Page 1

Risk Analysis, Vol. 32, No. 2, 2012

DOI: 10.1111/j.1539-6924.2011.01657.x

Resource Distribution in Multiple Attacks with Imperfect

Detection of the Attack Outcome

Gregory Levitin1,2,∗and Kjell Hausken3

This article extends the previous research of consecutive attacks strategy by assuming that

an attacker observes the outcome of each attack imperfectly. With given probabilities it may

wrongly identify a destroyed target as undestroyed, and wrongly identify an undestroyed tar-

get as destroyed. The outcome of each attack is determined by a contest success function that

depends on the amount of resources allocated by the defender and the attacker to each at-

tack. The article suggests a probabilistic model of the multiple attacks and analyzes how the

target destruction probability and the attacker’s relative resource expenditure are impacted

by the two probabilities of incorrect observation, the attacker’s and defender’s resource ratio,

the contest intensity, the number of attacks, and the resource distribution across attacks. We

analyze how the attacker chooses the number of attacks, the attack stopping rule, and the

optimal resource distribution across attacks to maximize its utility.

KEYWORDS: Attack; defense; detection; multiple attacks; optimization; protection; target;

vulnerability

1. INTRODUCTION

The September 11, 2001 attack illustrated that

major threats today involve strategic attackers. In

many decision situations a defender seeks to en-

sure the functionality of a system while an attacker

seeks to ensure the system’s dysfunctionality. The

conflict literature(1,2,3)has mostly considered static

contests where both contestants allocate their entire

resources in one-shot encounters. Sheeba(4)consid-

ered the optimal resource distribution among pro-

tections against different types of attacks based on a

Lanchester model(5)that describes the dynamics of

resource attrition. Accounting for such continuous

1Collaborative Autonomic Computing Laboratory, School of

Computer Science, University of Electronic Science and Tech-

nology of China, Chengdu, China.

2The Israel Electric Corporation Ltd., Haifa 31000, Israel.

3Faculty of Social Sciences, University of Stavanger, N-4036

Stavanger, Norway.

∗Address correspondence to Gregory Levitin, The Israel Elec-

tric Corporation Ltd., Po Box 10, Haifa 31000, Israel;

levitin@iec.co.il.

attrition, Taylor(6)considers this problem as a time

sequential resource allocation problem and presents

a solution as an optimal control problem. Research

on stochastic duels started in 1955,(7)and stochastic

analysis, reviewed by Taylor,(6)became more promi-

nent in the 1970s following breakthroughs in com-

puter technology (see Ref. 8 for a stochastic analysis

accounting for conditional probabilities).

Following(9)

Harris

Leininger,(12)

Budd et al.,(13)

Kovenock(14)investigated multibattle games assum-

ing that battles proceed sequentially. Yildirim(15)

considers consecutive contests where players ob-

serve their opponent’s previous resource allocation.

He finds that the Stackelberg outcome where the

underdog leads and the favorite follows cannot be

equilibrium.

Washburn and Kress(16)analyze shooting situa-

tions where there is information feedback between

shots and salvos of shots, where shooting is one

sided, with no return fire. This implies decreasing

the number of shots required, and increasing the kill

and Vickers,(10,11)

and Konrad and

304

0272-4332/12/0100-0304$22.00/1C ?2011 Society for Risk Analysis

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Resource Distribution in Multiple Attacks 305

probability. Glazebrook et al.(17)analyze situations

where a single Red wishes to shoot at a collection

of Blue targets, one at a time, to maximize some

measure of return obtained from Blues killed

before Red’s own (possible) demise. Deck and

Sheremeta(18)consider sequential attack and de-

fense. The attacker seeks to win at least one battle

while the defender seeks to win every battle. The

defender either folds immediately or, if his valuation

is sufficiently high and the number of battles is suffi-

ciently small, then he fights in each battle. Attackers

respond to defense with diminishing assaults over

time. The results are tested experimentally.

Hausken et al.(19)consider the optimal resource

distribution among different types of protections

for defending a single object against intentional

and unintentional attacks, but again in a one-shot

encounter. Soland(20)considers an ABM defense

problem with multiple targets but only one stage.

Levitin and Hausken(21)and Hausken and Levitin(22)

consider the system defense of parallel systems sub-

ject to two consecutive attacks. The existing works

on multiple attacks presume a fixed probability of

target destruction in each attack and do not consider

the influence of the defender’s and the attacker’s

efforts on this probability (see the review of the

shoot-look-shoot problem literature in Ref. 23).

Thus, the existing models do not answer such

questions as whether a few strong strikes are more

effective than a larger number of weaker strikes or

how to distribute the limited attack resource among

the consecutive strikes.

Levitin and Hausken(24)consider a situation

when an attacker attacks a system repeatedly.

Equipped with a fixed resource, the attacker chooses

the optimal number of attacks, and how to distribute

the resource across the attacks. The attacker can

choose one large initial attack, or decreasing, equal,

or increasing geometric resource distribution across

the attacks. The defender’s resource lasts through all

the attacks.

We consider a situation when the attacked

system constitutes a single concentrated target and

can be destroyed by any attack with some proba-

bility. The target can either survive or be destroyed

in a series of attacks (partial destruction and the

corresponding partial damage is not considered).

Thus, the attacker and the defender evaluate the

risk associated with the series of attacks as the

probability of target destruction. We assume that

the attacked system is of value to the defender and,

thus, we do not consider the cases of wrong target

identification that leads to wasting the attacker’s

resources in attacking the wrong object.

When the attacker attacks a system repeatedly, it

is often the case that it imperfectly observes the out-

come of each attack. Two prominent cases are that

the attacker wrongly identifies a destroyed target as

undestroyed, with a certain probability, and wrongly

identifies an undestroyed target as destroyed, with

a certain probability. A system can be anything of

value, such as a bridge, a power generator, a vot-

ing system, a detector, etc. This article considers a

model that reflects this situation. The outcome of

each attack is determined by a contest success func-

tion that depends on the amount of resources allo-

cated by the defender and attacker to each attack,

and on the intensity of the contest. Imperfect ob-

servation of each attack outcome means that the at-

tacker may incorrectly continue the attack sequence

despite having destroyed the target, or may incor-

rectly discontinue the attack sequence without hav-

ing destroyed the target, and thus in some cases not

use its entire resource. Following the models used in

the decision-making literature,(25,26)we assume that

the probabilities of the two possible types of the

wrong observation are independent and exogenously

given. The article analyzes how the target destruction

probability and the attacker’s relative resource ex-

penditure are impacted by these two probabilities of

incorrect observation, the attacker’s and defender’s

resource ratio, the contest intensity, the number

of attacks, and the resource distribution across at-

tacks. We analyze how the attacker chooses the

optimal number of attacks and the optimal distribu-

tion across attacks to maximize the target destruction

probability.

We assume that the defender statically seeks

to preserve the status quo by keeping the target

intact, while the attacker seeks to alter the status

quo by destroying the target. To achieve its goal, the

attacker needs to win at least one attack whereas the

defender must withstand all the attacks. We model

the common case that the protection is static and

cannot be changed over time. An optimal defense

strategy is often to allocate the entire protection

resource before the first attack. If the target is de-

stroyed, the protection is destroyed too. If the target

is not destroyed in a given attack, the protection

remains in place also for the subsequent attack.

This assumption is realistic for protections such as

bunkers, shielded premises, etc. In contrast, missiles

fired to prevent an attack are expendable and cannot

be used to prevent a subsequent attack. We thus do

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306 Levitin and Hausken

not consider expendable defense. For the attacker an

attack effort is commonly expendable. If the attack

is a missile or human effort, exerting such effort in a

given attack usually means that the same effort is not

available for the subsequent attack. A new resource

is needed for the subsequent attack. Because of

this asymmetry between defense and attack, we

model how the attacker distributes its resource over

time, while the defender’s resource remains in place

through all attacks until the target is destroyed.

As in Ref. 24 we use the geometric progression

to model the attacker’s effort variation since it is

simple and flexible. Only one parameter is needed,

which allows representing very different resource

distributions over K attacks. Finding the optimal

value of this parameter allows one to perform

qualitative analysis and compare increasing versus

decreasing attack effort options as well as single ver-

sus multiple attack strategies. One alternative such

as letting the attacker freely choose its resource for

each of the K attacks gives K−1 resource allocation

decisions for the attacker when its total resource is

fixed, and a much more complicated analysis.

It is assumed that the attacker is able to change

the attack effort by choosing different attack dura-

tion, amount of ammunition, or manpower involved.

Section 2 presents the model, Section 3 analyzes

the model, and Section 4 analyzes the optimal attack

strategy, and Section 5 concludes.

2. THE MODEL

For any single attack the vulnerability of a target

(probability that the target is destroyed in the attack)

is determined by a contest between the defender

exerting effort t and the attacker exerting effort T

in this attack. The contest is expressed as a contest

success function modeled with the common ratio

form(2)as:

Tm

Tm+ tm=

where ∂v/∂T > 0, ∂v/∂t < 0, and m ≥ 0 is a pa-

rameter that expresses the intensity of the contest.

Equation (1) is extensively used in the rent seek-

ing literature. See Refs. 2 for the use of m, 3 for

an axiomatization where m plays a role, 27 for

a review, 28 for an illustration of the usefulness

of the function for a variety of application areas,

and 1 for recent literature. If the attacker exerts

high effort, it is likely to win the contest, which

gives high vulnerability. If the defender exerts high

v(T,t) =

1

1 + (t/T)m,

(1)

effort, it is likely to win the contest, which gives low

vulnerability. When m = 0 and tT > 0, the efforts t

and T have no impact on the vulnerability regardless

of their size, which gives vulnerability 0.5. A value

of 0 < m < 1 gives a disproportional advantage of

investing less than one’s opponent. When m = 1,

the investments have proportional impact on the

vulnerability. A value of m >1 gives a dispropor-

tional advantage of investing more effort than one’s

opponent (economies of scale). Finally, m = ∞

gives a step function where “winner takes all.”

The parameter m is a characteristic of the

contest, which can be illustrated by the history of

warfare. Low intensity occurs for targets that are

predictable, and where the individual ingredients of

each target are dispersed, i.e., physically distant or

separated by barriers of various kinds. Neither the

defender nor the attacker can get a significant upper

hand. An example is the time prior to the emergence

of cannons and modern fortifications in the 15th cen-

tury. Another example is entrenchment combined

with the machine gun, in multiple dispersed loca-

tions, in World War I.(28)High m occurs for targets

that are less predictable, and where the individual

ingredients of each target are concentrated, i.e.,

close to each other or not separated by particular

barriers. This may cause “winner-take-all” battles

and dictatorship by the strongest. Either the de-

fender or the attacker may get the upper hand. The

combination of airplanes, tanks, and mechanized

infantry in World War II allowed both the offense

and defense to concentrate firepower more rapidly,

which intensified the effect of force superiority.

We assume that the defender uses the same pro-

tection (for example, bunkers or missile interception

systems) during the series of K attacks and allocates

its entire resource into this protection: t = r. On the

contrary, the attacker distributes its entire resource

R among K attacks such that the resource allocated

to attack i is Tiand?K

outcome of each attack and can stop the sequence

of attacks if the target is destroyed, preserving the

remaining attack resources. The outcome of each

attack can be estimated wrongly. The attacker can

wrongly identify the destroyed target as undestroyed

with probability a and the undestroyed target as

destroyed with probability b. Assuming that the

attacker knows the probabilities of the wrong iden-

tification, we can confine the analysis to a ≤ 0.5 and

b ≤ 0.5. Indeed, when 0.5 < a, b ≤ 1, the attacker

knows that it has an incorrect detection probability

i=1Ti= R.

We assume that the attacker observes the

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Resource Distribution in Multiple Attacks 307

greater than 0.5. It then replaces a with 1−a and

replaces b with 1−b and makes a decision opposite to

the detector’s advice. Hence a = b = 0.5 constitutes

maximum detection uncertainty for the attacker.

a = 0 gives certainty that a destroyed target is

destroyed. b = 0 gives certainty that an undestroyed

target is undestroyed.

The success probability of the ith attack accord-

ing to Equation (1) is vi= v(Ti,t).

As the attacker knows that it can wrongly iden-

tify the undestroyed target as destroyed, it may de-

cide to continue the sequence of attacks after it gets

the information about target destruction the first

time. We consider the rule when the attacker decides

that the target is destroyed when after n attacks it

identifies it as destroyed. The attacker can optimize n

to achieve the desired balance between the probabil-

ity of the target destruction and the attack resource

expenditure. Thus the attacker terminates the attack

sequenceeitherifitdepletesitsresourcesorifitiden-

tifies the target as destroyed n times.

The attacker decides that the target is destroyed

and stops the sequence of attacks after the jth attack

(n ≤ j ≤ K) in two cases:

1. The target is not destroyed in j attacks, but

is wrongly identified as destroyed after the

jth attack and was wrongly identified as de-

stroyed after n−1 out of j−1 previous attacks.

The probability of this event is:

p1(j) = bψ(j −1,n−1)bn−1(1−b)j−n

j?

j?

i=1

(1−vi)

= ψ (j − 1,n − 1)bn(1 − b)j−n

i=1

(1 − vi),

(2)

where

ψ(x, y) =

⎧

⎪⎪⎩

⎪⎪⎨

x!/(y!(x − y)!)

1

ifx ≥ y > 0

ify = 0

ifx < y.

0

For n = 1 we get:

p1(j) = b(1 − b)j−1

j?

i=1

(1 − vi).

(3)

2. The target is destroyed in the hth attack (h ≤

j), and this fact is correctly identified after the

jth attack, whereas during the j−1 previous

attacks the target destruction was identified

(correctly or wrongly) n−1 times.

The conditional probability that the target was

wrongly identified as destroyed exactly s times after

the h−1 first attacks given the target is not destroyed

in these attacks is:

?h − 1

s

The conditional probability that after the hth at-

tack, but before the jth attack, the target was cor-

rectlyidentifiedasdestroyedexactlyetimesgiventhe

target is destroyed in the hth attack is:

?j − h

e

The attacker stops the sequence of attacks af-

ter the jth attack if s + e = n−1 and after the jth

attack the target destruction is correctly identified.

Thus, the conditional probability of the attack se-

quence termination after the jth attack given the tar-

get is destroyed by the hth attack is:

x(h,s) =

?

bs(1 − b)h−1−s.

(4)

y(h,e) =

?

(1 − a)eaj−h−e.

(5)

(1 − a)

n−1

?

s=0

x(h,s)y(h,n − 1 − s).

(6)

Thus, the unconditional probability that the tar-

get is destroyed after the hth attack and the sequence

of attacks terminates after the jth attack is:

?h−1

i=1

n−1

?

?h−1

i=1

×(1 − b)h−1−sψ(j − h,n−1 − s)

×(1 − a)n−1−saj−h−n+1+s.

For n = 1 we get:

?h−1

i=1

(For example, if n = 2 and h = 3, the attacker

can terminate the sequence of attacks after the fifth

attack if it correctly identifies the undestroyed target

after attacks 1 and 2 with probability (1−b)2, wrongly

p2(j,h) = (1 − a)vh

?

(1 − vi)

?

×

s=0

x(h,s)y(h,n − 1 − s)

= (1 − a)vh

?

(1 − vi)

?n−1

s=0

?

ψ(h − 1,s)bs

(7)

p2(j,h) = (1 − a)vh

?

(1 − vi)

?

(1 − b)h−1aj−h.

(8)

Page 5

308 Levitin and Hausken

identifies the destroyed target as undestroyed after

attack 3 with probability a, and correctly identifies

the destroyed target after attacks 4 and 5 with prob-

ability (1−a)2. Alternatively the attacker can ter-

minate the sequence of attacks after fifth attack if

it wrongly identifies the undestroyed target as de-

stroyed after attack 1 with probability b, correctly

identifies the undestroyed target after attack 2 with

probability (1−b), destroys the target in the third at-

tack, wrongly identifies the destroyed target as unde-

stroyed after attacks 3 and 4 with probability a2, and

correctly identifies the destroyed target after attacks

5 with probability (1−a). The target vulnerabilities in

attacks 4 and 5 play no role since the target is de-

stroyed in attack 3.)

The overall probability that the attacker stops

the sequence of attacks after the jth attack is:

p(j) = p1(j) +

j

?

h=1

p2(j,h).

(9)

When the attacker stops the sequence of attacks

after the jth attack the attacker spends the resource

?j

attacks and considers the target to be undestroyed

after the series of K attacks in the two cases:

i=1Ti.

The attacker spends its entire resource R in K

1. The attacker fails to destroy the target and

wrongly identifies the target as destroyed less

than n times. The probability of this event is:

g1=

K

?

i=1

(1 − vi)

n−1

?

j=0

?K

j

?

bj(1 − b)K−j.

(10)

For n = 1:

g1= (1 − b)K

K

?

i=1

(1 − vi).

(11)

2. The attacker destroys the target in the hth at-

tack, but identifies the target as destroyed less

than n times.

The conditional probability that the target was

wrongly identified as destroyed exactly s times after

the h−1 first attacks given the target is not destroyed

in these attacks is x(h,s) defined in Equation (4).

The conditional probability that after attacks

h,...,K the target was correctly identified as de-

stroyed exactly e times given the target is destroyed

in the hth attack is:

?K − h + 1

z(h,e) =

e

?

(1 − a)eaK−h−e+1.

(12)

The attacker wrongly considers the target to be

undestroyed after the series of K attacks if s + e < n.

The probability of this event for 1 ≤ h ≤ K is:

?h−1

i=1

?h − 1

i=1

n−1−s

?

g2(h)= vh

?

(1−vi)

?n−1

s=0

?n−1

?

x(h,s)

n−1−s

?

e=0

z(h,e)

=vh

?

(1−vi)

?

s=0

ψ(h − 1,s)bs(1−b)h−1−s

×

e=0

ψ(K − h + 1,e)(1 − a)eaK−h−e+1.

(13)

For n = 1:

g2(h) = vh

?h−1

i=1

?

(1 − vi)

?

(1 − b)h−1aK−h+1. (14)

The overall probability that the attacker never

identifies the target as destroyed and spends its entire

resource R in K attacks is:

g = g1+

K

?

h=1

g2(h).

(15)

The expected attacker’s resource expenditure

can be obtained as:

e =

K

?

j=n

?

p1(j) +

j

?

h=1

p2(j,h)

?

j

?

i=1

Ti

+

?

g1+

K

?

h=1

g2(h)

?

R.

(16)

We will present the expected resource expendi-

ture as a fraction of the total attacker’s resource:

E = e/R

=

K

?

j=n

?

p1(j) +

j

?

h=1

p2(j,h)

?

×

j

?

K

?

i=1

?

τi+ g1+

K

?

h=1

g2(h)

=

j=n

ψ (j − 1,n − 1)bn(1 − b)j−n

j?

i=1

(1 − vi)

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Resource Distribution in Multiple Attacks 309

+

j

?

n−1

?

h=1

(1 − a)vh

?h−1

i=1

?

(1 − vi)

?

×

s=0

ψ (h − 1,s)bs(1 − b)h−1−s

×ψ (j − h,n−1 − s)(1 − a)n−1−saj−h−n+1+s

?

×

j

?

K

?

i=1

τi+

K

?

?

i=1

(1 − vi)

n−1

?

?

j=0

?K

j

?

bj(1 − b)K−j

+

h=1

vh

?h−1

i=1

(1 − vi)

×

n−1

?

s=0

ψ (h − 1,s)bs(1 − b)h−1−s

×

n−1−s

?

e=0

ψ (K − h + 1,e)(1 − a)eaK−h−e+1,(17)

where τi= Ti/R.

For n = 1, Equation (17) takes the form:

?

i=1

j?

+(1 − b)K

i=1

K ?

The probability that the target is not destroyed

by the attacker is equal to the sum of probabilities

that the attacker attacks K times (which means that

inthefirstK−1 attacks itwrongly identifies thetarget

asdestroyedlessthanntimes)andfailstodestroythe

target:

E =

K ?

j=1

b(1 − b)j−1

j?

(1 − vi)

(1 − vi)

+

h=1

(1 − a)vh

?h−1

?

i=1

?

(1 − b)h−1aj−h

?

j?

i=1

τi

K ?

?

(1 − vi)

+

h=1

vh

?h−1

i=1

(1 − vi)

?

(1 − b)h−1aK−h+1.

(18)

K

?

i=1

(1 − vi)

n−1

?

j=0

?K − 1

j

?

bj(1 − b)K−1−j, (19)

and that the attacker stops the sequence of attacks

after the jth (n ≤ j <K) attack (which means that it

wrongly identifies the target as destroyed n−1 times

after j−1 first attacks and wrongly identifies the tar-

get as destroyed after the jth attack):

K−1

?

The overall target destruction probability, thus,

takes the form:

?K − 1

K−1

?

For n = 1, Equation (21) takes the form:

j=n

ψ (j − 1,n − 1)bn(1 − b)j−n

j?

i=1

(1 − vi).(20)

V = 1 −

K ?

i=1

(1 − vi)

n−1

?

j=0

j

?

bj(1 − b)K−1−j

−

j=n

ψ (j − 1,n − 1)bn(1 − b)j−n

j?

i=1

(1 − vi).

(21)

V = 1 − (1 − b)K

K ?

Following Ref. 24, we assume that the attacker

allocates effort T1to the first attack and changes the

effort according to the geometric progression:

K ?

i=1

(1 − vi)

j?

−b

j=1

(1 − b)j−1

i=1

(1 − vi).

(22)

Ti= qTi−1for 1 < i ≤ K,

(23)

such that:

K

?

i=1

Ti= T1qK− 1

q − 1

= R.

(24)

The parameter q determines the strategy of ef-

fort variation through the K sequential attacks: q >1

corresponds to increasing the attack effort; q<1 cor-

responds to the decreasing the attack effort; q = 1

corresponds to even resource distribution across the

K attacks; q = 0 corresponds to a single attack with

T1= R. The attacker selects q by choosing differ-

ent amount of ammunition, attack duration, or man-

power involved.

For the given resource R and effort variation pa-

rameter q we obtain:

⎧

⎪⎪⎩

and

T1=

⎪⎪⎨

Rq − 1

qK− 1,q ?= 1

R

K,

q = 1

(25)

Ti= qi−1T1.

(26)

Inserting t = r and Equations (25) and (26) into

Equation (1) for t = tj and T = Tj we obtain the

Page 7

310 Levitin and Hausken

b=0.1

0.5

0.52

0.54

0.56

0.58

0.6

V

0 0.1 0.20.3

n=2

n=5

0.40.5

a

n=1

n=4

n=3

b=0.1

0.6

0.7

0.8

0.9

1

0 0.10.2 0.3

n=2

n=5

0.40.5

a

E

n=1

n=4

n=3

Fig. 1. E and V as functions of a for b = 0.1, q = r/R = m = 1, K = 5, and different n.

a=0.1

0.25

0.35

0.45

0.55

0 0.10.2 0.3

n=2

n=5

0.4 0.5

b

V

n=1

n=4

n=3

a=0.1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.10.20.3

n=2

n=5

0.40.5

b

E

n=1

n=4

n=3

Fig. 2. E and V as functions of b for a = 0.1, q = r/R = m = 1, K = 5, and different n.

probability of success in the jth attack as:

vj(K,q)

=

⎧

⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

Having Equations (25) and (26) we can also re-

place?j

⎪⎪⎨

K,

1

1 +

?

r(qK− 1)

R(q − 1)qj−1

1

?rK

?mif q ?= 1,

j = 1,..., K

1 +

R

?mif q = 1

.

(27)

i=1τiin Equation (4) with:

⎧

⎪⎪⎩

qj− 1

qK− 1,q ?= 1

j

q = 1

.

(28)

For the even resource distribution with q = 1,

Ti= T = R/K, and vi= v=

and (20) become:

1

1+(Kr/R)m, Equations (17)

E =1

K

K

?

j

?

j=n

j

?

ψ (j − 1,n − 1)bn(1 − b)j−n(1 − v)j

+

h=1

(1 − a)v(1 − v)h−1

n−1

?

s=0

ψ(h − 1,s)bs

×(1 − b)h−1−sψ(j − h,n−1 − s)

× (1 − a)n−1−saj−h−n+1+s?

+(1 − v)K

j=0

n−1

?

?K

j

?

bj(1 − b)K−j

Page 8

Resource Distribution in Multiple Attacks311

m=0.5

0.5

0.6

0.7

0.8

543210

q

V

n=1

n=4

n=2

n=5

n=3

m=0.5

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

543210

q

E

n=1

n=4

n=2

n=5

n=3

m=2

0.1

0.2

0.3

0.4

0.5

V

543210

q

n=1

n=4

n=2

n=5

n=3

m=2

0.6

0.7

0.8

0.9

1

543210

q

E

n=1

n=4

n=2

n=5

n=3

Fig. 3. E and V as functions of q for a = b = 0.1, r/R = 1, K = 5, and different n and m.

K

?

n−1−s

?

and

+

h=1

v(1 − v)h−1

n−1

?

s=0

ψ (h − 1,s)bs(1 − b)h−1−s

×

e=0

ψ (K − h + 1,e)(1 − a)eaK−h−e+1

(29)

V = 1 − (1 − v)K

n−1

?

j=0

?K − 1

j

?

bj(1 − b)K−1−j

−

K−1

?

j=n

ψ (j − 1,n − 1)bn(1 − b)j−n(1 − v)j.

(30)

For n = 1, Equations (29) and (30) take the form:

K

?

j

?

E =

1

K

j=1

j

?

b(1 − b)j−1(1 − v)j

+

h=1

(1 − a)v(1 − v)h−1(1 − b)h−1aj−h

?

+(1 − b)K(1 − v)K+

K

?

h=1

v(1 − v)h−1

×(1 − b)h−1aK−h+1

(31)

and

V = 1 − (1 − b)K(1 − v)K− b

K

?

j=1

(1 − b)j−1(1 − v)j.

(32)

3. ANALYZING THE MODEL

Figs. 1 and 2 plot the expected relative attacker’s

resource expenditure E, and the overall target de-

struction probability V, as functions of the wrong

identification probabilities a and b for q = r = R =

m = 1, K = 5, and different n where b = 0.1

when plotting as functions of a, and a = 0.1 when

plotting as functions of b. The target destruction

probability is independent of a, since a destroyed tar-

get is destroyed regardless of the attacker’s identi-

fication. The attacker’s resource expenditure is an

Page 9

312 Levitin and Hausken

m=0.5

0.5

0.6

0.7

0.8

0.9

1

05 10 1520

K

V

q=0.5 q=1q=2

m=0.5

0.35

0.5

0.65

0.8

0.95

E

05 101520

K

q=0.5 q=1q=2

m=2

0

0.1

0.2

0.3

0.4

0.5

V

05 101520

K

q=0.5 q=1q=2

m=2

0.7

0.75

0.8

0.85

0.9

0.95

1

05 101520

K

E

q=0.5 q=1 q=2

Fig. 4. E and V as functions of K for a = b = 0.1, r/R = 1, n = 3, and different m and q.

increasing function of a because the attacker spends

more resources on additional useless attacks if it can-

not correctly identify the destroyed target.

Both V and E decrease in b. The reasons are

that if the attacker wrongly identifies an undestroyed

target as destroyed, it incorrectly allocates less re-

sources believing its attacks were successful, and the

target enjoys lower destruction probability because

of the attacker’s incorrect identification.

Both V and E increase in n because when n in-

creases the probability that the attacker stops the

series of attacks before the target is destroyed de-

creases. When n = K the attacker always attacks K

times and spends its entire resource, which corre-

sponds to E = 1.

Fig. 3 plots E and V as functions of the attack

effort variation factor q for various n and m when

r/R = 1, K = 5, and a = b = 0.1 For low contest in-

tensity m = 0.5, the destruction probability has a dis-

tinct maximum when q is close to 1. This means that

for not intensive contests the optimal attacker’s strat-

egy is to distribute the resource evenly among the at-

tacks. For a highly intensive contest with m = 2, the

destruction probability is minimal when q is close to

1 and achieves its maximal value when q = 0, which

corresponds to concentrating the entire resource in a

single attack. For both low and high contest intensi-

ties the expected attacker’s resource expenditure E is

a U-shaped or decreasing function of q.

Fig. 4 plots V and E as functions of the number

of attacks for various q and m when r/R = 1, a = b =

0.1. For low contest intensity m the destruction prob-

ability increases with K when q = 1, which means

that the attacker should evenly distribute its resource

among as many attacks as possible. For highly inten-

sive contests the attacker prefers a single attack as

the target destruction probability decreases with K.

The expected resource expenditure is always a de-

creasing function ofK.Itshould bementioned that in

realistic situations the number of attacks is limited by

time constraints, by tactical reasons, by limited mini-

mal cost of a single attack, etc. Therefore, the upper

limit of K always exists.

4. THE OPTIMAL ATTACK STRATEGY

The complex interaction of the attacker’s free

choice variables q, K, and n makes the intuitive

Page 10

Resource Distribution in Multiple Attacks313

0

0.3

0.6

0.9

1.2

1.5

q

0.50.4 0.3 0.20.1

a=0.1,m=2

a=0.1,m=0.5

0

b

a=0.4,m=2

a=0.4,m=0.5

0

2

4

6

8

10

0.50.4 0.30.2 0.10

b

K

a=0.1,m=2

a=0.1,m=0.5

a=0.4,m=2

a=0.4,m=0.5

0

2

4

6

8

10

0.50.4 0.30.20.10

b

n

a=0.1,m=2

a=0.1,m=0.5

a=0.4,m=2

a=0.4,m=0.5

0.6

0.7

0.8

0.9

1

0.50.40.3 0.2 0.1

a=0.1,m=2

a=0.1,m=0.5

0

b

V

a=0.4,m=2

a=0.4,m=0.5

0

0.2

0.4

0.6

0.8

1

0.50.40.3 0.20.1

a=0.1,m=2

a=0.1,m=0.5

0

b

E

a=0.4,m=2

a=0.4,m=0.5

0.2

0.4

0.6

0.8

1

0.5 0.40.30.20.10

b

U

a=0.1,m=2

a=0.1,m=0.5

a=0.4,m=2

a=0.4,m=0.5

Fig. 5. Optimal strategy parameters and outcomes as functions of b for different a and m, r/R = 0.2, ε = 0.1.

choice of the optimal attack strategy impossible. This

section illustrates the application of the suggested

multiple attack model for numerical analysis of the

optimal attack strategy.

We assume that the attacker tries to maximize

the probability of the target destruction in a series of

attacks and to minimize its own expected resource

expenditure. If the target value for the attacker is

C and the value of its total resource is c, the total

attacker’s utility is CV−cE. Maximizing this utility

is equivalent to maximizing the function U = V−εE,

where ε = c/C. The attacker’s optimal strategy is:

(q, K,n) = arg

q,K,n

?U(q, K,n) → max?.

Figs. 5 and 6 plot the optimal strategy parame-

ters q, K, n (when 1 ≤ K ≤ 10), and the corresponding

values of V, E, and U as functions of the wrong iden-

tification probability b for different a and m when

r/R=0.2and r/R=0.6 forε =0.1 (inexpensive attack

Page 11

314 Levitin and Hausken

0

0.3

0.6

0.9

1.2

1.5

q

0.5 0.40.30.2 0.1

a=0.1,m=2

a=0.1,m=0.5

0

b

a=0.4,m=2

a=0.4,m=0.5

0

2

4

6

8

10

0.50.4 0.30.20.10

b

K

a=0.1,m=2

a=0.1,m=0.5

a=0.4,m=2

a=0.4,m=0.5

0

2

4

6

8

10

0.50.40.30.2 0.10

b

n

a=0.1,m=2

a=0.1,m=0.5

a=0.4,m=2

a=0.4,m=0.5

0.6

0.7

0.8

0.9

1

0.50.4 0.30.2 0.1

a=0.1,m=2

a=0.1,m=0.5

0

b

V

a=0.4,m=2

a=0.4,m=0.5

0

0.2

0.4

0.6

0.8

1

0.5 0.40.3 0.2 0.1

a=0.1,m=2

a=0.1,m=0.5

0

b

E

a=0.4,m=2

a=0.4,m=0.5

0.2

0.4

0.6

0.8

1

0.5 0.40.3 0.2 0.10

b

U

a=0.1,m=2

a=0.1,m=0.5

a=0.4,m=2

a=0.4,m=0.5

Fig. 6. Optimal strategy parameters and outcomes as functions of b for different a and m, r/R = 0.6, ε = 0.1.

resource). Figs. 7 and 8 present the same plots for ε =

0.5 (expensive attack resource).

Based on Figs. 5–8 one can make the follow-

ing conclusions about the optimal attack strategy.

When the contest intensity is low, the attacker ben-

efits from increasing the number of attacks indepen-

dently from the probabilities of incorrect observation

a and b. Indeed, the reduction of the attacker’s per-

attack effort has small effect on the success probabil-

ity of each attack when m is small, whereas the in-

creased number of attacks increases the probability

that at least in one of them the target is destroyed.

The number n of detected attack successes needed

to stop the sequence of attacks always increases with

b to compensate the probability that the attacker

wronglyidentifiesanundestroyedtargetasdestroyed

and provide the high probability of target destruction

V. On the other hand, n decreases with a to avoid

the excessive resource expenditure E. When m = 0.5

the attack effort variation factor q remains almost

Page 12

Resource Distribution in Multiple Attacks315

0

0.3

0.6

0.9

1.2

1.5

q

0 0.1

a=0.1,m=2

a=0.1,m=0.5

0.2 0.30.40.5

b

a=0.4,m=2

a=0.4,m=0.5

0

2

4

6

8

10

0 0.1 0.20.3 0.40.5

b

K

a=0.1,m=2

a=0.1,m=0.5

a=0.4,m=2

a=0.4,m=0.5

0

2

4

6

8

10

0 0.10.2 0.30.4 0.5

b

n

a=0.1,m=2

a=0.1,m=0.5

a=0.4,m=2

a=0.4,m=0.5

0.6

0.7

0.8

0.9

1

0 0.1

a=0.1,m=2

a=0.1,m=0.5

0.20.3 0.4 0.5

b

V

a=0.4,m=2

a=0.4,m=0.5

0

0.2

0.4

0.6

0.8

1

0 0.1

a=0.1,m=2

a=0.1,m=0.5

0.2 0.3 0.40.5

b

E

a=0.4,m=2

a=0.4,m=0.5

0.2

0.4

0.6

0.8

1

0 0.10.2 0.3 0.40.5

b

U

a=0.1,m=2

a=0.1,m=0.5

a=0.4,m=2

a=0.4,m=0.5

Fig. 7. Optimal strategy parameters and outcomes as functions of b for different a and m, r/R = 0.2, ε = 0.5.

insensitive to a and b and is slightly greater than 1,

which corresponds to increase of effort during the se-

ries of attacks. When n remains unchanged, q slightly

decreases with b.

When the contest is intensive, the attacker

prefers to reduce the number of attacks, concen-

trating more resources in each attack to achieve

the effort superiority, or avoid the effort inferior-

ity, compared with the defender. When the defender-

attacker resource ratio r/R is high (attacker is

inferior), the attacker prefers to concentrate all its

resources in the single attack (K = 1, n = 1, q =

0), which guarantees the total resource expenditure

E = 1. With decrease of the resource ratio r/R the

attacker may prefer several attacks with decreasing

efforts (q<1).

When the attack resource is inexpensive (ε =

0.1), the attacker maintains a high level of the target

destruction probability V by increasing the expected

resource expenditure E. Therefore, E increases with

Page 13

316Levitin and Hausken

0

0.3

0.6

0.9

1.2

1.5

q

0 0.1

a=0.1,m=2

a=0.1,m=0.5

0.2 0.3 0.4 0.5

b

a=0.4,m=2

a=0.4,m=0.5

0

2

4

6

8

10

0 0.10.20.3 0.40.5

b

K

a=0.1,m=2

a=0.1,m=0.5

a=0.4,m=2

a=0.4,m=0.5

0

2

4

6

8

10

0 0.10.2 0.30.40.5

b

n

a=0.1,m=2

a=0.1,m=0.5

a=0.4,m=2

a=0.4,m=0.5

0.6

0.7

0.8

0.9

1

0 0.1

a=0.1,m=2

a=0.1,m=0.5

0.2 0.3 0.40.5

b

V

a=0.4,m=2

a=0.4,m=0.5

0

0.2

0.4

0.6

0.8

1

0 0.1

a=0.1,m=2

a=0.1,m=0.5

0.2 0.3 0.40.5

b

E

a=0.4,m=2

a=0.4,m=0.5

0.2

0.4

0.6

0.8

1

0 0.10.20.30.4 0.5

b

U

a=0.1,m=2

a=0.1,m=0.5

a=0.4,m=2

a=0.4,m=0.5

Fig. 8. Optimal strategy parameters and outcomes as functions of b for different a and m, r/R = 0.6, ε = 0.5.

aandb,whereasV remainsalmostinsensitivetoboth

probabilities of incorrect observation.

When the attack resource is relatively expensive

(ε = 0.5), the desire to reduce the expected resource

expenditure E plays a much greater role in choos-

ing the attack strategy. This leads to increasing sensi-

tivity of the target destruction probability V to the

probabilities of incorrect observation a and b and

decreasing sensitivity of the expected resource ex-

penditure E to these probabilities. For the intensive

contest (m = 2) the resourceful attacker (r/R = 0.2)

increases the number of attacks K to compensate

the increase of b but provides moderate expected re-

source expenditure by decreasing the attack efforts

Page 14

Resource Distribution in Multiple Attacks 317

across the attacks (q<1). However when the attacker

is inferior with the resources (r/R = 0.6) it still prefers

the single-attack strategy.

5. CONCLUSION

We consider the situation when an attacker at-

tacks a system repeatedly to ensure its destruction,

while a defender protects the system to prevent its

destruction. The attacker chooses the optimal num-

ber of attacks, the attack stopping rule, and resource

distribution across the attacks. The defender’s re-

source lasts through all the attacks. The attacker im-

perfectly observes the outcome of each attack. With

probability a it wrongly identifies a destroyed tar-

get as undestroyed, and with probability b it wrongly

identifies an undestroyed target as destroyed. The

outcome of each attack is determined by a contest

success function that depends on the amount of re-

sources allocated by the defender and attacker to

each attack. The article suggests a model of the mul-

tiple attack and demonstrates how this model can

be used for analyzing the impact of the two prob-

abilities of incorrect observation, the attacker’s and

defender’s resource ratio and the contest intensity

on the optimal attack strategy, the target destruction

probability and the attacker’s relative resource ex-

penditure. Simulations are provided to illustrate the

model.

The complex interaction of parameters influenc-

ing the outcome of the multiple attacks makes impos-

sible its analysis based on intuition. The suggested

model provides a qualitative insight into the influ-

ence of probabilities of wrong identification on the

attack success probability and the resource expen-

diture. It can help the potential defender to evalu-

ate its chances for survival and justify the measures

of disguising the attack results. It can also help the

attacker to justify the effort of getting correct in-

formation about the outcomes of attacks (by using

surveillance techniques or applying intelligence ef-

fort).Italsogivestheattackerthepossibilitytoadjust

its strategy for any given attack conditions charac-

terized by wrong detection probability, contest inten-

sity, and available resources. For example, as shown

in Section 4, the inferior attacker should switch from

a multiple attack strategy to single attack when the

probability of wrong identification of an undestroyed

target as destroyed increases above a certain level in

the case of an intensive contest and expensive attack

resource. However, the superior attacker still prefers

multiple attacks in these conditions.

ACKNOWLEDGMENTS

We thank three anonymous referees of this jour-

nal for useful comments.

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