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Tactical7_030812
(Marginal spelling corrections 030813)
Optimal decentralized tactical decisions of isolated Blue
units behind Red lines
LOHMANDER Peter, SUAS, Sweden,
The paper has been presented as an invited paper at EURO/Informs Istanbul 2003, Military Operations Research III.
Keywords: Military Operations Research (46), Game Theory (30)
Abstract:
The tactical decision problems of isolated blue units (IBU) with limited supplies behind
enemy (Red) lines are analysed with mixed strategy game theory. Red has a limited
number of mobile units for randomised protection and defence of potential Red targets.
The Nash equilibrium mixed attack and defence strategies are determined. The sensitivity
of the Nash equilibrium to possible parameter changes is determined in general form. The
optimal Blue response to non-optimal Red strategies is described. The dynamics of
adaptive strategy frequency changes is analysed. Easily used INTERNET software has
been developed for simple decentralized optimisation of the mixed strategies.
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Optimal decentralized tactical decisions of isolated Blue
units behind Red lines
LOHMANDER Peter, SUAS, Sweden,
Keywords: Military Operations Research (46), Game Theory (30)
Abstract:
The tactical decision problems of isolated blue units (IBU) with
limited supplies behind enemy (Red) lines are analysed with
mixed strategy game theory.
Red has a limited number of mobile units for randomised
protection and defence of potential Red targets.
The Nash equilibrium mixed attack and defence strategies are
determined.
The sensitivity of the Nash equilibrium to possible parameter
changes is determined in general form.
The optimal Blue response to non-optimal Red strategies is
described.
The dynamics of adaptive strategy frequency changes is analysed.
Easily used INTERNET software has been developed for simple
decentralized optimisation of the mixed strategies.
www.Lohmander.com
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Briefing
The enemy already controls large parts of our
territory. The enemy has located the headquarters of
one army brigade to our area and has started using
our railroad, transporting troops and armed vehicles.
Most enemy resources are being transported to other
battle fields. In our neighbourhood, there is one enemy
tank, used for local protection of the brigade staff and
the railroad. The enemy tank commander has
optimized a randomized patrol schedule.
We constitute one ranger platoon and have very
limited contacts with higher units. We have sufficient
supplies for one qualified mission, including
explosives, ammunition, food and water. After such a
mission, we will have to try to retire to friendly units
behind our own lines.
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The decision problem
How should we act?
How will the enemy act?
What result can we expect if we act optimally?
These problems have been well described and
analyzed in classical game theory.
In this paper, we will start by defining the mixed
strategy Nash equilibrium.
Then, we will use comparative statics applied to the
mixed strategy Nash equilibrium.
We will investigate the qualitative effects of changes in
the revenues and costs, represented by the elements of
the game matrix, on the optimal frequences of our
strategy selections.
We will also investigate the qualitative effects of
changes in one game matrix element on the expected
value of the game under optimal play.
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If the "value" of one particular enemy target increases
(our expected net gain from attacking that target
increases if it is not protected by the enemy tank), then
it will be shown that our optimal frequency of
attacking that particular enemy decreases (ceteres
paribus).
In some cases, if the value (the element in the game
matrix) of one particular enemy target increases (if
not protected) then the value also increases if it
happens to be protected. These two value changes both
affect the optimal Nash equilibrium. In different cases,
the increasing target values imply that our optimal
attack frequences increase, are unchanged or
decrease.
It will be shown that the optimal frequency of one
strategy may increase, be unchanged of decrease if the
value of one enemy target increases. The direction of
change is a function of all the elements in the game
matrix.
Hence, it is hard or even impossible to construct
general tactical guidelines concerning target selection
that really are optimal in different situations.
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Ranger platoon commanders need to estimate the
elements of the game matrix in the field, using locally
available information together with target value
parameters sent from the headquarters.
Then, optimal local platoon level decisions have to be
based both on the locally obtained information and on
the centrally provided information.
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Analytical solution with comparative statics
11 12 


 21  22 
Assumption A1.
11  12
11   21
 22  12
 22   21
A1. guarantees that:
11  22  12  21  0
Probability
that Blue
attacks Red
target T1.
Probability
that Blue
attacks Red
target T2.
Probability
that the Red
tank
protects T1.
Probability
that the Red
tank
protects T2.
x
(1-x)
y
(1-y)
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Expected net gain of Blue:
  11 xy  12 x(1  y)  21 (1  x) y  22 (1  x)(1  y)
  11xy  12 x  12 xy  21 y  21 xy  22  22 x  22 y  22 xy
  22  (12  22 ) x  (21  22 ) y  (11  12  21  22 ) xy
The Nash equilibrium is determined by these
equations:

 (12   22 )  (11  12   21   22 ) y  0
x

 ( 21   22 )  (11  12   21   22 ) x  0
y
The Nash equilibrium is:
 22   21
x 
11  12   21   22
 22  12
*
y 
11  12   21   22
*
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 0  x*  1
A1  
*
0  y  1
Comparative statics:
In the following section, a number of derivatives are
determined via the quotient rule. In all cases, f denotes
the nominator and g denotes the denominator of the
expression in question. This simplified notation is used
in order to make all steps obvious to the reader and at
the same time to save space.
A1   g  (11  22  12  21 )  0
x
f ´g  fg´ ( 22   21 )


0
2
2
11
(g)
(g)
*
y
f ´g  fg´ ( 22  12 )


0
2
2
11
(g)
(g)
*
10
x
f ´g  fg´ ( 22   21 )(1)


0
2
2
12
(g)
(g)
*
y*
f ´g  fg´  g  ( 22  12 )(1)


2
2
12
(g)
(g)
y* (11  12   21   22 )  ( 22  12 )

12
( g )2
y
11  12   21   22   22  12

12
( g )2
*
y
 21  11

0
2
12
(g)
*
11
x*
f ´g  fg´ (1)(11  12   21   22 )  ( 22   21 )(1)


2
 21
(g)
( g )2
x* 11  12   21   22   22   21

2
 21
(g)
x
12  11

0
2
 21
(g)
*
y*
f ´g  fg´ ( 22  12 )(1)


2
 21
(g)
( g )2
y
( 22  12 )

0
2
 21
(g)
*
x*
f ´g  fg´ (11  12   21   22 )  ( 22   21 )


2
 22
(g)
( g )2
12
x* 11  12   21   22   22   21

 22
( g )2
x* 11  12

0
2
 22
(g)
y*
f ´g  fg´ (11  12   21   22 )  ( 22  12 )


2
 22
(g)
( g )2
y
11   21

0
2
 22
(g)
*
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Summary of the comparative statics analysis:
dx*  0
d11  0   *
dy  0
dx*  0
d12  0   *
dy  0
dx*  0
d 21  0   *
dy  0
dx*  0
d 22  0   *
dy  0
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Case
#1
#2
#3
#4
Optimal Blue
Response
The net gain of Blue from
Attack T1 with
attacking red target T1 (if T1 higher
is defended by Red)
probability.
increases.
(Attack T2 with
lower
probability.)
Optimal Red
Response
Allocate more
defence resource
time to T1.
The net gain of Blue from
Attack T1 with
attacking red target T1 (if T1 lower probability.
is not defended by Red)
increases.
(Attack T2 with
higher
probability.)
Allocate more
defence resource
time to T1.
The net gain of Blue from
Attack T2 with
attacking red target T2 (if T2 higher
is defended by Red)
probability.
increases.
(Attack T1 with
lower
probability.)
Allocate more
defence resource
time to T2.
The net gain of Blue from
Attack T2 with
attacking red target T2 (if T2 lower probability.
is not defended by Red)
increases.
(Attack T1 with
higher
probability.)
Allocate more
defence resource
time to T2.
(Allocate less
defence resource
time to T2.)
(Allocate less
defence time
resources to T2.)
(Allocate less
defence resource
time to T1.)
(Allocate less
defence time
resources to T1.)
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Expected Blue net gains from game matrix element
changes:

  x
 y



 ij  ij x  ij y  ij
*
*
*
 

0
However, in Nash equilibrium, x
.
y
Hence,
*


x
y

 (0)
 (0)
 ij  ij
 ij
 ij



 ij  ij
*
*
*
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Hence, in the two by to case,

 x* y *  0
11
*

 x* (1  y* )  0
12
*
 *
 (1  x* ) y*  0
 21

*
*
 (1  x )(1  y )  0
 22
*
We conclude that:

0
 ij
*
(i, j )
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Optimal Blue (and Red) response(s) to non-optimal
Red (and Blue) strategies

 (12   22 )  (11  12   21   22 ) y  0
x

 ( 21   22 )  (11  12   21   22 ) x  0
y
The dynamics of adaptive strategy frequency changes
x

sgn( )  sgn( x)  sgn( )
t
x
y

sgn( )  sgn( y )  sgn( )
t
y
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A quantitative solution to the decision problem
In order to make it possible for the ranger platoon
commander to optimize the decisions based on local
information, a "user friendly" tactical game optimizer
has been constructed.
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Results:
The tactical decision problems of isolated (Blue)
units with limited supplies behind enemy (Red) lines
have been analysed with mixed strategy game
theory.
The Nash equilibrium mixed attack and defence
strategies have been determined in general form.
The sensitivity of the Nash equilibrium to possible
parameter changes has been determined in general
form.
If the value of an enemy target increases, it may be
optimal to increase, leave unchanged or to decrease
the probability of attack. Detailed analysis is
necessary.
The optimal Blue (and Red) response to non-optimal
Red (and Blue) strategies has been described.
The dynamics of adaptive strategy frequency
changes has been analysed.
Easily used INTERNET software has been
developed for simple decentralized optimisation of
the mixed strategies.
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References:
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1974
Dresher, M., Games of Strategy, Theory and Application, Prentice-Hall, 1961
Försvarsmakten, Armereglemente del 2 Taktik, Stockholm, Sverige, M7741-100612, 1995
Försvarsmakten, Militärstrategisk doktrin, Stockholm, Sverige, M7740-774002, 2002
Isaacs, R., Differential Games, A Mathematical Theory with Applications to Warfare and
Pursuit, Control and Optimization, Wiley, 1965
Lohmander, P., The constrained probability orbit of mixed strategy games with marginal
adjustment: General theory and timber market application, Swedish Univerity of
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(Web-sida): http://www.sekon.slu.se/~plo/spel5.htm
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Sciences, U.S.A., 36, 48-49. 1950
von Neumann, J., A numerical method to determine optimum strategy, Naval Research
Logistics Quarterly, 1, 1954
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91-7125-028-X, 1994
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second edition, December 1994
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