Defender (Mis)coordination in Security Games

Transcription

1 Albert Xin Jiang University of Southern California Defener (Mis)coorination in Security Games Ariel D. Procaccia Carnegie Mellon University Yuni Qian University of Southern California Nisarg Shah Carnegie Mellon University Abstract We stuy security games with multiple efeners. To achieve maximum security, efeners must perfectly synchronize their ranomize allocations of resources. However, in real-life scenarios (such as protection of the port of Boston) this is not the case. Our goal is to quantify the loss incurre by miscoorination between efeners, both theoretically an empirically. We introuce two notions that capture this loss uner ifferent assumptions: the price of miscoorination, an the price of sequential commitment. Generally speaking, our theoretical bouns inicate that the loss may be extremely high in the worst case, while our simulations establish a smaller yet significant loss in practice. 1 Introuction Security games, a special class of Stackelberg games, have been eploye as ecision ais to scheule limite security resources at critical infrastructure sites incluing key airports an ports in the Unite States (e.g., Los Angeles, Boston, New York) as well as in protecting transportation infrastructure (international flights an metro trains) [Tambe, 2011; Shieh et al., 2012]. These eployments have fuele the security games research area, with focus on efficient computation for scale-up an hanling the significant uncertainty in this omain [Korzhyk et al., 2011a; Basilico et al., 2009; Yin et al., 2010; Jain et al., 2010]. Previous research in security games has assume a single efener agency with control over all the security resources even if there were multiple attacker types or multiple coorinate attackers [Paruchuri et al., 2008; Korzhyk et al., 2011b]. Yet at most major critical infrastructure sites, e.g., port of New York or the Los Angeles International Airport, multiple efener agencies incluing city police, state an feeral law enforcement agencies are responsible for security [Panel, 2011]; these agencies each have control over their own security resources an scheules. Jiang, Qian an Tambe were supporte by US Coast Guar grant HSHQDC-10-D an MURI grant W911NF Procaccia an Shah were supporte by NSF grant CCF an a gift from Microsoft. Milin Tambe University of Southern California tambe@usc.eu In theory, achieving perfect coorination is simple: efeners can pool their resources together, an employ a single centralize algorithm to allocate resources. Our work is motivate by the basic observation that this ieal is far from reality; in practice we observe limite coorination between efeners, which we believe can lea to significantly poorer performance. For example, it is well known that at major ports such as the port of New York, the US Coast Guar an ifferent police epartments each scheule their own boat patrols an security resources inepenently; there is no centralize scheuling. Our research question is therefore: how much o efeners lose ue to lack of coorination? While the ieal coorination mechanism may not be feasible from a policy perspective, our goal is to use it as a gol stanar in orer to inform policy makers an ultimately bring about a higher level of coorination. To that en, the paper offers four key contributions. First, we introuce an analyze the price of miscoorination (PoM) in simultaneous move settings an provie bouns on the worst case loss uner ifferent assumptions of target values. Secon, we introuce the price of sequential commitment (PoSC) when efeners move sequentially. We illustrate that even in situations with low PoM, PoSC may be unboune. Thir, we introuce techniques to compute PoM an PoSC. Finally, we present experimental results base on realistic port scenarios illustrating a smaller yet significant loss in practice. 2 Backgroun on Security Games A security game is a 5-tuple (T, S, R, A, U). T is the set of targets; we enote T = n. A scheule is a subset of the target set T ; S 2 T enotes the collection of feasible scheules. R is the set of resources that belong to the efener; we enote R = m. Every resource has a collection of scheules (a subset of S) to which it can be assigne; A : R 2 S is the function that maps a resource to its collection of possible scheules. In our theoretical results we make the stanar assumption that for any resource, any subset of a feasible scheule is itself a feasible scheule [Yin et al., 2010]. When a resource is assigne to a scheule, we say that all the targets in the scheule are covere by the resource. The payoffs are given by four ifferent utility functions. If target t is attacke, the efener s utility is U c (t) if t was covere, an U u (t) if t was not covere. Similarly,

2 the attacker s utility is Ua(t) c if t was covere, an Ua u (t) if t was not covere. We assume that U c(t) U u (t) an Ua(t) c Ua u (t). Note that it makes no ifference to the players utilities whether a target is covere by one resource or by more than one resources. A pure strategy of the efener is an assignment of resources to feasible scheules; a efener may opt to employ a mixe strategy, which selects at ranom from pure strategies accoring to a istribution. The solution of the security game is the optimal Stackelberg strategy for the efener. Given the efener s mixe strategy that ranomly allocates resources to scheules, the attacker selects a best response by choosing to attack a target that maximizes its expecte utility. The efener chooses its mixe strategy to maximize its own utility given that the attacker best respons. Given the emphasis on explicit game-theoretic moels an equilibrium analysis, our paper also complements research on strategies for multi-robot patrol [Agmon, 2010; Agmon et al., 2009]. 3 The Price of Miscoorination We exten the basic moel to the setting where multiple efeners, with their isjoint sets of resources, efen a set of targets against a single attacker. In particular, we analyze the nee for the efeners to coorinate their moves, an the incurre loss when coorination is lacking. 3.1 Our Moel Let D be the set of efeners; we enote D =. Let the set of resources R be partitione into {R i } i D where R i is the set of resources owne by efener i D. We take an optimistic point of view by assuming that all efeners are intereste in overall security, hence they are all enowe with the same utility functions U c an U u as before. The maximum utility is achieve by the efeners when they pool their resources together, an scheule the whole set R of resources as if they were owne by a single efener. We call this strategy profile the optimal correlate profile (OCP) as the mixe strategies of iniviual efeners are correlate. In other wors, the exact realizations of assignments of the resources of ifferent efeners may be epenent on each other. This represents the scenario with full coorination. Consier an alternative scenario, where the efeners choose their mixe strategies but these mixe strategies are uncorrelate. In other wors, the instantiations of the various mixe strategies are inepenent of each other. We efine the optimal uncorrelate profile (OUP) as the profile of uncorrelate mixe strategies for the efeners that yiels maximum utility among all profiles of uncorrelate mixe strategies. Clearly, the utility to the efeners uner OUP is no greater, an may be strictly smaller, than the utility uner OCP. We efine the supremum (over a given class of security games) of the ratio of the utility uner OCP to that uner OUP as the price of miscoorination (PoM). Example 1. Let there be two efeners, efener 1 with resource r 1 an efener 2 with resource r 2. Let T = {t 1, t 2, t 3 }, with U u (t) = U c a(t) = 0 an U c (t) = U u a (t) = 1 for all t T. Resource r 1 can cover either target t 1 or target t 2, an resource r 2 can cover either target t 2 or target t 3. The optimal correlate profile (OCP) uniformly ranomizes between assigning the resources to {t 1, t 2 }, {t 1, t 3 }, an {t 2, t 3 }. Each target is covere with probability 2/3, hence the efeners are guarantee utility 2/3. In contrast, uner the optimal uncorrelate profile (OUP) each efener covers its own target (t 1 or t 3 ) with probability ( 5 1)/2 = 0.618, an the share target t 2 with the complement probability (which equalizes the coverage probability of t 2 with t 1 an t 3 ). The efeners utility is therefore 0.618, an the ratio of the utilities uner OCP an OUP is 1.078, which is therefore a lower boun for the PoM. The PoM is relate to the notion of meiation value [Ashlagi et al., 2008; Braonjic et al., 2009] (which in turn is inspire by the price of anarchy [Koutsoupias an Papaimitriou, 1999; Roughgaren an Taros, 2002]). Briefly, the meiation value in a game is the ratio between the maximum social welfare (i.e., sum of utilities) in any correlate equilibrium an the maximum social welfare in any mixe-strategy Nash equilibrium. If we efine an artificial game between the efeners, where the utility function of each efener is the same as the common utility function, then the OCP is the welfare-maximizing correlate equilibrium an the OUP is the welfare-maximizing Nash equilibrium. Hence, the PoM coincies with the meiation value in this game. To guarantee that the PoM is meaningful, we assume hereinafter that the efener utilities are non-negative. General security games may have negative utilities; however, by shifting all efener utilities by the minimum efener utility we can obtain a security game with non-negative efener utilities, which is equivalent in terms of its optimal strategies for the efeners an the attacker. Moreover, the PoM of the shifte game has a natural interpretation in the original game: informally, it tells us what fraction of the gap between the worst possible outcome an the best possible correlate outcome is ue to miscoorination. 3.2 Bouns on the PoM Our first result shows that in some games coorination is crucial, as the price of miscoorination may be arbitrarily large. Theorem 1. The PoM is unboune in general security games. Proof. Let there be two efeners, efener 1 with resource r 1 an efener 2 with resource r 2. Let T = {t 1, t 2, t 3 }. Resource r 1 can cover either target t 1 or target t 2, an resource r 2 can cover either target t 2 or target t 3. In terms of structure, this is ientical to the game of Example 1. However, the utilities are efine ifferently. For the efeners, targets t 1 an t 3 are ientical. If either target is attacke, it gives the efeners utility x if covere an 1 if uncovere, where x > 1. Target t 2, if attacke, gives the efeners utility 0 whether it was covere or not. Formally, U c (t 1) = U c (t 3) = x, U u (t 1) = U u (t 3) = 1 an U c (t 2) = U u (t 2) = 0. For the attacker, targets t 1 an t 3 give utility 1 if uncovere an 0 if covere. Target t 2 gives the attacker utility z if uncovere an 0 if covere. Formally, U c a(t) = 0 for all t T, U u a (t 1 ) = U u a (t 3 ) = 1 an U u a (t 2 ) = z.

3 Irrespective of the values of x an z, the efeners never let the attacker attack target t 2 as it woul give them the worst possible utility (zero utility). Now fix x an increase z. As z, the utility of the attacker for a successful attack against target t 2 increases, hence it must be covere by the efeners with probability 1 in the limit to avoi such an attack. Thus, as z, the OCP for the efeners converges to assigning r 1 {t 1 } an r 2 {t 2 } with probability 1/2, an r 1 {t 2 } an r 2 {t 3 } with probability 1/2. This way, target t 2 is fully covere while both targets t 1 an t 3 are covere with probability 1/2, giving the efeners utility 1/2 + x/2. In the limit of the uncorrelate case, at least one efener must cover target t 2 with probability 1. The other efener can only cover one of targets t 1 an t 3. The attacker attacks the other target, resulting in a utility of 1 for the efeners. Thus, taking z in this security game implies that the ratio of utility uner OCP an OUP is at least (1/2 + x/2)/1. As x can be arbitrarily large, the PoM is unboune. The proof of Theorem 1 relies on an extreme asymmetry in the utilities for ifferent targets. It is natural to ask, what is the PoM when the targets are ientical? Specifically, the utility functions of the efeners satisfy U c(t) = x an U u(t) = 0 for all targets t T an some x 0, where the utility for targets being uncovere is zero since we shift the utility functions to make the worst-case utility zero. Similarly, the utility functions of the attacker satisfy Ua(t) c = 0 an Ua u (t) = x for all targets t T an some x 0. We believe that although this is a special case, it is useful for practitioners to explore the extreme points of the problem. It is also worth noting that the case of ientical targets is still rich, even from a complexity-theoretic point of view: it immeiately follows from existing proofs [Korzhyk et al., 2010, Theorem 5] that computing the optimal efener strategy for this case is N P- har, even for one efener an scheules of size at most 3. It is easy to check that in the case of ientical targets, the attacker woul attack the target that is covere with minimum probability in orer to maximize his own utility. Therefore, the utility of the efeners is proportional to the minimum probability with which any target is covere. Let c t enote the coverage probability of target t T in a strategy profile, an let c min = min t T c t enote the minimum coverage probability. Then we are intereste in the ratio c min (OCP )/c min (OUP ). We show that in this case the PoM is upper-boune by a small constant. Theorem 2. The PoM of security games with ientical targets e is at most e Proof. Given a security game with ientical targets an efeners, we claim that there exists an OCP where no target is covere by more than one efeners simultaneously in any pure strategy realization of the OCP. Inee, if a target is covere by more than one efener in a realization, we remove resources of all but one efener assigne to that target in that realization (the assumption that any subset of a feasible scheule is a feasible scheule plays a key role here). Next, consier any such OCP. Let c t,i enote the probability with which efener i covers target t in the OCP. Since a target is not covere by more than one efener at a time, the coverage probability of target t is c t,i, so c min (OCP ) = min t T c t,i. Now consier the marginal uncorrelate profile (MUP) where each efener i scheules its resources as they were scheule in the OCP, but the mixe strategies of ifferent efeners are now uncorrelate. 1 In other wors, each efener follows the mixe strategy that is obtaine as the marginal of the OCP. In this case, target t is still covere by efener i with probability c t,i. However, a target may be covere by more than one efener simultaneously, so the probability of target t being covere overall is 1. Thus, [ ] c min (OUP ) c min (MUP ) = min 1 (1 c t,i ), t T an it follows that min t T P om c t,i ( min t T 1 ). We claim that for all t T, 1 (1 c t,i ) ( 1 1 ) e c t,i. (1) Inee, if c t,i = 0, then Equation (1) follows trivially. Hence, assume that c t,i > 0. Now, 1 c t,i ct,i 1 e c t,i 1 1 e, where the first transition hols because 1 x e x, an the last transition is true since f(x) = (1 e x )/x is a ecreasing function in (0, 1], so f(x) f(1) for all x (0, 1]. Also, c t,i is the coverage probability of target t in the OCP, so 0 < c t,i 1. We have thus establishe (1). We can now conclue that min t T P om c t,i [ min t T 1 ] [ ] min t T = max c t,i t T 1 max c t,i t T 1 e e 1, where the last transition is ue to Equation (1). Is it possible that the upper boun for ientical targets is in fact much closer to 1? Our next theorem answers this question in the negative. Theorem 3. The PoM of security games with ientical targets is at least 4/3. 1 The mixe strategy assignments of various resources of the same efener are still correlate as before.

4 Proof. Consier the following security game with ientical targets. There are efeners, each efener i has a single resource r i. There are + 1 ientical targets, t 1,..., t +1. For 1 i, resource r i can cover either target t i or target t i+1. Consier the following + 1 pure strategies where strategy i leaves only target i uncovere. This is uniquely achieve by assigning resource r j to target t j for j < i an resource r j to target t j+1 for j i. Since at least one target must be uncovere in any pure strategy, the OCP uniformly ranomizes over these + 1 pure strategies to achieve the optimal minimum coverage probability c min (OCP ) = 1 1/(+1). We next prove that c min (OUP ) < Suppose for contraiction that c min (OUP ) 0.75, so all targets are covere with probability at least First, we prove by inuction that for all 1 i, resource r i covers target t i with probability at least 0.5. For the base case of resource r 1, this is obvious since target t 1 is covere with probability at least 0.75 an r 1 is the only resource that can cover it. Suppose it is true for resource r k. Observe that resource r k covers target t k with probability at least 0.5, hence it covers target t k+1 with probability at most 0.5. If resource r k+1 covers target t k+1 with probability less than 0.5, then ue to the lack of correlation between the assignments of resources r k an r k+1, the probability of coverage of t k+1 woul be less than = 0.75, which contraicts the assumption. Hence, r k+1 must cover t k+1 with probability at least 0.5. Therefore, the inuction hypothesis hols. In particular, resource r covers target t with probability at least 0.5, an hence target t +1 with probability at most 0.5. It follows that the coverage probability of target t +1 is at most 0.5, which yiels a contraiction. Hence, c min (OUP ) < We conclue that P om (1 1/( + 1))/0.75. Taking, we get that P om 4/3. 4 The Price of Sequential Commitment The PoM is optimistic in a sense, because it compares the optimal correlate utility with the optimal uncorrelate utility. Even achieving the optimal uncorrelate utility requires some level of coorination among the efeners: they are not pooling their resources together, but they are coorinating their mixe strategies. Below we consier a more pessimistic moel of efener commitment, an show that in this moel the loss is unboune even when targets are ientical. 4.1 Our Moel We assume that the efeners commit sequentially. The first efener commits to a mixe strategy that optimizes the joint utility function in the absence of the other efeners. Subsequently, each efener chooses a mixe strategy that maximizes the joint utility function given the strategies of the earlier efeners, in the absence of the later efeners; note that there may be more than one optimal strategy. The mixe strategies of various efeners are still uncorrelate. 2 In such a moel, we can now consier the loss in utility ue to sequential commitment compare to the OCP. We e- 2 Our results hol even if the efeners commit to mixe strategies that are correlate with the strategies of the earlier efeners. fine two prices: the price uner the best orer of commitment (PoSC b ) an the price uner the worst orer of commitment (PoSC w ). Formally, PoSC b (resp. PoSC w ) is the supremum (over all security games) of the ratio of the utility uner the OCP to the maximum (resp. minimum) utility over all orers of sequential commitment. 3 It is easy to check that the PoM is a lower boun for the PoSC b, which in turn is a lower boun for the PoSC w. 4.2 Bouns on the PoSC While Theorem 2 shows that the PoM in security games with ientical targets is upper-boune by a small constant, we show that this is not the case even for PoSC b in security games with ientical targets. Theorem 4. The PoSC b (even) in security games with ientical targets is unboune. Proof. Consier a security game with two efeners: efener 1 owns resource r 1 an efener 2 owns resource r 2. The target set T = T 1 T 2 consists of 2 k ientical targets, where T 1 = T 2 = k. Resource r 1 can either cover all the targets in T 1 simultaneously or any single target in T 2, an resource r 2 can either cover all the targets in T 2 simultaneously or any single target in T 1. The efener who commits first has k + 1 feasible scheules for its resource. To maximize the minimum coverage probability, it uniformly ranomizes over the feasible scheules to cover each target with ientical probability 1/(k + 1). Due to the uniform coverage by the first efener, the efener committing later also uniformly ranomizes over its k + 1 feasible scheules. Hence, each target is covere with probability at most 2/(k +1). Note that the optimal strategies are unique. In contrast, we have c min (OCP ) = 1 because efeners 1 an 2 can respectively cover targets in T 1 an T 2 simultaneously. Thus, the price uner the best orer of commitment in this particular security game is at least (k + 1)/2. Hence, PoSC b is Ω(k), an k can be arbitrarily large. The proof of Theorem 4 uses a very simple construction: only two symmetric efeners an ientical targets. However, to obtain the lower boun the efeners nee to be able to cover an increasingly larger number of targets simultaneously. It turns out that we can obtain a matching upper boun that scales linearly with the maximum number of targets any efener can cover simultaneously; call this parameter the max-simultaneous-coverage. In the security game constructe in the proof of Theorem 4, the max-simultaneouscoverage is k. We aitionally assume that every efener can cover every target otherwise an optimal strategy for a efener who commits first an cannot cover all targets woul be to not cover anything, as any strategy woul yiel a utility of 0. We call this property complete iniviual coverage. Theorem 5. Denote the max-simultaneous-coverage by k. Then the PoSC w is O(k) in security games with ientical targets an complete iniviual coverage. 3 Our lower boun on PoSC b in Theorem 4 (resp. upper boun on PoSC w in Theorem 5) works even with the best (resp. worst) tie-breaking among the set of all optimal strategies at each step.

5 Proof. Let be the number of efeners an n be the number of targets. Since any efener can cover at most k targets simultaneously, the total number of targets covere simultaneously in any pure strategy is at most k. It follows that the minimum coverage probability in any mixe strategy (an hence in the OCP) is at most ( k)/n. Moreover, the minimum coverage probability cannot be more than 1. Hence, c min (OCP ) min( k/n, 1). Next, consier sequential commitment with the worst orer. The first efener commits to a mixe strategy that maximizes the minimum coverage probability in absence of the other efeners; if this probability is p 1, then it is easy to see that the worst choice of optimal strategy for the efener is the strategy that covers every target with probability exactly p 1. Such a mixe strategy is feasible since one can reuce the coverage probability of a target as much as require by removing it from the scheules in some of the pure strategy assignments (recall that any subset of a scheule is a feasible scheule). Inuctively, each successive efener also commits to a mixe strategy that covers all targets with ientical probability, maximizing this probability. Denote the ientical coverage probability by efener i as p i. Due to complete iniviual coverage, p i 1/n for all i as the efeners can achieve uniform coverage probability of at least 1/n by uniformly ranomizing over pure strategies that cover ifferent targets. Hence, every target is covere with probability at least 1 (1 p i) 1 (1 1/n) 1 e /n. Thus, min( k/n, 1) P osc w k 1 e /n 1 k = O(k), 1 e 1 min(/n, 1) 1 e /n where the thir transition hols because the function f(x) = min(x, 1)/(1 e x ) achieves its maximum at x = 1. From the proof of Theorem 4 an the statement of Theorem 5, we obtain a complete picture regaring the price of sequential commitment. Corollary 1. Denote the max-simultaneous-coverage by k. Both PoSC b an PoSC w are Θ(k) in security games with ientical targets an complete iniviual coverage. Note that PoSC b PoM, an Theorem 1 shows that the PoM can be unboune even with k = 1 if targets are nonientical. However, that proof crucially uses a game that oes not have the complete iniviual coverage property. We can actually construct an example with k = 1 an complete iniviual coverage where the price uner the best orer scales with the utilities, showing that PoSC b is unboune for nonientical targets even with k = 1. 5 Experimental Results We now turn to an empirical investigation of the PoM an PoSC; our goal is to quantify the loss cause by efeners miscoorination in realistic security games. We compute the PoM an PoSC in these games as follows. The OCP can be compute by solving the traitional security game (with a single efener) where all the resources are assume to belong to one efener. Computing PoSC is also easy; the optimal sequential strategies can be compute by rolling the coverage of earlier efeners into the utility functions, an at each step solving a traitional security game with a single efener using existing computational tools [Tambe, 2011]. The computation of the OUP is trickier; we formulate a nonlinear program whose solution is the OUP, an solve it using the YALMIP toolbox of MATLAB [Lofberg, 2004]; we omit the nontrivial etails ue to lack of space. We worry about the computational efficiency (or lack thereof) of our algorithms only insofar as it restricts our simulations, because while our simulations can inform policy ecisions these are not computations that we expect efeners to perform on a regular basis. For realistic security games, we use the port patrolling problem where a patrolling boat (resource) goes aroun an checks multiple targets a ay [Shieh et al., 2012]. Specifically, we use the actual map of Boston Port shown in Figure 1(a). The figure also shows the time require for the boat to move between various noes. We assume that the boat requires two time units at each target it visits for the check proceure. Due to the extreme computational buren, we restrict ourselves to the case of two efeners, each with one boat. Figure 1(b) shows the locations of homebases of the two efeners (marke in black). The other noes are consiere potential targets. In any scheule, a boat starts from its homebase, visits some targets an finally returns back to its homebase. All visite targets are consiere to be covere by the boat in that scheule. The boat must return within a patrolling time limit. (a) Map (b) Graph Structure Fig. 1: Boston Port, homebases, an targets. We consier two istributions of target valuations, both of which create non zero-sum security games with non-ientical targets. The importance of non-zero-sum security games in realistic omains has been emphasize in the risk analysis literature [Powell, 2007]. Denote by U[a, b] the uniform istribution over all integers between a an b. Homogeneous istribution: Uner this istribution, every target s utility to the efeners is in U[20, 39] if covere, an in U[0, 9] if uncovere (an vice versa for the attacker). Thus, there is homogeneity in target valuations although the actual realizations may create nonientical targets. Heterogeneous istribution: This istribution is relate to the target valuation of the worst case instance of Theorem 1, an captures realistic scenarios where some tar-

6 gets are far more valuable than others. For the gray targets (which are in some sense share between the two efeners), the utility to the efeners is in U[6, 10] if covere, an in U[1, 5] if uncovere. The utility to the attacker for these targets is 0 if covere an in U[30, 59] if uncovere. These targets are highly valuable; the efeners o not want them to be attacke whether or not they are covere, an the attacker has high utility for a successful attack. For the remaining targets, the utility to the efeners is in U[30, 59] if covere, an in U[10, 15] if uncovere. The utility to the attacker is 0 if covere, an in U[1, 5] if uncovere. These represent targets which the attacker is almost inifferent about, an the efeners o not have rastically low utility even in case of a successful attack. Note that these are shifte utility functions as explaine in Section 3.1. Figures 2 an 3 show the PoM an PoSC for the homogeneous an heterogeneous istributions, respectively. 4 All values are average over 100 ranom trials on target utilities. In both graphs, the x-axis shows the patrolling time limit for both efeners. Legens PoSC b an PoSC w show the average loss (uner the 100 ranom trials) where the average is taken over the best an the worst sequence respectively in every instance, whereas legen P osc 01 (resp. P osc 10 ) shows the average loss uner the fixe commitment sequence where efener 0 (resp. efener 1) commits first. PoSC or PoM PoSC w PoSC 01 PoSC 10 PoSC b PoM Patrolling Time Limit Fig. 2: PoM an PoSC uner the Homogeneous Distribution. Interestingly, the PoM in the homogeneous case (Figure 2) is almost 1 (the worst patrol times give an average PoM of 1.038), but the PoSC w can be significant (as high as 1.749). As expecte, the loss ue to miscoorination is much higher in the heterogeneous case (Figure 3), with the PoM reaching 1.434, an the PoSC w achieving a whopping Corollary 1 shows that both the PoSC b an PoSC w are Θ(k) where k is the max-simultaneous-coverage parameter, which reuces to the maximum scheule size (over all resources) when every efener has one resource. The maximum scheule size is a monotonic function of the patrolling time limit. Hence, the graphs show various losses as functions of the maximum scheule size. While the theoretical 4 Technically the figures show average ratios rather than the PoM an PoSC themselves, which are efine as supremums. PoSC or PoM PoSC w PoSC 01 PoSC 10 PoSC b PoM Patrolling Time Limit Fig. 3: PoM an PoSC uner the Heterogeneous Distribution. results preict a monotonic increment in PoSC b an PoSC w in the worst-case, they seem to have an inverse U shape relationship with the maximum scheule size in our empirical analysis. The ifference arises because the lower boun of Theorem 4 uses an example with an increasing number of targets while the targets are fixe in our case. Hence at a threshol patrolling time limit, the efeners can fully protect all targets in the OCP, an increasing it further maintains the utility in the OCP while increasing the utility in the uncorrelate case for both PoSC an PoM this results in the ownwar curves when time limits are large. The low PoM an PoSC for low patrol time limits is ue to the small overlap between the scheules of the two efeners, which entails less nee for coorination among their strategies. Another interesting observation is that in our particular graph structure, the commitment sequence where efener 1 commits first generally outperforms the sequence where efener 0 commits first. Further, always using the former sequence helps avoi the extremely high PoSC in the isastrous special cases. Thus, the commitment sequence (even if fixe) has great influence on the PoSC. 6 Discussion Our theoretical results suggest that the loss ue to miscoorination can be extremely high or relatively low, epening on the assumptions that are mae. In general though, the PoM (which in the worst case is arbitrarily high) can be significant (more than 30% loss) in realistic simulations. The PoSC is consistently significant, with a loss of as much as 70% in our simulations. We view the PoSC as a better proxy of reality, an in fact one can argue that even the PoSC is optimistic. We therefore interpret our results as suggesting a nee for greater coorination among efeners in critical infrastructure sites. As we hinte in our introuction, the challenge for future work is not purely computational. Inee, the simplest an most effective solution is scheuling the joint pool of resources, but it seems unrealistic to expect organizations as inepenent as, e.g., the New York Police Department an the Coast Guar to aopt such a policy. Hence, to esign realistic coorination mechanisms, it is necessary to etermine what feasible form of coorination can overcome most of the loss while keeping policy makers in the loop.

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