Modeling Resources Allocation in Attacker‐Defender Games ...jzhuang/Papers/GZ_RA2016.pdf · the “warm up” effect. We acknowledge that the term “warm up” could also be interpreted

Risk Analysis, Vol. 36, No. 4, 2016 DOI: 10.1111/risa.12502

Modeling Resources Allocation in Attacker-DefenderGames with “Warm Up” CSF

Peiqiu Guan and Jun Zhuang∗

Like many other engineering investments, the attacker’s and defender’s investments mayhave limited impact without initial capital to “warm up” the systems. This article studies such“warm up” effects on both the attack and defense equilibrium strategies in a sequential-movegame model by developing a class of novel and more realistic contest success functions. Wefirst solve a single-target attacker-defender game analytically and provide numerical solutionsto a multiple-target case. We compare the results of the models with and without consider-ation of the investment “warm up” effects, and find that the defender would suffer higherexpected damage, and either underestimate the attacker effort or waste defense investmentif the defender falsely believes that no investment “warm up” effects exist. We illustrate themodel results with real data, and compare the results of the models with and without consid-eration of the correlation between the “warm up” threshold and the investment effectiveness.Interestingly, we find that the defender is suggested to give up defending all the targets whenthe attack or the defense “warm up” thresholds are sufficiently high. This article provides newinsights and suggestions on policy implications for homeland security resource allocation.

KEY WORDS: Attacker-defender games; contest success functions (CSFs); game theory; subgame-perfect Nash equilibria (SPNE); “warm up” threshold

1. INTRODUCTION

Hundreds of billions of dollars have beenspent on homeland security since September 11,2001,(1) and numerous models(2–5) have been de-veloped to study the strategic interactions betweenthe governments (defenders) and the terrorists(attackers). In order to help the government tomake better decisions in alocating the limiteddefense resources among multiple targets, binarydefense allocation,(6–12) such as defending or notdefending, may not be significantly informative tosupport the real decision making.

When the defense and the attack effortsare modeled as continuous, instead of binary,

Department of Industrial and Systems Engineering, SUNY at Buf-falo, Buffalo, NY, USA.∗Address correspondence to Jun Zhuang, Department of In-dustrial and Systems Engineering, University at Buffalo, TheState University of New York, Buffalo, NY 14260-2050, USA;[email protected].

Hirshleifer(13) and Skaperdas(14) introduce two formsof contest success functions (CSFs) among theplayers in the field of rent seeking, tournaments,and conflict.(15) One is the ratio form P(A, D) =

k1 Am

k1 Am+k2 Dm+C , and the other is the exponential form

P(A, D) = exp [k1 A]exp [k1 A]+exp [k2 D]+C , where m > 0 and ki >

0 (i = 1, 2) are the mass effect parameters, A andD represent the attacker’s and the defender’s in-vestment efforts, and C is the inherent defenselevel.

The CSFs capture the essential relationshipsamong the probability of a successful attack, defenseand attack efforts, and the inherent defense levels.The CSFs are normally assumed to be continuous,twice differentiable, and with diminishing marginalreturns with respect to both the defense and attackefforts. Table I summarizes the CSFs in the counter-terrorism literature. For example, for the func-tion form of CSFs P(D) = e−λD, the probability ofsuccessful attack P(D) decreases exponentially in

776 0272-4332/16/0100-0776$22.00/1 C© 2015 Society for Risk Analysis

Attacker-Defender Games with “Warm Up” CSF 777

Table I. CSFs in the Attacker-Defender Game Literature, Where A, D, and C Represent the Attack Effort, Defense Effort, and theInherent Defense Level, Respectively

Function References ∂ P∂ D

∂ P∂ A

∂2 P∂ D2

∂2 P∂ A2

Binary attack and continuous defense efforts (Exponential)Bier et al.,(16) Hao et al.(17)

e−kD Wang and Bier(18) ≤ 0 NA ≥ 0 NAShan and Zhuang(19)

Continuous attack and defense efforts (Ratio)A

k(A+D+C) Zhuang and Bier(2) ≤ 0 ≥ 0 ≥ 0 ≤ 0Am

Am+Dm Hausken(24) ≤ 0 ≥ 0 ≥ 0 ≤ 0A

A+D+C Hausken and Zhuang(21) ≤ 0 ≥ 0 ≥ 0 ≤ 0k1 A

k1 A+k2 D+C Guan et al.(22) ≤ 0 ≥ 0 ≥ 0 ≤ 01 − e−kA/D Nikoofal and Zhuang(23) ≤ 0 ≥ 0 ≥ 0 ≤ 0

the defender’s effort D,(16–19) but it does not de-pend on the attack effort from the attacker. For an-other body of the literature, ratio-form CSFs,(2,20–22)

the probability of successful attack decreases con-vexly in defender’s efforts and the inherent defenselevel, and increases concavely in the attacker’s ef-forts. Nikoofal and Zhuang(23) combine both the ra-tio and the exponential forms of the CSFs. As shownin Table I, the probability of successful attack de-creases in the defender’s effort ( ∂ P

∂ D ≤ 0) and in-creases in the attacker’s effort ( ∂ P

∂ A ≥ 0), both with di-

minishing marginal returns ( ∂2 P∂ D2 ≥ 0, ∂2 P

∂ A2 ≤ 0).Although the property of diminishing marginal

returns may hold when the attacker and defender in-vestments are sufficiently high, such property wouldnot hold in practice when the investments are small.For example, depending on specific context, thedefender may need to spend millions or billions ofdollars to purchase, set up, and test a new securityprogram (e.g., new software to track millions of vis-itors to the United States). The first several millionsof dollars spent may not decrease the probability ofa successful attack at all. Similarly, the attacker mayhave to spend a significant amount of resources toprepare for the attacks, and the initial thousands ofdollars (or even millions of dollars in a larger-scaleattack plot) may not increase the probability ofsuccessful attack.

To our best knowledge, none of the previous lit-erature studies the realistic phenomenon where thediminishing marginal returns over continuous invest-ment levels do not apply. We call such a phenomenonthe “warm up” effect. We acknowledge that the term“warm up” could also be interpreted as the time win-dow before the main activity (e.g., security systemrestarted from a shutdown or sleep mode), and could

be modeled in multiple-period games. By contrast,this article considers “warm up” effects statically, andproposes a new functional form of CSF to model the“warm up” effects in a single-period game, as an ex-tension to the literature provided in Table I.

The rest of the article is organized as follows:Section 2 introduces the notations and assumptionsin this article; Section 3 proposes a sequential gamemodel between the attacker and the defender withthe “warm up” CSFs, and solves a single-target gamemodel analytically; Section 4 solves for the multiple-target case numerically, illustrates the results withreal data, compares the results of the models with andwithout consideration of the correlation between thelevel of “warm up” threshold and the investment ef-fectiveness, and provides policy implications for thehomeland security resource allocation; Section 5 con-cludes the article, and the Appendix provides theproofs for propositions.

2. NOTATIONS, ASSUMPTIONS,AND MODELS

2.1. Notations and Assumptions

The notations used throughout the article are de-fined as follows:

� n > 0: The number of targets.� Gi ≥ 0: Defense resource allocations to the ith

target, ∀i = 1, 2, . . . , n.� G ≡ (G1, G2, . . . , Gn): Vector for defense re-

source allocation.� Ti ≥ 0: The attack resources to target

i , ∀i = 1, 2, . . . , n.

778 Guan and Zhuang

� T ≡ (T1, T2, . . . , Tn): Vector for attack re-sources.

� Ai > 0: The inherent defense level of thetarget i .

� WTi ≥ 0 and WGi ≥ 0: The “warm up” thresh-olds for the attack and defense investments fortarget i , respectively; such threshold is definedas the minimal level of investment before theinvestment becomes impactful; zero or minimal“warm up” effects could be accounted for bysetting the thresholds to be zero.

� Pi (Ti , Gi ) ∈ [0, 1]: The probability of a success-ful attack for target i , which is continuousand decreasing in defensive resource, Gi (whenGi > WGi ) with diminishing marginal effect,and increasing in attack resource, Ti (when Ti >

WTi ) with diminishing marginal effects, ∀i =1, 2, . . . , n:

∂ Pi (Ti , Gi )∂Gi

≤ 0,∂2 Pi (Ti , Gi )

∂Gi2 ≥ 0,

∂ Pi (Ti , Gi )∂Ti

≥ 0,∂2 Pi (Ti , Gi )

∂Ti2 ≤ 0. (1)

� kT and kG: The effectiveness coefficients of theattacker’s and the defender’s investment “warmup” thresholds, respectively.

� βi ≥ 0 and αi ≥ 0: Effectiveness coefficients ofthe attack and defense investments to target i ,respectively.

� Vi ≥ 0: Valuation of target i , ∀i = 1, 2, . . . , n.For simplicity, we use the same target valua-tions for both the defender and the attacker(4)

in this article.� LG(T, G) and LT(T, G): The objective functions

of the defender and the attacker, respectively.� T̂(G) = (T̂1(G), T̂2(G), . . . , T̂n(G)): Attacker’s

best responses in the true model.� T̄(G) = (T̄1(G), T̄2(G), . . . , T̄n(G)): Attacker’s

best responses in the defender’s false beliefmodel.

� (T∗, G∗): Subgame-perfect Nash equilibria(SPNE) in the true model.

� (T∗∗, G∗∗): SPNE in the defender’s false beliefmodel.

Note that all the subscripts i will be omitted for thenotations in the case of n = 1.

Following Azaiez and Bier,(25) Wang andBier,(18) and Shan and Zhuang,(26) both the defenderand the attacker are assumed to be rational. The in-

teraction between the attacker and the defender ismodeled as a sequential game, and the attacker is as-sumed to be the second mover. The attacker is as-sumed to choose not to attack if he is indifferent be-tween attacking and not attacking.

2.2. Contest Success Functions

Following Hirshleifer,(13) Zhuang and Bier,(2)

Hausken and Zhuang,(20) and Hausken andZhuang,(21) we consider the ratio-form CSF inthis article. Most of the CSFs in the literature assumethat the CSFs increase in attack investment anddecrease in defense investment, which may not holdin practice when the attack and defense systemsneed to “warm up.” Different from the literature assummarized in Table I, the “warm up” CSF in thisarticle is defined as a piece-wise ratio function withconsideration given to the defense and the attackinvestment “warm up” effects:

Pi (Ti , Gi ) (2)

= βi (Ti − WTi )+

βi (Ti − WTi )++αi (Gi − WGi )++ Ai

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0, if˜Ti ≤ WTi

βi (Ti −WTi )βi (Ti −WTi ) +Ai

, if˜Ti > WTi ˜&˜Gi ≤ WGi

βi (Ti −WTi )βi (Ti −WTi ) +αi (Gi −WGi ) +Ai

, if˜Ti > WTi ˜&˜Gi > WGi

which has the following properties:

� If Gi is smaller than or equal to the defense“warm up” threshold (Gi ≤ WGi ), the probabil-ity of a successful attack would not be changedby the increase of the defense investment Gi .For example, if the defense investment for anairport screening system is less than the cost ofpurchasing backscatter machines, the probabil-ity of successful attack would remain the sameor just slightly decreased in the defense invest-ment.

� If Gi is larger than the defense “warm up”threshold (Gi > WGi ), we have ∂ Pi (Ti ,Gi )

∂Gi≤ 0,

which means that the probability of a success-ful attack decreases in the defense investment.

� If the attack effect Ti is smaller than or equal tothe attack “warm up” threshold (Ti ≤ WTi ), weassume that Pi (Ti , Gi ) = 0, which means thatno attack would be successfully launched. Forexample, an attacker could not launch a bombattack successfully if he does not have enough


0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1(a)

WG

P(T

,G)

Defender investment G0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1(b)

Attacker investment T

WT

0

1

2

0

1

20

0.5

1

Defender investment G

(c)

Attacker investment T

P(T

,G)

0.10.1

0.2

0.2

0.3

0.3

0.4

0.40.5

0.5

0.6

0.6

0.7

0.70.8

0.9

WG

WT

(d)


Att

acke

r in

ve

stm

en

t T

0 0.5 1 1.5 20

0.5

1

1.5

2

Fig. 1. CSF with consideration of “warm up” effects.

resources to acquire, produce, or use suchbomb.

� If Ti is larger than the attack “warm up” thresh-old (Ti > WTi ), we have ∂ Pi (Ti ,Gi )

∂Ti≥ 0, which

means the success attack probability increasesin the attack effort.

Fig. 1 illustrates the “warm up” CSF as a func-tion of the attack and defense investments. We onlyconsider a single target in this illustration; i.e., n = 1.The baseline values of the model parameters are:β0 = 1, α0 = 1, kG = kT = 0, A= 0.5, G = 1, T = 1,and WG = WT = 0.5.

Fig. 1(a) shows that the probability of a suc-cessful attack decreases in the defender investmentG with diminishing marginal effects in the intervalof (WG,∞), while the “warm up” CSF remains thesame within the defense “warm up” threshold (G ∈[0, WG]) as WG increases. Fig. 1(b) shows that theprobability of a successful attack increases in the at-tacker’s investment T with diminishing marginal ef-fects when T > WT . When the attacker’s resource isless than or equal to the attack “warm up” thresh-old, T ∈ [0, WT], the probability of successful attackequals to zero. Figs. 1(c) and (d) show how the prob-ability of a successful attack changes as both the at-tack and defense investments vary using a 3-D plotand contour, respectively.

2.3. Modeling Investment Effectiveness Depends on“Warm Up” Threshold

Now we model the scenario in which ahigher start-up (“warm up”) cost leads to higherefficiency.(27) For example, the backscatter machinesfor airport screening cost $250,000 to $2,000,000

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

Defender Investment G

CS

Fs w

ith

Wa

rm U

p E

ffe

cts

: P

(T,G

)

(a)

WG

1W

G

2

WG

1

WG

2

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

WT

1 WT

2

Attacker Investment T

(b)

WT

1

WT

2

Fig. 2. Probability of a successful attack as functions of defenseand attack investments for two levels of defense and attack “warmup” thresholds, α0 = β0 = 1, ˜kT = kG = 1, T = G = 1.

each,(28) which is more expensive than the pat-down(no equipment cost), but is less invasive and moreefficient. Other examples of such investment include:(1) purchasing of vehicles and unmanned aerial vehi-cles, and weapons for the border patrollers; (2) pur-chasing and installing advanced monitors and secu-rity systems for federal buildings. This is also true forattack efforts. For example, the dirty bomb attackcould cause more damages than the regular bombattack(15) and it costs the attacker much more “warmup” investment than that in the regular bomb attack.In this article, we model relationships between theinvestment “warm up” thresholds and their effective-ness coefficients linearly:

βi = β0 + kTWTi , (3)

αi = α0 + kGWGi , (4)

where the initial effectiveness coefficients of attackand defense investments are denoted as β0 and α0,and corresponding correlation coefficients are de-noted as kT and kG (kT, kG ≥ 0), respectively; whenkT = 0 or kG = 0, the investment effectiveness coef-ficients are assumed to be insensitive to the changesof the investment “warm up” threshold; the amountsof the “warm up” thresholds do not impact the ef-ficiency of the attack or the defense systems. WhenkT > 0, kG > 0, higher investment “warm up” thresh-olds lead to higher investment effectiveness.

Fig. 2 illustrates the relationships between the“warm up” thresholds and the investment effective-ness. Let W1

G, W2G˜(W1

G < W2G) represent two levels

of defense “warm up” thresholds; Fig. 2(a) showsthat although the probability of a successful attackremains the same for both lines in the intervals ofG ∈ [0, W1

G] and G ∈ [0, W2G], respectively, the prob-

ability of a successful attack decreases at a sharper

780 Guan and Zhuang

rate in the case of W2G than in the case of W1

G whenG > W1

G and G > W2G.

Similarly, Fig. 2(b) shows that the probability ofa successful attack remains zero when the attack in-vestment is less than or equal to the attack “warmup” threshold (see the line with when T ≤ W1

T , andthe line with when T ≤ W2

T). The probability of asuccessful attack increases at a higher rate in thecase with attack “warm up” threshold W2

T than inthe case with attack “warm up” threshold W1

T , whereW2

T > W1T .

2.4. Optimization Models

In a sequential game model, the attacker is as-sumed to be a second mover, who can decide theattack strategy T after observing the defender’s re-source allocation G over the n targets. The goal ofthe attacker is to maximize the total expected dam-age to the defender (CSF weighted by the target val-uations), and to minimize the attack costs:

LT(T, G)

= maxT

n∑i=1

⎛⎜⎜⎝ Pi (Ti , Gi )Vi︸︷︷︸

Expected damage

− Ti︸︷︷︸Attack costs

⎞⎟⎟⎠ . (5)

As the first mover, the defender considers the at-tacker’s best response T̂(G) to the defender’s strat-egy G, which is defined as

T̂(G) ≡ arg maxT

LT(T, G), (6)

before making the decision. The objective of the de-fender is to minimize the total expected damage anddefense costs:

LG(T̂(G), G)

= minG

n∑i=1

⎛⎜⎜⎝ Pi (T̂i (G), Gi )Vi︸︷︷︸

Expected damage

+ Gi︸︷︷︸Defense costs

⎞⎟⎟⎠.(7)

Thus, the SPNE is defined as follows:

Definition 1. We call a collection of strategy (T∗, G∗)an SPNE, if and only if both Equations (8) and (9)are satisfied:

T∗ = T̂(G∗), (8)

G∗ = arg minG

LG(T̂(G), G). (9)

According to the attacker’s best response de-fined in Equation (6), the SPNE can be solvedthrough backward induction.

3. MODEL SOLUTION AND ANALYSISFOR n = 1

With the “warm up” CSF defined in Equation(2), this section solves the equilibrium strategies forboth the attacker and the defender in the sequentialgame by using backward induction. We first study thecase with a single target n = 1.

3.1. Attacker’s Best Response Function

As the second mover in the sequential gamemodel, the attacker’s best response function is givenin Proposition 1.

Proposition 1. For n = 1, the attacker’s best responsefunction is given by:

T̂(G) = arg maxT

LT(T, G) =⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Case a :√βV A−A

β+ WT, G ≤ WG&V > A

β&

WT <√

βV A−A√βV A V + A−√

βV Aβ

Case b :√βV�−�

β+ WT, G > WG&V > �

β&

WT <√

βV�−�√βV�

V + �−√βV�

β

Case c :0, G ≤ WG& V ≤ A

β

or G ≤ WG&V > Aβ

&

WT ≥√

βV A−A√βV A

V + A−√βV A

β

Case d :0, G > WG&V ≤ �

β

or G > WG&V > �β

&

WT ≥√


V + �−√βV�

β

(10)

where � = α(G − WG) + A, β = β0 + kTWT, and α =α0 + kGWG.

Remarks: The attacker’s best response functionfor a single target in Equation (10) shows that the at-tacker would attack the target in Cases a and b, andput forth zero effort in Cases c and d as his best re-sponses. In Case a, the attacker’s best response effortlevel does not depend on the value of the defender’sstrategy G, while the attacker’s best response effort


0 2 4 60

0.5

1

1.5

2

WG


(a) Low Target Valuation: V=0.01

Case c

Case d

Att

acke

r B

est

Re

sp

on

se

0 2 4 60

0.5

1

1.5

2

WG


(b) Moderate Target Valuation: V=0.6

Case a

Case d

0 2 4 60

0.5

1

1.5

2

WG


(c) High Target Valuation: V=6

Case a

Case b

Case d

Fig. 3. Best response of the attacker with consideration given tothe “warm up” threshold.

first increases then decreases in the defense effort Gin Case b.

Cases c and d provide the conditions to deterattacks, including high attack “warm up” threshold,high inherent defense level, high defense investment,or low target valuation.

With the baseline values of model parametersset at β0 = α0 = 1, kT = kG = 0, A= 0.5, n = 1, andWG = WT = 0.5, Fig. 3 illustrates the attacker’s bestresponse function in Proposition 1 with considera-tion given to both the defender’s and the attacker’sinvestment “warm up” thresholds, for three levels oftarget valuations.

Fig. 3(a) shows that the attacker would not at-tack a low-value target, which illustrates Cases c andd in Proposition 1. Fig. 3(b) shows the attacker at-tacks a moderate target with the amount of attackresource T̂(·) = 0.55 (Case a), which is greater thanthe attack “warm up” amount WT = 0.5, during thedefender’s “warm up” period G ≤ WG, and does notattack otherwise (Case d). Fig. 3(c) illustrates thecase for the high-value target. The attacker’s best re-sponse remains constant (but at a higher level com-pared to that in Fig. 3(b) for moderate-value target)within the defender’s “warm up” period (G ≤ WG)and does not depend on the defense effort G (Casea). When the defense effort G is in a moderate inter-val (WG = 0.5 < G < 6), the attacker’s best responsefirst increases and then decreases in the defense ef-fort (Case b). We also note that an attack is de-terred by a high defense effort G (G ≥ 6), which isthe Case d of the attacker’s best response function inEquation (10).

3.2. Subgame-Perfect Nash Equilibrium (SPNE)

According to Definition 1, substituting the at-tacker’s best response function in Equation (10)into the defender’s optimization problem (Equation

0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

12

14

16

Case BCase A

(a)

Defense warm up threshold WG

0 0.5 1 1.50

2

4

6

8

10

12

Attack warm up threshold WT

(b)

Case B Case D

T*

G*

LT

LG

P(T*,G*)Baseline Value

Fig. 4. The sensitivity analyses of WT and WG without consider-ing the relationships between the “warm up” thresholds and theinvestment effectiveness (kG = kT = 0).

(7)), the SPNE solution is solved and provided inProposition 2.

Proposition 2. There are five cases of SPNE so-lutions (T∗, G∗) for a single-target model. All theSPNE solutions (T∗, G∗), the corresponding feasibleset Fk, optimal conditions Ok, ˜∀k = A, B, . . . , E, theCSF P(T∗, G∗), and attacker’s and defender’s objec-tive functions LT(T∗, G∗) and LG(T∗, G∗) are pro-vided in Table II.

3.3. Sensitivity Analyses

This section studies how the probability of asuccessful attack, the defender’s and the attacker’sequilibrium strategies, and their objective functionschange when the model parameters vary. Based ona set of baseline values, WT = 0.1, WG = 0.5, β0 =α0 = 1, A= 0.1, and V = 15, we change the samemodel parameters one at a time and keep the othersconstant.

We first conduct the sensitivity analyses with-out consideration of the relationships between the“warm up” thresholds and the investment effective-ness (i.e., kG = kT = 0).

Fig. 4(a) shows that the defender first increasesthe defense effort as the defense “warm up” thresh-old (WG) increases, and then gives up defending thetarget if WG is sufficiently high (WG > 2.7). The de-fender’s expected damage and costs increase in WG.From Fig. 4(b), we note that the attack would be de-terred by a high attack “warm up” threshold. Thedefender reduces the defensive resource, even downto zero defense resource, to the target if the at-tack “warm up” threshold WT is sufficiently high.The probability of a successful attack slightly de-creases as the attack “warm up” threshold increases

782 Guan and ZhuangT

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II.

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T∗

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P(T

∗ ,G

∗ )L

T(T

∗ ,G

∗ )L

G(T

∗ ,G

∗ )

AO

A

√ βV

A−A

β+

WT

0√ β

VA

−A√ β

VA

√ βV

A−A

√ βV

AV

−√ β

VA

−Aβ

−W

T

√ βV

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√ βV

AV

BO

BV

(α 2β

−α

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)+W

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2βV

−α2β

V−α

+αV

(2β

V−α

2βV

−α+α

V−

βα−α

2

4β)V

−W

T(

2βV

−α2β

V−α

+αV

−α

V 4β)V

−A α

+W

G

CO

C0

00

00

DO

D0

1 α(β

(V−√ V

WT

)2

V−

A)+

WG

00

1 α(β

(V−√ V

WT

)2

V−

A)+

WG

EO

E0

Vβ−A α

+W

G0

0Vβ

−A α+

WG

OA

≡

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩{{FA

∩F

B∩

{√ βV

A−A

√ βV

AV

≤(

2βV

−α2β

V−α

+αV

−α

V 4β)V

+W

G}}

∪{F

A∩

F̄B}}

⋂ {{FA

∩F

C∩

{√ βV

A−A

√ βV

AV

≤0}

}∪{F

A∩

F̄C}}

⋂ {{FA

∩F

D∩

{√ βV

A−A

√ βV

AV

≤1 α

(β(V

−√ VW

T)2

V−

A)+

WG}}

∪{F

A∩

F̄D}}

⋂ {{FA

∩F

E∩

{√ βV

A−A

√ βV

AV

≤β

V−A α

+W

G}}

∪{F

A∩

F̄E}}

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

FA

≡{V

>A β

˜&˜W

T<

√ βV

A−A

√ βV

AV

+A

−√ βV

Aβ

}F

B≡{ V

>m

ax(α

2V

4β2

,4β

Aα

2)˜

&˜W

T<

V(1

−α β

+α

2

4β2

)}F

C≡{ V

>A β

˜&˜W

T≥

√ βV

A−A

√ βV

AV

+A

−√ βV

Aβ

˜or˜

V≤

A β

}

FD

≡

⎧ ⎪ ⎨ ⎪ ⎩V>

WT,V

>(V

−√ VW

T)2

V+

A β

WT

≥1

−(V

−√ VW

T)2

V(V

−√ VW

T−1

V−√ V

WT

)−V

+√ V

WT

⎫ ⎪ ⎬ ⎪ ⎭F

E≡{ V

>A β

}

OB

≡

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩{{FB

∩F

A∩

{(2β

V−α

2βV

−α+α

V−

αV 4β

)V+

WG

≤√ β

VA

−A√ β

VA

V}}

∪{F

B∩

F̄A}}

⋂ {{FB

∩F

C∩

{(2β

V−α

2βV

−α+α

V−

αV 4β

)V+

WG

≤0}

}∪{F

B∩

F̄C}}

⋂ {{FB

∩F

D∩

{(2β

V−α

2βV

−α+α

V−

αV 4β

)V+

WG

≤1 α

(β(V

−√ VW

T)2

V−

A)+

WG}}

∪{F

B∩

F̄D}}

⋂ {{FB

∩F

E∩

{(2β

V−α

2βV

−α+α

V−

αV 4β

)V+

WG

≤β

V−A α

+W

G}}

∪{F

B∩

F̄E}}

OC

≡

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩{{FC

∩F

A∩

{0≤

√ βV

A−A

√ βV

AV

}}∪

{FC

∩F̄

A}}

⋂ {{FC

∩F

B∩

{0≤

(2β

V−α

2βV

−α+α

V−

αV 4β

)V+

WG}}

∪{F

C∩

F̄B}}

⋂ {{FC

∩F

D∩

{0≤

1 α(β

(V−√ V

WT

)2

V−

A)+

WG}}

∪{F

C∩

F̄D}}

⋂ {{FC

∩F

E∩

{0≤

βV

−A α+

WG}}

∪{F

C∩

F̄E}}

OD

≡

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩{{FD

∩F

A∩

{1 α(β

(V−√ V

WT

)2

V−

A)+

WG

≤√ β

VA

−A√ β

VA

V}}

∪{F

D∩

F̄A}}

⋂ {{FD

∩F

B∩

{1 α(β

(V−√ V

WT

)2

V−

A)+

WG

≤(

2βV

−α2β

V−α

+αV

−α

V 4β)V

+W

G}}

∪{F

D∩

F̄B}}

⋂ {{FD

∩F

C∩

{1 α(β

(V−√ V

WT

)2

V−

A)+

WG

≤0}

}∪{F

D∩

F̄C}}

⋂ {{FD

∩F

E∩

{1 α(β

(V−√ V

WT

)2

V−

A)+

WG

≤β

V−A α

+W

G}}

∪{F

D∩

F̄E}}

OE

≡

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩{{FE

∩F

A∩

{βV

−A α+

WG

≤√ β

VA

−A√ β

VA

V}}

∪{F

E∩

F̄A}}

⋂ {{FE

∩F

B∩

{βV

−A α+

WG

≤(

2βV

−α2β

V−α

+αV

−α

V 4β)V

+W

G}}

∪{F

E∩

F̄B}}

⋂ {{FE

∩F

C∩

{βV

−A α+

WG

≤0}

}∪{F

E∩

F̄C}}

⋂ {{FE

∩F

D∩

{βV

−A α+

WG

≤1 α

(β(V

−√ VW

T)2

V−

A)+

WG}}

∪{F

E∩

F̄D}}


0 5 10 15 20 25 300

5

10

15

20

25

Case A

Case ECase B

(a)

Target Valuation V0 2 4 6 8 10 12 14

0

2

4

6

8

10

12

Inherent Defense Level A

(b)

Case B

Case A

Case C

T*

G*

LT

LG

P(T*,G

*)

Baseline Value

Fig. 5. The sensitivity analysis of V and Awithout considering therelationships between the “warm up” thresholds and the invest-ment effectiveness (kG = kT = 0).

0 2 4 6 80

5

10

15

20

Case B Case E Case A

Defense warm up threshold WG

(a)

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

Attack warm up threshold WT

(b)

Case BCase D

T*

G*

LT

LG

P(T*,G

*)

Baseline Value

Fig. 6. The sensitivity analysis of WT and WG with considerationgiven to the relationships between the “warm up” thresholds andthe investment effectiveness (kG = kT = 0.1).

and drops to zero when the attack is deterred (forWT > 0.3).

Fig. 5 shows how the model results are sensitiveto the change of the target valuation V and the inher-ent defense level A. In particular, Fig. 5(a) shows that(a) the target with low valuation (0 < V < 2) wouldnot be defended but attacked (Case A); (b) the tar-get with moderate valuation (2 ≤ V ≤ 5) would bedefended and not be attacked (Case E); and (c)both defender and attacker increase their investment(Case B) if the target’s valuation is large (V > 5).From Fig. 5(b), we note that attack could be deterredby a high inherent defense (A> 12.8) even when nodefense investment is allocated to the target, whichimplies that no attack would be launched on a targetwith a high inherent defense level.

Fig. 6 studies the sensitivity analyses of the de-fense and attack “warm up” thresholds with consid-eration given to the relationships between the valueof the thresholds and the investment effectiveness.In the analyses, we set kT = kG = 0.1, which meansone unit of the “warm up” investments can increase0.1 units of effectiveness of the attack and defenseinvestments: β = β0 + kTWT and α = α0 + kGWG.

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

Case BCase E

k

(a) (b)

0 100 200 300 4000

2

4

6

8

10

12

14

16

k

Case B

Case A

T*

G*

LT

LG

P(T*,G

*)

Baseline Value

Fig. 7. The sensitivity analyses of kG, kT .

Comparing to the case without consideration ofthe relationships between the defense “warm up”thresholds and the investment effectiveness inFig. 4, the defender gives up defending with a higherdefense “warm up” threshold (WG > 6 in Fig. 6(a),and WG > 2.7 in Fig. 4(a)). Similar to the case with-out consideration of the relationship between the at-tack “warm up” threshold and the investment effec-tiveness in Fig. 4(b), the attack is also finally deterredby some significant level of WT , which is WT > 0.3.

Fig. 7(a) shows that the defense equilibrium in-vestment G∗ first increases and then decreases inkG (the defense investment becomes more effective).We also note that an attack would be deterred if kG

is high (kG > 0.3) because of the increased effective-ness of the defense investment. Fig. 7(b) shows thatboth the attack and defense investments (T∗ and G∗)decrease as the attack investment becomes more ef-fective, and the defender would give up defending asthe kT is sufficiently high (kT ≥ 140).

3.4. Comparison of the Models With and WithoutConsideration of the Investment“Warm Up” Effects

To study the importance of the novel “warmup” model, we compare the results of the hypo-thetical model (defender’s false belief model) whenthe defender does not consider the “warm up” ef-fects (WG = WT = 0), but in fact the thresholds exist(WG > 0, WT > 0), with the model proposed in thisarticle (true model). We study the consequence dueto this hypothetical belief. In the defender’s belief,her optimal strategy should be defined as follows:

G∗∗ = arg minG

LG(T̄(G), G)

= P(T, G|WT = 0, WG = 0)V + G,

784 Guan and Zhuang

0 2 4 6 80

0.5

1

1.5

2


(a) Low Target Valuation: V=0.01

Attacker

Best R

esponse

0 2 4 6 80

0.5

1

1.5

2


(b) Moderate Target Valuation: V=0.6

0 2 4 60

0.5

1

1.5

2


(c) High Target Valuation: V=6

Fig. 8. Best response of the attacker in the hypothetical modelwhere the defender does not consider the “warm up” threshold.

where T̄(G) = arg max LT(T, G) = P(T, G|WT =0, WG = 0)V − T is denoted as the attacker’s bestresponse function according to the defender’s belief.

However, the attacker makes his best responsewith consideration given to the investment “warmup” effects. The defender’s payoff in the hypotheticalmode depends on the defender’s equilibrium strategyin the hypothetical model G∗∗ and the true attackerequilibrium strategy T∗, which is defined in Equation(8). Thus, due to the false belief, the SPNE (T∗∗, G∗∗)is given as follows:

T∗∗ = T̂(G∗∗)

= arg max LT(T, G∗∗)= P(T, G∗∗)V−T,(11)

G∗∗ = arg minG

LG(T̄(G), G)

= P(T, G|WT = 0, WG = 0)V + G, (12)

where T̄(G) = arg max LT(T, G) = P(T, G|WT =0, WG = 0)V − T. Note that the defender’s and theattacker’s objective functions at the equilibriumpoints are denoted as L∗∗

G and L∗∗T , respectively.

Fig. 8 shows the results of the attacker’s best re-sponse T̄(G) without considering the “warm up” ef-fects of the attack and defensive investments, WT =WG = 0.

Comparing the results in Fig. 3, we note that theattacker chooses to not attack a low valued targetas his best response, regardless of the defender’s in-vestment in both Figs. 3(a) and 8(a). Fig. 8(b) showsthat the attacker would only attack a moderate val-ued target with low attack effort (T = 0.05) whenthe defender’s investment is low (G = 0), and the at-tacker would choose not to attack the target if thedefender’s investment G is large. However, the at-tacker’s attack effort is about 10 times less in Fig. 8(b)as compared to Fig. 3(b) (T = 0.55), which impliesthat the defender would underestimate the attacker’sattack effort if she does not consider the “warm up”effects of the investment. Fig. 8(c) shows that theattacker’s best response first increases and then de-

0 0.5 1 1.5 24

5

6

7

Defense ‘‘warm up" threshold: WG

Exp

ecte

d Lo

sses

LG**

LG*

0 1 2 30

2

4

6

Attack ‘‘warm up" threshold: WT

Exp

ecte

d Lo

sses

LG**

LG*

0 0.5 1 1.5 20

0.5

1

1.5

2

Equ

ilibr

ium


G**

G*

0 1 2 30

0.5

1

1.5

2

Equ

ilibr

ium


G**

G*

0 0.5 1 1.5 21

1.5

2

Equ

ilibr

ium


T**

T*

0 1 2 30

1

2

3

4


Equ

ilibr

ium

T**

T*

(a) (b)

(c) (d)

(e) (f)

Fig. 9. Comparison of defender’s expected payoff, defender’sequilibrium strategy, and attacker’s equilibrium strategy in the hy-pothetical model and the true model.

creases as the defender’s investment increases fora high valued target case. Comparing the results inFigs. 3(c) and 8(c), we note that the overall attacker’sbest response would be underestimated if “warm up”effects are not considered. From Fig. 3(c), we alsonote that the attacker would not attack when his bestresponse level is lower than the attack “warm up”threshold, but in Fig. 8(c) the attacker’s best responseis positive even when it is below the attack “warmup” threshold, which would lead to the waste of thedefense effort.

Figs. 9(a) and (b) show that the defender’s pay-offs (expected damage and loss) in both models arevery close to each other when the defense and attack“warm up” thresholds (WG and WT) are small, butthe differences enlarge as WG and WT increase. Inparticular, Fig. 9(a) shows that the defender wouldsuffer up to 1.27 (6.62/5.22) times higher expecteddamage and cost than that in the model proposedin this article (true model) as the defense “warm up”threshold increases if the defender fails to considerthe “warm up” effects. In Fig. 9(b), we note that thedefender suffers up to 5 units more expected damageand cost than that in the true models if she fails toconsider the “warm up” effects.

Figs. 9(c) and (d) show that the defender’s equi-librium strategy does not depend on the defense and


attack “warm up” thresholds in the defender’s falsebelief model, but it does in the true model.

Figs. 9(e) and (f) show that the attacker’sequilibrium strategy is dependent on the defenseand attack “warm up” thresholds in both the hy-pothetical model and the true model. In particular,Fig. 9(e) shows that in the hypothetical model (de-fender’s false belief model), the attacker’s equilib-rium decreases as the defense “warm up” thresholdincreases when WG is in a small range (WG ≤ 1.4),while in the true model, the attacker’s equilibriumstrategy first increases and then decreases in WG

when WG is in a small range (WG ≤ 0.9). The at-tacker’s equilibrium strategy remains the same inboth models when WG is high (when WG > 0.9 inthe true model, and WG > 1.4 in the hypotheticalmodel). Fig. 9(f) shows that the attacker wouldgive up attacking in a higher “warm up” threshold(WT = 2.2) due to the defender’s wrong false beliefthan that in the true model (WT = 1.5).

4. NUMERICAL ILLUSTRATION FORMULTIPLE-TARGET MODEL

In this section, we study the equilibrium solu-tion for the model with multiple targets. According tothe complex analytical solution for the single-targetmodel in Proposition 2, we expect an intractable so-lution for the multiple-target case. Instead of obtain-ing the analytical solution, we focus on the numericalsolutions in this section.

4.1. Numerical Illustration

The numerical illustration for the multiple-targetmodel is generated in this section based on theheuristic search algorithm. To illustrate the modelwith multiple targets, following Bier et al.,(16) Haoet al.,(17) Hu et al.,(5) Nikoofal and Zhuang,(23) andShan and Zhuang,(19) we use the expected prop-erty damage caused by the terrorist attack for ur-ban areas in the United States to estimate the targetvaluations.(29)

In particular, we select the top five most valu-able urban areas in the United States, which are NewYork City, Chicago, San Francisco, Washington, DC,and Los Angeles, as shown in Table III.

For the baseline values of other model pa-rameters, we set WG = WT = 0.5, β0 = α0 = 1, andA= 0.1 for all targets.

We study how the defensive allocation strategieschange as the level of defense “warm up” threshold

Table III. Expected Property Damage for the Top Five UrbanAreas in the United States

# Urban Area Expected Property Losses (Vi $M)

1 New York City 413.02 Chicago 115.03 San Francisco 57.04 Washington, DC 36.05 Los Angeles 34.0

50 1000

50

100

150

200

Allo

ca

tio

n

New York City

ChicagoSan FranciscoWashingtonLos Angeles

0 20 40 60 80 100

400

500

600

700

WG

De

fen

de

r L

osse

s

50 100 150 2000

100

200

300

New York City

Chicago

San FranciscoWashington

Los Angeles

0 50 100 150 200

400

500

600

700

WG

De

fen

de

r L

osse

s

(a1) k

T=0, k

G=0

(a2) k

T=0, k

G=0

(b1) k

T=5%, k

G=5%

(b2) k

T=5%, k

G=5%

Fig. 10. Defense allocations as functions of the levels of defense“warm up” thresholds levels.

WG increases, when the relationship of the defense(attack) “warm up” threshold and defense (attack)investment effectiveness coefficients kG (kT) are con-sidered in Fig. 10.

Fig. 10 shows that the defender moves thedefense resources from the less valuable to morevaluable targets as the defense “warm up” thresholdincreases and finally gives up defending any of thetargets if the defense “warm up” thresholds aresufficiently high. Fig. 10 compares optimal defensiveresource allocations between two cases: (a) no corre-lation between “warm up” threshold and investmenteffectiveness kT = kG = 0, and (b) low correlationkT = kG = 5%. We find that the defender gives updefending targets from the low-valuation targets tothe high-valuation target in both cases as the “warmup” threshold increases. For example, the mostvaluable target “New York City” is the last targetthat the defender would give up. In Fig. 10(a2), thedefender’s expected damage and cost increase asthe defense “warm up” threshold increases whenkT = kG = 0. From Fig. 10(b2), we note that thedefender’s expected damage and cost first decreaseand then increase in the case of kT > 0, kG > 0. Sincethe high defense (attack) “warm up” threshold caninduce high defense investment effectiveness in thesecond case through α = α0 + kGWG, as defenseinvestment “warm up” threshold increases, thedefender’s investment becomes more effective,

786 Guan and Zhuang

Fig. 11. Defense allocations as functions of the levels of attack“warm up” thresholds levels.

leading to less damage. The expected damage andcost decrease. However, as the defense “warm up”threshold increases, it costs the defender more to“warm up,” leading to a high investment cost. Thus,the defender’s expected damage and costs increase.

Fig. 11 studies how the defender’s defense allo-cation strategy changes as the attacker’s “warm up”threshold WT increases. We also compare the optimalattack resource allocations between two cases: (a) nocorrelation between “warm up” threshold and invest-ment effectiveness kT = kG = 0, and (b) low correla-tion kT = kG = 5%.

Since a high attack “warm up” threshold candeter the attack, the defender would stop defend-ing the target if the attack is deterred by the attack“warm up” threshold. The defender stops defend-ing targets one by one from the low-valued target tothe high-valued target. For example, since the mostvaluable target, “New York City,” expects the high-est attack “warm up” threshold to deter an attack,the defender would stop defending it last. The de-fender’s expected damage and cost decrease as theattack “warm up” threshold increases for case (a),which is shown in Fig. 11(a2). In case (b), the de-fender overall suffers higher expected loss and costthan in case (a) when the correlation between “warmup” threshold and investment effectiveness is consid-ered in Fig. 11(b2). There are a lot of ways to in-crease the attack “warm up” threshold, such as in-creasing the terrorists’ training cost, making it moredifficult for them to pass through the security, or ac-quiring weapons. Though an attack with high attack“warm up” threshold would cause more damage tothe defender because of high effectiveness, the attackwould be deterred if the attack “warm up” thresh-old is sufficiently high such that the defender doesnot need to defend some or all targets. Thus, the de-fender’s expected damage and cost decrease to zero.

5. CONCLUSIONS AND FUTURE RESEARCHDIRECTIONS

In this article, we study a novel class of contestsuccessful functions (CSF) to capture the “warm up”effects in the attack and defense investments in coun-terterrorism. To our knowledge, such investment“warm up” effects in attacker-defender resource al-location problems have not been studied in the lit-erature. This article fills the gap by studying attackand defense “warm up” effects in a game-theoreticalmodel.

This article solves the equilibrium solution an-alytically for the single-target model, identifies fivecases of SPNE in the attacker-defender game, andsolves the multiple-target model numerically. Inter-estingly, we find that the defender would give up de-fending all the targets as either the attack or defense“warm up” thresholds are sufficiently high. For a highdefense “warm up” threshold, the defender gives updefending some or all targets because it is too costlyto defend the targets. For a high attack “warm up”threshold, the defender stops defending some or allof the targets because the attacks are deterred by thehigh attack “warm up” threshold, which leads to zeroexpected damage to the defender. We also find thatthe defender’s expected damage first decreases andthen increases as the defender’s defense “warm up”threshold increases, and it first increases and then de-creases as the attacker’s attack “warm up” thresholdincreases.

This article also provides suggestions on how toallocate limited defense resources to various targetswhen the defense “warm up” thresholds are consid-ered. We find that not only high inherent defense lev-els could deter an attack, but also high attack “warmup” thresholds for the attacker. There are scenarioswhere these are correlated to each other; e.g., for awell-constructed defense system, it usually costs theattacker a high price (attack “warm up” threshold) topass through the security of the defense system andlaunch an attack.

In the future, we could consider cooperationsamong decentralized defenders. For example, if twocities have similar defense needs and are close by,but cannot afford the defense because of the highdefense “warm up” threshold, they could sharesome common defense. Thus, a cooperative defendergame among multiple targets with overarching ef-fects could be an interesting extension.

This article considers a complete-informationgame without deception. In practice, the attacker


may be uncertain about the “warm up” threshold andthus we can study when and how deception and se-crecy (12,30,31) could be used by the defender to mis-lead the attackers.

In this article, we consider a one-period gamewhere the “warm up” effects are embedded in theCSF, and such effects would not fail. In the future,we could use multi-period or continuous-time gamesto study more general “warm up” effects, including:(a) a period of heightened vulnerability to the tar-get during the “warm up” period; and (b) the use ofbackup/standby systems (32) after a potential failureof “warm up.”

ACKNOWLEDGMENTS

This research was partially supported by the U.S.Department of Homeland Security (DHS) throughthe National Center for Risk and Economic Analysisof Terrorism Events (CREATE) under award num-ber 2010-ST-061-RE0001. This research was also par-tially supported by the U.S. National Science Foun-dation under award numbers 1200899 and 1334930.However, any opinions, findings, and conclusions orrecommendations in this document are those of theauthors and do not necessarily reflect views of theDHS, CREATE, or NSF. We thank the editors andanonymous referees for their helpful comments. Theauthors assume responsibility for any errors.

APPENDIX

Proof of Proposition 1 A.1

First, we prove the concavity of the attacker’s ob-jective function. For n = 1, we have:

maxT

LT(T, G) = P(T, G)V − T.

From the assumption in Equation (1), we know that∂2 P(T,G)

∂T2 ≥ 0. Then we have ∂2 LT(T,G)∂T2 = ∂2 P(T,G)

∂T2 V ≥ 0.

Thus, the attacker’s maximization problem isconcave in T. In order to find a T maximizing theattacker’s objective function, we let the first-orderderivative of the attacker’s objective function equalto 0, ∂LT(T,G)

∂T = 0, and solve for T.

For n = 1, according to Equation (2), we rewritethe CSFs as follows:

P(T, G) =

⎧⎪⎪⎨⎪⎪⎩

0, if T ≤ WT

β(T−WT)β(T−WT)+A, if T > WT & G ≤ WG

β(T−WT)β(T−WT)+α(G−WG)+A, if T > WT & G > WG.

Then, we rewrite the objective function of the at-tacker’s optimization problem as follows:

maxT≥0

LT(T, G)

=

⎧⎪⎪⎨⎪⎪⎩

−T, if T ≤ WT

β(T−WT)β(T−WT)+AV − T, if T > WT & G ≤ WG

β(T−WT)β(T−WT)+α(G−WG)+AV − T, if T > WT & G > WG.

When G > WG, we have:

maxT≥0

LT(T, G)

={−T, if T ≤ WT

β(T−WT)β(T−WT)+α(G−WG)+AV − T, if T > WT.

(A1)

In order to maximize the attacker’s objective whenT ≤ WT , from Equation (A1), we note that the opti-mizer is TC1 = 0 and the attacker’s objective is LC1

T =0.

If T > WT , according to the attacker’s objec-tive function defined in Equation (A1), we have∂LT(T,G)

∂T = 0 ⇒ βV[α(G−WG)+A][β(T−WT)+α(G−WG)+A]2 V − 1 = 0.

Then, we have:

T̂(G) =⎧⎪⎪⎨⎪⎪⎩

0, if V ≤ α(G−WG)+Aβ

1β

(√

βV[α(G − WG) + A] if V >α(G−WG)+A

β.

−[α(G − WG) + A]) + WT

Let � ≡ α(G − WG) + A; we have:{T̂(G)C2 = 0, if V ≤ �

β

T̂(G)C3 =√

βV�−�

β+ WT, if V > �

β.

(A2)

Substituting Equation (A2) into the attacker’s objec-tive function in Equation (A1), we have:⎧⎪⎪⎪⎨

⎪⎪⎪⎩LC2

T = 0, if V ≤ �β

LC3T =

(1 − �√

βV�

)V

−(

1β

(√βV� − �

)+ WT

)if V > �

β.

788 Guan and Zhuang

If all inequality conditions in Equation (A3) holdwhen G > WG, then T̂(G) is the attacker’s best re-sponse function:

T̂(G) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

0, if V > �β

& LC1T ≥ LC3

T

T̂(G)C3 , if V > �β

& LC3T > LC1

T

0, if V ≤ �β.

(A3)

Thus,

T̂(G) =⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0, if V > �β

& G > WG &

WT ≥√


V + �−√βV�

β√βV�−�

β+ WT, if V > �

β& G > WG &

WT <√


V + �−√βV�

β

0, if V ≤ �β.

(A4)

Similarly, for G ≤ WG, according to the CSFs definedin Equation (2), we write the attacker’s objectivefunction for n = 1 as follows:

maxT≥0

LT(T, G)

={ −T, if T ≤ WT

β(T−WT)β(T−WT)+AV − T, if T > WT.

(A5)

From Equation (A5), we note that TC4 = 0 is the op-timizer for the attacker when T ≤ WT , and the corre-sponding attacker’s objective is LC4

T = 0.If T > WT , according to the attacker’s objective

function defined in Equation (A5), we have:

∂LT(T, G)∂T

= 0

⇒ βV A[β(T − WT) + A]2

V − 1 = 0

⇒

⎧⎪⎪⎨⎪⎪⎩

T̂(G)C5 = 0, if V ≤ Aβ

T̂(G)C6 if V > Aβ.

= 1β

(√βV A− A

)+ WT,

(A6)

Substituting the local optimizers in Equation (A6)into the attacker’s objective function in Equation

(A5), we have:

⎧⎪⎪⎪⎨⎪⎪⎪⎩

LC5T = 0, if V ≤ A

β

LC6T =

(1 − A√

βV A

)V if V > A

β.

−(

1β

(√βV A− A

)+ WT

) (A7)

The attacker’s best response function T̂(G) under thecondition of G ≤ WG is defined as follows:

T̂(G)

=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

0, if V > Aβ

& LC4T ≥ LC6

T

T̂(G)C6, if V > Aβ

& LC6T > LC4

T , when G ≤ WG

0, if V ≤ Aβ

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0, if G ≤ WG, V > Aβ

&

WT ≥√

βV A−A√βV A V+ A−√

βV Aβ√

βV A−Aβ

+ WT, if G ≤ WG, V > Aβ

&

WT <√

βV A−A√βV A

V+ �−√βV A

β

0, if G ≤ WG, V ≤ Aβ.

(A8)

Combining Equations (A4) and (A8), the attacker’sbest response function is summarized as follows:

T̂(G) = arg maxT≥0

LT(T, G)

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

√βV A−A

β+ WT, G ≤ WG & V > A

β&

WT <√


βV Aβ√

βV�−�

β+ WT, G > WG & V > �

β&

WT <√


V + �−√βV�

β

0, G ≤ WG & V ≤ Aβ

or

G ≤ WG & V > Aβ

&

WT ≥√


βV Aβ

0, G > WG & V ≤ �β

or

G > WG & V > �β

&

WT ≥√


V + �−√βV�

β

where � ≡ α(G − WG) + A, β ≡ β0 + kTWT .


Proof of Proposition 2 A.2

We solve the equilibrium solution for the single-target model by substituting the attacker’s best re-sponse function (Equation (10)) into the defender’soptimization model in Equation (7) under differentconditions:

� Case A: Under the conditions of G ≤WG, V > A

β& WT <

√βV A−A√

βV AV + A−√

βV Aβ

,we have the attacker’s best response functionT̂(G) =

√βV A−A

β+ WT . After substituting T̂(G)

into the defender’s optimization model inEquation (7), we have:

minG≥0

LG(T̂(G), G)

=β((√

βV A−Aβ

+ WT

)− WT

)β((√

βV A−Aβ

+ WT

)− WT

)+ A

V + G

=√

βV A− A√βV A

V + G.

The minimizer that minimizes LG(T̂(G), G) =√βV A−A√

βV AV + G is GA = 0. The feasible set is

F A ≡ {V > Aβ

& WT <√

βV A−A√βV A

V + A−√βV A

β}.

The corresponding attacker’s strat-egy is T A =

√βV A−A

β+ WT The CSF is

P(T A, GA) =√

βV A−A√βV A

.

And the corresponding payoffs of the at-tacker and the defender are LA

T =√

βV A−A√βV A V −

√βV A−A

β− WT, and LA

G =√

βV A−A√βV A V

� Case B: Under conditions of G > WG & V >�β

& WT <√


V + �−√βV�

β, we have

T̂(G) =√

βV�−�

β+ WT . Substitute T̂(G) to the

defender’s optimization model in Equation (7),we have:

minG≥0

LG(T̂(G), G)

=βV

[(√βV�−�

β+ WT

)− WT

]β[(√

βV�−�β

+ WT

)− WT

]+ α(G − WG) + A

V + G

= V

⎛⎝1 −

√α(G − WG) + A

βV

⎞⎠+ G, (A9)

where � = α(G − WG) + A.

Let ∂LG(T̂(G),G)∂G = 0, then we have:

∂LG(T̂(G), G)∂G

= V

(−√

1βV

× α

2√

α(G − WG) + A

)+ 1 = 0

⇒ G ={

αV4β

− Aα

+ WG, V >4β Aα2

0, V ≤ 4β Aα2

.

According to the conditions WG ≥ 0 and G >

WG, we have G > 0. Then the local optimizerunder this condition is GB = αV

4β− A

α+ WG.

The corresponding feasible set is F B ≡ {V >

max( α2V4β2 ,

4β Aα2 ) & WT < V(1 − α

β+ α2

4β2 )}.The corresponding attacker’s strategyis TB = V( α

2β− α2

4β2 ) + WT. The CSF is

P(TB, GB) = 2βV−α

2βV−α+αV . And the corre-sponding attacker’s and defender’s payoffs are:LB

T = ( 2βV−α

2βV−α+αV − βα−α2

4β)V − WT, and LB

G =( 2βV−α

2βV−α+αV − αV4β

)V + WG.� Case C: Under the conditions of G ≤

WG & V > Aβ

& WT ≥√

βV A−A√βV A

V + A−√βV A

β,

or G ≤ WG & V ≤ Aβ

, we have the attacker’s

best response function T̂(G) = 0 such thatminG≥0 LG(T̂(G), G) = G.

We note that the minimizer GC3 = 0 mini-mizes the defender’s minimization problem,and the corresponding feasible set is FC ≡ {V >Aβ

& WT ≥√


βV Aβ

or V ≤ Aβ}. The

corresponding attacker’s strategy is TC = 0.

The CSF is P(TC, GC) = 0. And the attacker’sand defender’s corresponding objectives areLC

T = 0, and LCG = 0

� Case D: Under the condition of G >

WG & V > �β

& WT ≥√


V + �−√βV�

β,

or G > WG & V ≤ �β

, we have the attacker’s

best response function T̂(G) = 0 such thatminG≥0 LG(T̂(G), G) = G.

- Under the condition of G > WG & V >�β

& WT ≥√


V + �−√βV�

β,

* If V > WT , from condition WT ≥√


V +�−√

βV�

β, we have 1

α( β(V−√

VWT)2

V − A) + WG ≤G ≤ 1

α( β(V+√

VWT)2

V − A) + WG. Then the

790 Guan and Zhuang

minimizer is GD = 1α

( β(V−√VWT)2

V − A) + WG.

The corresponding feasible set is:

F D ≡

⎧⎪⎪⎪⎨⎪⎪⎪⎩

V > WT , V >(V−

√VWT )2

V + Aβ

WT ≥ 1 − (V−√

VWT )2

V (V−

√VWT−1

V−√

VWT) − V + √

VWT

⎫⎪⎪⎪⎬⎪⎪⎪⎭ .

The corresponding attacker’s strategy is TD =0. The CSF is P(TD, GD) = 0 And the de-fender’s and defender’s corresponding objec-tives are LD

T = 0, and LDG = 1

α( β(V−√

VWT)2

V −A) + WG.

* If V ≤ WT , from condition WT ≥√


V +�−√

βV�

β, we have 0 ≤ G ≤ 1

α( β(V+√

VWT)2

V −A) + WG. Then the minimizer is G = 0. It con-flicts with the conditions G > WG, ∀WG > 0.So, G = 0 is not a feasible solution under thiscondition.

- Case E: Under the condition ofG > WG & V ≤ �

β= α(G−WG)+A

β, from

condition V ≤ �β

= α(G−WG)+Aβ

, we have

G ≥ βV−Aα

+ WG. Thus, the minimizer of thecase under the condition G > WG & V ≤ �

β

is GE = βV−Aα

+ WG. The corresponding

feasible set is F E ≡{

V > Aβ

}. The corre-

sponding attacker’s strategy is TE = 0. TheCSF is P(TE, GE) = 0. And the attacker’sand defender’s corresponding objectives areLE

T = 0, and LEG = βV−A

α+ WG.

According to the feasible set Fk, case k is opti-mal if Fk ∩ F j = ∅, or if Fk ∩ F j �= ∅ and Lk

G ≤ LjG,

, ∀k, j = A, B, . . . , E, k �= j . Therefore, the optimalrange of case i is defined as:

Ok ≡⋂

j=A,B,...,E, j �=k{{Fk ∩ F j ∩ {LkG ≤ Lj

G}} ∪ {Fk

∩F̄ j }}, k = A, B, . . . , E.

Thus, for Cases A–E, we have the optimal set asfollows:

OA =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

{{F A ∩ F B ∩ {√

βV A−A√βV A

V ≤(

2βV−α2βV−α+αV − αV

4β

)V + WG}} ∪ {F A ∩ F̄ B}}

⋂{{F A ∩ FC ∩ {√

βV A−A√βV A

V ≤ 0}} ∪ {F A ∩ F̄C}}⋂{{F A ∩ F D ∩ {

√βV A−A√

βV AV ≤ 1

α

(β(V−√

VWT )2

V − A)

+WG}} ∪ {F A ∩ F̄ D}}⋂{{F A ∩ F E ∩ {

√βV A−A√

βV AV ≤ βV−A

α + WG}} ∪ {F A ∩ F̄ E}}

OB =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

{{F B ∩ F A ∩ {(


4β

)V + WG ≤

√βV A−A√

βV AV}} ∪ {F B ∩ F̄ A}}⋂{{F B ∩ FC ∩ {

(2βV−α


)V + WG ≤ 0}} ∪ {F B ∩ F̄C}}⋂{{F B ∩ F D ∩ {

(2βV−α


)V

+WG ≤ 1α

(β(V−√

VWT )2

V − A)

+ WG}} ∪ {F B ∩ F̄ D}}⋂{{F B ∩ F E ∩ {(


4β

)V+WG ≤ βV−A

α + WG}} ∪ {F B ∩ F̄ E}}

OC =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

{{FC ∩ F A ∩ {0 ≤√

βV A−A√βV A

V}} ∪ {FC ∩ F̄ A}}⋂{{FC ∩ F B ∩ {0 ≤(


4β

)V + WG}} ∪ {FC ∩ F̄ B}}

⋂{{FC ∩ F D ∩ {0 ≤ 1α

(β(V−√

VWT )2

V − A)

+ WG}} ∪ {FC ∩ F̄ D}}⋂{{FC ∩ F E ∩ {0 ≤ βV−Aα + WG}} ∪ {FC ∩ F̄ E}}

OD =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

{{F D ∩ F A ∩ { 1α

(β(V−√

VWT )2

V − A)

+ WG ≤√

βV A−A√βV A

V}} ∪ {F D ∩ F̄ A}}⋂{{F D ∩ F B ∩ { 1

α

(β(V−√

VWT )2

V − A)

+ WG ≤(


4β

)V

+WG}} ∪ {F D ∩ F̄ B}}⋂{{F D ∩ FC ∩ { 1

α

(β(V−√

VWT )2

V − A)

+ WG ≤ 0}} ∪ {F D ∩ F̄C}}⋂{{F D ∩ F E ∩ { 1

α

(β(V−√

VWT )2

V − A)

+ WG ≤ βV−Aα + WG}}

∪{F D ∩ F̄ E}}

OE =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

{{F E ∩ F A ∩ { βV−Aα + WG ≤

√βV A−A√

βV AV}} ∪ {F E ∩ F̄ A}}⋂{{F E ∩ F B ∩ { βV−A

α + WG ≤(


4β

)V+WG}} ∪ {F E ∩ F̄ B}}⋂{{F E ∩ FC ∩ { βV−A

α + WG ≤ 0}} ∪ {F E ∩ F̄C}}⋂{{F E ∩ F D ∩ { βV−A

α + WG ≤ 1α

(β(V−√

VWT )2

V − A)

+WG}} ∪ {F E ∩ F̄ D}}

Thus, the five cases of optimal strategies and theircorresponding optimal ranges and payoffs are provedas documented in Table II.

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Modeling Resources Allocation in Attacker‐Defender Games ...jzhuang/Papers/GZ_RA2016.pdf · the “warm up” effect. We acknowledge that the term “warm up” could also be interpreted