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Stochastic versions of Lanchester equations in wargaming
41
European Journal of Operational Research 24 (1986) 41-45 North-Holland
Stochastic versions of Lanchester equations in wargaming M. A M A C H E R
Swiss Military Department, Bern, Switzerland
D. M A N D A L L A Z
Swiss Federal Institute of Technology, Z~rich, Switzerland
Abstract: This paper investigates a stochastic differential equation derived from the simple square-law Lanchester model and compares its solution with the corresponding Markovian death process. Keywords: Differential equations, military, stochastic processes, gaming, Markov processes
where the exponential-operator for matrices is defined as usual by
1. I n t r o d u c t i o n
The simplest model for the time evolution of the number of survivors in a battle is given by the following equations, known as 'Lanchester equations' in the context of wargaming: dX(t) dt
arY(t) '
dr(t)_ dt
abX(t)
(1)
Y(t) > O. Using the matrix notation ~ :=
(0 :) ~
ab
the solution of (1) can be written as Z(t) = exp(-t~}.2(0) Received September 1983
j=o
for any square matrix A. The system (1) has the following integral:
ab(X2(O)-XZ(t))=ar(y2(o)
- YZ(t)),
(3)
i.e. the well-known Lanchester square-law, which leads immediately to the parity condition
where X(t), Y(t) denote the number of 'blue' and 'red' survivors at time t, and a b, a r the attrition coefficients of blue and red respectively. The system of ordinary differential equations (1) is restricted to the first quadrant, i.e. X(t)>~O and
,
exp(A) := ~ A J/j!
(2)
abX2(0) = arr2(0).
(4)
Numerous generalisation of (1) have been proposed such as heterogeneity of forces, time dependent attrition and non-linearity. The main shortcomings of (1) and related models are their rather unrealistic simplicity and, more important, their completely deterministic approach. Further, the authors of the present article have not been convinced by the statistical validations based on historical data which have been given up to now. In the following we shall consider a stochastic version of (1), in which the attrition coefficients are subject to time dependent random disturbances. Of course this model had to be kept unrealistically simple in order to ensure analyti-
0377-2217/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
M. Amaeher, D. Mandallaz / Stochastic versions of Lanchester equations
42
cally tractable results. Conceptually, however, it could be used as a building block for the simulation of complex systems.
since only f~dq'(u)Z(u) is mathematically meaningful. Using It6 calculus (1), it can be shown that the solution of (6) is given by
2. Stochastic differential equation (S.D.E.)
Z ( t , to) = exp( - t ~ - g'(t, t o ) ) - Z ( 0 ) ,
Let us consider the stochastic difference equation corresponding to the differential system (1), where the deterministic attrition coefficients za • ab, A. a r are disturbed by random shocks 86, 6/.
(8)
'/'(t, to) being a reahsation of the 2-dimensional Brownian matrix:
to),=(0
Wr(t,to))
Wb(t, to)
0
X(jA)--x((j--1)A) = --[A .a r + 6/] V ( ( j -
y(jA)- y((j-
1)A),
(5a)
1)a)
= -[/1.a b + 6J] X ( ( j -
I)A),
(5b)
where j = 1 , 2 , 3 . . . . . K; A = T/K: step length for the time interval (0, T); 86, 8/: i.i.d, random variables with mean 0 and variances A. o~,, A. Or2. Remark. For technical reasons 8 6 , 8 / a r e of order A1/2 in probability and not A. This is to ensure convergence towards 'white noise' when A ~ 0. Letting A ~ 0 in (5) yields formally: dX(t) = dY(t) =
d Z ( t ) = - ~ - Z ( t ) - d ~ ( / ) - Z'(t)
(6)
where
(0
d q , ( t ) : = (0 dWb(t )
dW,(t) ) 0 "
ab
ar) 0
- Z'(u)
= exp( - * d u - ( g'(u + du) - '/'( u))}
•2 ( u ) -
i'(u),
since we can take 2~(u) as new initial condition by using the semi-group properties of e x p { - ~ . u -
dZ(u) = ( - ~du - d~(u))2(u),
Using matrix notation, this yields
~"=
az(u) = Z(u + du)
Taylor development up to linear term yields
- a r Y ( t ) d t - Y(t)dWr(t), - a b X ( t ) d t - X(t)dWb(t ).
~(t):=[X(t)l Y(t)]'
The stochastic process t ~ Z(t, to) does not actually describe the real battle, which is represented by the restriction to X(t), Y(t) > 0. The unrestricted process Z(t, to) is a good approximation only as long as the majority of the trajectories have not led to battle termination. A heuristic justification of (8) can be derived as follows:
which is eq. (6). A rigorous proof using It6 calculus requires quadratic terms and a study of convergence. dq'(u) being independent of Z(u) and E dg'(u) = 0 we obtain at once Ed Z ( u ) = - ~ E Z ( u)du,
'
W~(t), Wb(t ) are independent Brownian motions with variances oft, oft. Strictly speaking (6) is a short hand notation for
Z( t ) - Z(O) = - fot~ • Z(u)du - fotdXp( u)Z( u)
(7)
(9)
i.e. the expected value EZ(u) satisfies the deterministic differential equation. Tedious but straight-forward application of It6 formulae lead to simple expressions for the moments of the non-restricted process Z(t, to). Let
m ( t ) ,=
ex:(t) er (t) I , EX(t)Y(t)
M. Amacher, D. Mandallaz / Stochastic versions of Lanchester equations
:=
0
aft
-2a r )
o2
0
-2a b •
--a b
-at
43
The waiting times between events are exponentially destributed with parameter )~(t) := arY( t ) + abX( t ).
0
Then
M(t) = exp( t~2 )" M(0).
(10)
Since EX(t), EY(t) are known (by (2) and (9)), the variance-covariance structure of Z(t) can be easily calculated.
The event is an X-jump with probability p = arY(t)/~(t ) and Y-jump with probability 1 - p . For further details see i.e. [3, Chapter 4]. Tables 1 and 2 below summarize the results for 2 examples. All simulations are based on 1000 runs.
Models: I
Theoretical moments of nonrestricted solution. II Empirical moments of restricted solution. Ill Empirical moments of Markov process.
3. Simulation To illustrate the model (6) we simulated the exact solution (8) for the restricted process, i.e. stopping as soon as X(t) or Y(t) reached zero. The empirical moments of these simulations are compared with the theoretical moments of the non-restricted process as obtained by (2), (9), (10). For completeness we also give the empirical moments for the well-known bivariate Markovian death process defined by the following properties:
X(t+dt)=X(t)-I with probability at" Y(t )d t,
Y(t + d / ) = Y ( t ) - 1 with probability a b • X ( t ) d t .
Remarks. (1) The simulation and analysis were carried out with the statistical package SAS [2]. (2) Deviations from the normal distribution are based on the Shapiro-Wilks and Kolmogorof-Smirnof tests. (3) In order to ensure monotone decreasing functions for X(t), Y(t) in (8), the coefficient of variation cannot be to large. (4) Simulations were also carried out for a randomly disturbed Markov-death process, i.e. by allowing ar(t ), ab(t ) to be random variables with means ar, a b. The results were essentially equivalent to Model III.
Table 1 P a r i t y case Models
I Mean
II Std
Time
N u m b e r of red survivals
T = 1 T = 2 T = 4 T = 6 T = 8 T=10 T=12
24.6 20.1 13.5 9.0 6.1 4.1 2.8
Median battle d u r a t i o n 2 5 % - 7 5 % r a n g e o f battle d u r a t i o n P r o b a b i l i t y of red victory
-
Deviation from normal distribution
1.4 1.9 2.8 4.0 5.9 8.8 13.2
III
Mean
Std
Mean
Std
24.6 20.1 13.4 8.9 5.7 4.7 3.8
1.4 1.9 2.8 3.3 3.2 2.2 1.4
24.6 19.8 13.6 10.1 6.4 4.9 4.3
2.6 3.7 5.0 5.7 3.6 2.9 2.4
10 9-12 0.53
7.1 6-9.4 0.46
moderate up to T --- 4
not significant
P a r a m e t e r s : blue = red = 30 units at t i m e 0; attrition coefficients a b = a r = 0.20. R a n d o m structure: coefficient of variation: o b / a b = o r / a r = 0.25
M. Amacher, D. Mandallaz /Stochastic versions of Lanchester equations
44
Table 2 Non-parity case Models
I
II
Time
Number of red survivals
T=I T=2 T=4 T=6 T=8 T=10 T=12
27.6 23.4 17.5 14.5
Mean
Std 1.4 1.8 2.7 4.1
_ 1 -
Median battle duration 25%-75% range of battle duration Probability of red victory Deviation from normal distribution
III
Mean
Std
Mean
Std
27.6 23.3 17.6 13.8 9.8
1.3 1.8 2.5 2.8 2.7
28.0 23.8 17.6 11.6 8.4
5.6 4.7
2.3 1.9
7.0
2.4 3.3 5.2 5.9 4.4 3.2 2.8
5.3
8.0
6.5
7--9
5.4-8.8
0.99
0.74
moderate for T<6
significant departures
1 Blue negative. Parameters: blue = 30 red = 33 at time 0; attrition coefficients a b = a r = 0.20. Random structure: coefficient of variation o b / a b = or/a r = 0.25.
4. Discussion
of results
The previous examples and several other simulations under a wide range of conditions (while keeping the coefficient of variation below 25% and considering only small or moderate initial sample sizes of up to 60 units) suggest the following conclusions: (1) The battle duration has a highly skew distribution of g a m m a type. The mean battle duration is not recommended as a s u m m a r y statistics. (2) The theoretical m o m e n t s of the nonrestricted solution of S.D.E. yield reasonable estimates only as long as the majority of battle have not terminated (preferably 75% or more). A p p r o a c h i n g parity the approximation improves for mean values, whereas it gets worse for the standard deviations. (3) For both the S.D.E. and the Markov-death process the n u m b e r of survivors can depart substantially from the normal distribution, especially in the early and late stage of the battle. Confidence interval based on normal theory can be very misleading. (4) Simulation of difference equation instead of S.D.E. yields similar pattern (for a reasonable choice of the step length).
(5) The Markov-death processes display larger variance for the distribution of survivors and shorter battle duration. (6) The winning probabilities for the S.D.E. are highly sensitive to small departures from parity. This concerns less the Markov-death process. (7) The S.D.E. being a continuous model of a discrete process, bias due to rounding off can be severe in small samples. (8) In large scale complex battle with heterogeneous forces the simulation of stochastic difference equations requires shorter computing time than the simulation of the Markov-death process.
5. Further
research
Models of S.D.E. allowing for much larger coefficient of variation as well as general waiting time distribution for Markov-death processes could be promising, though there is little hope for analytically tractable solutions. The influence of the termination criterion (i.e. not necessarily 'fight to the finish') on the quality of approximations by non-restricted S.D.E. deserves special attention. Finally, statistical validation based on battle outcome (as c o m p a r e d with
M. Amacher, D. Mandallaz / Stochastic versions of Lanchester equations
the winning probabilities as predicted by S.D.E. or death process) could be attempted in order to assess the stochastic Lanchester models from a more qualitative view point.
45
References [1] Arnold, L.A., Stochastische Differentialgleichungen, Oldenbourg, Munich-Vienna, 1973. [2] SAS User's Guide, SAS Institute Inc., Carry, NC, 1979. [3] Taylor, J.G., "Lanchester-type models of warfare", Naval Postgraduate School, Monterey, CA, 1980; or by NTIS, Signature AD A 90842.
European Journal of Operational Research 24 (1986) 41-45 North-Holland
Stochastic versions of Lanchester equations in wargaming M. A M A C H E R
Swiss Military Department, Bern, Switzerland
D. M A N D A L L A Z
Swiss Federal Institute of Technology, Z~rich, Switzerland
Abstract: This paper investigates a stochastic differential equation derived from the simple square-law Lanchester model and compares its solution with the corresponding Markovian death process. Keywords: Differential equations, military, stochastic processes, gaming, Markov processes
where the exponential-operator for matrices is defined as usual by
1. I n t r o d u c t i o n
The simplest model for the time evolution of the number of survivors in a battle is given by the following equations, known as 'Lanchester equations' in the context of wargaming: dX(t) dt
arY(t) '
dr(t)_ dt
abX(t)
(1)
Y(t) > O. Using the matrix notation ~ :=
(0 :) ~
ab
the solution of (1) can be written as Z(t) = exp(-t~}.2(0) Received September 1983
j=o
for any square matrix A. The system (1) has the following integral:
ab(X2(O)-XZ(t))=ar(y2(o)
- YZ(t)),
(3)
i.e. the well-known Lanchester square-law, which leads immediately to the parity condition
where X(t), Y(t) denote the number of 'blue' and 'red' survivors at time t, and a b, a r the attrition coefficients of blue and red respectively. The system of ordinary differential equations (1) is restricted to the first quadrant, i.e. X(t)>~O and
,
exp(A) := ~ A J/j!
(2)
abX2(0) = arr2(0).
(4)
Numerous generalisation of (1) have been proposed such as heterogeneity of forces, time dependent attrition and non-linearity. The main shortcomings of (1) and related models are their rather unrealistic simplicity and, more important, their completely deterministic approach. Further, the authors of the present article have not been convinced by the statistical validations based on historical data which have been given up to now. In the following we shall consider a stochastic version of (1), in which the attrition coefficients are subject to time dependent random disturbances. Of course this model had to be kept unrealistically simple in order to ensure analyti-
0377-2217/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
M. Amaeher, D. Mandallaz / Stochastic versions of Lanchester equations
42
cally tractable results. Conceptually, however, it could be used as a building block for the simulation of complex systems.
since only f~dq'(u)Z(u) is mathematically meaningful. Using It6 calculus (1), it can be shown that the solution of (6) is given by
2. Stochastic differential equation (S.D.E.)
Z ( t , to) = exp( - t ~ - g'(t, t o ) ) - Z ( 0 ) ,
Let us consider the stochastic difference equation corresponding to the differential system (1), where the deterministic attrition coefficients za • ab, A. a r are disturbed by random shocks 86, 6/.
(8)
'/'(t, to) being a reahsation of the 2-dimensional Brownian matrix:
to),=(0
Wr(t,to))
Wb(t, to)
0
X(jA)--x((j--1)A) = --[A .a r + 6/] V ( ( j -
y(jA)- y((j-
1)A),
(5a)
1)a)
= -[/1.a b + 6J] X ( ( j -
I)A),
(5b)
where j = 1 , 2 , 3 . . . . . K; A = T/K: step length for the time interval (0, T); 86, 8/: i.i.d, random variables with mean 0 and variances A. o~,, A. Or2. Remark. For technical reasons 8 6 , 8 / a r e of order A1/2 in probability and not A. This is to ensure convergence towards 'white noise' when A ~ 0. Letting A ~ 0 in (5) yields formally: dX(t) = dY(t) =
d Z ( t ) = - ~ - Z ( t ) - d ~ ( / ) - Z'(t)
(6)
where
(0
d q , ( t ) : = (0 dWb(t )
dW,(t) ) 0 "
ab
ar) 0
- Z'(u)
= exp( - * d u - ( g'(u + du) - '/'( u))}
•2 ( u ) -
i'(u),
since we can take 2~(u) as new initial condition by using the semi-group properties of e x p { - ~ . u -
dZ(u) = ( - ~du - d~(u))2(u),
Using matrix notation, this yields
~"=
az(u) = Z(u + du)
Taylor development up to linear term yields
- a r Y ( t ) d t - Y(t)dWr(t), - a b X ( t ) d t - X(t)dWb(t ).
~(t):=[X(t)l Y(t)]'
The stochastic process t ~ Z(t, to) does not actually describe the real battle, which is represented by the restriction to X(t), Y(t) > 0. The unrestricted process Z(t, to) is a good approximation only as long as the majority of the trajectories have not led to battle termination. A heuristic justification of (8) can be derived as follows:
which is eq. (6). A rigorous proof using It6 calculus requires quadratic terms and a study of convergence. dq'(u) being independent of Z(u) and E dg'(u) = 0 we obtain at once Ed Z ( u ) = - ~ E Z ( u)du,
'
W~(t), Wb(t ) are independent Brownian motions with variances oft, oft. Strictly speaking (6) is a short hand notation for
Z( t ) - Z(O) = - fot~ • Z(u)du - fotdXp( u)Z( u)
(7)
(9)
i.e. the expected value EZ(u) satisfies the deterministic differential equation. Tedious but straight-forward application of It6 formulae lead to simple expressions for the moments of the non-restricted process Z(t, to). Let
m ( t ) ,=
ex:(t) er (t) I , EX(t)Y(t)
M. Amacher, D. Mandallaz / Stochastic versions of Lanchester equations
:=
0
aft
-2a r )
o2
0
-2a b •
--a b
-at
43
The waiting times between events are exponentially destributed with parameter )~(t) := arY( t ) + abX( t ).
0
Then
M(t) = exp( t~2 )" M(0).
(10)
Since EX(t), EY(t) are known (by (2) and (9)), the variance-covariance structure of Z(t) can be easily calculated.
The event is an X-jump with probability p = arY(t)/~(t ) and Y-jump with probability 1 - p . For further details see i.e. [3, Chapter 4]. Tables 1 and 2 below summarize the results for 2 examples. All simulations are based on 1000 runs.
Models: I
Theoretical moments of nonrestricted solution. II Empirical moments of restricted solution. Ill Empirical moments of Markov process.
3. Simulation To illustrate the model (6) we simulated the exact solution (8) for the restricted process, i.e. stopping as soon as X(t) or Y(t) reached zero. The empirical moments of these simulations are compared with the theoretical moments of the non-restricted process as obtained by (2), (9), (10). For completeness we also give the empirical moments for the well-known bivariate Markovian death process defined by the following properties:
X(t+dt)=X(t)-I with probability at" Y(t )d t,
Y(t + d / ) = Y ( t ) - 1 with probability a b • X ( t ) d t .
Remarks. (1) The simulation and analysis were carried out with the statistical package SAS [2]. (2) Deviations from the normal distribution are based on the Shapiro-Wilks and Kolmogorof-Smirnof tests. (3) In order to ensure monotone decreasing functions for X(t), Y(t) in (8), the coefficient of variation cannot be to large. (4) Simulations were also carried out for a randomly disturbed Markov-death process, i.e. by allowing ar(t ), ab(t ) to be random variables with means ar, a b. The results were essentially equivalent to Model III.
Table 1 P a r i t y case Models
I Mean
II Std
Time
N u m b e r of red survivals
T = 1 T = 2 T = 4 T = 6 T = 8 T=10 T=12
24.6 20.1 13.5 9.0 6.1 4.1 2.8
Median battle d u r a t i o n 2 5 % - 7 5 % r a n g e o f battle d u r a t i o n P r o b a b i l i t y of red victory
-
Deviation from normal distribution
1.4 1.9 2.8 4.0 5.9 8.8 13.2
III
Mean
Std
Mean
Std
24.6 20.1 13.4 8.9 5.7 4.7 3.8
1.4 1.9 2.8 3.3 3.2 2.2 1.4
24.6 19.8 13.6 10.1 6.4 4.9 4.3
2.6 3.7 5.0 5.7 3.6 2.9 2.4
10 9-12 0.53
7.1 6-9.4 0.46
moderate up to T --- 4
not significant
P a r a m e t e r s : blue = red = 30 units at t i m e 0; attrition coefficients a b = a r = 0.20. R a n d o m structure: coefficient of variation: o b / a b = o r / a r = 0.25
M. Amacher, D. Mandallaz /Stochastic versions of Lanchester equations
44
Table 2 Non-parity case Models
I
II
Time
Number of red survivals
T=I T=2 T=4 T=6 T=8 T=10 T=12
27.6 23.4 17.5 14.5
Mean
Std 1.4 1.8 2.7 4.1
_ 1 -
Median battle duration 25%-75% range of battle duration Probability of red victory Deviation from normal distribution
III
Mean
Std
Mean
Std
27.6 23.3 17.6 13.8 9.8
1.3 1.8 2.5 2.8 2.7
28.0 23.8 17.6 11.6 8.4
5.6 4.7
2.3 1.9
7.0
2.4 3.3 5.2 5.9 4.4 3.2 2.8
5.3
8.0
6.5
7--9
5.4-8.8
0.99
0.74
moderate for T<6
significant departures
1 Blue negative. Parameters: blue = 30 red = 33 at time 0; attrition coefficients a b = a r = 0.20. Random structure: coefficient of variation o b / a b = or/a r = 0.25.
4. Discussion
of results
The previous examples and several other simulations under a wide range of conditions (while keeping the coefficient of variation below 25% and considering only small or moderate initial sample sizes of up to 60 units) suggest the following conclusions: (1) The battle duration has a highly skew distribution of g a m m a type. The mean battle duration is not recommended as a s u m m a r y statistics. (2) The theoretical m o m e n t s of the nonrestricted solution of S.D.E. yield reasonable estimates only as long as the majority of battle have not terminated (preferably 75% or more). A p p r o a c h i n g parity the approximation improves for mean values, whereas it gets worse for the standard deviations. (3) For both the S.D.E. and the Markov-death process the n u m b e r of survivors can depart substantially from the normal distribution, especially in the early and late stage of the battle. Confidence interval based on normal theory can be very misleading. (4) Simulation of difference equation instead of S.D.E. yields similar pattern (for a reasonable choice of the step length).
(5) The Markov-death processes display larger variance for the distribution of survivors and shorter battle duration. (6) The winning probabilities for the S.D.E. are highly sensitive to small departures from parity. This concerns less the Markov-death process. (7) The S.D.E. being a continuous model of a discrete process, bias due to rounding off can be severe in small samples. (8) In large scale complex battle with heterogeneous forces the simulation of stochastic difference equations requires shorter computing time than the simulation of the Markov-death process.
5. Further
research
Models of S.D.E. allowing for much larger coefficient of variation as well as general waiting time distribution for Markov-death processes could be promising, though there is little hope for analytically tractable solutions. The influence of the termination criterion (i.e. not necessarily 'fight to the finish') on the quality of approximations by non-restricted S.D.E. deserves special attention. Finally, statistical validation based on battle outcome (as c o m p a r e d with
M. Amacher, D. Mandallaz / Stochastic versions of Lanchester equations
the winning probabilities as predicted by S.D.E. or death process) could be attempted in order to assess the stochastic Lanchester models from a more qualitative view point.
45
References [1] Arnold, L.A., Stochastische Differentialgleichungen, Oldenbourg, Munich-Vienna, 1973. [2] SAS User's Guide, SAS Institute Inc., Carry, NC, 1979. [3] Taylor, J.G., "Lanchester-type models of warfare", Naval Postgraduate School, Monterey, CA, 1980; or by NTIS, Signature AD A 90842.