of covariance, as also expressedin (12), that Cov(Nit, NitNjt) � E(Njt)Var(Nit). If we alsoassume Cov(Nit, NitNjt) � 0, it follows that Var(Nit) � 0since E(Njt) 0 when no group is extinct in expectation.Hence with the three assumptions (1) independence be-tween N1t and N2t, which implies Cov(N1t, N2t) � 0, (2)Cov(Nit, NitNjt) � 0, and (3) E(wi(N1t, N2t, t)) �wi(E(N1t), E(N2t), t) and E(ri(N1t, N2t, t)) � ri(E(N1t),E(N2t), t), the first stochastic differential equation in (8) ismathematically equivalent to the classical Lanchester linearwar equations in (1) when the variables are interpreted asexpected values.

The first two of these three assumptions are very restric-tive and contribute to the inaccurate prediction of the de-terministic Lanchester model. We consider the first assump-tion of independence between N1t and N2t as especiallyrestrictive. Quite intuitively, two groups waging war are notindependent of each other, as the manner in which onegroup wages war affects the other and vice versa. None ofthe four approximations analyzed below assume indepen-dence between N1t and N2t, but consider the four combi-nations where Cov(N1t, N2t) equals or does not equal zero,and where Cov(Nit, NitNjt) equals zero or equals E(Njt)Var(Nit).

The simplest Approximation 1 (A1) we analyze assumesCov(N1t, N2t) � 0, which implies Cov(N1t, N2t) � 0, andCov(Nit, NitNjt) � 0. Inserting into (8) gives

E�N1t� � ��1�t�E�N1t�E�N2t� � �1�t�E�w1�N1t, N2t, t��

� �1�t�E�r1�N1t, N2t, t��,

Var�N1t� � �1�t�E�N1t�E�N2t� � �1�t��E�w1�N1t, N2t, t��


� 2Cov�N1t, r1�N1t, N2t, t��, Cov�N1t, N2t� � 0.

(15)

We refer to A1 as the “extended” Lanchester equationssince also a variance equation is included, differing fromVar(Nit) � 0 for the classical Lanchester equations. A1 isthe approximation closest to Lanchester’s spirit when ac-counting for variances of group sizes.

The second Approximation 2 (A2) assumes as for A1 thatCov(N1t, N2t) � 0, but assumes Cov(Nit, NitNjt) �E(Njt)Var(Nit). Inserting into (8) gives

E�N1t� � ��1�t�E�N1t�E�N2t� � �1�t�E�w1�N1t, N2t, t��

� �1�t�E�r1�N1t, N2t, t��,

Var�N1t� � �2�1�t�E�N2t�Var�N1t� � �1�t�E�N1t�E�N2t�

� �1�t��E�w1�N1t, N2t, t�� 2Cov�N1t, w1�N1t, N2t, t��

� �1�t��E�r1�N1t, N2t, t�� 2Cov�N1t, r1�N1t, N2t, t��,

Cov�N1t, N2t� � 0. (16)

Although independence between N1t and N2t impliesCov(Nit, NitNjt) � E(Njt)Var(Nit), the reverse implication isnot correct. A2 is similar to A1 but Var(Nit) is different.These two choices of Cov(Nit, NitNjt) in A1 and A2 seemmost intuitive to close the system. Without reinforcementand withdrawal, i.e., �i(t) � �i(t) � 0, the expected valuesE(Nit) for A1 and A2 in (15) and (16) equal the determin-istic values ni(t) for the deterministic Lanchester linear warsolution in (1).

The third Approximation 3 (A3) assumes Cov(N1t,N2t) � 0 and Cov(Nit, NitNjt) � 0. Inserting into (8) gives



Var�N1t� � �1�t��E�N1t�E�N2t� � Cov�N1t, N2t�� 1�t�

� �E�w1�N1t, N2t, t�� 2Cov�N1t, w1�N1t, N2t, t��

� �1�t��E�r1�N1t, N2t, t�� 2Cov�N1t, r1�N1t, N2t, t��,

Cov�N1t, N2t� � ��1�t�Cov�N2t, w1�N1t, N2t, t�� 2�t�

� Cov�N1t, w2�N1t, N2t, t�� 1�t�Cov�N2t, r1�N1t, N2t, t��

� �2�t�Cov�N1t, r2�N1t, N2t, t��. (17)

A3 brings us further afield from the Lanchester equations byassuming actual covariance in warfare between N1t and N2t,but avoids complications associated with Cov(Nit, NitNjt).

The fourth Approximation 4 (A4) assumes Cov(N1t,N2t) � 0 and Cov(Nit, NitNjt) � E(Njt)Var(Nit). Insert-ing into (8) gives



Var�N1t� � �2�1�t�E�N2t�Var�N1t� � �1�t��E�N1t�E�N2t�

� Cov�N1t, N2t�� 1�t��E�w1�N1t, N2t, t��


� 2Cov�N1t, r1�N1t, N2t, t��,

Cov�N1t, N2t� � ��1�t�E�N1t�Var�N2t� � �2�t�E�N2t�

� Var�N1t� � �1�t�Cov�N2t, w1�N1t, N2t, t�� 2�t�

� Cov�N1t, w2�N1t, N2t, t�� 1�t�Cov�N2t, r1�N1t, N2t, t��

� �2�t�Cov�N1t, r2�N1t, N2t, t��. (18)


Since A4 is the only approximation where Cov(N1t, N2t) �0 � Cov(Nit, NitNjt), it is the only approximation withoutreinforcement that can furnish E(N1t) � 0 through theargument allowing Cov(N1t, N2t) � 0 in the first featureafter (8).

Inserting (15)–(18) into (10) gives the correspondingapproximated war loss equations, that is,

EL�N1t� � ��1�t�E�N1t�E�N2t�,

VarL�N1t� � �1�t�E�N1t�E�N2t�,

CovL�N1t, N2t� � 0, (19)

EL�N1t� � ��1�t�E�N1t�E�N2t�,

VarL�N1t� � �2�1�t�E�N2t�Var�N1t� � �1�t�E�N1t�E�N2t�,

CovL�N1t, N2t� � 0, (20)

EL�N1t� � ��1�t��E�N1t�E�N2t� � Cov�N1t, N2t��

VarL�N1t� � �1�t��E�N1t�E�N2t�

� Cov�N1t, N2t��, CovL�N1t, N2t� � 0, (21)

EL�N1t� � ��1�t��E�N1t�E�N2t� � Cov�N1t, N2t��,

VarL�N1t� � �2�1�t�E�N2t�Var�N1t� � �1�t��E�N1t�E�N2t�

� Cov�N1t, N2t��,

CovL�N1t, N2t� � ��1�t�E�N1t�Var�N2t�

� �2�t�E�N2t�Var�N1t�. (22)

Analogous expressions for group 2 are found by permutingthe indices in (15)–(22), though not permuting withinri(N1t, N2t, t) or wi(N1t, N2t, t). Without reinforcement/withdrawal, �i(t) � �i, and E(Nit0

) � ni0, Var(Nit0) �

�i0, Cov(N1t0, N2t0

) � c0, EL(Nit0) � nLi0, VarL(Nit0

) ��Li0, CovL(N1t0

, N2t0) � cL0, the analytical solution of A1

in (15) and (19) is

E�N1t� ��1n20 � �2n10�n10

�1n20e��1n20��2n10�t � �2n10

,

Var�N1t� � �E�N1t� � n10 � �10,

Cov�N1t, N2t� � 0, EL�N1t� � E�N1t� � nL10 � n10,

VarL�N1t� � Var�N1t� � �L10 � �10,

CovL�N1t, N2t� � cL0. (23)

The analytical solution of (A2) in (16) and (20) is

E�N1t� ��1n20 � �2n10�n10

�1n20e��1n20��2n10�t � �2n10

,

Cov�N1t, N2t� � 0, CovL�N1t, N2t� � cL0,

Var�N1t� ��1n20 � �2n10�

2��10 � n10�

��1n20e��1n20��2n10�t � �2n10�

2

��1n20 � �2n10�n10

�1n20e��1n20��2n10�t � �2n10

,

EL�N1t� � E�N1t� � nL10 � n10,

VarL�N1t� � Var�N1t� � �L10 � �10. (24)

If cL0 � 0, assumed hereafter, A3 reduces to A1. A4 has noanalytical solution. Analogous expressions for group 2 arefound by permuting the indices in (23)–(24).

4. SIMULATION OF AN EXAMPLE

Without reinforcement/withdrawal, Var(Nit0) � 0 �

Cov(N1t0, N2t0

), and systematically varying �1 and �2 andthe initial conditions E(N1t0

) and E(N2t0), Monte Carlo

simulations of the Kolmogorov linear war equations im-plies5

MC: Cov�N1t, N2t� 0, (25)

MC: Cov�Nit, NitNjt� E�Njt�Var�Nit�

N Cov�Nit2 , Njt� E�Nit�Cov�Nit, Njt�. (26)

The negative covariance in (25) is intuitive. Consider twogroup sizes E(N1t) and E(N2t). If group 1 loses one agent,group 1 gets reduced capacity and wages war less success-fully against group 2. Group 2 can expect a lower loss, andis less likely to lose one agent. War loss for group 1 makeswar loss for group 2 less likely. The more group 1 decreasesin the size, the less group 2 decreases in size. The negativecovariance is not large in magnitude. Without reinforcementno group can increase in size, which would cause a morenegative covariance. The equivalence of the two inequali-ties in (26) follow from the definition of covariance, whereCov(Nit

2 , Njt) is far more negative than Cov(Nit, Njt).

5 (25)–(26) seem impossible to prove analytically. Scrutiny sug-gests no instance where (25)–(26) do not hold.


Consider linear warfare with initial conditions E(Ni0) �10 agents at time t � 0. We set �1(t) � 0.3 and �2(t) �0.2 so that group 1 is 50% worse off with respect to combateffectiveness than group 2, and assume Cov(N1t0

, N2t0) �

0 and Var(Nit0) � 0. Figure 1 presents four curves for each

group. The first curve (filled box for group 1, unfilled boxfor group 2) gives the comparison standard based on 3000MC simulations.6 The second curve shows E(Nit) � ni(t)which is equal for A1, A2, and the deterministic Lanchesterlinear war solution. The third curve shows E(Nit) for (34),while the fourth shows one stochastic realization N1t. Fig-ure 2 shows the associated four standard deviations Std-(Nit) � �Var(Nit) for each group. Table 1 evaluates thecorrectness of the Approximations relative to (8).

The MC curve for E(Nit) approaches a strictly positiveasymptote. Although group 2 is expected to win the war, thestochastic nature of war is such that group 1 wins somebattles in the long run. In contrast, E(Nit) for A1 and A2 areslightly lower and approaches zero asymptotically throughassuming Cov(N1t, N2t) � 0 instead of (25). The varianceof (A1) is mostly too large through ignoring Cov(N1t,N2t) � 2Cov(Nit, NitNjt), which is mostly negative. (Nofurther precision is possible in that regard.) Equation (26)causes the variance and covariance of A4 to be too small,and hence E(Nit) is too large. An interesting tradeoff is atplay for the variance of A2. First, Cov(N1t, N2t) � 0causes a too large RHS of (16) due to (25). Second,

E(N2t)Var(N1t) causes a too small RHS of (16) due to (26).These two effects partly cancel each other making A2 thepotentially best approximation. The least squares methodwas applied to determine the fit between A1, A2, and A4, onthe one hand, and the MC solution, on the other hand. Equalweight was assigned to deviations of N1 and N2. The sum ofthe least squares is systematically largest for A4, normalizedto LS � 1 in Table 1. The sum of the least squares is lowestfor A2 for Std(Nit), and equally low for A1 and A2 forE(Nit) and Cov(N1t, N2t).

5. SIMULATION OF THE ARDENNESCAMPAIGN

This section compares the approximations with empiricsfrom the December 15, 1944–January 16, 1945 WWIIArdennes campaign and Monte Carlo simulations. Section 2assumes stochastic reinforcement. Empirics for the WWIIArdennes campaign for the Allied force (group 1) and theGerman force (group 2), as presented by Bracken [1], sug-gests conceptualizing reinforcement as exogenous and de-terministic. The expected value of a deterministic variableequals the deterministic variable, and the covariance be-tween a stochastic variable and a deterministic variable iszero. With deterministic time-dependent reinforcementri(N1t, N2t, t) � ri(t) allowed to be both positive andnegative, �i(t) � 0, (8) simplifies to

E�N1t� � ��1�t��E�N1t�E�N2t� � Cov�N1t, N2t�� 1�t�r1�t�,

Var�N1t� � �1�t��E�N1t�E�N2t� � Cov�N1t, N2t�

� 2Cov�N1t, N1tN2t�� 1�t�r1�t�,

Cov�N1t, N2t� � ��1�t�Cov�N2t, N1tN2t�

� �2�t�Cov�N1t, N1tN2t�, (27)

6 The approximation pih � �i(t)hnj is valid when �i(t)hnj �� 1.

The sample path generating methodology is such that the timeperiod from t � 0 to t � 5 is divided into 1600 time steps, whichgives a time increment of h � 5/1600. Inserting �i(t) � 0.3 andnj � 10 gives �i(t)hnj � 0.009375 �� 1. Inserting into theexponential function in (3) gives 1 � Exp[�0.009375] �0.00933119, which acceptably gives O(h2) � �0.00004381.

Figure 1. Expected values: MC simulations, ApproximationsA1, A2, A4 and realizations N1t and N2t for groups 1 and 2, n10 �n20 � 10, �1(t) � 0.3, �2(t) � 0.2. A2 is a good approxima-tion.

Figure 2. Standard deviations: MC simulations and Approxima-tions A1, A2, A4 for groups 1 and 2, n10 � n20 � 10, �1(t) �0.3, �2(t) � 0.2. A2 is a good approximation.


and the four approximations A1, A2, A3, A4 in (15)–(18)

E�N1t� � ��1�t�E�N1t�E�N2t� � �1�t�r1�t�,

Var�N1t� � �1�t�E�N1t�E�N2t� � �1�t�r1�t�,

Cov�N1t, N2t� � 0, (28)

E�N1t� � ��1�t�E�N1t�E�N2t� � �1�t�r1�t�,

Cov�N1t, N2t� � 0,

Var�N1t� � �2�1�t�E�N2t�Var�N1t�

� �1�t�E�N1t�E�N2t� � �1�t�r1�t�, (29)


� �1�t�r1�t�,

Var�N1t� � �1�t��E�N1t�E�N2t� � Cov�N1t, N2t��

� �1�t�r1�t�, Cov�N1t, N2t� � 0, (30)


� �1�t�r1�t�,

Var�N1t� � �2�1�t�E�N2t�Var�N1t� � �1�t�

� �E�N1t�E�N2t� � Cov�N1t, N2t��

� �1�t�r1�t�,

Cov�N1t, N2t� � ��1�t�E�N1t�Var�N2t�

� �2�t�E�N2t�Var�N1t�. (31)

The approximated war loss equations (19)–(22) remain un-changed. Reinforcement �1(t)r1(t) is given exogenouslyand deterministically, calculated from Bracken’s data asforce size at day t � 1 minus force size at day t plus forcelosses at day t. Using the definition in (10), the simulationspresent the incremented expected values IEL(Nit) �

EL(Nit)h, where h � 1 day,7 that is the expected value ofthe loss of agents in group i on day t, plus versus minus theincremented standard deviation IStdL(Nit) � �IVarL(Nit)of this loss on day t, and plus versus minus the accumulativestandard deviation StdL(Nit) � �VarL(Nit) of this lossfrom day t � t0 � 1 to day t � t. A band between plusversus minus x times the standard deviation, that is �x�bands, is quite indicative of probabilistic behavior. Thisband is in this article intended as heuristics for discussion.The band spanned out by IStdL(Nit) captures in a sense thestandard deviation in the scatter on a day-to-day basis, andis our main focus below. The broader band spanned out byStdL(Nit) captures the accumulative standard deviation es-timated before the war starts.8

With firepower scores and weights 20, 5, 40, 1 for tanks,APCs, artillery, manpower,9 applying the “least square fit”technique (Appendix B), the optimal combat effectivenesscoefficients for the 33-day Campaign are (�1(t), �2(t)) �(5.38 � 10�9, 8.22 � 10�9). These sizes are of the sameorder of magnitude as those of Bracken [1], Chen and Chu[3], and Fricker [7].10 Simulations (not included due tospace constraints) reveal that the fit between the model andthe empirics is not equally satisfactory throughout the Cam-paign. The historical record, with multifarious events oc-curring throughout the 33-day Campaign, suggests that thecombat effectiveness coefficients �1(t) and �2(t) are in factvariable over the 33 days. Hence let us divide the Campaigninto phases. A plausible choice is three phases: an initialphase of surprise, a middle phase of “standard” warfare, and

7 We can ignore O(h2) and set pih � �i(t)hnj in (3) since h � 1,

�i(t) � 9.97 � 10�9, and nj � 786189 give �i(t)hnj � 0.0078�� 1. Inserting into the exponential function in (3) gives 1 �Exp[�0.0078] � 0.00776966, which acceptably gives O(h2) �0.0000303411.8 StdL(Nit) should for exhaustive presentation be added and sub-tracted from the accumulative expected value �EL(Nit), whichwould require additional graphs believed to be intuitive.9 The weights are derived from standard U.S. Army ConceptsAnalysis Agency practices, as utilized by Bracken [1], Chen andChu [3], Fricker [7]. Fricker [7, p. 16] attaches weight 30 to airpower.10 �1(t) and �2(t) of the same order of magnitude can be consid-ered as a consistency requirement. We plausibly believe that theAllied and German forces are comparably equipped w.r.t. capabil-ities, weapons, training.

Table 1. Evaluation of the approximations A1, A2, and A4.

A1 A2 A4

E(Nit) Too small: (25), LS � 0.47 Too small: (25), LS � 0.47 Too large: (26), LS � 1

Std(Nit) Mostly too large: LS � 0.87 Tradeoff OK: (25)–(26), LS � 0.46 Too small: (26), LS � 1

Cov(N1t, N2t) 0: Too large: (4.1), LS � 0.30 0: Too large: (25), LS � 0.30 Too small: (26), LS � 1


a final phase preparing for withdrawal. First there is aninitial phase 1 lasting from day 1 through day 10, withconsiderable surprise and disruption. In the first part ofphase 1 the German force attacked and in the last days ofphase 1 the Allied force attacked. After the initial learningand surprise in phase 1, there is a middle phase 2 of warfarefrom day 11 through day 20, where both forces are engag-ing. Finally there is a final phase 3 where the German forceprepares for withdrawal. The “least square fit” technique isused to estimate the combat effectiveness coefficients �1(t)and �2(t) separately for each of the three phases, as shownin Appendix B. The transition points of day 11 and day 21between phases 1 and 2, and between 2 and 3, are deter-mined by visual inspection of the empirics. See Section 7for further discussion of phases and transition points. Forphase 1 running from days 1 through 10 this gives (�1(t),�2(t)) � (7.655 � 10�9, 9.97 � 10�9). For phase 2running from days 11 through 20 this gives (�1(t),�2(t)) � (4.32 � 10�9, 8.44 � 10�9). For phase 3running from days 21 through 33 this gives (�1(t),�2(t)) � (4.82 � 10�9, 6.8 � 10�9). The combateffectiveness coefficient �1(t) is lower than �2(t) through-out the Campaign. The Allied force wages war in a superiormanner compared with the German force. This is in agree-ment with Bracken [1] and Chen and Chu [3], but differsfrom Dupuy [6], discussed below for tank warfare.

It is of interest to assess how the “scatter” or fluctuationsin available empirics fare with respect to the standard de-viations in the approximations, where we set Cov(N1t0

,N2t0

) � 0, CovL(N1t0, N2t0

) � 0, Var(Nit0) � 0,

VarL(Nit0) � 0. The first curve in Figures 3–4 presents

Bracken’s [1] data for the Ardennes campaign for combatforces, hereafter called forces. The second curve is the

incremented expected value IEL(Nit) of the loss of forces11

according to Lanchester linear war, applying the “leastsquare fit” technique for the three phases (Appendix B). Thethird and fourth curves are �IEL(Nit) plus versus minus theincremented standard deviation IStdL(Nit) of this loss ac-cording to A2, which we in accordance with the earlieranalysis and Figures 1–2 take to be the best approximation,where t is time in days. The band spanned out by these twostandard deviations captures only 3 of the 33 data points forthe Allied force and 6 of the 33 data points for the Germanforce. The large fluctuation due to Allied reinforcement atday 5 falls outside the band. The fifth and sixth curves are�IEL(Nit) plus versus minus the accumulative standarddeviation StdL(Nit). This band broadens up considerably astime elapses, suggesting increasing uncertainty in the esti-mation of war loss further into the future. The broader bandcaptures 4 of the 33 data points for the Allied force and 12of the 33 data points for the German force.

The high combat effectiveness �1(t) � 7.655 � 10�9

during phase 1 captures to a considerable extent the elementof surprise exercised by the German force in the initialphase of the Campaign, combined with poor weather. Theweather cleared and the Allied force quickly learned how todefend itself and improve its attack capability, reducing�1(t) to �1(t) � 4.32 � 10�9 in the second phase, mostlikely joined by some fatigue setting in on the Germanforce’s ability to wage war. The Allied force improved itsability to wage war more successfully in phase 2, combinedwith the German force learning to improve its defense fromphase 1 to phase 2, which decreased �2(t) from �2(t) �9.97 � 10�9 to �2(t) � 8.44 � 10�9. Now it was the

11 The incremental expected war loss IEL(Nit) is inevitably neg-ative. The minus sign is to facilitate plotting along the positivevertical axis.

Figure 3. Allied loss of forces: Empirics, the Lanchester linearversion plus versus minus Std for A2. Three sets of deterministic�1(t) and �2(t) over three phases; days 1–10, 11–20, 21–33. Theband does not capture all the scatter in empirics.

Figure 4. German loss of forces: Empirics, the Lanchester linearversion plus versus minus Std for A2. Three sets of deterministic�1(t) and �2(t) over three phases; days 1–10, 11–20, 21–33. Theband does not capture all the scatter in empirics.


Allied force’s turn to exercise surprise to some limitedextent. In the third phase both forces have learned to im-prove their defense, and some fatigue was likely setting incombined with preparation for withdrawal. This caused�1(t) to increase marginally and �2(t) to decrease fromphase 2 to phase 3. The Lanchester linear solution improveswith time dependent on �1(t) and �2(t) in the three phases.The stochastic nature of warfare suggests possibly increasedrealism of considering �1(t) and �2(t) as stochasticallyvarying rather than as deterministic. Due to changing con-ditions pertaining to weather, political factors, and otherfluctuations, it is of interest to substitute �1(t) and �2(t)with �1s(t) and �2s(t), which vary stochastically accordingto �is(t) � Random[RayleighDistribution[�i(t)�2/�]],where �i(t) is the value estimated with the “least square fit”technique for each of the three phases. This means that�is(t) is drawn from a Rayleigh distribution with expectedvalue �i(t) and standard deviation �i(t)�(4 � �)/�. Al-lowing such increased variation in �is(t) gives the band inFigures 5–6, based on 7000 MC simulations.12 Much, butnot all, of the scatter in the empirics is captured by the band.To determine the confidence with which the relevant meanfalls within the band, we have for each day i, i � 1, . . . ,33, counted up which fraction of the 7000 realizations fallswithin the band for day i. Averaging over all 33 days gives0.68 for both Figures 5 and 6. Virtually equivalent resultsfollow from the Lognormal and Gamma distributions withthe same expected values and variances. These two distri-butions allow for adjusting the breadth of the band upwardsand downwards.13

Although the firepower score approach with weights 20,5, 40, 1 is reasonably well established today, this maychange in the future. The weights may change with newempirics, and alternative methods of analysis may emerge.Tank warfare is most common among the ten ways in whichtanks, APCs, artillery, manpower interact (Simpkin [14],pp. 69–83), and artillery attacking tanks is second mostcommon.14 The “least square fit” technique gives (�1(t),�2(t)) � (1.1 � 10�5, 0.74 � 10�5), and with threephases, (�1(t), �2(t)) � (1.37 � 10�5, 0.59 � 10�5),(�1(t), �2(t)) � (0.89 � 10�5, 0.55 � 10�5), (�1(t),�2(t)) � (0.57 � 10�5, 0.24 � 10�5). The coefficients�2(t) for the German force is lower than �1(t) for theAllied force throughout the Campaign. The German forcewages tank warfare in a superior manner compared with theAllied force, contrary to the result above for firepowerscores. The German force was well trained for the panzerstrategy outlined by De Gaulle before WWII, (Speer [15],

12 The time period from t � 0 to t � 33 is divided into 33 timesteps, which gives a time increment of h � 1, which is sufficientto ensure stability and precision.13 The Normal distribution cannot be used since it also generatesnegative values.

14 Taylor [17, p. I:21] describes three “approaches for predictingthe effectiveness of combat units,” i.e., firepower scores, MCsimulations, and analytical models. The Kolmogorov model, MCsimulations, and analytical models facilitate analyzing basic eventsmicroscopically with arbitrarily high precision. The approximativeaspect is introduced in the Approximations. Firepower scoresinvolves introducing the approximative aspect at a more basiclevel. Taylor [17, p. I:24] suggests that “although the firepower-score approach has been widely used for top-level planning, it hasreceived increasing criticism in recent years . . . . It does not mea-sure the accomplishment of unit missions . . . , it ignores most ofthe significant factors that affect mission accomplishment . . . , itoftentimes bears little relation to the physical combat or otherprocesses under study. Stockfish [16, p. 90] claims that no satis-factory simple technique for aggregating modern conventionalforces currently exists. Although the firepower-score approach hasbeen thus far much criticized, conventional forces must be aggre-gated in many analyses, and until a better alternative is developed,firepower scores will continue to be used.”

Figure 5. Replication of Figure 3 with stochastic �is(t) � Ran-dom[RayleighDistribution[ � ]]. The band captures almost all thescatter in empirics.

Figure 6. Replication of Figure 4 with stochastic �is(t) � Ran-dom[RayleighDistribution[ � ]]. The band captures almost all thescatter in empirics.


pp. 204–266), which we believe explains this result com-patible with Dupuy’s [6] analysis. On the other hand, westipulate that the Allied force performed the science ofoperational analysis superiorly, more successfully linkingtanks, APCs, artillery, manpower into a total force. Incontrast, operational analysis was new and not fully under-stood by the German military. This likely explains thereverse result above for firepower scores.

Figures 7–8 show tank warfare. �1s(t) and �2s(t) varystochastically according to � i s( t) � Random[Ray-leighDistribution[�i(t)�2/�]]. The two last curves de-noted “A2D” provides the bands for deterministic �1(t) to�1(t) for the three phases. For the first two phases stochasticwarfare captures the scatter more convincingly. For the thirdphase the bands are comparable. A band for stochasticwarfare more realistically captures large fluctuations.Counting up which fractions of the 7000 realizations fallwithin the bands, we get 0.68 for both Figures 7 and 8 forthe broader band, and 0.38 for Figure 7 and 0.57 for Figure8 for the narrow band.

The high combat effectiveness �1(t) � 1.37 � 10�5

during phase 1 captures to a considerable extent the elementof surprise exercised by the German force in the initialphase of the Campaign, combined with poor weather. Theweather cleared and the Allied force quickly learned how todefend itself and improve its attack capability, reducing�1(t) to �1(t) � 0.89 � 10�5 in the second phase, mostlikely joined by some fatigue setting in on the Germanforce’s ability to wage war. The Allied force improved itsability to wage war more successfully in phase 2, combinedwith the German force learning to improve its defense fromphase 1 to phase 2, which decreased �2(t) marginally from�2(t) � 0.59 � 10�5 to �2(t) � 0.55 � 10�5. Now itwas the Allied force’s turn to exercise surprise to somelimited extent. In the third phase both forces have learned toimprove their defense, and some fatigue was likely setting

in combined with preparation for withdrawal especially onthe German side. This caused both �1(t) and especially�2(t) to decrease from phase 2 to phase 3.

Stochastically varying reinforcement coefficients �1(t)and �2(t) are possible to scrutinize in future research, wherea few points are worth noting. First, reinforcement dependson political will, which is very hard to model, deploymentdecisions by military leadership, and on the logistics ofbringing the reinforcement to the battlefield, which aresubject to other kinds of fluctuations than those for �1(t)and �2(t) where fluctuations on the battlefield itself play arole. Second, reinforcement typically occurs in units (that is,groups of individuals). The variance of the probability dis-tribution for drawing �1(t) and �2(t), estimated from em-pirics for reinforcement, seems appropriately chosen con-siderably larger than the variance of the probability distri-bution for drawing �1(t) and �2(t).

We hope that improved data gathering from warfare willallow future stochastic analyses to be supported more thor-oughly on empirical grounds. We consider the possibility ofapplying a data analytic approach to accounting for uncer-tainty in the combat effectiveness coefficients �1(t) and�2(t) to be especially promising.15 Compiling data frommultiple wars, these wars can be classified according tovarious criteria such as linear warfare, square warfare, etc.,influenced by factors such as politics, culture, terrain,weather, and the extent to which artillery, tanks, APCs,infantry, and air force are employed. A class of warfare canconsist of different wars at a given location at differentpoints in time, realizing that some warring parties wage waron and off decade after decade, century after century. Manyfactors, such as terrain and weather, typically remain con-stant through time, whereas other factors, such as weaponry

15 We thank an anonymous referee for providing several of thesuggestions in this paragraph.

Figure 7. Allied tank warfare: Stochastic �is(t) � Ran-dom[RayleighDistribution[�i(t)�2/�]]. A band for stochasticwarfare more realistically captures large fluctuations.

Figure 8. German tank warfare: Stochastic �is(t) � Ran-dom[RayleighDistribution[�i(t)�2/�]]. A band for stochasticwarfare more realistically captures large fluctuations.


change over time. A class of warfare can also consist ofdifferent wars at different locations at similar points intime, where factors such as terrain and weather varyacross wars, while weaponry and other factors remainmore similar. Having compiled n wars within a givenclass of warfare, the “least square fit” technique appliedin this article can be used to estimate n different sets of(�1(t), �2(t)). These n sets can either be used directly toconstruct a probability distribution p(�1(t), �2(t)) over�1(t) and �2(t) according to conventional statistical tech-niques, which may possibly give something akin to the Ray-leigh distribution used in this article, or draws can be madedirectly from the n sets according to statistical theory. Suchknowledge of the distribution of the combat effectivenesscoefficients �1(t) and �2(t) would be especially valuable tocombat planners during a war.

6. FORECASTING OF POSSIBLE FUTURES

An interesting use of the Kolmogorov model and theassociated approximations is to incorporate information upto the current time in a conflict and attempt to forecastpossible futures, with the intention being to determine whatdegree of combat effectiveness or reinforcements are nec-essary to achieve a positive outcome given various scenar-ios (for example, worst case). Whereas Section 5 applied themodel to past data, this section applies the model as a toolto evaluate ongoing performance and to suggest necessaryadjustments during an active campaign.16

Scenario 1 runs the assumptions of Figures 3–4 beyondday 33, that is from day 1 through day 80. Figures 9–10

quantify how both forces grow weaker, and how the lossesdecline, as time elapses. The German force was relativelysuccessful over the first few days of the Campaign. A relevantscenario for the German force to evaluate is the implications ofincreasing the combat effectiveness �1(t) considerably at somepoint during the Campaign. Scenario 2 keeps the assumptionsof Scenario 1 for days 1–20, but increases �1(t) five times to�1(t) � 2.41 � 10�8 for days 21–80. The implication is thatthe Allied force at day 80 suffers a reduction from 684,520 to406,093, while the German force at day 80 enjoys an increasefrom 381,408 to 406,945, for Scenario 2 relative to Scenario 1.Under Scenario 2 the weaker Allied force represents a lowerthreat to the German force. Figure 10 shows how this large�1(t) inflicts considerable loss on the Allied force, althoughthis loss decreases from day 21 and thereafter as especially theAllied force gets reduced. The German force enjoys somewhatreduced losses for days 21–80 for Scenario 2 compared withScenario 1.

A relevant scenario for the German force to evaluate isthe implications of somehow injecting successfully forcesfrom the Eastern Front into the Ardennes Campaign at somepoint in December 1944. A modest number of 850 Germantanks and 1 million German soldiers were located at theEastern Front toward the end of WWII (Polygram VideoInternational [13]). Scenario 3 keeps the assumptions ofScenario 1 for days 1–20, but injects a reinforcement of�2(t)r2(t) � 200,000 forces to the German force on dayt � 21.17 Figure 11 shows how this reduces the Allied force

16 We thank one referee for some of these formulations and sug-gestions.

17 Evaluating the realism of such predictions depends on the avail-ability of tanks, APCs, artillery, combat manpower, and on theweights 20, 5, 40, 1. 200,000 forces assumes high weight forcombat manpower. Note that on day 10 the Allied forces had 4062tanks, 7486 APCs, 3847 artillery, and 426,360 combat manpower.With the weights 20, 5, 40, 1, the influences are 11.6238%,5.35548%, 22.0171%, 61.0036%, respectively.

Figure 9. Total number of forces. S1 (Scenario 1) is a forecast ofFigures 3–4 for days 34–80. S2 (Scenario 2) increases �1(t) to�1(t) � 2.41 � 10�8 for days 21–80, causing at day 80 theAllied to suffer a substantial reduction and the German to enjoy amarginal increase.

Figure 10. Loss of forces. S1 (Scenario 1) is a forecast ofFigures 3–4 for days 34–80. S2 (Scenario 2) increases �1(t) to�1(t) � 2.41 � 10�8 for days 21–80, causing the Allied forceto endure larger losses than the German force.


to 651,190 on day 80, while the German force gets reducedto 534,385 on day 80. The losses for the Allied force fordays 21–80 for Scenario 3 in Figure 12 decline similarly,though with somewhat smaller magnitude, to the losseswhen �1(t) was increased five times to �1(t) � 2.41 �10�8 in Scenario 2. The losses for the German force aresimilar, and substantially larger than the losses under Sce-nario 2 since the German force is much larger under Sce-nario 3.

Moving 200,000 forces for availability on one given dayis a difficult endeavor, even with today’s means. Scenario 4reinforces �2(t)r2(t) � 5000 forces to the German forceper day between day 21 and day 60, for a total of 200,000.Figure 11 shows how this reduces the Allied force to661,216 on day 80, while the German force gets reduced to548,530 on day 80. Both these two forces are larger underScenario 4 than under Scenario 3. Noninjected forces arenot subject to combat loss, and the delayed injection causesboth forces to be smaller during the first days after day 21under Scenario 4 compared with Scenario 3. The losses forboth forces, shown in Figure 12, increase gradually fromday 21 through day 60 concomitant with force injection, andthereafter decline for days 61–80 when force injectionceases.

7. DISCUSSION OF RUN TIMES ANDSUGGESTIONS FOR FURTHER WORK

For Approximations A1, A2, A3, A4 the set of 10coupled differential equations referred to after (10) aresolved directly for the given time period of the war underanalysis. Both running Monte Carlo simulations and solv-

ing the Kolmogorov equations involve computing theunderlying probability distribution p(N1t � n1, N2t � n2)in (7). These two methods are approximately equally timeconsuming. Using the Monte Carlo method, 10 corre-sponding equations are determined for the given timeperiod, and are successively updated sufficiently manytimes to achieve correctness gradually. If we reasonablyassume that the 10 equations for the Approximations areas time-consuming to determine as the 10 equations inthe Monte Carlo simulations are to update, then the MCmethod is as many more times time-consuming than theApproximations as the number of Monte Carlo runs.Hence a rule of thumb for the time saving potential ofapproximations is the number of MC runs. There existsno hard theoretical specification of the “correct” numberof Monte Carlo simulations to run. The number should beas high as possible, but the necessary or desired numbervaries considerably from phenomenon to phenomenon.For Figures 1–2 satisfactory results were obtained by3000 MC runs, while Figures 5– 6 are based on 7000 MCruns. Often 500 MC runs are indicative of “correct re-sults,” but seldom constitute scientific validity. Table 2illustrates run times applying Mathematica 4.0 (WolframResearch Inc.) on a 2.53 GHz PC.

For the Ardennes Campaign choosing the max num-bers of tanks, APCs, artillery, manpower gives the fire-power scores n1 � 786,189, n2 � 583,569. Applying theBinomial Distribution, one MC run takes 28 seconds (s).For all the war scenarios we assume �1(t) � 9.97 � 10�9,�2(t) � 7.655 � 10�9, and h � 1 which gives pi

h ��i(t)hnj � O(h2) �� 1. For the January 17–February 28,1991 Gulf war Desert Storm (www.desert-storm.com,Blair [2], Chap. 4) analogous calculations gives 33 s, and

Figure 11. Total number of forces. S3 reinforces 200,000 Ger-man forces on day 21. S4 reinforces 5000 German forces per daybetween days 21 and 60, which is a total of 200,000 forces.Gradual reinforcement (S4) causes larger forces at day 80 thaninstantaneous reinforcement.

Figure 12. Loss of forces. S3 reinforces 200,000 German forceson day 21. S4 reinforces 5000 German forces per day betweendays 21 and 60, which is a total of 200,000 forces. Gradualreinforcement (S4) causes increasing and thereafter decreasinglosses.


the 1950 Korean war (www.koreanwar.com) takes 5 s.18

A hypothetical war scenario with 28,000 tanks,19 10,000APCs, 5000 artillery, and one million manpower for bothforces takes 1 min and 5 s. With 1500 MC runs theKorean war takes above 2 hours (h) to simulate.20 Thesesimulation times suggest that the Kolmogorov and MCapproaches are too time-consuming. Furthermore, a warfighter may wish to consider competing forecasting sce-narios, and may wish to run simulations over extendedtime periods, in situations where time is a scarce re-source. Time considerations like these motivate the needfor and provide compelling arguments for approxima-tions.

Extending the analysis of approximations beyond A1,A2, A3, A4 amounts to closing (8) at a higher level ofcomplexity. The first such step is to provide approximationsfor E(N1t

2 N2t) and E(N1tN2t2 ) in terms of E(N1t), E(N2t),

Var(N1t), Var(N2t), Cov(N1t, N2t), without unboundedlyintroducing ever more complex expressions. Instead of fivelinear war equations this gives seven linear war equations,which together with seven linear war loss equations gives aset of 14 coupled differential equations. Further extensionsof approximations are to square warfare and other forms ofwarfare.

Section 5 assumed three phases and determined the tran-sition points at days 11 and 21 by visual inspection. Alter-natively the “least square fit” technique could be used todetermine the transition points, accounting for all the dif-

ferent manners in which 33 days can be divided into threephases. Sufficiently many phases would allow for capturingall the scatter in the empirics, but cannot be justified theo-retically. The more phases assumed, the less explanatoryand predictive power. At the limit, with 33 phases, themodel would match the empirics of the Ardennes Campaignexactly. Such a model is rendered inoperative and the anal-ysis reduces to tabulation of �1(t) and �2(t). The numberof phases should be kept at a minimum, but not too small.The challenge amounts to estimating �1(t) and �2(t) intel-ligently without using the empirics so exhaustively that themodel is rendered inoperative. This paper has consideredthree ways of doing that. The first is to use the “least squarefit” technique to estimate �1(t) and �2(t) for the entire33-day Campaign. The second is to use the “least square fit”technique to estimate �1(t) and �2(t) when the 33-dayCampaign is divided into three phases. The third is to let�1(t) and �2(t) vary stochastically through using the Ray-leigh distribution. Phenomena of the kind discussed abovefrequently arise in the statistical literature. The idea is tofind an appropriate number of parameters in the model sothat it provides a good fit to the observed data while retain-ing predictive capability (avoiding overfitting). The stan-dard approach is to add a penalty term to the fitting criterion(in this case, the least squares criterion) that prevents addingtoo many parameters. For example, the AIC or BIC criteriaoften used in model comparison add a penalty to the like-lihood that discourages overfitting. Future work may con-sider such alternatives to determine the number of stages,location of cut points, choice of penalized criterion function,etc.21

Section 5 also analyzed a band between plus versusminus x times the standard deviation. These bands areindicative of probabilistic behavior, and are in this articleintended as heuristics for discussion. Future research may

18 This is schematic, confines attention to North and South Korea,and considers the first 33 days of the June 25, 1950–July 27, 1953war. The U.S. presence increased from 48,268 men July 31, 1950to 276,581 men July 31, 1953, with 500 tanks by mid-September1950. The Chinese eventually lost 900,000 men (www.army.mil).19 The U.S. had 28,000 tanks at the end of WWII, of which 6000remained in 1950 together with 591,000 army soldiers (Hoffmann[10], p. 9).20 Applying the Bernoulli distribution according to (5), computingthe more restrictive requirement �i max(t)hn1n2 � 0.05 �� 1gives the considerably lower h in Table 2, and longer simulationtimes.

21 We thank an anonymous referee of this journal for suggestingthe last five sentences of this paragraph, and also suggesting thecontent of the last paragraph of this section.

Table 2. Evaluation of run times for five war scenarios on a 2.53 GHz PC.

1991 Gulf War 1950 Korean War 1944 Ardennes Campaign Hypothetical War

Duration 43 days Assume 33 days 33 days Assume 33 daysActors Coal. Iraq North South Allied German Blue RedTanks 5258 4230 120 0 4761 747 28,000 28,000APCs 300 2870 0 0 8629 2065 10,000 10,000Artillery 2780 809 182 89 4153 4885 5000 5000Manpower 865,501 500,000 135,000 95,000 481,704 362,904 1,000,000 1,000,000Firepower 1,083,361 631,310 144,680 98,560 786,189 583,569 1,810,000 1,810,000Bernoulli 15 min 45 s 15 s 10 min 22 s 58 min 20 sh 43/5,864,209 33/93,831 33/3,018,966 33/21,557,393Binomial 33 s 5 s 28 s 1 min, 5 spi max

h 0.011 0.0014 0.0078 0.018


quantify the confidence by which the relevant mean iscontained within other bands than those considered in thisarticle. More generally, future research may generate rigor-ous quantitative uncertainty quantification for the stochasticmethodology proposed in this article.

8. CONCLUSION

The article develops a theorem which shows that theLanchester equations are not in general equal to the Kol-mogorov war equations. The latter are time consuming tosolve even on a time-efficient computer. The time factorplays a crucial role in real life warfare where simulatedresults of multiple scenarios are in need of quick delivery.This article presents four time efficient approximations inthe form of ordinary differential equations for the expectedsizes and variances of each group, and the covariance,accounting for reinforcement. The approximations are com-pared with “exact” Monte Carlo simulations and empiricsfrom the WWII Ardennes campaign. The band spanned outby plus versus minus the incremented standard deviationsfrom day to day captures some of the scatter in the empirics,but not all. Allowing for stochastically varying combateffectiveness coefficients, a substantial part of the scatter inthe empirics can be contained. The results therefore indi-cate, first, that the effectiveness coefficients are stochastic inreal warfare, and second, that the average effectivenesscoefficients tend to change over time, an initial value, amiddle value and a final value.

Just as the Lanchester equations can be extended toaccount for reinforcement and withdrawal, the Kolmogorovequations also account for reinforcement and withdrawal.We show how reinforcement and withdrawal play a role inthe expected value, variance, and covariance, both as anexpected value of the reinforcement/withdrawal, and as acovariance between the reinforcement/withdrawal and thetwo group sizes. Reinforcement/withdrawal is a typical phe-nomenon of resource allocation where the political domainmay be influential. Although Clausewitz [4] has suggestedwar as an extension of policy with other means, the politicaland military domains are often intertwined, constraining therole of the military commander. To the extent a model of thepolitical factors affecting reinforcement/withdrawal is orbecomes available, this article specifies how such a model islinked to the linear war equations through reinforcement/withdrawal.

The model is used to forecast possible futures. The im-plications of increasing with a factor five the combat effec-tiveness coefficient governing the size of the Allied forcefrom day 21 until a hypothetical day 80 is evaluated. Anal-ogously, the implications of injecting during the Campaign1000 tanks to the German force, transferred from the East-ern Front, are evaluated, with variance assessments.

APPENDIX A: PROOF OF EQ. (8)

The definition of expected value specifies

E�N1t� �def �

n1�0

�n2�0

n1p�N1t � n1, N2t � n2�. (32)

We assume there exist values n1 max and n2 max such that if the group sizesn1 or n2 exceed these values, then the joint probability p(N1t � n1, N2t �n2) of this event is zero, that is,

p�N1t � n1, N2t � n2� �mod

0 if n1 n1 max or n2 n2 max,

(33)

where we hereafter shorten the notation by writing p(N1t � n1, N2t �n2) � p(n1, n2). Equation (33) implies that all summations are finite.Hence all expectations exist and the time derivative can be taken inside thesummation sign. Defining hi(n1, n2) � n1n2 � wi(n1, n2, t)�i(t)/�i(t),and suppressing the time dependence on ri(n1, n2, t) � ri(n1, n2), (7),(32), (33) imply

E�N1t� � �n1�0

n1 max �n2�0

n2 max

n1p�n1, n2�

� �n1�0

n1 max �n2�0

n2 max

�1�t�n1��p�n1, n2�h1�n1, n2� � p�n1 � 1, n2�h1�n1 � 1, n2��

� �n1�0

n1 max �n2�0

n2 max

�1�t�n1��p�n1, n2�r1�n1, n2� � p�n1 � 1, n2�r1�n1 � 1, n2��

� �n1�0

n1 max �n2�0

n2 max

�2�t�n1��p�n1, n2�h2�n1, n2� � p�n1, n2 � 1�h2�n1, n2 � 1��

� �n1�0

n1 max �n2�0

n2 max

�2�t�n1��p�n1, n2�r2�n1, n2� � p�n1, n2 � 1�r2�n1, n2 � 1��.

(34)

LEMMA 1

l1 �def �

n2�0

n2 max

��p�n1, n2�h2�n1, n2� � p�n1, n2 � 1�h2�n1, n2 � 1�� 0. (35)

PROOF:

l1 � �n2�0

n2 max

� p�n1, n2�h2�n1, n2� � �n2�1

n2 max�1

p�n1, n2�h2�n1, n2�

� �p�n1, 0�h2�n1, 0� � 0. �

LEMMA 2:

l2 �def �

n2�0

n2 max

��p�n1, n2�r2�n1, n2� � p�n1, n2 � 1�r2�n1, n2 � 1�� 0.

PROOF:

l2 � �n2�0

n2 max

� p�n1, n2�r2�n1, n2� � �n2��1

n2 max�1

p�n1, n2�r2�n1, n2�

� �p�n1, n2 max�r2�n1, n2 max� � 0, (36)


where r2(n1, n2 max) � 0 since n2 max is the maximum size of group 2, andhence group 2 can receive no further reinforcement. �

LEMMA 3:

l3 �def �

n1�0

n1 max

n1��p�n1, n2�h1�n1, n2� � p�n1 � 1, n2�h1�n1 � 1, n2��

� � �n1�0

n1 max

p�n1, n2�h1�n1, n2�. (37)

PROOF:

l3 � ��p�1, n2�h1�1, n2� � p�2, n2�h1�2, n2�� 1 � ��p�2, n2�h1�2, n2�

� p�3, n2�h1�3, n2�) � 2 � · · · � ��p�n1 max � 1, n2�h1�n1 max � 1, n2�

� p�n1 max, n2�h1�n1 max, n2�) � �n1 max � 1�

� ��p�n1 max, n2�h1�n1 max, n2�� n1 max � � �n1�1

n1 max

p�n1, n2�h1�n1, n2�

� � �n1�0

n1 max

p�n1, n2�h1�n1, n2�, (38)

where the last equality follows since h1(0, n2) � h2(n1, 0) � w1(0, n2,t) � w2(n1, 0, t) � 0. �

LEMMA 4:

l4 �def �

n1�0

n1 max

n1��p�n1, n2�r1�n1, n2� � p�n1 � 1, n2�r1�n1 � 1, n2��

� �n1�0

n1 max

p�n1, n2�r1�n1, n2�. (39)

PROOF:

l4 � ��p�1, n2�r1�1, n2� � p�0, n2�r1�0, n2�� 1 � ��p�2, n2�r1�2, n2�

� p�1, n2�r1�1, n2�� 2 � ��p�3, n2�r1�3, n2� � p�2, n2�r1�2, n2�� 3 � · · ·

� ��p�n1 max, n2�r1�n1 max, n2� � p�n1 max � 1, n2�r1�n1 max � 1, n2�� n1 max

� �n1�0

n1 max

p�n1, n2�r1�n1, n2�, where r1�n1 max, n2� � 0. � (40)

Inserting Lemmas 1–4 into (34) gives

E�N1t� � ��1�t� �n1�0

n1 max �n2�0

n2 max

p�n1, n2�h1�n1, n2� � �1�t� �n1�0

n1 max �n2�0

n2 max

p�n1, n2�

� r1�n1, n2� � ��1�t�E�h1�N1t, N2t�� 1�t�E�r1�N1t, N2t��

� ��1�t�E�N1tN2t� � �1�t�E�w1�N1t, N2t�� 1�t�E�r1�N1t, N2t��. (41)

E�N1t2 � �

def �n1�0

n1 max �n2�0

n2 max

n12p�n1, n2�

� �n1�0

n1 max �n2�0

n2 max

�1�t�n12��p�n1, n2�h1�n1, n2� � p�n1 � 1, n2�h1�n1 � 1, n2��

� �n1�0

n1 max �n2�0

n2 max

�1�t�n12��p�n1, n2�r1�n1, n2� � p�n1 � 1, n2�r1�n1 � 1, n2��

� �n1�0

n1 max �n2�0

n2 max

�2�t�n12��p�n1, n2�h2�n1, n2� � p�n1, n2 � 1�h2�n1, n2 � 1��

� �n1�0

n1 max �n2�0

n2 max

�2�t�n12��p�n1, n2�r2�n1, n2� � p�n1, n2 � 1�r2�n1, n2 � 1��.

(42)

Lemmas 1 and 2 imply that the two last lines equal zero. Writing out gives

E�N1t2 � � �1�t� �

n2�0

n2 max

��p�1, n2�h1�1, n2� � p�2, n2�h1�2, n2�� 12

� ��p�2, n2�� h1�2, n2� � p�3, n2�h1�3, n2�� 22 � · · ·

� ��p�n1 max, n2�h1�n1 max, n2�� n1 max2 � � �1�t� �

n2�0

n2 max

��p�1, n2�r1�1, n2�

� p�0, n2�r1�0, n2�� 12 � ��p�2, n2�r1�2, n2� � p�1, n2�r1�1, n2�� 22 � · · ·

� ��p�n1 max, n2�r1�n1 max, n2� � p�n1 max � 1, n2�r1�n1 max � 1, n2�� n1 max2 �,

(43)

E�N1t2 � � �1�t� �

n1�0

n1 max �n2�0

n2 max

p�n1, n2�h1�n1, n2��1 � 2n1� � �1�t� �n1�0

n1 max �n2�0

n2 max

� p�n1, n2�r1�n1, n2��1 � 2n1� � �1�t�E�h1�N1t, N2t��1 � 2N1t�� 1�t�

� E�r1�N1t, N2t��1 � 2N1t�� 1�t��E�N1tN2t� � 2E�N1t2 N2t�� 1�t�

� �E�w1�N1t, N2t�� 2E�w1�N1t, N2t�N1t�� 1�t��E�r1�N1t, N2t�� 2

� E�r1�N1t, N2t�N1t��. (44)

E�N1tN2t� �def �

n1�0

n1 max �n2�0

n2 max

n1n2p�n1, n2�

� �n2�0

n2 max

n2 �n1�0

n1 max

�1�t�n1��p�n1, n2�h1�n1, n2� � p�n1 � 1, n2� � h1�n1 � 1, n2��

� �n2�0

n2 max

n2 �n1�0

n1 max

�1�t�n1��p�n1, n2�r1 � �n1, n2� � p�n1 � 1, n2�r1�n1 � 1, n2��

� �n1�0

n1 max

n1 �n2�0

n2 max

�2�t�n2 � ��p�n1, n2�h2�n1, n2� � p�n1, n2 � 1�h2�n1, n2 � 1��

� �n1�0

n1 max

n1 �n2�0

n2 max

� �2�t�n2 � ��p�n1, n2�r2�n1, n2�

� p�n1, n2 � 1�r2�n1, n2 � 1��, (45)


Using Lemmas 3 and 4 for lines 2 and 3, and with permuted indices forlines 4 and 6, gives

E�N1tN2t� � ��1�t� �n2�0

n2 max

n2 �n1�0

n1 max

p�n1, n2�h1�n1, n2�

� �2�t� �n1�0

n1 max

n1 �n2�0

n2 max

p�n1, n2�h2�n1, n2�

� �1�t� �n2�0

n2 max

n2 �n1�0

n1 max

p�n1, n2�r1�n1, n2� � �2�t� �n1�0

n1 max

n1 �n2�0

n2 max

p�n1, n2�r2�n1, n2�,

(46)

E�N1tN2t� � ��1�t�E�h1�N1t, N2t�N2t� � �2�t�E�h2�N1t, N2t�N1t�

� �1�t�E�r1�N1t, N2t�N2t� � �2�t�E�r2�N1t, N2t�N1t� � ��1�t�E�N1tN2t2 �

� �2�t�E�N1t2 N2t� � �1�t�E�w1�N1t, N2t�N2t� � �2�t�E�w2�N1t, N2t�N1t�

� �1�t�E�r1�N1t, N2t�N2t� � �2�t�E�r2�N1t, N2t�N1t�. (47)

Var�N1t� � E�N1t2 � � E�N1t�

2, Var�N1t� � E�N1t2 � � 2E�N1t�E�N1t�,

Cov�N1t, N2t� � E�N1tN2t� � E�N1t�E�N2t�,

Cov�N1t, N2t� � E�N1tN2t� � E�N1t�E�N2t� � E�N1t�E�N2t�. (48)

Equations (41), and (44) and (47) inserted into (48), give (8) when t isreintroduced.

APPENDIX B: THE “LEAST SQUARE FIT”TECHNIQUE

The values of the combat effectiveness coefficients �1(t) and �2(t) aredetermined by the “least square fit” technique. First �1(t) � �2(t) �10�11, written as (�1(t), �2(t)) � (10�11, 10�11), are chosen, and thesimulation is run from day t � 1 through day t � 33. For each day thedifference between the war loss EL(N1t) for the Allied force, determinedby the first equation in (10) setting Cov(N1t, N2t) � 0, and the empiricalwar loss for the Allied force determined by Bracken’s [1] data is calculatedand squared. Then the difference between the war loss EL(N2t) for theGerman force and the empirical war loss for the German force is calculatedand squared. These two squares are summed, yielding a sum of twosquared differences for each day, that is, one difference for the Allied forceand one difference for the German force. Summing these two squareddifferences for each day from day 1 through day 33 gives a numberLS(�1(t), �2(t)) which measures the fit of (�1(t), �2(t)) � (10�11,10�11). Thereafter, in increments of 10�11 along each dimension, (�1(t),�2(t)) � (10�11, 2 � 10�11) is chosen, and a new number LS(�1(t),�2(t)) measuring the fit is calculated in the same manner. Letting bothcoefficients �1(t) and �2(t) vary between 10�11 and a sufficiently highvalue in steps of 10�11, gives as many numbers LS(�1(t), �2(t)) as theproduct of the number of values chosen along the �1(t) dimension and thenumber of values chosen along the �2(t) dimension. The lowest of thesenumbers determines the combination (�1(t), �2(t)) that gives the best fit,which is (�1(t), �2(t)) � (5.38 � 10�9, 8.22 � 10�9), which givesLS(�1(t), �2(t)) � 6.77 � 107. For Figures 3–4 the same procedure isapplied from day 1 through day 10 to yield (�1(t), �2(t)) � (7.655 �10�9, 9.97 � 10�9) and LS(�1(t), �2(t)) � 3.44 � 107, from day 11

through day 20 to yield (�1(t), �2(t)) � (4.32 � 10�9, 8.44 � 10�9)and LS(�1(t), �2(t)) � 1.06 � 107, and from day 21 through day 33 toyield (�1(t), �2(t)) � (4.82 � 10�9, 6.8 � 10�9) and LS(�1(t),�2(t)) � 5.59 � 106. The sum of the least squares for the three phasesis naturally lower than for one phase.

ACKNOWLEDGMENTS

We thank the associate editor and two anonymous refer-ees of this journal for extensive and useful comments.

REFERENCES

[1] J. Bracken, Lanchester models of the Ardennes Campaign,Naval Res Logist 42 (1995), 559–577.

[2] A.H. Blair, At war in the Gulf: A chronology, Texas A&MUniversity Press, College Station, 1992.

[3] P.S. Chen and P. Chu, Applying Lanchester’s linear law tomodel the Ardennes Campaign, Naval Res Logist 45 (2001),1–22.

[4] C.V. Clausewitz, On war, Princeton University Press, Prince-ton, NJ, 1984 [original edition 1832].

[5] D.R. Cox and H.D. Miller, The theory of stochastic pro-cesses, Chapman and Hall, New York, 1965.

[6] T.N. Dupuy, Understanding war, history and theory of com-bat, A Ginger Book, Paragon House, New York, secondprinting, 1987.

[7] R.D. Fricker, Attrition models of the Ardennes Campaign,Naval Res Logist 45 (1998), 1–22.

[8] K. Hausken and J.F. Moxnes, The microfoundations of theLanchester War equations, Mil Oper Res 5(3) (2000),79 –99.

[9] K. Hausken and J.F. Moxnes, Stochastic conditional andunconditional warfare, European J Oper Res 140(1) (2002),61–87.

[10] G.F. Hoffmann, Tanks and the Korean War: A case in un-preparedness, ARMOR 109(5) (September/October 2000),7–12.

[11] F.W. Lanchester, Aircraft in warfare: The dawn of the FourthArm, Tiptree Constable, London, 1916.

[12] P.M. Morse and G.E. Kimball, Methods of operations re-search, MIT Press, Cambridge, MA, 1951.

[13] Polygram Video International Ltd., Battlefield: The Battle forBerlin, c/o Universal Studios, Hagedoornplein 2, 1031 BV,Amsterdam, The Netherlands, 1994.

[14] R.E. Simpkin, Red Armour: An examination of the Sovietmobile force concept, second printing, Brassey’s DefencePublishers, Oxford, 1984.

[15] A. Speer, Inside the Third Reich: Memoirs, second printing,Macmillan, New York, 1970.

[16] J. Stockfish, Models, data, and war: A critique of the study ofconventional forces, R-1526-PR, RAND, Santa Monica, CA,1975.

[17] J.G. Taylor, Lanchester models of warfare, Operations Re-search Society of America, Naval Postgraduate School,Monterey, CA, 1983.

[18] Y.S. Ventisel, Introduction to operations research, SovietRadio Publishing House, Moscow, 1964.


approximations and empirics for stochastic war equations

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