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Incorporating risk seeking attitude into defense strategy
Reliability Engineering and System Safety 123 (2014) 104–109
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Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress
Incorporating risk seeking attitude into defense strategy Ferenc Szidarovszky a, Yi Luo b,n a b
Senior Researcher, ReliaSoft Corporation, 1450 S. Eastside Loop, Tucson, AZ 85710, United States Postdoctoral Research Fellow, University of Michigan Health System, Ann Arbor, MI 48103, United States
art ic l e i nf o
a b s t r a c t
Article history: Received 4 January 2013 Received in revised form 21 October 2013 Accepted 2 November 2013 Available online 11 November 2013
Optimal resource allocation is first found in defending possible targets against random terrorist attacks subject to budget constraint. The mathematical model is a nonconvex optimization problem which can be transformed into a convex problem by introducing new decision variables, so standard methods can be used for its solution. Without budget constraint the simplified model can be solved by a very simple algorithm which requires the solution of a single variable monotone equation. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Nonconvex optimization Certainty equivalent Homeland security
1. Introduction Defending objectives, which can be the targets of terrorist attacks, is one of the most important goals of homeland security. The outcomes of the actions of the defender are uncertain because they also depend on the random actions of the attacker. Game theory is the most appropriate approach to model the interactions between the attacker and the defender. Attacker–defender games have been intensively studied in recent years. Some researchers consider the players' payoffs as deterministic values and assume that the defender seeks to minimize the damage, while the attacker tries to maximize it [5,7]. However, the players' payoffs are usually random due to the uncertainty in the game, and therefore classic equilibrium approach has its limitations to find the solutions under this situation. Risk analysis is often used to capture the uncertainty resulting by the presence of random variables in the players' payoff functions. A production and conflict model is introduced and analyzed in [6] when two agents are fighting for as large as possible shares of the total production, which is determined by their contest success function. A twoperson conflict model is discussed in [22] when the agents can select between converting resources into arms or into useful production. The wining probabilities of the agents depend on their armament levels and the obtained reward depends on the amount of the useful production. The Nash equilibrium of this two-person game is determined and its dependence on the risk-taking attitudes of the agents is examined. A new contest
n
Correspondence author. E-mail address: [email protected] (Y. Luo).
0951-8320/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ress.2013.11.002
model is introduced in [4] which is an extension and generalization of the rent-seeking games where contest functions determine the winning probabilities and exponential utility function are assumed. The existence and uniqueness of the equilibrium is proved in the special case when every player has a constant degree of absolute risk aversion. Comparative static results are proved showing how the utility dissipation is affected by the risk-taking attitude of the agents and the precise nature of the technology. The central moments describe the nature of the distribution of a random variable mathematically, and any distribution can be characterized by the mean, the variance, the skewness, etc [19]. The first moment is usually considered as the payoffs of the players in attacker–defender games. For instance, Bier et al. [2] develop optimal strategies to allocate resources among possible defensive investments based on the assumptions that the attacker and the defender will maximize and minimize the expected damage of an attack on the system, respectively. In order to find best strategic defensive allocation against an unknown attacker, Bier et al. [3] consider cases when the attacker seeks to maximize the expected payoff from launching an attack and the defender tries to minimize the expected loss of an attack. Hausken and Zhuang [11,12] employ contest success function to describe the probability of damage and compute the government's and terrorists' expected utilities in the attacker–defender game. Azaiez and Bier [1] claim that the defender maximizes the expected cost of the attack by considering an investment to strengthen the defense capability of the object. Levitin [13] suggests optimal defense strategy that presumes separation and protection of system elements based on the consideration that the attacker tries to maximize the expected damage of an attack. Levitin and Hausken [14] propose that the defender can enhance system reliability by
F. Szidarovszky, Y. Luo / Reliability Engineering and System Safety 123 (2014) 104–109
either protecting a subset of the genuine system elements or deploying separated redundant elements and false elements. The agents' optimal strategies can be obtained from the expected damage when the cumulative performance of the system elements cannot meet a demand. Hausken and Levitin [8,9] consider the interaction of the attacker and the defender as a two-person simultaneous game where damages might occur to several elements of the system according to a binomial distribution, and the possible damage is determined with a general contest function including the intensity of the contest. Wang et al. [24] discuss the attacker–defender problem and analyze how to allocate resources to maximize the probability of core services availability by considering all kinds of components in the entire system. Since the values of the core services can be estimated from the demand, the objective function of the defender is equivalent to the expected value of the available services under the attacks. Similarly, the expected payoffs to the attacker and to the defender are also employed by several scholars in conducting research on systems defense and attack models [25,26,18,20]. To our best knowledge, in all earlier studies expected value of the random objective is considered and optimized [10]. Clearly, the characterization of a random variable becomes more accurate if higher moments are also considered. However, the complexity of the computation increases as well. Modeling the uncertainties in this game should include the risk seeking attitude of the players. As it is common in the economic literature, uncertain outcomes are substituted with their certainty equivalents [21] including the first two central moments of the random variables. The certainty equivalent of the payoff of the defender is a linear combination of its expectation and variance, where a risk attitude coefficient is assigned to the variance. In this paper it is assumed that there are several possible targets which can be attacked, and the defender has the assessment of the probability for each possible target to be attacked. These probability values can be obtained by using actual data from previous interactions with the attacker, or from information about its capabilities or from other sources. The question is to find optimal resource allocation strategy of the defender prior to the attack. This paper offers a mathematical model and solution algorithm for this problem before an actual attack occurs. Instead of computing the amount of the damage by a contest function, we determine the proportion of the maximum possible damage which can be avoided by the protecting actions of the defender. For the sake of mathematical simplicity we use a simple form of the contest function, more general forms (such as used in [6]) including intensity can be applied in a similar way. We also incorporate risk by including the variance of the random payoff of the defender into the objective function. After an attack occurs, the defender responds to it, then newer attack occurs, the defender responds again, and so on. A possible model and solution procedure are offered for the resulting multistage stochastic game for example, in our earlier work [16] and in the other papers mentioned earlier. In developing the mathematical model, we will first derive the payoff function of the defender including the random elements, and then its certainty equivalent will be determined based on its expectation and variance. However, this payoff function is not concave in general, so standard methodology cannot be used. By introducing new decision variables both the objective function and the budget constraint become concave, so the model is transformed into a convex programming problem. The rest of the paper is organized as follows. Section 2 introduces the mathematical model and the transformation into a convex programming problem is given in Section 3. In the case of unlimited or very high available budget an unconstrained
105
optimization problem is obtained, its special solution algorithm is introduced in Section 4. An illustrative example is given in Section 5. The last Section 6 concludes the paper with future research directions.
2. The mathematical model Suppose there are I independent possible targets, and let i be the index of them (i¼1, 2,…, I). The intruder is assumed to attack one target at each time. Combined attacks can be considered as single attacks, since we can consider the combinations of targets as new targets, and so combined attacks as separate attacks. Let ni (i¼1, 2,…, I) be the effort of the attacker to attack target i and let pi (i¼1, 2,…, I) be the probability of the actual attack. Let vi (i¼ 1, 2, …, I) be the highest possible damage in object i if it is attacked and the object is unprotected, and let mi (i¼ 1, 2,…, I) be the effort of the defender to protect target i against the attack. In addition, let ci (i¼1, 2,…, I) be the unit cost of this effort. It is assumed that the defender is able to avoid mi =ðmi þ ni Þ proportion of the possible highest damage if it is actually attacked. This simple formula can be replaced by more sophisticated expressions, which would not significantly modify the model. This expression is very similar to the contest function concept from economics. So in this case the defender's payoff is zi ¼ vi mi =ðmi þ ni Þ ci mi , which occurs with probability pi. Here we assume that defending action is made only in the case of an attack. If a possible target is not attacked, then the defending resources remain idle there. The defender's payoff is a discrete random variable with possible values zi and occurring probabilities pi (i¼1, 2,…, I). Therefore the expectation and variance of the defender's payoff z are as follows: I mi EðzÞ ¼ ∑ vi ci mi pi and mi þ n i i¼1 2 I mi ci mi pi ðEðuÞÞ2 ð1Þ VarðzÞ ¼ ∑ vi mi þ ni i¼1 If r denotes the risk seeking attitude of the defender, then the certainty equivalent ([21]) of its random payoff is given as 2 I I mi mi c i mi p i r ∑ v i c i mi D ¼ ∑ vi mi þ n i mi þni i¼1 i¼1 " #2 I mi pi þ r ∑ v i c i mi p i : ð2Þ mi þni i¼1 The value r ¼0 refers to risk neutral attitude, r 40 to risk aversion and r o0 to risk seeking behavior. In this study, we assume that r Z 0, that is, the risk seeking behavior is excluded, which is not realistic in our case, as it will be explained later. The decision variables are mi (i¼ 1, 2,…, I) and the values of vi, ni, ci, pi (i¼1, 2,…, I) and r are assumed to be known by the defender. Let B denote the defender's available budget, then the budget constraint can be formulated as I
∑ mi r B:
i¼1
ð3Þ
Hence our model is to maximize the objective function (2) subject to the budget constraint (3). Notice that the objective function (2) can be interpreted as using the weighting method in a multiobjective programming problem, where the first objective is to maximize expectation and the second objective is to minimize variance. Notice that constraint (3) is linear, however the objective function (2) is non-concave in general, so standard iteration methods [15] cannot be used to find optimum. In the next section
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new decision variables will be introduced which will transform the model into a convenient convex programming problem, so standard methods can be successfully applied.
The two possible shapes of the function zi(mi) are shown in Fig. 1, where the continuous curve corresponds to the case of msi 40 and the broken downward sloping curve shows the case when msi r 0. Proposition 1. Function (4) is concave as an I-variable function.
3. The transformed model
Proof. By differentiation
Notice first that I
I
i¼1
i¼1
D ¼ ∑ zi pi r ∑ z2i pi þr
I
∑ z i pi
i¼1
!2 ;
ð4Þ
where we use again the notation z i ¼ vi
mi c i mi : mi þni
ð5Þ
I ∂D ¼ pl 2rzl pl þ 2rð ∑ zi pi Þpl ; ∂zl i¼1
ð12Þ
∂2 D ¼ 2rpl þ 2rp2l ∂z2l
ð13Þ
and for ja l;
We will first examine zi as function of mi. Clearly, zi0
¼
vi ni
ci ðmi þ ni Þ2
ð6Þ
and z″i ¼
ð14Þ
So the Hessian matrix of D is the following: 2vi ni ðmi þ ni Þ
o 0; 3
ð7Þ
so zi is strictly concave for mi Z 0. The stationary point is the solution of equation zi0 ¼
∂2 D ¼ 2rpl pj : ∂zl ∂zj
vi ni
ci ¼ 0 ðmi þ ni Þ2
ð8Þ
0 B B B B B H¼B B B B @ 0
which is given as rffiffiffiffiffiffiffiffi vi ni ni : msi ¼ ci
ð9Þ
msi r 0,
If then zi strictly decreases in mi for mi 4 0. If then zi strictly increases for mi o msi and is strictly decreasing for mi 4 msi . In both cases zi ð0Þ ¼ 0 and zi ðmi Þ- 1 as mi -1. Let ( 0 if ni Z vcii zni ¼ ð10Þ pffiffiffiffi pffiffiffiffiffiffiffiffi 2 ð vi ni ci Þ otherwise;
B B B B B ¼B B B B @
0 2 p1 B C B p2 p1 C B C B C B C C þ 2r B B C B C B C @ A 2rpI pI p1 1
2rp1 2rp2
1
2rp1
pI p2
1 p1 pI C p2 pI C C C C C C C C A 2 pI
1
p1 B C C B p2 C C B C C B C C B C C C þ2r B C p1 B C C B C C B C C @ A A 2rpI pI
2rp2
msi 4 0,
p2
pI :
ð15Þ T
If u ¼ ðu1 ; u2 ; :::; uI Þ is any vector then !2 I
I
uT Hu ¼ ∑ ð 2rpi Þu2i þ 2r i¼1
2 I
Then the range of zi(mi) is the interval ( 1; zni ), since in the first case zi(mi) has its maximal value at 0 with zi(0) ¼0 and in the second case the maximum occurs at msi with pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi vi ni =ci ni zi ðmsi Þ ¼ vi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ci vi ni =ci ni vi ni =ci pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ¼ vi vi ni ci vi ni ci þ ci ni pffiffiffiffi pffiffiffiffiffiffiffiffi2 ¼ vi ni c i : ð11Þ
0
p1 p2 p22
¼ 2r 4 ∑
i¼1
u2i pi
∑ pi ui
i¼1
!2 3 ∑ ui pi 5 r 0; I
i¼1
ð16Þ
since the bracketed term is the variance of a discrete random variable with values ui and occurring probabilities pi. Notice first that if zi o 0, then mi 40 with negative objective function term, but mi ¼ 0 generates zi ¼ 0. Therefore rational decision maker does not select negative zi value. If msi r 0, then mi ¼ 0 is selected and if msi 4 0, then we have three possibilities: (a). If zi 4zni , then no solution exists for mi ; (b). If zi ¼ zni , then the only solution is mi ¼ msi ; (c). If zi ozni , then the smaller solution is chosen.
zi
z*i
The terms with mi ¼ zi ¼ 0 can be omitted from both the objective function and the constraint.
msi
mi
Proposition 2. The budget constraint is also transformed into a concave constraint.
Proof. From Eq. (5) we have Fig. 1. Shape of zi ðmi Þ in two cases.
ci m2i þ mi ðci ni þ zi vi Þ þ zi ni ¼ 0;
ð17Þ
F. Szidarovszky, Y. Luo / Reliability Engineering and System Safety 123 (2014) 104–109
so the smaller solution is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 vi zi ci ni ðci ni þzi vi Þ2 4ci zi ni : mi ¼ 2ci By differentiation 2
2
solutions of equations ð18Þ
3
∂mi 16 ci ni zi þvi 7 ¼ 4 1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5: ∂zi 2ci 2 ðci ni þ zi vi Þ 4ci zi ni
ð19Þ
mi =∂z2i
has the same sign as and clearly, ∂ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc n z þ vi Þ½2ðci ni þ zi vi Þ 4ci ni ðci ni þ zi vi Þ2 4ci zi ni i i qi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðci ni þ zi vi Þ2 4ci zi ni ð20Þ which has the same sign as the following expression: ðci ni þ zi vi Þ2 þ 4ci zi ni þðci ni zi þ vi Þ2 ¼ 4ci ni vi 4 0
ð21Þ
Therefore mi ðzi Þ is convex in zi , so the budget constraint I
B ∑ mi ðzi Þ Z 0
ð22Þ
i¼1
is concave. Hence, maximizing function (4) subject to constraints (22) and zi r zni (i¼ 1, 2,…, I) is a convex optimization problem, the solution of which can be found by standard gradient-type algorithms [15]. An alternative approach can be suggested based on the additive structures I
I
∑ zi pi ; ∑ z2i pi
i¼1
I
and
i¼1
∑ mi ðzi Þ
i¼1
as a dynamic programming problem [23] where the state variables are the partial sums of the above expressions. In the case of a general contest function mki i =ðmki i þ nki i Þ we have to introduce the new variable zi ðmi Þ ¼ vi mki i =ðmki i þ nki i Þ ci mi which does not change the methodology significantly.
4. Special algorithm in a special case Assume now that there is no budget constraint. In this special case we can derive a simple optimization algorithm which requires the solution of a single-variable, piece-wise linear, monotonic equation. The idea of this method is basically the same as it is known from oligopoly theory [17]. Notice that ∂D ¼ pi 2rpi zi þ 2rQ pi ∂zi
ð23Þ
where Q ¼ ∑Ii ¼ 1 zi pi , so the optimum value zi ðQ Þ of zi with given value of Q is ( 1 1 if Q r zni 2r Q þ 2r zi ðQ Þ ¼ ð24Þ n zi otherwise;
vi
mi ci mi ¼ zi ðQ Þ mi þ n i
I
i¼1
ð25Þ
Let H(Q) denote the left hand side and G(Q) the first term. At Q¼0, HðQ Þ Z 0 and as Q -1, HðQ Þ- 1, since G(Q) is bounded from above. Therefore there is at least one solution Q for Q, and the corresponding decisions mi can be obtained as the smaller
ð26Þ
for i¼1, 2,…, I. We can finally show the uniqueness of the solution of Eq. (12) by considering the following three cases: 1 (i) If zni r 2r for all i, then zi ðQ Þ ¼ zni regardless of the value of Q, so it is the unique solution with Q ¼ ∑Ii ¼ 1 zni pi . 1 (ii) If zni 4 2r for all i, then G(0) 40 and for small values of Q (until the second case of (24) occurs with some value of i), G′ðQ Þ ¼ ∑Ii ¼ 1 pi ¼ 1. And then G′ðQ Þ becomes less than unity implying that there is a unique intercept of the curve of GðQ Þ and the 451 line. 1 1 (iii) If zni 4 2r and znj r 2r with some i and j, then G(0)4 0 and G′ðQ Þ o 1 for all Q 40 implying again the uniqueness of the intercept of the curve of G(Q) and the 451 line.
The uniqueness of Q implies the uniqueness of the optimal solution, since in the case of two solutions for mi , the smaller solution is taken by any rational decision maker. It was assumed that the defender is not risk seeker, that is r Z 0. If r o0, then fix all zj (ja i) values and notice that D is quadratic in zi with positive quadratic coefficient rð pi þ p2i Þ as pi o 1. Therefore if zi -1, then D also tends to infinity, so D is unbounded, no optimal solution exists. We can briefly analyze how the value of the risk seeking attitude affects the optimal decision. Assume r o r and all other model parameters remain the same, including the value of zni . 1 1 1 1 Then zni 2r o zni 2r and Q þ 2r 4 Q þ 2r , so zi ðQ Þ with r is greater 1 than the value of zi ðQ Þ with r in the interval [0, zni 2r ), and they 1 are equal for Q Z zni 2r . That is, with increasing value of r, function zi ðQ Þ is non-increasing. Therefore in Eq. (25) the first term is also non-increasing in r but non-decreasing in Q. So the solution Q of 1 Eq. (25) is also non-increasing in r, and the same holds for Q þ 2r as well as for zi ðQ Þ. The definition of Q implies that it is the expected payoff of the defender, so increasing the value of r decreases the expected payoff at the optimum. Since zi ðQ Þ is non-increasing in r and the smaller solution of Eq. (26) is non-decreasing in zi ðQ Þ, the optimal effort mi of the defender is also non-increasing in r.
5. Illustrative example Consider the case of I ¼3, v1 ¼ v2 ¼ v3 ¼4, n1 ¼n ¼1, 2 ¼n p ffiffiffi 3p ffiffiffi2c1 ¼ c2 ¼1, c3 ¼ 4 and r ¼ 1. Then from (10), zn1 ¼ zn2 ¼ 4 1 ¼1, zn3 ¼ 0, and so from relation (24) we have ( Q þ 12 if Q r 12 z1 ðQ Þ ¼ z2 ðQ Þ ¼ ð27Þ 1 otherwise and z3 ðQ Þ ¼ 0 for all nonnegative values of Q which are shown in
z i (Q)
1 , then zi ðQ Þ ¼ zni for all values of Q, and since D is concave. If zni r 2r always zi ðQ Þ r zni . And finally, the value of Q at the optimum is the solution of the single-variable equation
∑ pi zi ðQ Þ Q ¼ 0:
107
z 1 (Q) , z 2 (Q)
1 1 2
z 3 (Q)
1 2 Fig. 2. The graph of zi(Q) for i¼ 1–3.
Q
108
F. Szidarovszky, Y. Luo / Reliability Engineering and System Safety 123 (2014) 104–109
G(Q)
and z3 ðQ Þ ¼ 0. Fig. 4 shows these function, and from (25), ( 2 1 if Q r0:95 3 Q þ 20 GðQ Þ ¼ 2 otherwise: 3
Q
2 3 1 3
G(Q)
This function is illustrated in Fig. 5, and the solution of Eq. (25) is Q ¼ 0:1. Then the zi values are as follows: z1 ¼ z2 ¼
Q
1Q 2 2 3
ð31Þ
3 ¼ 0:15 20
and
z3 ¼ 0
implying that the optimal decisions are
Fig. 3. Solution of Eq. (25) with r¼ 1.
m1 ¼ m2 ¼ 0:0536
and
m3 ¼ 0:
zi(Q)
6. Conclusions This paper introduced an optimization model and a special solution algorithm to determine the optimal resource allocation strategy of a defender against random attacks. By introducing new variables both the objective function and the budget constraint become concave, so standard computer package could be used to find the optimal solution. As an alternative approach, dynamic programming also can be used to find the solution. In the special case without budget constraint, we could introduce a very simple, efficient method which requires the computation of the unique solution of a single-variable, piece-wise linear, monotonic equation. The dynamic extensions of the method and considering multiple attacks will be the main focus of our future research, which will give the sequence of optimal decisions of the defender in the cases of repeated multi-stage and multiple attacks.
z1(Q), z2(Q)
1
z3(Q)
0.05
Q
0.95 Fig. 4. The graph of zi(Q) for i¼ 1–3.
G(Q) Q
2 3
G(Q)
1 30
Acknowledgements The authors wish to express their gratitude to two referees for useful comments and suggestions. The authors also acknowledge the constructive criticism of an earlier version of this paper by Michael L. Valenzuela, whose comments were very helpful in revising this paper.
Q 0.1
0.95 Fig. 5. Solution of Eq. (25) with r ¼10.
References Fig. 2. By assuming equal probabilities, p1 ¼p2 ¼p3 ¼1=3, Eq. (25) can be rewritten as GðQ Þ ¼ Q with GðQ Þ ¼
ð28Þ
( 2Q þ 1 3 2 3
if Q r 12
ð29Þ
otherwise
The two sides of Eq. (28) are shown in Fig. 3. The unique solution is Q ¼ 2=3 and from (27), z1 ¼ z2 ¼ 1 and z3 ¼ 0. The corresponding optimal decisions are m1 ¼ m2 ¼ 1 and m3 ¼ 0 from Eq. (26). Assume next that the defender is risk-neutral, that is, r ¼0 with all other model parameters remaining the same. From (24) we see that for all i, zi ðQ i Þ ¼ zni , since the first case of (24) cannot occur. This is also clear from (2), where only the first summation is nonzero and its terms are independent of each other. So z1 ¼ z2 ¼ 1 and z3 ¼ 0 giving the same solution as before. Select finally a large value r ¼10. Then from (24), ( z1 ðQ Þ ¼ z2 ðQ Þ ¼
1 Q þ 20
1
if
Q r 0:95
otherwise
ð30Þ
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Contents lists available at ScienceDirect
Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress
Incorporating risk seeking attitude into defense strategy Ferenc Szidarovszky a, Yi Luo b,n a b
Senior Researcher, ReliaSoft Corporation, 1450 S. Eastside Loop, Tucson, AZ 85710, United States Postdoctoral Research Fellow, University of Michigan Health System, Ann Arbor, MI 48103, United States
art ic l e i nf o
a b s t r a c t
Article history: Received 4 January 2013 Received in revised form 21 October 2013 Accepted 2 November 2013 Available online 11 November 2013
Optimal resource allocation is first found in defending possible targets against random terrorist attacks subject to budget constraint. The mathematical model is a nonconvex optimization problem which can be transformed into a convex problem by introducing new decision variables, so standard methods can be used for its solution. Without budget constraint the simplified model can be solved by a very simple algorithm which requires the solution of a single variable monotone equation. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Nonconvex optimization Certainty equivalent Homeland security
1. Introduction Defending objectives, which can be the targets of terrorist attacks, is one of the most important goals of homeland security. The outcomes of the actions of the defender are uncertain because they also depend on the random actions of the attacker. Game theory is the most appropriate approach to model the interactions between the attacker and the defender. Attacker–defender games have been intensively studied in recent years. Some researchers consider the players' payoffs as deterministic values and assume that the defender seeks to minimize the damage, while the attacker tries to maximize it [5,7]. However, the players' payoffs are usually random due to the uncertainty in the game, and therefore classic equilibrium approach has its limitations to find the solutions under this situation. Risk analysis is often used to capture the uncertainty resulting by the presence of random variables in the players' payoff functions. A production and conflict model is introduced and analyzed in [6] when two agents are fighting for as large as possible shares of the total production, which is determined by their contest success function. A twoperson conflict model is discussed in [22] when the agents can select between converting resources into arms or into useful production. The wining probabilities of the agents depend on their armament levels and the obtained reward depends on the amount of the useful production. The Nash equilibrium of this two-person game is determined and its dependence on the risk-taking attitudes of the agents is examined. A new contest
n
Correspondence author. E-mail address: [email protected] (Y. Luo).
0951-8320/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ress.2013.11.002
model is introduced in [4] which is an extension and generalization of the rent-seeking games where contest functions determine the winning probabilities and exponential utility function are assumed. The existence and uniqueness of the equilibrium is proved in the special case when every player has a constant degree of absolute risk aversion. Comparative static results are proved showing how the utility dissipation is affected by the risk-taking attitude of the agents and the precise nature of the technology. The central moments describe the nature of the distribution of a random variable mathematically, and any distribution can be characterized by the mean, the variance, the skewness, etc [19]. The first moment is usually considered as the payoffs of the players in attacker–defender games. For instance, Bier et al. [2] develop optimal strategies to allocate resources among possible defensive investments based on the assumptions that the attacker and the defender will maximize and minimize the expected damage of an attack on the system, respectively. In order to find best strategic defensive allocation against an unknown attacker, Bier et al. [3] consider cases when the attacker seeks to maximize the expected payoff from launching an attack and the defender tries to minimize the expected loss of an attack. Hausken and Zhuang [11,12] employ contest success function to describe the probability of damage and compute the government's and terrorists' expected utilities in the attacker–defender game. Azaiez and Bier [1] claim that the defender maximizes the expected cost of the attack by considering an investment to strengthen the defense capability of the object. Levitin [13] suggests optimal defense strategy that presumes separation and protection of system elements based on the consideration that the attacker tries to maximize the expected damage of an attack. Levitin and Hausken [14] propose that the defender can enhance system reliability by
F. Szidarovszky, Y. Luo / Reliability Engineering and System Safety 123 (2014) 104–109
either protecting a subset of the genuine system elements or deploying separated redundant elements and false elements. The agents' optimal strategies can be obtained from the expected damage when the cumulative performance of the system elements cannot meet a demand. Hausken and Levitin [8,9] consider the interaction of the attacker and the defender as a two-person simultaneous game where damages might occur to several elements of the system according to a binomial distribution, and the possible damage is determined with a general contest function including the intensity of the contest. Wang et al. [24] discuss the attacker–defender problem and analyze how to allocate resources to maximize the probability of core services availability by considering all kinds of components in the entire system. Since the values of the core services can be estimated from the demand, the objective function of the defender is equivalent to the expected value of the available services under the attacks. Similarly, the expected payoffs to the attacker and to the defender are also employed by several scholars in conducting research on systems defense and attack models [25,26,18,20]. To our best knowledge, in all earlier studies expected value of the random objective is considered and optimized [10]. Clearly, the characterization of a random variable becomes more accurate if higher moments are also considered. However, the complexity of the computation increases as well. Modeling the uncertainties in this game should include the risk seeking attitude of the players. As it is common in the economic literature, uncertain outcomes are substituted with their certainty equivalents [21] including the first two central moments of the random variables. The certainty equivalent of the payoff of the defender is a linear combination of its expectation and variance, where a risk attitude coefficient is assigned to the variance. In this paper it is assumed that there are several possible targets which can be attacked, and the defender has the assessment of the probability for each possible target to be attacked. These probability values can be obtained by using actual data from previous interactions with the attacker, or from information about its capabilities or from other sources. The question is to find optimal resource allocation strategy of the defender prior to the attack. This paper offers a mathematical model and solution algorithm for this problem before an actual attack occurs. Instead of computing the amount of the damage by a contest function, we determine the proportion of the maximum possible damage which can be avoided by the protecting actions of the defender. For the sake of mathematical simplicity we use a simple form of the contest function, more general forms (such as used in [6]) including intensity can be applied in a similar way. We also incorporate risk by including the variance of the random payoff of the defender into the objective function. After an attack occurs, the defender responds to it, then newer attack occurs, the defender responds again, and so on. A possible model and solution procedure are offered for the resulting multistage stochastic game for example, in our earlier work [16] and in the other papers mentioned earlier. In developing the mathematical model, we will first derive the payoff function of the defender including the random elements, and then its certainty equivalent will be determined based on its expectation and variance. However, this payoff function is not concave in general, so standard methodology cannot be used. By introducing new decision variables both the objective function and the budget constraint become concave, so the model is transformed into a convex programming problem. The rest of the paper is organized as follows. Section 2 introduces the mathematical model and the transformation into a convex programming problem is given in Section 3. In the case of unlimited or very high available budget an unconstrained
105
optimization problem is obtained, its special solution algorithm is introduced in Section 4. An illustrative example is given in Section 5. The last Section 6 concludes the paper with future research directions.
2. The mathematical model Suppose there are I independent possible targets, and let i be the index of them (i¼1, 2,…, I). The intruder is assumed to attack one target at each time. Combined attacks can be considered as single attacks, since we can consider the combinations of targets as new targets, and so combined attacks as separate attacks. Let ni (i¼1, 2,…, I) be the effort of the attacker to attack target i and let pi (i¼1, 2,…, I) be the probability of the actual attack. Let vi (i¼ 1, 2, …, I) be the highest possible damage in object i if it is attacked and the object is unprotected, and let mi (i¼ 1, 2,…, I) be the effort of the defender to protect target i against the attack. In addition, let ci (i¼1, 2,…, I) be the unit cost of this effort. It is assumed that the defender is able to avoid mi =ðmi þ ni Þ proportion of the possible highest damage if it is actually attacked. This simple formula can be replaced by more sophisticated expressions, which would not significantly modify the model. This expression is very similar to the contest function concept from economics. So in this case the defender's payoff is zi ¼ vi mi =ðmi þ ni Þ ci mi , which occurs with probability pi. Here we assume that defending action is made only in the case of an attack. If a possible target is not attacked, then the defending resources remain idle there. The defender's payoff is a discrete random variable with possible values zi and occurring probabilities pi (i¼1, 2,…, I). Therefore the expectation and variance of the defender's payoff z are as follows: I mi EðzÞ ¼ ∑ vi ci mi pi and mi þ n i i¼1 2 I mi ci mi pi ðEðuÞÞ2 ð1Þ VarðzÞ ¼ ∑ vi mi þ ni i¼1 If r denotes the risk seeking attitude of the defender, then the certainty equivalent ([21]) of its random payoff is given as 2 I I mi mi c i mi p i r ∑ v i c i mi D ¼ ∑ vi mi þ n i mi þni i¼1 i¼1 " #2 I mi pi þ r ∑ v i c i mi p i : ð2Þ mi þni i¼1 The value r ¼0 refers to risk neutral attitude, r 40 to risk aversion and r o0 to risk seeking behavior. In this study, we assume that r Z 0, that is, the risk seeking behavior is excluded, which is not realistic in our case, as it will be explained later. The decision variables are mi (i¼ 1, 2,…, I) and the values of vi, ni, ci, pi (i¼1, 2,…, I) and r are assumed to be known by the defender. Let B denote the defender's available budget, then the budget constraint can be formulated as I
∑ mi r B:
i¼1
ð3Þ
Hence our model is to maximize the objective function (2) subject to the budget constraint (3). Notice that the objective function (2) can be interpreted as using the weighting method in a multiobjective programming problem, where the first objective is to maximize expectation and the second objective is to minimize variance. Notice that constraint (3) is linear, however the objective function (2) is non-concave in general, so standard iteration methods [15] cannot be used to find optimum. In the next section
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new decision variables will be introduced which will transform the model into a convenient convex programming problem, so standard methods can be successfully applied.
The two possible shapes of the function zi(mi) are shown in Fig. 1, where the continuous curve corresponds to the case of msi 40 and the broken downward sloping curve shows the case when msi r 0. Proposition 1. Function (4) is concave as an I-variable function.
3. The transformed model
Proof. By differentiation
Notice first that I
I
i¼1
i¼1
D ¼ ∑ zi pi r ∑ z2i pi þr
I
∑ z i pi
i¼1
!2 ;
ð4Þ
where we use again the notation z i ¼ vi
mi c i mi : mi þni
ð5Þ
I ∂D ¼ pl 2rzl pl þ 2rð ∑ zi pi Þpl ; ∂zl i¼1
ð12Þ
∂2 D ¼ 2rpl þ 2rp2l ∂z2l
ð13Þ
and for ja l;
We will first examine zi as function of mi. Clearly, zi0
¼
vi ni
ci ðmi þ ni Þ2
ð6Þ
and z″i ¼
ð14Þ
So the Hessian matrix of D is the following: 2vi ni ðmi þ ni Þ
o 0; 3
ð7Þ
so zi is strictly concave for mi Z 0. The stationary point is the solution of equation zi0 ¼
∂2 D ¼ 2rpl pj : ∂zl ∂zj
vi ni
ci ¼ 0 ðmi þ ni Þ2
ð8Þ
0 B B B B B H¼B B B B @ 0
which is given as rffiffiffiffiffiffiffiffi vi ni ni : msi ¼ ci
ð9Þ
msi r 0,
If then zi strictly decreases in mi for mi 4 0. If then zi strictly increases for mi o msi and is strictly decreasing for mi 4 msi . In both cases zi ð0Þ ¼ 0 and zi ðmi Þ- 1 as mi -1. Let ( 0 if ni Z vcii zni ¼ ð10Þ pffiffiffiffi pffiffiffiffiffiffiffiffi 2 ð vi ni ci Þ otherwise;
B B B B B ¼B B B B @
0 2 p1 B C B p2 p1 C B C B C B C C þ 2r B B C B C B C @ A 2rpI pI p1 1
2rp1 2rp2
1
2rp1
pI p2
1 p1 pI C p2 pI C C C C C C C C A 2 pI
1
p1 B C C B p2 C C B C C B C C B C C C þ2r B C p1 B C C B C C B C C @ A A 2rpI pI
2rp2
msi 4 0,
p2
pI :
ð15Þ T
If u ¼ ðu1 ; u2 ; :::; uI Þ is any vector then !2 I
I
uT Hu ¼ ∑ ð 2rpi Þu2i þ 2r i¼1
2 I
Then the range of zi(mi) is the interval ( 1; zni ), since in the first case zi(mi) has its maximal value at 0 with zi(0) ¼0 and in the second case the maximum occurs at msi with pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi vi ni =ci ni zi ðmsi Þ ¼ vi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ci vi ni =ci ni vi ni =ci pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ¼ vi vi ni ci vi ni ci þ ci ni pffiffiffiffi pffiffiffiffiffiffiffiffi2 ¼ vi ni c i : ð11Þ
0
p1 p2 p22
¼ 2r 4 ∑
i¼1
u2i pi
∑ pi ui
i¼1
!2 3 ∑ ui pi 5 r 0; I
i¼1
ð16Þ
since the bracketed term is the variance of a discrete random variable with values ui and occurring probabilities pi. Notice first that if zi o 0, then mi 40 with negative objective function term, but mi ¼ 0 generates zi ¼ 0. Therefore rational decision maker does not select negative zi value. If msi r 0, then mi ¼ 0 is selected and if msi 4 0, then we have three possibilities: (a). If zi 4zni , then no solution exists for mi ; (b). If zi ¼ zni , then the only solution is mi ¼ msi ; (c). If zi ozni , then the smaller solution is chosen.
zi
z*i
The terms with mi ¼ zi ¼ 0 can be omitted from both the objective function and the constraint.
msi
mi
Proposition 2. The budget constraint is also transformed into a concave constraint.
Proof. From Eq. (5) we have Fig. 1. Shape of zi ðmi Þ in two cases.
ci m2i þ mi ðci ni þ zi vi Þ þ zi ni ¼ 0;
ð17Þ
F. Szidarovszky, Y. Luo / Reliability Engineering and System Safety 123 (2014) 104–109
so the smaller solution is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 vi zi ci ni ðci ni þzi vi Þ2 4ci zi ni : mi ¼ 2ci By differentiation 2
2
solutions of equations ð18Þ
3
∂mi 16 ci ni zi þvi 7 ¼ 4 1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5: ∂zi 2ci 2 ðci ni þ zi vi Þ 4ci zi ni
ð19Þ
mi =∂z2i
has the same sign as and clearly, ∂ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc n z þ vi Þ½2ðci ni þ zi vi Þ 4ci ni ðci ni þ zi vi Þ2 4ci zi ni i i qi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðci ni þ zi vi Þ2 4ci zi ni ð20Þ which has the same sign as the following expression: ðci ni þ zi vi Þ2 þ 4ci zi ni þðci ni zi þ vi Þ2 ¼ 4ci ni vi 4 0
ð21Þ
Therefore mi ðzi Þ is convex in zi , so the budget constraint I
B ∑ mi ðzi Þ Z 0
ð22Þ
i¼1
is concave. Hence, maximizing function (4) subject to constraints (22) and zi r zni (i¼ 1, 2,…, I) is a convex optimization problem, the solution of which can be found by standard gradient-type algorithms [15]. An alternative approach can be suggested based on the additive structures I
I
∑ zi pi ; ∑ z2i pi
i¼1
I
and
i¼1
∑ mi ðzi Þ
i¼1
as a dynamic programming problem [23] where the state variables are the partial sums of the above expressions. In the case of a general contest function mki i =ðmki i þ nki i Þ we have to introduce the new variable zi ðmi Þ ¼ vi mki i =ðmki i þ nki i Þ ci mi which does not change the methodology significantly.
4. Special algorithm in a special case Assume now that there is no budget constraint. In this special case we can derive a simple optimization algorithm which requires the solution of a single-variable, piece-wise linear, monotonic equation. The idea of this method is basically the same as it is known from oligopoly theory [17]. Notice that ∂D ¼ pi 2rpi zi þ 2rQ pi ∂zi
ð23Þ
where Q ¼ ∑Ii ¼ 1 zi pi , so the optimum value zi ðQ Þ of zi with given value of Q is ( 1 1 if Q r zni 2r Q þ 2r zi ðQ Þ ¼ ð24Þ n zi otherwise;
vi
mi ci mi ¼ zi ðQ Þ mi þ n i
I
i¼1
ð25Þ
Let H(Q) denote the left hand side and G(Q) the first term. At Q¼0, HðQ Þ Z 0 and as Q -1, HðQ Þ- 1, since G(Q) is bounded from above. Therefore there is at least one solution Q for Q, and the corresponding decisions mi can be obtained as the smaller
ð26Þ
for i¼1, 2,…, I. We can finally show the uniqueness of the solution of Eq. (12) by considering the following three cases: 1 (i) If zni r 2r for all i, then zi ðQ Þ ¼ zni regardless of the value of Q, so it is the unique solution with Q ¼ ∑Ii ¼ 1 zni pi . 1 (ii) If zni 4 2r for all i, then G(0) 40 and for small values of Q (until the second case of (24) occurs with some value of i), G′ðQ Þ ¼ ∑Ii ¼ 1 pi ¼ 1. And then G′ðQ Þ becomes less than unity implying that there is a unique intercept of the curve of GðQ Þ and the 451 line. 1 1 (iii) If zni 4 2r and znj r 2r with some i and j, then G(0)4 0 and G′ðQ Þ o 1 for all Q 40 implying again the uniqueness of the intercept of the curve of G(Q) and the 451 line.
The uniqueness of Q implies the uniqueness of the optimal solution, since in the case of two solutions for mi , the smaller solution is taken by any rational decision maker. It was assumed that the defender is not risk seeker, that is r Z 0. If r o0, then fix all zj (ja i) values and notice that D is quadratic in zi with positive quadratic coefficient rð pi þ p2i Þ as pi o 1. Therefore if zi -1, then D also tends to infinity, so D is unbounded, no optimal solution exists. We can briefly analyze how the value of the risk seeking attitude affects the optimal decision. Assume r o r and all other model parameters remain the same, including the value of zni . 1 1 1 1 Then zni 2r o zni 2r and Q þ 2r 4 Q þ 2r , so zi ðQ Þ with r is greater 1 than the value of zi ðQ Þ with r in the interval [0, zni 2r ), and they 1 are equal for Q Z zni 2r . That is, with increasing value of r, function zi ðQ Þ is non-increasing. Therefore in Eq. (25) the first term is also non-increasing in r but non-decreasing in Q. So the solution Q of 1 Eq. (25) is also non-increasing in r, and the same holds for Q þ 2r as well as for zi ðQ Þ. The definition of Q implies that it is the expected payoff of the defender, so increasing the value of r decreases the expected payoff at the optimum. Since zi ðQ Þ is non-increasing in r and the smaller solution of Eq. (26) is non-decreasing in zi ðQ Þ, the optimal effort mi of the defender is also non-increasing in r.
5. Illustrative example Consider the case of I ¼3, v1 ¼ v2 ¼ v3 ¼4, n1 ¼n ¼1, 2 ¼n p ffiffiffi 3p ffiffiffi2c1 ¼ c2 ¼1, c3 ¼ 4 and r ¼ 1. Then from (10), zn1 ¼ zn2 ¼ 4 1 ¼1, zn3 ¼ 0, and so from relation (24) we have ( Q þ 12 if Q r 12 z1 ðQ Þ ¼ z2 ðQ Þ ¼ ð27Þ 1 otherwise and z3 ðQ Þ ¼ 0 for all nonnegative values of Q which are shown in
z i (Q)
1 , then zi ðQ Þ ¼ zni for all values of Q, and since D is concave. If zni r 2r always zi ðQ Þ r zni . And finally, the value of Q at the optimum is the solution of the single-variable equation
∑ pi zi ðQ Þ Q ¼ 0:
107
z 1 (Q) , z 2 (Q)
1 1 2
z 3 (Q)
1 2 Fig. 2. The graph of zi(Q) for i¼ 1–3.
Q
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F. Szidarovszky, Y. Luo / Reliability Engineering and System Safety 123 (2014) 104–109
G(Q)
and z3 ðQ Þ ¼ 0. Fig. 4 shows these function, and from (25), ( 2 1 if Q r0:95 3 Q þ 20 GðQ Þ ¼ 2 otherwise: 3
Q
2 3 1 3
G(Q)
This function is illustrated in Fig. 5, and the solution of Eq. (25) is Q ¼ 0:1. Then the zi values are as follows: z1 ¼ z2 ¼
Q
1Q 2 2 3
ð31Þ
3 ¼ 0:15 20
and
z3 ¼ 0
implying that the optimal decisions are
Fig. 3. Solution of Eq. (25) with r¼ 1.
m1 ¼ m2 ¼ 0:0536
and
m3 ¼ 0:
zi(Q)
6. Conclusions This paper introduced an optimization model and a special solution algorithm to determine the optimal resource allocation strategy of a defender against random attacks. By introducing new variables both the objective function and the budget constraint become concave, so standard computer package could be used to find the optimal solution. As an alternative approach, dynamic programming also can be used to find the solution. In the special case without budget constraint, we could introduce a very simple, efficient method which requires the computation of the unique solution of a single-variable, piece-wise linear, monotonic equation. The dynamic extensions of the method and considering multiple attacks will be the main focus of our future research, which will give the sequence of optimal decisions of the defender in the cases of repeated multi-stage and multiple attacks.
z1(Q), z2(Q)
1
z3(Q)
0.05
Q
0.95 Fig. 4. The graph of zi(Q) for i¼ 1–3.
G(Q) Q
2 3
G(Q)
1 30
Acknowledgements The authors wish to express their gratitude to two referees for useful comments and suggestions. The authors also acknowledge the constructive criticism of an earlier version of this paper by Michael L. Valenzuela, whose comments were very helpful in revising this paper.
Q 0.1
0.95 Fig. 5. Solution of Eq. (25) with r ¼10.
References Fig. 2. By assuming equal probabilities, p1 ¼p2 ¼p3 ¼1=3, Eq. (25) can be rewritten as GðQ Þ ¼ Q with GðQ Þ ¼
ð28Þ
( 2Q þ 1 3 2 3
if Q r 12
ð29Þ
otherwise
The two sides of Eq. (28) are shown in Fig. 3. The unique solution is Q ¼ 2=3 and from (27), z1 ¼ z2 ¼ 1 and z3 ¼ 0. The corresponding optimal decisions are m1 ¼ m2 ¼ 1 and m3 ¼ 0 from Eq. (26). Assume next that the defender is risk-neutral, that is, r ¼0 with all other model parameters remaining the same. From (24) we see that for all i, zi ðQ i Þ ¼ zni , since the first case of (24) cannot occur. This is also clear from (2), where only the first summation is nonzero and its terms are independent of each other. So z1 ¼ z2 ¼ 1 and z3 ¼ 0 giving the same solution as before. Select finally a large value r ¼10. Then from (24), ( z1 ðQ Þ ¼ z2 ðQ Þ ¼
1 Q þ 20
1
if
Q r 0:95
otherwise
ð30Þ
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