Military Applications and Sensitivity Analysis of Coupling
Game Management
Mo Weia, Jose B. Cruz, Jrb, Genshe Chena, Erik Blaschc, and Martin Krugerd
a
- Intelligent Automation, Inc., 15400 Calhoun Dr, Suite 400, Rockville, MD 20855, {mwei, gchen}@i-a-i.com
b
- The Ohio State University, 2015 Neil Ave, Columbus, OH 43202, [email protected]
c
- AFRL, WPAFB, OH, USA, [email protected]
d
- The Office of Naval Research, [email protected]
Abstract—Coupling game theory can be used to formulate
cases “lying between” non-cooperative games and
cooperative games and it can provide more reasonable
control strategies for players. Traditional Transferable
Utility (TU) cooperative games, non-cooperative games,
and two-person zero-sum games are special cases of
coupling game. This paper describes military applications of
a coupling game. Approaches to determine coupling factors
in a coupling game are discussed. Sensitivity functions
about coupling factors are provided. UAV experiments
confirm the benefits of applying coupling game theory in
non-ideal complex military situations. 1 2
and then apply the corresponding game theory. In this way,
the main part is taken and some useful information may be
discarded.
Unfortunately, sometimes the discarded non-mainstream
information can lead to very different results. We use the
following two prisoner’s dilemma game tables to illustrate
the importance of discarded non-mainstream information.
Example 1 Game Table 1 (for game 1) and Game Table 1
(for game 2) are logic-homogeneous. According to current
game theories, for both games, cell 1 is always the noncooperative solution, and cell 4 is always the Transferable
Utility (TU) cooperative solution. When traditional game
theory is used to analyze the two games, exactly the same
suggestions for the two games will be obtained. But
appealing to common sense, in game 2, for both players, the
inducement for non-cooperation is very large and the
cooperation risk is very small when compared with game 1.
In real world situations, two ordinary friends will be more
likely to reach cell 1 in game 2 compared with the case
when they are in game 1.
TABLE OF CONTENTS
1. INTRODUCTION ......................................................1
2. TECHNICAL APPROACH ........................................2
3. SIMULATIONS ........................................................7
4. CONCLUSIONS .......................................................8
REFERENCES .............................................................8
BIOGRAPHY ...............................................................9
1. INTRODUCTION
Table 1 Smaller inducement for non-cooperation
In recent years game theory has received substantial
attention. In applications of game theory [1-4], a first step is
to model the real world cases as precisely as possible.
Currently cooperative games, non-cooperative games, and
constant-sum games are used to describe real-world
situations.
Game models which utilize only one of cooperation, noncooperation and adversary models are called ideal game
models. However, there are cases “lying between” noncooperative games and cooperative games. In these “nonideal” cases, there exists more than one kind of factors
among non-cooperation, cooperation, and adversary
models. Imperfect coordination among UAVs, or
cooperation benefit/risk among companies/countries might
result in this category of game models. Experiences and
instincts are used to choose the closest ideal game model,
Table 2 Larger inducement for non-cooperation
1
1
2
1-4244-1488-1/08/$25.00 ©2008 IEEE.
IEEEAC paper #1507, Version 4, Updated 2007:12:04
1
This implies that there might be some important factor that
has not been modeled by current game theories, causing
users to instinctively distinguish between these two games
while current game models could not. The factor is the
information discarded by ideal game models, which are
mainly
based
on
logistics.
Since 3.0001 > 3 and
300000 > 3 are exactly the same from the perspective of
logistics, existing ideal game models discard the
information about the large difference between the numbers.
2. TECHNICAL APPROACH
Summary of Coupling Game
In this section, we summarize coupling game theory. Details
are in [6].
Suppose that there is a game in which there are D decision
makers and the decision makers’ objectives are payoffs. We
will use U d = {u1d , u2d ," , u d } to represent player d ’s
nd
The “coupling game model” (CGT) proposed in [6], which
introduces coupling factors to model non-ideal information,
can describe and analyze such complex situations more
precisely. However, the estimation of coupling factors is not
easy. In [6], a fuzzy estimation approach was suggested.
Another approach is to apply emotion-rationality
mechanism to analyze coupling factors. Both approaches
are experimented in a project that tries to research highly
intelligent threat prediction mechanisms.
decision space. Here, d = 1, 2," , D. nd represents the
number of possible choices or decisions for player d . We
will use P d (u1 , u 2 ," , u D ) or simply P d
to represent
i i "i
i i "i
i
i
i
1 2
D
1
2
d
1 2
D
the payoff for player d when each player k
chooses u k = u k . Here ik is the k -th player’s choice index
i
k
and take one of the numbers 1, 2," , nk . For simplification,
JG
we use P = ( P1 , P 2 ," , P D )T to represent the
i i "i
i i "i
i i "i
1 2
d
1 2
d
1 2
d
original payoffs for the D players, respectively.
Coupling game theory can find widely applications in
battlefields. The first possible application is the imperfect
cooperation between UAVs (Unmanned Aerial Vehicles).
When there is imperfect coordination in a team, the
relationship between UAVs in this team is neither ideally
cooperative nor ideally non-cooperative. To avoid
discarding non-ideal information, coupling factor and
coupling game could be applied.
Players face non-ideal game situations. Players know that
Pi di "i is the original payoff but they want to maximize a
1 2
D
yet-to-be-determined modified payoff, which is based on
the original payoffs and quantified relationships among the
players. Players want to model modified payoffs. Players do
not want to discard any known information, thus reluctant
to choose the closest ideal model. They need models to
accommodate these non-ideal situations.
A second application includes non-combatant civilians in
battlefields. Civilians often play an active role in wars. They
are not just passively static but might emotionally and/or
purposefully take actions to help one side in a battle. This is
to say, they can make choices like a game player, too.
Unfortunately, existing two-player (Blue and Red) game
theoretic models [5, 8-10] usually do not consider this
situation, even though collateral damage has been
considered in a paper on a two-player game model [20].
Extending such two-player models to three-player (Blue,
Red, and White civilian) game model can capture more
battlefield information and analyze more complex
situations. Since emotion and rationality will determine the
civilian player’ attitudes toward Blue or Red, which can be
modeled as coupling factors in a multi-player coupling
game.
A game reconstruction equation set is as follows.
⎡ Pi1ic "i ⎤ ⎡ w w
12
⎢ 1 2 D ⎥ ⎢ 11
⎢ P 2c ⎥ w
w22
= ⎢ i1 i2 "iD ⎥ = ⎢⎢ 21
⎢#
⎥ ⎢" "
⎢ Dc ⎥ ⎢ w
wD 2
P
⎣⎢ i1 i2 "iD ⎦⎥ ⎣ D1
JJK
P c i1 i2 "iD
JJK
1
w1D ⎤ ⎡⎢ Pi1 i2 "iD ⎤⎥ ⎡ w1 ⎤
⎢ JJK ⎥
⎥ 2
JK
" w2 D ⎥ ⎢ Pi1 i2 "iD ⎥ ⎢ w2 ⎥ JK
⎢
⎥ = ⎢ ⎥ P i1 i2 "iD = W P i1 i2 "iD
" " ⎥ ⎢#
# ⎥
⎥
⎢
⎥
JJJK
" wDD ⎥⎦ ⎢⎢ P D ⎥⎥ ⎢⎣ wD ⎥⎦
⎣ i1 i2 "iD ⎦
"
(1)
where
−∞ ≤ w j
1 j2
≤ +∞, j1 = 1, 2," , D; j2 = 1, 2," , D
Definition 1 The game array whose i1 i2 " iD -element is
JJK
P c i1 i2 "iD is defined as Coupling Game Array (CGA). The
JK
corresponding game array whose elements are Pi1 i2 "iD is
defined as Original Game Array (OGA).
This paper is organized as follows. In Section 2, we will
summarize the basic idea of coupling game theory. Section
3 describes the two approaches to the estimation problem of
coupling factors. Section 4 provides sensitivity analysis on
coupling factors. Section 5 discusses the experimental
results and explanations. Section 6 has conclusions for the
paper.
Definition 2 If we apply current non-cooperative game
theory to the coupling game array, the game strategy is
defined as Non-Cooperative Coupling Game Strategy
(NCCGS). The Nash equilibria obtained from the coupling
game array are called Non-Cooperative Coupling Game
Equilibria (NCCGE). The coupling game array plus the
Non-Cooperative Game strategy will be called NonCooperative Coupling Game (NCCG).
Pi di "i
1 2
2
D
is called the Raw Payoff of player d . P dc
is
i i "i
1 2
D
JK
called the Coupling Payoff of player d . Pi1 i2 "iD is called
JJK
the Raw Payoff Vector. P c i1 i2 "iD is called the Coupling
Payoff Vector. W is called the Coupling Factor Matrix.
JJK
is called the Coupling factor, or Weight. wd is called
w
as Q d
i1 i2 "id −1id +1"iD
→
qk = ⎡ Pi1i "i "i
⎣ 12 d D
j1 j2
JJK JJK
JJJK T
= ⎡ q1T q2T " qnT ⎤ ,
d ⎦
⎣
" Pi Di "i "i ⎤ = ⎡ Pi1i "k"i
⎣ 12
1 2
d
D ⎦
D
Pi 2i "i "i
1 2
d
D
where
Pi 2i "k"i
1 2
D
" Pi Di "k"i ⎤
1 2
D ⎦
.
Theorem 1: If an original game array satisfies the
conditions:
player d ’s coupling factor vector. (1) is called the
Coupling Game Transformation Equation (CGTE). The
whole approach is called Coupling Game Model.
(1) The number of choices of each player nd is less than
the number of players D
Many popular traditional games are special cases of
coupling game. We use the simplest 2-player case
nd < D for all possible player indices d
(3)
P =w P +w P
1c
ij
1
11 ij
2
12 ij
Pij2 c = w21 Pij1 + w22 Pij2
(2)
in Q d
(2)
After removing any individual column
, the remaining matrix is still full rank
i1 i2 "id −1id +1"iD
where
−∞ ≤ wij ≤ +∞, i = 1, 2; j = 1, 2
then for any cell of the original game array, there always
exist at least one D -player rational coupling factor
matrix W , such that the corresponding cell in the
transformed game is a non-cooperative coupling game
(NCCG) equilibrium.
to illustrate some meanings of CGTE (Fig. 1).
In some cases, a higher social payoff [12] in a game can be
guaranteed by a higher set of coupling factors. That is to
say, under some conditions better cooperation can guarantee
higher social payoff. This is stated by the following rational
cooperation theorem.
Theorem 2 Assume that for two twice differentiable
bivariate functions P1 ( x a , xb ) and P 2 ( x a , x b ) , where x a
and xb are scalar input variables such that ( x a , xb ) ∈ Χ , Χ
is a convex set, and Χ ⊆ R 2 hold, it is known that
and
P1 ( x a , xb ) = h1a ( x a ) + h1b ( xb )
2
a
b
2a
a
2b
b
1
a
P ( x , x ) = h ( x ) + h ( x ) hold, and both P ( x , xb )
and P 2 ( x a , xb ) are bivariate concave in Χ .
(1)
(2) 0 ≤ w1a ≤ w2a ≤ 1, 0 ≤ w1b ≤ w2b ≤ 1 .
Fig. 1. Game plane: the relationship between coupling
game and traditional games
(3) There exist ( x1a , x1b ) ∈ Χ and ( x2a , x2b ) ∈ Χ such
that
In some games, decision makers can make any desired cell
the equilibrium of coupling game. This is stated in Theorem
to denote a
1, whose proof is in [6]. We use Q d
∂P1 ( x a , xb )
∂x a
i1 i2 "id −1id +1"iD
nd × D matrix for player d along the direction determined
i1 i2 "id −1id +1"iD
The
w1b
i1 i2 "id −1 xid +1"iD
matrix
can
and
be
∂P ( x , x )
∂xb
1
by the subscripts i1 i2 " id −1id +1 " i D . The element for the x -th
row and
y -the column in this matrix is
d
.
Q
( x, y ) = Qxyd = P y
written
3
x a = x1a
xb = x1b
a
+ w1a
∂P 2 ( x a , xb )
∂x a
+
∂P ( x , x )
∂xb
b
2
x = x1a
xb = x1b
a
a
x a = x1a
xb = x1b
=0
(4)
b
x a = x1a
xb = x1b
=0
∂P1 ( x a , xb )
∂x a
∂P ( x , x )
∂xb
1
w2b
x a = x2a
xb = x2b
a
+ w2a
∂P 2 ( x a , xb )
∂x a
+
∂P ( x , x )
∂xb
b
2
x a = x2a
xb = x2b
a
x a = x2a
xb = x2b
If input is Perfect, output should be One.
=0
(5)
Each UAV assigns corresponding outputted coupling
factors as the coupling factors for the other UAVs via these
fuzzy rules.
b
x a = x2a
xb = x2b
=0
hold simultaneously.
P1 ( x1a , x1b ) + P 2 ( x1a , x1b ) ≤ P1 ( x2a , x2b ) + P 2 ( x2a , x2b )
holds, and the equality holds if and only if
( x1a , x1b ) = ( x2a , x2b ) .
Then
The proof can be found in [12].
Estimation of Coupling Factors
To apply coupling game to battlefields, estimating the
coupling factors is the first step. It is often difficult to
provide the exact values for coupling factors. However,
many real world situations accept approximate estimations
[6], and it is often not difficult to know the signs, the
approximate ranges and amplitudes of coupling factors. In
this section we describe two possible suboptimal
approaches to the estimation of coupling factors. Both
approaches are being experimented and assessed for
different applications in research projects.
Fig. 2. A simplified set of fuzzy membership functions
To ensure real time application, the chosen combination rule
of the input recommendations is Center-Average Rule [26],
which is stated in the following formula.
O out =
∑tO
∑t
i
i
(6)
i
where ti is the vertical coordinate of the input and the i th
The first approach is called Fuzzy approach and can be
applied in imperfectly cooperating teams. It applies fuzzy
mechanism [25] to provide the estimation.
input membership function. Oi is the center of the i th
output membership function. Oout is the combined fuzzy
output, that is, the coupling factor. Another possible
combination rule is Center-Of-Gravity (COG) [26], which
takes more calculation when the membership functions are
complex.
According to different precision requirement, different input
fuzzy membership functions and output fuzzy membership
functions can be constructed. In the experiments, a
simplified set of fuzzy membership functions for inputs and
outputs are stated in Fig. 2. The input is the extent of
imperfectness of coordination among two UAVs. The
output is the value of coupling factor. The extent of
imperfectness of coordination can be identified as five
classes: “None”, “Bad”, “Ordinary”, “Good”, and “Perfect”.
Their corresponding membership function shapes are
illustrated as the subplot (a) in Fig. 2. The output
membership functions are stated in the subplot (b) in Fig. 2.
The fuzzy rules are as follows:
The second approach is called Civilian Emotion approach.
It can be applied in multi-player emotional environments,
such as battlefields with White civilians. Traditionally,
battlefield games with civilians are modeled as follows.
J B (k ) = B f (k ) + Bb (k ) − 0.2 R f (k ) − 0.3Rb (k ) + W ( k )
(7)
J R (k ) = −1.5B f (k ) − 2.3Bb ( k ) + R f (k ) + Rb (k )
where k is the index of time step. J B (k ) and J R (k ) are the
normalized payoff functions being maximizing by Blue and
Red (the labels of two fighting forces), respectively. B f ( k )
and Bb ( k ) are the values of undestroyed Blue fighters and
If input is None, output should be Zero.
If input is Bad, output should be Low.
If input is Ordinary, output should be Medium.
Blue bombers, respectively. R f ( k ) and Rb (k ) are the
counterparts of the Red side. W (k ) is the value of
undestroyed (neutral) civilian. The differences between the
If input is Good, output should be High.
4
game represented by (8)
absolute values of the coefficients of B f (k ) ’s stand for the
first asymmetry and implies that Red cares more about
destroying the enemy than keeping themselves safe. The
coefficients of W (k ) stands for the fact that Red might not
care about damages of the civilian properties while Blue
does care. Blue and Red can choose inputs from different
sets, say u B (k ) and u R (k ) , respectively, to update B f (k ) ,
J B (k ) = B f (k ) + Bb (k ) − 0.2 R f (k ) − 0.3Rb (k ) + W (k )
(8)
J R (k ) = −1.5 B f (k ) − 2.3Bb (k ) + R f (k ) + Rb (k )
J W (k ) = −0.001B f (k ) − 0.001Bb (k ) + 0.001R f (k ) + 0.001Rb (k ) + W (k )
where J W (k ) is the payoff of the civilian player and the
civilian player also deserves an input set uW (k ) . In this
way, civilians are treated as a player that is at the same level
with two opposing forces, which are often labeled as Blue
(Force) and Red (Force).
Bb (k ) , R f (k ) , Rb (k ) , and J B (k ) and J R (k ) . Their own
input sets might be different according to different
technology levels, financial status, morale, etc., which stand
for the second asymmetry. Blue and Red will assess J B (k )
and J R (k ) according to different standard, which will stand
In addition, as many researchers [22-24] pointed out,
emotion plays an important role in decision making almost
everywhere. No one is absolutely rational, especially for
civilians in battlefield. This is because civilians are not
militarily organized and their emotions can vary greatly if
they experience collateral damages from different forces.
However, in the model stated in (8), civilians are ideally
rational, that is to say, without any emotions. It does not
analyze situations such as what a civilian family will think
or do when a member is killed or their house is destroyed by
one force in an attack. If Blue or Red collaterally damages a
civilian family, the relatives might deeply hate Blue or Red,
respectively, and take biased actions to affect the battle and
avenge for their loss. Such actions might be telling one side
(say Red) everything they know about the other side (say
Blue) while telling the other side “slightly misleading”
information or simply keeping silent when asked by the
other side. If White civilians distrust Red more than distrust
Blue, they would feel happy if Red loses something. Note
that civilians who take biased actions will continue to be
civilians. However, when civilians are agitated too much,
some of them might directly join one army in the battle even
if this implies that they are not civilians any more.
for the third asymmetry. For example, a 10,000
soldier/platform casualty might be intolerable for Blue
while Red can easily ignore such casualty.
It seems that the modeling methodology illustrated by (7) is
reasonable because it is easy to deal with classical game
framework. However, as pointed out in [21], it has a key
disadvantage: civilians are treated as “passive objects”
without any desires or capabilities to take actions to affect
Blue or Red. In this way, the strategies of Blue and Red
have effect on neutral people, while the neutral civilians can
only passively watch the battle process and do not take an
active role. In a coarse scenario, this might be reasonable.
However, in modern counter-strike wars, such as the Iraqi
War, it has been observed that the “neutral” civilians either
have intent and capabilities or threatened by insurgents so
that they would do or have to do something to maximize
their own benefits. For example, if civilian people guess
that there might be battles around their block, they might
consolidate or build up more and higher walls to protect
themselves, which might cause more obstacles for videoguided missiles and heavy weapons. Moreover, some
people might think the existence of US forces in Iraq is
undesirable no matter how US forces do favors for them.
Therefore, they show some sympathy to insurgents. Suicide
bomber commanders might threaten civilians to help
terrorists hide and attack, too.
For such complex situations, applying coupling game and
updating some coupling factors according to emotion
dynamics is a reasonable approach. Considering this, an
even more comprehensive model might be as follows
⎡ J Bc ( k ) ⎤
⎡ J B ( k ) ⎤ ⎡W11c (k ) W12c (k ) W13c (k ) ⎤ ⎡ J B ( k ) ⎤
⎥ ⎢ R
⎢ Rc ⎥
⎢ R
⎥ ⎢ c
⎥
c
c
c
⎢ J ( k ) ⎥ = W ( k ) × ⎢ J ( k ) ⎥ = ⎢W21 ( k ) W22 (k ) W23 (k ) ⎥ × ⎢ J ( k ) ⎥
⎢ J Wc ( k ) ⎥
⎢ J W ( k ) ⎥ ⎢W c (k ) W c (k ) W c (k ) ⎥ ⎢ J W ( k ) ⎥
32
33
⎣
⎦
⎣
⎦ ⎣ 31
⎦
⎦ ⎣
Under such situations, civilians are slightly adversarial to
US force and not exactly “neutral”. Although US forces
know this perfectly, US forces cannot treat them as enemies
due to social-political-military constraints. However, US
forces should at least be able to consider the effect of
courses of actions (COAs) of such “slightly adversarial
civilian” and base decision-making on the combination of
COAs from both enemy and such civilians. In other words,
a more accurate tool should be developed to model neutral
civilian as a “White” player and considering the coupling
between civilian’s actions and Blue/Red payoffs. This will
allow Blue player to make decisions on a more
comprehensive and accurate background and greatly reduce
unnecessary losses. Based on the above argument, a more
accurate model for such a scenario might be a 3-player
(9)
where the “ c ” in superscript stands for “comprehensive”.
Among the elements of W c (k ) , W31c (k ) and W32c (k ) reflect
the civilian attitudes toward Blur and Red, respectively. If
White civilians hate Red more than they hate Blue, which
means W31c (k ) > 0 > W32c (k ) , they would feel happy if Red
loses something or Blue gets something. Since civilians are
not militarily trained or organized and their emotion varies
much faster than opposing forces in battlefields, it is
reasonable to assume that opposing forces’ attitudes toward
each other or civilians are relatively stable and will not
5
P1c ( x a , xb ) = P1 ( x a , x b ) + wa P 2 ( x a , x b )
emotionally change like civilians do during a battle.
Emotion of civilians will affect coupling factor matrix
according
to
following
formulas.
W c (k )
⎧Wijc 0 ,
if i ≤ 2 or j = 3 or k = 0
⎪
N W ( k −1)
⎪
if i = 3 and j ≠ 3 and k > 0 and ∑ vWh (k − 1)) = 0
⎪0,
h =1
⎪
N W ( k −1)
⎪ c0
⎪W31 +(W31c (k − 1) − W31c 0 ) μ31 + W21c 0 ( Loss X ( k − 1) / ∑ vhW ( k − 1)),
h =1
⎪
⎪
N W ( k −1)
Wijc ( k ) = ⎨
if i = 3 and j = 1 and k > 0 and ∑ vWh ( k − 1)) ≠ 0
⎪
h =1
⎪
N W ( k −1)
⎪
c0
c
c0
c0
X
⎪W32 +(W32 (k − 1) − W32 ) μ32 + W12 ( Loss (k − 1) / ∑ vhW ( k − 1)),
⎪
h =1
⎪
N W ( k −1)
⎪
if i = 3 and j = 2 and k > 0 and ∑ vWh ( k − 1)) ≠ 0
⎪⎩
h =1
where
W i jc 0
For simplification, we use P1 for P1 ( x a , x b ) and P 2 for
P 2 ( x a , x b ) in later induction. What we want to know is how
much the optimal strategy ( x a , x b ) will change when wa
and wb change. In other words, we wish to know the
sensitivity function matrix
(10)
⎡ S xa
⎢ w
⎢ S xba
⎣ w
a
S wx a =
exponential decay factor. v (k ) is the value of i th
platform. Loss X (k ) is the collateral damage caused by X .
wa ∂x a
x a ∂wa
(13)
and other elements in the sensitivity function matrix are
1
1
2
2
defined similarly. Assuming ∂P , ∂P , ∂P , ∂P are not
a
b
a
b
∂x ∂x ∂x ∂x
zero, we do the following induction according to definition
of Nash equilibrium.
X can take values from {B, R} . If the i th platform is
killed, viW (k ) will be calculated as part of loss and the
value of i th platform will be set to zero for later time steps.
Note that if we use Loss(k ) to denote the loss of white due
to
the
actions
launched
at
time
k,
B
R
holds.
This
is
because
there
Loss (k ) + Loss (k ) ≥ Loss (k )
might be cases weapons from both Blue and Red
simultaneously kill some White lives/properties. For
civilians, it is often not feasible to estimate the weapon from
which side has larger collateral killing probability thus
blame more to that side. The most direct way is to blame
W
two sides simultaneously. N ( k − 1) is the number of White
∑
(12)
a
where
W
i
N W ( k −1)
S wxb ⎤
⎥
b
S wxb ⎥
⎦
a
is the initial coupling factor [21], μij is the
civilians platforms. Note that
(11)
Pi 2 c ( x a , x b ) = wb P1 ( x a , x b ) + P 2 ( x a , xb )
2
∂P1
a ∂P
+
=0
w
∂x a
∂x a
wb
2
⎛ ∂P1 ⎞ ⎛ ∂P 2 ⎞
wa = − ⎜ a ⎟ / ⎜ a ⎟
⎝ ∂x ⎠ ⎝ ∂x ⎠
⇒
vWh (k − 1)) = 0 implies
(14a)
∂P ∂P
+
=0
∂xb ∂x b
1
(14b)
⎛ ∂P ⎞ ⎛ ∂P ⎞
wb = − ⎜ b ⎟ / ⎜ b ⎟
⎝ ∂x ⎠ ⎝ ∂x ⎠
1
h =1
that all White civilians are collaterally killed. In this case
although the dead White civilians still appear in the
coupling factor updating equations, they will not be able to
perform any biased actions in the game.
2
a
a
b
⇒ x = g1 ( w , w ) = g1
(14c)
xb = g 2 ( wa , wb ) = g 2
Sensitivity Analysis
We have
Based on the multiple player game models with civilian
player, results from modern optimal control and dynamic
game theories can be readily applied to calculate optimal
control according to the global maximization equilibrium of
objective functions, or suboptimal control according to
various approaches. For details, see [11-13, 15-16, 20]. In
this section we provide several sensitivity functions about
estimation of coupling factors in a typical scenario
experimented. The coupling game in the experimental
scenario is normalized so that for a two-player game there
are only two coupling factors, wa and wb , to estimate. This
a
S xwa =
∂ 2 P1
x a ∂wa
xa
∂( x a )2
=
wa ∂x a ⎛ ∂P1 ⎞ ⎛ ∂P 2 ⎞
⎜ a ⎟/⎜ a ⎟
⎝ ∂x ⎠ ⎝ ∂x ⎠
∂P 2 ∂P1 ∂ 2 P 2
−
∂x a ∂x a ∂ ( x a ) 2
⎛ ∂P 2 ⎞
⎜ a⎟
⎝ ∂x ⎠
2
(15)
=
⎛ ∂ 2 P1 ∂P 2 ∂P1 ∂ 2 P 2 ⎞
xa
−
⎜
⎟
∂P1 ∂P 2 ⎝ ∂ ( x a ) 2 ∂x a ∂x a ∂ ( x a ) 2 ⎠
∂x a ∂x a
and
a
S wx a =
two-player coupling game is stated as follows.
1
S
wa
xa
=
∂P1 ∂P 2
∂x a ∂x a
(16)
⎛ ∂ P ∂P ∂P ∂ P ⎞
xa ⎜
− a
⎟
a 2
a
∂x ∂ ( x a ) 2 ⎠ xa = g1 ( wa , wb )
⎝ ∂ ( x ) ∂x
b
a
b
2
1
2
1
2
2
x = g2 ( w , w )
6
reaching Cell 2, Cell 4, Cell 6, or Cell 8. If we apply
coupling game and use fuzzy theory to estimate coupling
factors, although there might exist estimation errors, in the
long run, we can get better results better than applying noncooperative game theory or applying TU cooperative game
theory.
Other elements in the sensitivity function matrix can be
calculated similarly. Based on the sensitivity function
matrix, we have the payoff sensitivity function as follows.
1
S wPa =
wa ∂P1 wa ∂P1 ∂x a wa ∂P1 xa x a
=
=
S a
P1 ∂wa P1 ∂x a ∂wa P1 ∂x a w wa
(17)
Fig. 3 is the simulation result of applying coupling game to
UAV battlefield with imperfect coordination. Fuzzy
approach is applied to estimate the coupling factors. It is
seen that the coupling game algorithm (the red up most
points) has advantages over traditional game theories (the
green bottom points are for traditional cooperative game
approach and the blue medium points are for traditional
non-cooperative game approach) in such situations.
Other combinations can be calculated similarly. For social
payoff P1 + P 2 , we have
S wPa + P =
1
2
2
2 P
⎞ (18)
wa ⎛ ∂P1 ∂P 2 ⎞
wa ⎛ P1 P1
S
+ a ⎟= 1
+ S wPa a ⎟
2 ⎜
a
2 ⎜ wa
a
P + P ⎝ ∂w ∂w ⎠ P + P ⎝
w
w ⎠
1
Other combinations can be calculated similarly.
comparison of the team payoff gotten from different game theories
9.4
3. SIMULATIONS
9.2
9
the average team payoff
We experimented with the two estimation approaches for
two types of scenarios, respectively.
Table 3 A 3 × 3 game
1
ij
UAV 2
2
ij
(P , P )
UAV 1
B
C
8.6
NonCooperative
Cooperative
Coupling Game
8.4
8.2
8
7.8
7.6
A
A
8.8
B
C
5, 5
12, 3
15, 0
(cell 1)
(cell 2)
(cell 3)
3, 12
9, 9
14, 4
(cell 4)
(cell 5)
(cell 6)
0, 15
4, 14
10, 10
(cell 7)
(cell 8)
(cell 9)
7.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Index of tries (each point is 1000 times run)
1.6
1.8
Fig. 3. Comparison of coupling game, non-cooperative
game, and cooperation game in UAV situations
In a second application scenario, civilian emotion approach
is applied to estimate the coupling game factors. There are
Blue, Red, and White civilians in a battlefield. White
civilian properties might be collaterally destroyed by Blue
or Red, thus causing the change of civilian emotion. If the
civilian anger does not exceed some limit (predefined
according to long-term statistical data), when civilian anger
towards Red is higher than that toward Blue, civilians might
purposefully perform some biased actions (such as the
actions mentioned in section 3) to improve kill probabilities
of Blue weapon, and vice versa. Note that in this case the
civilians still keep the “civilian” identity. However, if the
civilian anger overwhelms that limit, some of them might
directly join one army to revenge their loss. In this case
these civilians lose “civilian” identity thus are not civilians
any more.
The first application scenario describes an imperfect
cooperation among team UAVs. In a battlefield there will
be high-value but harder targets which need more than one
UAV to coordinate to destroy, and there will be low-value
but easier targets which need only one UAV to destroy.
When perfect coordination does not exist, the UAVs will
face situations similar to prisoner’s dilemmas stated in
Table 3 (maximizing the payoffs). The non-cooperative
equilibrium is Cell 1, which will be reached by applying
traditional non-cooperative game theory. The TU
cooperative equilibrium is Cell 9, which is best and will be
reached by applying traditional TU cooperative game
theory. However, if a player wishes to use TU cooperative
game theory to reach cell 9 it will experience the risk of
Fig. 4 and Fig. 5 are two typical scenarios for this situation.
Since the plots are relatively small, we clarify the labels on
them: 1) the labels of horizontal axis are “time”; 2) the
labels of vertical axis are “team value”; 3) The titles are
“Emotional-Rational scope”. In Fig. 4, since the civilians
distrust Red much more than distrust Blue, some of them
become Blue at time step 3. Similarly, in Fig. 5 the great
7
anger toward Blue leads some civilians to join Red. Clearly,
civilians benefit the side they join and help them to win the
battle.
[3] Hughes, W. P., Jr., (Ed.), Military modeling, The
Military Operations Research Society, 1984.
[4] Kott, A., Ground, L., and Langston, J., “Estimation of
battlefield attrition in a course of action analysis
decision support system”, Presented at The Military
Operations Research Society Workshop on Land and
Expeditionary Warfare, June 1999.
Emotional-Rational scope
400
BlueTeamValue
RedTeamValue
WhiteTeamValue
350
300
team value
250
200
[5] Przemieniecki, J. S., Mathematical methods in defense
analysis, 3rd ed., (Education Series), New York: AIAA,
2000.
150
100
50
0
1
2
3
4
time
5
6
[6] Wei, Mo, Reconstruction Theories of Non-Ideal
Games, Ph.D. dissertation, The Ohio State University,
2006.
7
Fig. 4. Some civilians become Blue
[7] Nash, J. F., “The Bargaining Problem”, Econometrica
18, pp.155-162, 1950.
Emotional-Rational scope
400
BlueTeamValue
RedTeamValue
WhiteTeamValue
350
[8] Lanchester, F. A., Aircraft in warfare: The dawn of the
fourth arm, London: Constable, pp. 39-46, 1916.
300
[9] Taylor, J. G., Lanchester models of warfare, Operation
Research Society of America, Arlington, VA, 1983.
team value
250
200
150
[10] Helmbold, R. L., “A modification of Lanchester’s
equations”, Operations Research, 13, pp. 857-859,
1965.
[11] Athans, M., and Falb, O. L., Optimal Control, New
York: Mcgraw Hill, 1966.
100
50
0
1
1.5
2
2.5
3
3.5
time
4
4.5
5
5.5
6
[12] Wei, M. and Cruz, J. B., Jr., “Role of Cooperation in
Coupling Game Theory”, International Journal of
Control, Vol. 80, No. 4, pp. 611-623, April 2007.
Fig. 5. Some civilians become Red
4. CONCLUSIONS
[13] Simaan, M., and Cruz, J. B., Jr., “On the Stackelberg
strategy in nonzero sum games”, Journal of
Optimization Theory and Applications, 11, 5, pp. 533555, 1973.
In this paper, we investigated the feasibility of coupling
game application in intelligent threat analysis and
prediction. Different approaches to estimation of coupling
factors are discussed. They are suitable for different areas.
Several useful sensitivity functions about the estimation of
coupling factors are provided. Experiments with the Fuzzy
and Civilian Emotion approaches show that coupling game
can analyze more complex military scenarios and provide
more satisfying results than traditional game theories.
[14] Cruz, J. B., Jr., and Simaan, M. A., “Multi-agent
control strategies with incentives”, Proceedings of the
First DARPA/JFACC Symposium on Advances in
Enterprise Control, San Diego, CA, Nov. 15-18, pp.
177-182, 1999.
[15] Nash, J. F., Jr., “Equilibrium points in N-person
games”, Proceedings of the U.S. National Academy of
Sciences, 36, pp. 48-59, 1950.
REFERENCES
[16] Cruz, J. B., Jr., Simaan, M. A., Gacic, A., Jiang, H.,
Letellier, B., Li, M., and Liu, Y., “Game-Theoretic
Modeling and Control of a Military Air Operation”,
IEEE Transactions on Aerospace and Electronic
Systems, Vol. 37, No. 4, October, 2001.
[1] Brackney, H., “The dynamics of military combat”,
Operations Research, Vol. 7, pp. 30-49, 1959.
[2] Taylor, J. G., “Force-on-force attrition modeling”, The
Military Operations Research Society, 1980.
8
[17] Brooks, H., DeKeyser, T., Jaskot, D., Sibert, D.,
Sledd, R., Stilwell, W., and Scherer, W., “Using
agent-based simulation to reduce collateral damage
during military operations”, Proceedings of the 2004
IEEE Systems and Information Engineering Design
Symposium, pp. 71-77, 2004.
interests include game control and corresponding
applications, such as non-ideal games, coupling game
theory, sharing creditability game theory and bottom-line
game theory, multiplayer games with civilian players,
decentralized multiplayer pursuer-evader games, game
application in networking, etc.
[18] Foster, K.R., and Lerch, I.A., “Collateral damage:
American science and the war on terrorism”,
Technology and Society Magazine, IEEE, Vol. 24,
Issue 3, pp. 43-53, 2005.
Jose B. Cruz, Jr. received his B.S. degree in electrical
engineering (summa cum laude)
from the University of the
Philippines (UP) in 1953, the S.M.
degree in electrical engineering from
the Massachusetts Institute of
Technology (MIT), Cambridge in
1956, and the Ph.D. degree in
electrical engineering from the
University of Illinois, UrbanaChampaign, in 1959. He is currently
a Distinguished Professor of Engineering and Professor of
Electrical and Computer Engineering at The Ohio State
University (OSU), Columbus. Previously, he served as
Dean of the College of Engineering at OSU from 1992 to
1997, Professor of electrical and computer engineering at
the University of California, Irvine (UCI), from 1986 to
1992, and at the University of Illinois from 1965 to 1986.
He was a Visiting Professor at MIT and Harvard University,
Cambridge, in 1973 and Visiting Associate Professor at the
University of California, Berkeley, from 1964 to 1965. He
served as Instructor at UP in 1953–1954, and Research
Assistant at MIT from 1954 to 1956. He is the author or
coauthor of six books, 21 chapters in research books, and
numerous articles in research journals and refereed
conference proceedings.
[19] Toedtmann, B., Riebach, S., and Rathgeb, E.P., ”The
Honeynet quarantine: Reducing collateral damage
caused by early intrusion response”, Systems, Man and
Cybernetics (SMC) Information Assurance Workshop,
Proceedings from the Sixth Annual IEEE, pp. 464-465,
2005.
[20] Cruz, J. B., Jr., Chen, G., Garagic D., Tan, X., Li, D.,
Shen, D., Wei, M., Wang, X., “Team dynamics and
tactics for mission planning,” Proceedings, IEEE
Conference on Decision and Control, pp. 3579-3584,
December 2003.
[21] Wei, M., Chen, G., Cruz, J. B., Jr., Kwan, C., and
Kruger, M., “Game-Theoretic Control of Military Air
Operations with Civilian Players”, AIAA, Guidance,
Navigation, and Control Conference and Exhibit, 2006.
[22] Minsky, M., The Society of Mind, New York, Simon
and Schuster, pp. 163, 1986.
[23] Gratch, J. and Marsella, S., “A Domain-Independent
Framework for Modeling Emotion”, Journal of
Cognitive Research, Vol. 5, Issue 4, pp. 269-306, 2004.
Dr. Cruz was elected as a member of the National Academy
of Engineering (NAE) in 1980. In 2003, he was elected a
Corresponding Member of the National Academy of
Science and Technology (Philippines). He is a Fellow of
IEEE elected in 1968,
a Fellow of the American
Association for the Advancement of Science (AAAS),
elected 1989, a Fellow of the American Society for
Engineering Education (ASEE), elected in 2004, and a
Fellow of IFAC elected in 2007. He received the Curtis W.
McGraw Research Award of ASEE in 1972 and the
Halliburton Engineering Education Leadership Award in
1981. He is a Distinguished Member of the IEEE Control
Systems Society and received the IEEE Centennial Medal in
1984, the IEEE Richard M. Emberson Award in 1989, the
ASEE Centennial Medal in 1993, and the Richard E.
Bellman Control Heritage Award, American Automatic
Control Council (AACC), 1994. In addition to membership
in NAE, ASEE, AAAS, and IEEE, he is a Member of the
Philippine American Academy for Science
and
Engineering (Founding member, 1980, President 1982, and
Chairman of the Board, 1998–2000), Philippine Engineers
and Scientists Organization (PESO), National Society of
Professional Engineers, Sigma Xi, Phi Kappa Phi, and Eta
Kappa Nu. He served as a Member of the Board of
[24] Senglaub, M. and Harris, D., “A Modified Perspective
of Decision Support in C2”, SAND2005-1701C,
February 2002.
[25] Zadeh, L.A., “Fuzzy Sets”, Information and Control,
Vol. 8, pp. 338-353, 1965.
[26] Passino, K. M. and Yurkovich, S., "Fuzzy Control",
Addison Wesley Longman, Menlo Park, CA, 1998.
BIOGRAPHY
Mo Wei has a BS degree in EE from
Northern JiaoTong University and a
master’s degree in EE from Tsinghua
University. He received his Ph. D.
degree in
Electrical and Computer
Engineering from the
Ohio State
University in 2006. His research
9
Examiners for Professional Engineers for the State of
Illinois, from 1984 to 1986. He served on various
professional society boards and editorial boards, and he
served as an officer of professional societies, including
IEEE, where he was President of the Control Systems
Society in 1979, Editor of the IEEE TRANSACTIONS ON
AUTOMATIC CONTROL, a Member of the Board of
Directors from 1980 to 1985, Vice President for Technical
Activities in 1982 and 1983, and Vice President for
Publication Activities in 1984 and 1985. He served as Chair
(2004–2005) of the Engineering Section of the American
Association for the Advancement of Science (AAAS).
Lead for the Air Force Research Laboratory, Adjunct
Professor at WSU, and a reserve Maj with the Air Force
Office of Scientific Research.
Dr. Blasch was a founding member
of the International Society of
Information Fusion (ISIF) and the
2007 ISIF President. Dr, Blasch has
many military and civilian career
awards; but engineering highlights
include team member of the winning ‘91 American Tour del
Sol solar car competition, ’94 AIAA mobile robotics
contest, and the ’92 AUVs competition where they were
first in the world to automatically control a helicopter. Since
that time, Dr. Blasch has foused on Automatic Target
Recognition, Targeting Tracking, and Information Fusion
research compiling 160+ scientific papers and book
chapters. He is active in IEEE and SPIE including regional
activities, conference boards, journal reviews and
scholarship committees.
Genshe Chen received his B. S. and M. S. in electrical
engineering, Ph. D in aerospace engineering, in 1989, 1991
and 1994 respectively, all from Northwestern Polytechnical
University, Xian, P. R. China. He
did postdoctoral work at the Beijing
University of Aeronautics and
Astronautics and Wright State
University from 1994 to 1997. He
worked at the Institute of Flight
Guidance and Control of the
Technical
University
of
Braunshweig (Germany) as an
Alexander von Humboldt research fellow and at the Flight
Division of National Aerospace Laboratory of Japan as a
STA fellow from 1997 to 2001. He was a Postdoctoral
Research Associate in the Department of Electrical and
Computer Engineering of The Ohio State University from
2002 to 2004. Since February 2004, Dr. Chen has been with
the Intelligent Automation, Inc., Rockville, MD. He has
served as the Principal Investigator/Technical lead for more
than 15 different projects, including maneuvering target
detection and tracking, joint ATR and tracking, cooperative
control for teamed unmanned aerial vehicles, a stochastic
differential pursuit-evasion game with multiple players,
multi-missile interception, asymmetric threat detection and
prediction, space situation awareness, and cyber defense,
etc. He is currently the program manager in Networks,
Systems and Control, leading research and development
efforts in target tracking, information fusion and
cooperative control. His research interests include guidance
and control of aerospace vehicle, GPS/INS/image integrated
navigation systems, target tracking and information fusion,
cooperative control and optimization for military operations,
computational intelligence and data mining, hybrid system
theory and Markov chain, signal processing and computer
vision, cooperative and non-cooperative game theory,
Bayesian networks, , Influence Diagram, and GIS.
Martin Kruger is currently serving as the Intelligence,
Surveillance and Reconnaissance Thrust Area Manager for
the Expeditionary Warfare Maneuver Warfare & Combating
Terrorism Science and Technology Department at the
Office of Naval Research. In that capacity, he is responsible
for maturing and transitioning applicable technology.
Research interests include sensing, data fusion &
visualization, resource management and information
dissemination. The overall objective of the program is to
increase the efficiency and effectiveness of the translation
of intelligence requirements to actionable intelligence
relevant to the Global War on Terror.
Before coming to ONR, Mr. Kruger served as a
research and development manager for the Future Theater
Air and Missile Defense program office at the Naval Sea
Systems Command. He has also worked for the Marine
Corps Warfighting Laboratory and for the Naval Surface
Warfare Center Indian Head Division. Mr Kruger started his
career as a Naval Officer, serving as an instructor at the
Naval Nuclear Propulsion School.
After leaving active duty, CAPT Martin Kruger
has continued serving the Navy as a drilling reservist.
Reserve assignments have included four command tours,
one each at a shipyard, a SUPSHIP, a NAVSEA field
activity and a Weapon Station. He is currently serving as a
Chief Ordnance Inspector.
Martin Kruger holds a bachelor in engineering in
Chemical Engineering, a Masters of Science in Industrial
Chemistry and a Masters in Business Administration. He is
also a graduate of the Naval War College and is Level 3
Certified in Program Management.
Erik Blasch received his B.S. in mechanical engineering
from MIT and Masters in mechanical and industrial
engineering from Georgia Tech and MBA, MSEE, from
Wright State University and a PhD from WSU in EE. Dr.
Blasch also attended Univ of Wisconsin for an MD/PHD in
Mech. Eng until being called to Active Duty in the United
States Air Force. Currently, he is a Fusion Evaluation Tech
10