Minimum Support Size o the Deender s Strong Stackeberg Equiibrium Strategies in Security Games Jiarui Gan University o Chinese Academy o Sciences The Key Lab o Inteigent Inormation Processing, ICT, CAS Beijing 100190, China ganjr@ics.ict.ac.cn Bo An Nanyang Technoogica University Singapore 639798 boan@ntu.edu.sg Abstract Stackeberg security games have been appied to address chaenges in security resource aocation o reaword inrastructure protection tasks. The key to such an appication is to eicienty compute the deender s optima strategy in consideration o the attacker s surveiance capabiity and best response. Experimenta resuts show that the deender s optima strategy oten uses ony a sma subset o pure strategies, as compared with the entire pure strategy set which can be exponentiay arge. A number o agorithms in the iterature have aready expoited this sma support size observation. This paper anayzes a number o widey studied security games and provides bounds on the minimum support size o the deender s Strong Stackeberg Equiibrium SSE strategies in security games. Introduction Stackeberg security games have been used to mode many rea-word scenarios where a deender commits to a strategy and an attacker makes its attacking decision with knowedge o the deender s commitment. Systems appying Stackeberg game modes to assist with randomized resource aocation decisions have been deveoped and are currenty in use, such as: ARMOR, deveoped at the Los Angees Internationa Airport LAX to randomize checkpoints on the roadways entering the airport and canine patro routes within the airport terminas Pita et a. 2008; IRIS, used by the Federa Air Marshas Service FAMS as a scheduer or randomized depoyment Tsai et a. 2009; PROTECT, used by the US Coast Guard USCG to randomize patroing at the port o Boston Shieh et a. 2012; and GUARDS, used by the United States Transportation Security Administration T- SA to scheduing resources to protect airports in the USA An et a. 2011b; Pita et a. 2011. The core o the above appications is computing o the deender s optima strategy, where diicuty ies in the arge scae o the deender s strategy space. Speciicay, a deender pays a mixed strategy which is a probabiistic distribution over a set o pure strategies. The pure strategy set can be very arge due to combinatoria exposion. For exampe, in a security game where the deender aocates securi- Copyright c 2013, Association or the Advancement o Artiicia Inteigence www.aaai.org. A rights reserved. ty resources to cover a set o n targets, the deender coud have 2 n pure strategies. How to optimize the deender s s- trategy over such a arge scae pure strategy space is thus a big chaenge. Athough agorithms have been designed to speed up the computing process or to scae up the mode Paruchuri et a. 2008; Kiekintved et a. 2009; Jain et a. 2010; Korzhyk, Conitzer, and Parr 2010; An et a. 2011c; Jain et a. 2011 it sti remains as an open research area to design more eicient agorithms to dea with scaabiity and uncertainty An et a. 2011a; 2012; Brown et a. 2012; Yang, Ordonez, and Tambe 2012; Yin and Tambe 2012; An et a. 2013. It has been noticed that ony pure strategies with non-zero probabiities in a mixed strategy contribute to the impementation o the mixed strategy whie the others are unused. We ca the set o pure strategies with non-zero probabiities the support o the mixed strategy. Experimenta resuts show that the deender s optima strategy oten has a sma support, as compared with the entire pure strategy set which can be exponentiay arge Shieh et a. 2012. A number o agorithms in the iterature have aready expoited this sma support size observation, and particuary avoided enumerating the entire set o pure strategies whie computing the deender s optima strategy. For exampe, a coumn generation method adds one best pure strategy into the support each time, unti the soution cannot be improved e.g., Jain et a. 2010; and a doube-orace based approach Jain et a. 2011; Jain, Conitzer, and Tambe 2013, where the payers pay a series o games iterativey using a subset o pure strategies unti convergence. One interesting question ies behind these appications that is yet not answered to the best o our knowedge is how sma the support o the deender s optima strategy can be. To answer the above question, this paper anayzes the structure o the deender s strategy in a number o widey s- tudied security games and provides bounds on the minimum support size o the deender s Strong Stackeberg Equiibrium SSE strategies in security games. The rest o the paper is organized as oows. We irst introduce the Stackeberg game and Stackeberg equiibria. Then starting with a typica Stackeberg security game which can be compacty represented, we present the bound o the minimum support size o the deender s SSE strategies or a genera Stackeberg game and, more generay, or a Bayesian Stackeberg game
which aows or mutipe types o attackers. In the end, we appy the bound to other types o speciaized Stackeberg security games. Stackebeg Games and Equiibria A Stackeberg game is payed by two payers: a eader and a oower. The eader acts irst and the oower observes the eader s strategy beore taking an action. Each payer has a set o pure strategies, denoted as S and T or the eader and the oower, respectivey. Let s j be the j th pure strategy in S and t i be the i th pure strategy in T. When the eader pays s j and the oower pays t i, they receive payos U j,i and, respectivey. Furthermore, the eader commits a mixed U j,i strategy x = x j which is a probabiistic distribution over the set S o pure strategies. Simiary, the attacker commits a mixed strategy y = y i. In this case, the expected payo or the eader and the oower are deined respectivey as oows: U x, y = T y i i=1 U x, y = T y i i=1 x j U j,i 1 x j U j,i 2 Stackeberg Equiibria In a Stackeberg equiibrium, the oower observes the eader s strategy x and responds with strategy x : x y that is optima with respect to his expected payo, i.e., U x, x U x, y or a other easibe strategies y; the eader, knowing the payo-driven behavior o the oower, chooses an optima strategy x that maximizes his payo, i.e., U x, x U x, x or a other easibe strategies x. Typicay, there might be more than one optima strategy or the oower, and each o these strategies may induce a dierent payo or the eader. Two types o Stackeberg equiibrium, the Strong Stackeberg Equiibrium SSE and the Weak Stackeberg Equiibrium WSE, were introduced Leitmann 1978; Breton, Aj, and Haurie 1988, where SSE assumes the oower breaks ties in avor o the deender and chooses the one that is optima or the eader, whie WSE assumes that the oower chooses the worst one or the eader. Most exiting work takes SSE as the soution concept as most iteratures in this research ied did, since 1 SSE exists in a Stackeberg games, whie WSE may not Basar et a. 1995; 2 SSE can aways be induced by the eader through deviating ininitesima rom the optima strategy Von Stenge and Zamir 2004. Deinition 1. A pair o strategies x, x orms a Strong Stackeberg Equiibrium SSE i it satisies the oowing: 1. The eader pays a best-response: U x, x U x, x, or a the eader strategies x. 2. The oower pays a best-response: U x, x U x, y, or a the oower strategies y. 3. The oower breaks ties in avor o the eader: U x, x U x, y, or a optima oower strategies y. Speciay, we restrict the attacker s strategy to pure strategies, i.e., x : x t, t T, because according to E- q. 2 i one mixed strategy is optima or the attacker, then a the pure strategies with non-zero probabiities shoud be optima, so that the attacker aways has a best pure-strategy response. It oows that the payo unctions deined in E- qs. 1 and 2 can be rewritten as assuming the attacker attacks t i : U x, t i = x j U ji, 3 U x, t i = x j U ji. 4 Bound the Minimum Support Size o the Leader s SSE Strategies In this section, we expoit the structure o a eader strategy and presents an upper bound on the size o the minimum support deined beow o the eader s SSE strategies. For ease o description, we use Φ to represent the upper bound on the minimum support size o the eader s SSE strategies. Denote x as the support size o a mixed strategy x, we have min x Φ, x X where X is the set o eader s SSE strategies. In other words, the eader aways has an SSE strategy that can be impemented by no more than Φ pure strategies. Deinition 2. A pure strategy is a support strategy o a mixed strategy i it is assigned with a non-zero probabiity by this mixed strategy. The set o a support strategies is caed the support o the mixed strategy. We start rom a specia Stackeberg security game mode presented beow, where a eader s strategy can be represented compacty as a coverage vector, since it provides an easyto-oow exampe o how arge the pure strategy set coud be and how tight the support coud be bounded. We ca this game mode compact game mode so as to distinguish it rom other types o Stackeberg security games. Then with a simiar idea that Φ or a compact game is obtained, we generaize it and present Φ or a genera Stackeberg game and urthermore a Bayesian Stackeberg game. Compact Game Mode The compact game mode appies to many security domains e.g., Pita et a. 2008; Tsai et a. 2009. In this game mode, the eader is a deender who protects a set o targets, and the oower is an attacker who wants to attack a target. Let the set o targets be 1,..., n. The deender aocates security resources to protect the targets, and his pure strategy is an aocation o the recourses. Typicay, in the compact game mode, a target is either covered or uncovered by security resources, and when it is covered, adding more resources to
it makes no dierence. In this case, a rationa deender assigns at most one resource to a target, and a pure strategy o the deender can thus be deined as a 0/1 coverage vector with the j th pure strategy s j = s ji {0, 1} n, where s ji = 1 represents that target i is covered and s ji = 0 uncovered. Correspondingy, the attacker s pure strategy is to choose one target to attack. Let t i be the attacker s pure s- trategy o attacking target i. When the attacker pays t i and target i is uncovered, he receives utiity U 0 at i, and the deender receives utiity U 0 d t i. Simiary, when the attacker pays t i and target i is covered, he receives utiity U 1 at i, and the deender receives U 1 d t i. It oows that when the deender pays a mixed strategy x, the targets are covered with probabiities cx = c i = s j S x js j with c i or target i, and are uncovered with 1 c. We reer to c as the coverage vector. For a strategy proie x, t i, the expected utiity or the deender and the attacker can be deined respectivey as oows: U d c, ti = ci U 1 d t i + 1 c i U 0 d t i, 5 U a c, ti = ci U 1 at i + 1 c i U 0 a t i. 6 Generay, some resource restrictions, such as scheduing constraints, can be enorced on the deender s pure strategy set S. However, even i in the presence o such restrictions, the size o S may sti be exponentiay arge in terms o the number o targets. For exampe, when there are m avaiabe resources, the size o S is at east n m. Φ or a Compact Game According to Eqs. 5 and 6, t- wo deender strategies x 1 and x 2 resuts in the same attacker response and moreover the same utiity or each payer, i they induce the same coverage vector, i.e., cx 1 = cx 2. We utiize this observation to seek equivaent deender s- trategies with smaer supports. For exampe, when there are three targets and the deender pays a mixed strategy where pure strategies 1, 0, 0, 1, 1, 0, 0, 1, 0 and 0, 1, 1 are assigned with probabiities 0.25 or each, a coverage vector 0.5, 0.5, 0.5 is induced. In this case, it is aso possibe to use ony pure strategies 1, 0, 0 and 0, 1, 1 with probabiity 0.5 or each and a the others with probabiity 0, which induces the same coverage vector. In act, a deender strategy irst induces a coverage vector and then aects the game through this coverage vector. Thereore, rather than speciying an exact mixed strategy, the deender coud irst cacuate an optima coverage vector that is impementabe by his easibe mixed strategies, i.e., c P = x j s j x 0, 1T x = 1, such that s j S U d c, c U d c, c, c P, and then impement c with a mixed strategy which is optima or the deender in this case. Note that the impementabe coverage vector set P is a convex hu in an n- dimensiona space deined by points in S. According to the Carathéodory s theorem Danninger-Uchida 2001, any Figure 1: Carathéodory s theorem: in a pane i.e., a 2- dimensiona space, any point x in a convex hu aso ies in a 2-simpex. point, in particuar c, in the convex hu ies in an r-simpex with vertices in S, where r n, and any point in the r- simpex can be represented as a convex combination o the r + 1 vertices o the simpex. Thereore, c can aways be impemented by no more than n + 1 pure strategies corresponding to the vertices o the simpex where c ies in, and an upper bound Φ = n+1 is obtained or the compact game mode Coroary 2. Theorem 1 Carathéodory s theorem Danninger-Uchida 2001. I a point x R d ies in the convex hu o a point set P, there is a subset P o P consisting o d + 1 or ewer points such that x ies in the convex hu o P, i.e., x ies in an r-simpex with vertices in P, where r d Figure 1. Coroary 2. The minimum support size o the deender s SSE strategies in a compact game is n + 1, where n is the number o targets. Φ or a Genera Stackeberg Game The compact game mode is a specia case o Stackeberg security games. In other appication o Stackeberg game modes, the deender s strategies over the targets may be more compex than being simpy either covering or not covering. For exampe, in some rea-word scenarios, it makes a dierence when dierent number or dierent types o security resources are assigned to a target Pita et a. 2011; Shieh et a. 2012; and in a network-based game Washburn and Wood 1995; Tsai et a. 2010; Jain et a. 2011; Jain, Conitzer, and Tambe 2013, the deender paces security resources on the edges e.g., streets, roads that ead to the target, instead o directy on the targets more concrete exampes are presented oowing the deinitions o the modes in the next section. In these cases, the deender strategy cannot be represented as coverage over the targets, and Coroary 2 is thus not appicabe. In this section, we present Φ or a genera Stackeberg game using a simiar idea that Φ or a compact game is obtained. This genera Φ appies to a security games derived rom Stackeberg games, which we exempiy in the next section. According to Eqs. 3 and 4, i two eader strategies x 1 and x 2 satisy U x 1, t i = U x 2, t i and U x 1, t i = U x 2, t i, i = 1,..., T, the oower woud respond with the same pure strategy according to U when they are payed in particuar, when there are mutipe optima pure x
strategies or the oower, he reers to U and aso has the same response and receive the same payo according to U ; and the oower s response woud induce the same eader payo; x 1 and x 2 are thus equivaent or both payers in terms o the payos induced. We write the above conditions as oows: x 1j u j = x 2j u j, where u j = U j1,..., U j T, U j1,..., U j T T is a 2 T - dimensiona vector. Thereore, i some strategy proie x, t i orms SSE, another strategy proie x, t i, such that x j u j = x j u j, aso orms SSE. This is equivaent to impementing the 2 T -dimensiona vector x j u j with the set o points {u 1,..., u } as their convex combination. Simiar to the anaysis in the ast section, according to the Carathéodory s theorem, there is aways an impementation where no more than 2 T +1 points have non-zero weights, which corresponds to an SSE strategy or the eader with a support o ess than 2 T + 1 pure strategies, namey, Φ = 2 T + 1 or a genera Stackeberg game. A Tighter Φ Deeper anaysis indicates that Φ = 2 T + 1 can be even tighter. To present the tighter Φ, we irst assume that there is ony one optima strategy or the oower, and we reax this assumption ater. In this case, or t- wo eader strategies x 1 and x 2, the oower responds with the same optima strategy i U x 1, t i = U x 2, t i, i = 1,..., T note that the condition U x 1, t i = U x 2, t i, i = 1,..., T is not necessary since there is no tie by assumption; and the eader receives the same payo i U x 1, t = U x 2, t, where t is the oower s optima strategy. Now given the SSE strategy proie x, t i, any strategy proie x, t i, such that x ju j = x j u j, 7 aso orms SSE, where u j1 j = U,..., U j T, U ji T. The rest is the same with the proo o the previous Φ, so that Φ = T + 2 given the assumption that there is ony one optima strategy or the oower. Next, we reax the assumption on the oower s optima strategies and show that the above concusion sti hods. We show that a strategy proie x, t i with x satisying E- q. 7 sti orms SSE. When the assumption is reaxed and the eader pays x, the oower may choose another pure strategy t i that is aso optima or him. There are oowing cases: 1 x, t i gives the eader the same payo as x, t i, then it is aso optima or the eader and satisies the SSE condition; 2 x, t i gives the eader a dierent payo, which is ater a impossibe because i it gives the eader a higher payo, then it is better than x, t i, which contradicts that x, t i orms SSE; i it gives the eader a ower payo, then the oower shoud not choose it because he break ties in avor o the eader. Thereore, x is the eader s optima strategy, a tighter Φ o T + 2 is obtained Coroary 3. Coroary 3. The minimum support size o the eader s SSE strategies in a genera Stackeberg game is T + 2, where T is the set o oower s pure strategies. Generaize Attacker Types: Φ or a Bayesian Stackeberg Game A Bayesian Stackeberg game aows mutipe types o eaders and oowers. Typicay, the eader type is restricted to one or the security game interest. The Bayesian Stackeberg games arise in scenarios where the eader has uncertain knowedge about dierent types o oowers she may ace Paruchuri et a. 2008; Jain, Kiekintved, and Tambe 2011. Athough Coroary 3 can be appied to a Bayesian Stackeberg game by using the Harsanyi transormation to transorm mutipe oowers to a singe oower Harsanyi and Seten 1972, the singe oower s pure strategy space is the cross product o each oower type s pure strategy set, which can be exponentiay arge. The Φ obtained may thus be meaningess. In the oows we present a tighter Φ or a Bayesian Stackeberg game, not appying Coroary 3 directy but using the same core idea. Let there be Λ types o oowers. A oower o type λ occurs with probabiity p λ and has a set T λ o pure s- trategies indexed by I λ = {1,..., T λ }. Given a strategy proie x, I, where x is the eader s mixed strategy, and I = i λ I 1 I Λ represents the indices o the oowers pure strategies, the expected payos or oower λ and the eader are deined respectivey as oows: U λ x, I = x j U ji λ λ, 8 U Λ x, I = λ Λ p λ x j U ji λ, 9 where U ji λ λ and U ji λ are respectivey the payos or oower λ and the eader when they choose pure strategies s j S and t iλ T λ. Given an SSE strategy proie x, I, where I = i λ, x, I aso orms SSE i x satisy the oowing equation which is a variant o Eq. 10: x ju λj = x j u λj, λ Λ, 10 where u j1 λj = Uλ,..., U j T λ λ, U ji λ T is speciied or each type o oowers. Obviousy, each oower makes identica responses under x and x according to Eq. 8, and the eader receives the same expected payo according to Eq. 9. Combining u λj or a λ Λ makes a vector o dimension λ Λ T λ + 1. By appying the Carathéodory s theorem, Φ = λ Λ T λ + 1 + 1 is then obtained or a Bayesian Stackeberg game Coroary 4. Coroary 4. The minimum support size o the eader s SSE strategies in a Bayesian Stackeberg game is λ Λ T λ + 1 + 1, where T λ is the pure strategy set o oower type λ.
Φ or Other Stackeberg Security Games In this section, we review other types o Stackeberg security game modes discussed in the iterature and appy Coroary 3 to these security game modes. Mutipe Protection Types or a Target In some rea-word scenarios, it makes a dierence when d- ierent number or types o security resources are assigned to a target. For exampe, an attacker woud be more ikey captured at a target protected by ten security guards than one that is protected by ony one guard. The deender is abe to execute a variety o security activities on each target. Each activity π requires m π security resources and provides d- ierent payos U π d t i and U π a t i when it is executed on target i and the attacker attacks this target. For exampe, as shown in Tabe 1, the deender can execute three activities {π 0, π 1, π 2 }, where π k assigns k security guards on a target. Assigning more security guards on a target provides a higher/ower payo or the deender/attacker. The deender s pure strategy is thus an assignment o a m avaiabe security resources to execute activities on the n targets, which is more compex than a 0/1 vector. Furthermore, we can construct a counter exampe as oows, which impies that the game cannot be represented through a coverage vector. Suppose the attacker attacks target 1, and the deender executes π 0 and π 2 on target 1 with probabiity 0.5 or each, the expected payos or both payers are 10 and 5, respectivey; however, by executing π 1 on target 1 with probabiity 1.0, the same coverage rate can be attained, whie the payos or the payers, being 20 and 15 respectivey, are dierent rom the previous case. π 0 π 1 π 2 t 1-10, 30 20, -15 30, -20 t 2-15, 40 20, -10 30, -15 Tabe 1: Mutipe protection types or each target each entry shows the deender s/attacker s payo. Despite o the more compex deender strategy, the number o attacker s pure strategies, being equa to the number n o targets, remains unchanged as compared with the compact game mode. Appying Coroary 3, we obtain Φ = n + 2. A Network-based Security Game In a network-based game mode Washburn and Wood 1995; Tsai et a. 2010; Jain et a. 2011; Jain, Conitzer, and Tambe 2013, a deender takes action on a graph G = V, E with a set V o nodes and a set E o edges. The attacker is abe to start at a node s S V and traves through a path P in an attempt to reach one o the targets t T V. The deender paces security resources on the edges, instead o directy on the targets, to capture the attacker. I an edge is covered by a resource, and the attacker traves through this edge, the attacker is captured; otherwise, the attacker traves through this edge successuy. Thereore, the deender s pure strategy is an aocation o resources on the edges denoted by the s e 1 e 2 t a Coverage over targets s e 1 e 2 e 3 e 4 b Coverage over edges Figure 2: Counter exampes where coverage vectors do not make sense. set L o covered edges; the attacker s pure strategy is an s-t path P rom a start node to a target node. Denote the set o easibe deender and attacker pure strategies as L and P, respectivey. The network-based security game cannot be represented through a coverage vector over the targets. A counter exampe is shown in Figure 2a. I the deender has one resource and pays a mixed strategy with 0.5 probabiity or each o the pure strategies {0, 1} i.e., pacing one resource on e 1 and no resource on e 2 and {1, 0}, then a coverage o 0 is induced since the target is not covered by neither o the pure strategies, but the attacker has 0.50 chance o being caught no matter which path he chooses. Furthermore, a coverage vector over the edges does not make sense, either. This is shown by a counter exampe in Figure 2b. I the deender has two resources and pays {1, 0, 1, 0} and {0, 1, 0, 1} with 0.5 probabiity or each, a coverage vector o 0.5, 0.5, 0.5, 0.5 T over the edges is induced, which is the same with paying {1, 1, 0, 0} and {0, 0, 1, 1} with 0.50 probabiity or each. However, the attacker wi aways be caught under the ormer deender strategy whie he has 0.50 chance o not being caught under the atter one no matter which path he chooses. By appying Coroary 3, Φ = P + 2 is obtained or a network-based security game. Speciicay, in a networkbased security game, P can be very arge, such that P + 2 > L, which corresponds to the case where the vertex number i.e., L o a convex hu is even smaer than the dimension i.e., P + 2 o the space. Since obviousy the support size cannot be arger than the number o pure strategies, a more accurate Φ is min{ P + 2, L }. Note that this specia case is not discussed when we obtain the previous Φs because the pure strategy set o the deender is more ikey to be much arger than that o the attacker. Concusion In this paper, we anayze the structure o the deender s s- trategy in a number o widey studied security games and provides Φ or these games. Φ = T + 2 is obtained or a genera orm two-payer Stackberg game, and more generay Φ = λ Λ T λ + 1 + 1 or a Bayesian Stackeberg game. Apparenty, Φ depends ony on the oower s pure strategy space, so that the support o the deender s SSE s- trategy can be bounded very tighty when the pure strategy space o the attacker is drasticay smaer than that o the deender. This happens in the compact game mode, and the mode with mutipe protection types or each target. When t
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