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a17d1c228b8cb73378e0e28718817863164fd0a7 | subsection | 43 | 129 | Comparison of Equilibrium Utilities | Secondly, the attack and defense costs jointly determine the set of facilities that are targeted or secured in equilibrium. On one hand, the set of vulnerable facilities increases as the cost of attack decreases. On the other hand, when the cost of defense is sufficiently high, the attacker tends to conduct an attack w... | {
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390be81da2fa85284fda33423e9860ea5192535f | subsection | 44 | 129 | First Mover Advantage | We now focus on identifying parameter ranges in which the defender has the first mover advantage, i.e., the defender in SPE has a strictly higher payoff than in NE. To identify the first mover advantage, let us recall the expressions of type I regimes for \Gamma in (REF )–(REF ) and type \widetilde{\mathrm {I}} regimes... | {
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f08777ab8fb1b2c7bae87929dc7e4093f3bea706 | subsection | 45 | 129 | First Mover Advantage | That is,
H\stackrel{\Delta }{=}\left\lbrace \left(p_a, p_d\right)|p_d>\widetilde{p}_d(p_a)\right\rbrace =\cup _{j=1}^{K} \widetilde{\Lambda }_j.We next compare the properties of NE and SPE for cost parameters in each set based on Theorems REF and REF , and Propositions REF .Set L:
Attacker: In \Gamma , the total attac... | {
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5c4339771dfcc3400089eca436f1d3a59ddce780 | subsection | 46 | 129 | First Mover Advantage | In set L, the defender secures facilities in \widetilde{\Gamma } with the same level of effort as that in \Gamma , and the attacker is still deterred with probability 1.On the other hand, in set H, the defense cost is so high that the defender is not able to secure all targeted facilities with an adequately high level ... | {
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f62f0e508e11c35c10b42f1d112b6e21e726b6d7 | subsection | 47 | 129 | First Mover Advantage | Thus, for all i=1, \dots , K, the type I regime \Lambda ^i in \Gamma is a proper subset of the type \widetilde{\mathrm {I}} regime \widetilde{\Lambda }^i in \widetilde{\Gamma }. Consequently, for any \left(p_a, p_d\right) \in \mathbb {R}_{>0}^2, we can have one of the following three cases:0<p_d< \bar{p}_d(p_a): The de... | {
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f9c8273924efc16a173bbf08464828c0b421f158 | subsection | 48 | 129 | First Mover Advantage | Set M:
Attacker: In \Gamma , the attacker conducts an attack with probability 1, whereas in \widetilde{\Gamma } the attacker is fully deterred. The attacker's equilibrium utility is lower in \widetilde{\Gamma } in comparison to that in \Gamma , i.e., U_a>\widetilde{U}_a.
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f6b6594b59b83da957a5de19892b9c693587ff47 | subsection | 49 | 129 | First Mover Advantage | We define the usage cost in this case as the average cost of travelers in Wardrop equilibrium . Therefore, the usage costs corresponding to attacks to different edges are C_1=20, C_2=19, C_3=18 and the pre-attack usage cost is C_{\emptyset }=17. From (), K=3, and \bar{\mathcal {E}}_{(1)}=\lbrace e_1\rbrace , \bar{\math... | {
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04f39c06e31cee7e099a4710e126a7912b79c4f8 | subsection | 50 | 129 | Model Extensions and Dynamic Aspects | In this section, we first discuss how relaxing our modeling assumptions influence our main results. Next we introduce a dynamic setup in which the users of the infrastructure system face uncertainty about the outcome of attacker-defender interaction (i.e., identity of the compromised facility), and follow a repeated le... | {
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38fcf2e8076b5699febb0b10855d14685a0e4990 | subsection | 51 | 129 | Relaxing Model Assumptions | Our discussion centers around extending our results when the following modeling aspects are included: facility-dependent cost parameters, less than perfect defense, and attacker's ability to target multiple facilities.Facility-dependent attack and defense costs.
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1ba481a91053de9455ac7ba71041f50776532696 | subsection | 52 | 129 | Relaxing Model Assumptions | Again our results on NE and SPE in Sec. – Sec. can be readily extended to this case. However, the expressions for thresholds for attack probability and security effort level need to be modified.
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d8e69c8fe444f8646710b3cd045cfce073a5b1f0 | subsection | 53 | 129 | Relaxing Model Assumptions | In this case, if the defense cost p_{d, \bar{e}} is sufficiently low, then by proactively securing the facility \bar{e} with the threshold effort \widehat{\rho }_{\bar{e}}, the defender can deter the attack completely and obtain a strictly higher utility in \widetilde{\Gamma } than that in \Gamma . Thus, for such cost ... | {
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97eb16549aca0c49de9540f8f631dc1bbf756834 | subsection | 54 | 129 | Relaxing Model Assumptions | The main conclusion of our analysis also holds: the defender obtains a higher utility by proactively defending all vulnerable facilities when the facility-dependent cost parameters lie in type \widetilde{\mathrm {I}} regimes.
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40c008f26d87b86982c2c2f739dcd740c0a0e487 | subsection | 55 | 129 | Relaxing Model Assumptions | In particular, consider that the attacker's cost parameters \left(p_{a,e}\right)_{e\in \mathcal {E}} in this game are such that there is only one vulnerable facility \bar{e}\in \mathcal {E} such that C_{\bar{e}}-C_{\emptyset }>p_{a, \bar{e}}, and the threshold effort on that facility \widehat{\rho }_{\bar{e}}=\left(C_{... | {
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c903be8cade2e3fa468c66b27106f5dec851cde8 | subsection | 56 | 129 | Rational Learning Dynamics | We now discuss an approach for analyzing the dynamics of usage cost after a security attack. Recall that the attacker-defender model enables us to evaluate the vulnerability of individual facilities to a strategic attack for the purpose of prioritizing defense investments. One can view this model as a way to determine ... | {
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267d3a5e65c013d5f0d4ad019418081ae90781bf | subsection | 57 | 129 | Rational Learning Dynamics | In particular, in each stage \in , they maintain a belief about the state t. The initial belief 0 can be different from the prior state distribution . However, we require that 0 is absolutely continuous with respect to (\cite {kalai1993subjective}):
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e2a746e00c61705d73322451d271c07aac79bfd2 | subsection | 58 | 129 | Rational Learning Dynamics | REF for the example network) plus a random variable \epsilon _e:c_e^s(q^{t*}(\theta ^t))=\left\lbrace
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8f8ba336c602b07a36fee8b6862d90eb866c0318 | subsection | 59 | 129 | Rational Learning Dynamics | However, such experiments are in general not costless.As a final remark, we note another implication of proactive defense strategy in ranges of attack/ defense cost parameters where the first-mover advantage holds. In particular, when the cost parameters are in the sets L and M as given in (REF )-(REF ), the attack is ... | {
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b39ccb85340d9e487e63f3c1d07888405286f471 | subsection | 60 | 129 | Rational Learning Dynamics | Examples include cyber-security attacks to transportation facilities that can result in hard-to-detect effects such as compromised traffic signals of a major intersection, or tampering of controllers governing the access to a busy freeway corridor. Then, one can study the problem of learning by rational but imperfectly... | {
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1ea02e3e587f0d7bab5e148b28b6b2896a659010 | subsection | 61 | 129 | Rational Learning Dynamics | The equilibrium routing strategy q^{t*}(\theta ^t) can be computed efficiently for this game. Moreover, in each stage, the equilibrium is essentially unique in that the equilibrium edge load is unique for a given belief ().The realized cost on each edge e\in \mathcal {E}, denoted c_e^s(w_e^{t*}(\theta ^t)), equals to t... | {
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7dfd2a1283361a4802dd84ee422e1b59713b5282 | subsection | 62 | 129 | Rational Learning Dynamics | REF respectively.
[Figure: Learning leads to long-run inefficiency \mathbf {s}=\emptyset : (a) Equilibrium routing strategies; (b) Beliefs.]These cases illustrate that if the post-attack state is not perfectly known by the users of the system, then the cost experienced by the users depend on the learning dynamics induc... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.2307/2951717",
"end": 719,
"openalex_id": "https://openalex.org/W2143100276",
"raw": "Drew Fudenberg and David K Levine. Steady state learning and nash equilibrium. Econometrica: Journal of the Econometric Society, pages 547–573, 1993.",... | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.029061470180749893,
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-0.017674989998340607,
-0.014912319369614124... | |
de5d8dfb3dc0201705e2dda539ee790211cac90e | subsection | 63 | 129 | Proofs of Section | Proof of Lemma REF .
We first show that the strategy in (REF ) is feasible. Since \rho _{(1)}\le 1, and for any i=1, \dots , m-1, \rho _{(i)}-\rho _{(i+1)}>0, \sigma _d(s_d) is non-negative for any s_d\in S_d. Additionally,\sum _{s_d\in S_d}\sigma _d(s_d)&=\sigma _d\left(\emptyset \right)+\sum _{i=1}^{m-1} \sigma _d\le... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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0.024870356544852257,
-0.03603912889957428,
0.0... | |
81e247700e21a89ae0079211d1d95c8b3fa78ec8 | subsection | 64 | 129 | Proofs of Section | Consider any pure strategy of the attacker, s_a\in \mathcal {E}, the utilities of the defender with strategy s_d and s_d^{^{\prime }} are as follows:u_d(s_d, s_a)&=-C(s_d, s_a)-|s_d| p_d=-C(s_d, s_a)-(|s_d^{^{\prime }}|+|s_d\setminus \bar{\mathcal {E}}|) p_d, \\
u_d(s_d^{^{\prime }}, s_a)&=-C(s_d^{^{\prime }}, s_a)-|s_... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.02125799097120762,
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0.019960841163992882,
-0.003775086486712098,
0.002868989482522011,
-0.016344083473086357,
-0.039... | |
3a3fd7a59565ca76abc50dd4c099eea51dfcd751 | subsection | 65 | 129 | Proofs of Section | However, C(s_d, \emptyset ) = C_{\emptyset } and p_a>0. Therefore, U_a(s_d, \emptyset )> U_a(s_d, s_a). Hence, any s_a\in \mathcal {E}\setminus \bar{\mathcal {E}} is strictly dominated. Hence, in equilibrium, the probability of the attacker choosing facility e\in \mathcal {E}\setminus \bar{\mathcal {E}} is 0 in \Gamma ... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.00462312251329422,
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0.04736030474305153,
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0.03786993399262428,
-0.027464095503091812,
-0.... | |
c65f916e550377afec44692e7cf78297335ca7b3 | subsection | 66 | 129 | Proofs of Section | \quad &\sum _{e\in \bar{\mathcal {E}}} \sigma _a(e)+ \sigma _a(\emptyset )=1, \\
&\sigma _a(\emptyset ) \ge 0, \quad \sigma _a(e) \ge 0, \quad \forall e\in \bar{\mathcal {E}}.Given any s_d\in S_d, we can express the objective fucntion in (REF ) as follows:&\sum _{e\in \bar{\mathcal {E}}} \left(C(s_d, e)+|s_d| p_d-p_a\r... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.022745629772543907,
0.021265866234898567,
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0.05104418098926544,
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0.04869486764073372,
-0.006639862433075905,
-... | |
ad321b544cda4a16ee0d2b530550e509602d06fd | subsection | 67 | 129 | Proofs of Section | We first argue that the defender's best response is in (REF ). For edge e\in \mathcal {E} such that \sigma _a(e)<\frac{p_d}{C_{e}-C_{\emptyset }}, we have \left(C_{\emptyset }-C_{e}\right)\sigma _a(e)+p_d>0. Since \rho \in BR(\sigma _a) maximizes U_d(\sigma _d, \sigma _a) as given in (REF ), \rho _e must be 0. Addition... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.01955655962228775,
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0.02510927990078926,
0.00520567549392581,
0.03407905995845795,
-0.02520080842077732,
-0.0246668... | |
6ee7bdc59d89f51a495429469707dba6414a4b74 | subsection | 68 | 129 | Proofs of Section | We obtain:
V(\sigma _a^{^{\prime }})-V(\sigma _a)\stackrel{}{=}\epsilon \left(C_{\widehat{e}}-C_{\emptyset }\right)>0
The last inequality holds from () and \widehat{e}\in \bar{\mathcal {E}}. Therefore, \sigma _a cannot be an attacker's equilibrium strategy.
If there does not exist such \bar{e} as defined in case (a)... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.047884054481983185,
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0.03405902162194252,
0.02691761404275894,
-0.023667357861995697,
-0.... | |
0dc0fe0c8377ea823a67c023566af9f95821c839 | subsection | 69 | 129 | Proofs of Section | There is no vulnerable facility, and thus \sigma _a^*(\emptyset )=1.
i=1, \dots , K:
Since p_d satisfies (REF ) or (REF ), we obtain:
\sum _{e\in \cup _{k=1}^{i} \bar{\mathcal {E}}_{(k)}}\frac{p_d}{C_{e}-C_{\emptyset }}=\sum _{k=1}^{i} \frac{p_d\cdot E_{(k)}}{C_{(k)}-C_{\emptyset }} < 1
Therefore, the set of feasibl... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.008148585446178913,
0.017228001728653908,
-0.05575830116868019,
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0.01805201731622219,
0.025956450030207634,
-0.015175596810877323,
-0.... | |
559583a83eda7548b8c1e091d274f616a6eb5ac2 | subsection | 70 | 129 | Proofs of Section | By definition of Nash equilibrium, the probability vector \rho ^{*} is induced by an equilibrium strategy if and only if it satisfies the following two conditions:\rho ^{*} is a best response to any \sigma _a^*\in \Sigma ^{*}_a.
Any attacker's equilibrium strategy is a best response to \rho ^{*}, i.e. the attacker has... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.0129086310043931,
-0.01863054186105728,
-0.015990832820534706,
0.00913979858160019,
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0.04598890617489815,
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0.007518590893596411,
-0.01829485595226288,
-0.028... | |
776da85552fb4d4dbefabe7d039dff0a172353f2 | subsection | 71 | 129 | Proofs of Section | Therefore, \rho ^{*} in (REF )-() satisfies both conditions (1) and (2). \rho ^{*} is the unique equilibrium strategy.
Type II regimes \Lambda _j:
If j=0:
Consider an attacker's strategy \sigma _a such that:
\sigma _a(e)&=\frac{1}{E_{(1)}}, \quad \forall e\in \bar{\mathcal {E}}_{(1)}, \\
\sigma _a(e)&=0, \quad \for... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.009736181236803532,
0.008584016002714634,
-0.026171710342168808,
0.0069587756879627705,
-0.005451804026961327,
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0.03644200786948204,
0.04361443221569061,
0.02171565219759941,
0.04962705820798874,
0.011491135694086552,
0.017946315929293633,
-0.030032610520720482,
-0... | |
9e064f75b74c68b0f97261bb3801d91905e92f85 | subsection | 72 | 129 | Proofs of Section | Therefore, for any e\in \cup _{k=1}^{j-1}\bar{\mathcal {E}}_{(k)}, \rho ^{*} satisfies:
&& &&U_a(\rho ^{*}, e)&=C_{(j)}-p_a, \\ &&\stackrel{(\ref {Ua_rewrite})}{\Rightarrow } \quad &&\rho _e^{*}\left(C_{\emptyset }-p_a\right)+(1-\rho _e^{*}) \left(C_{(k)}-p_a\right)&=C_{(j)}-p_a, \\&&\stackrel{\text{ }}{\Rightarrow } ... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.01712198741734028,
0.046818412840366364,
-0.04257606714963913,
-0.006237622816115618,
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0.04993150010704994,
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0.008850938640534878,
-0.008980650454759598,
-0... | |
535f55356a619c44b463fff9f5d90bb8b8d825dc | subsection | 73 | 129 | Proofs of Section | Hence, the set of best response strategies of the attacker is \Delta (\bar{\mathcal {E}}^{*}\cup \lbrace \emptyset \rbrace ), where \bar{\mathcal {E}}^{*} is the set defined in (REF ).Otherwise, if there exists a facility e\in \lbrace \mathcal {E}| C_{e}-p_a>C_{\emptyset }\rbrace such that \tilde{\rho }_e<\widehat{\rho... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.005161856301128864,
0.03064303658902645,
-0.04266829043626785,
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0.003944833297282457,
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0.012567002326250076,
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0.028369225561618805,
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0.06000419333577156,
-0.023134881630539894,
0.00... | |
52e379f7c731adb35208fcb163c5c026eed59584 | subsection | 74 | 129 | Proofs of Section | For the sake of contradiction, we assume that in SPE, there exists a facility e\in \bar{\mathcal {E}}^{*} such that \widetilde{\sigma }_a^*(e, \tilde{\rho }^{*})>0, i.e. \widetilde{\sigma }_a^*(\emptyset , \tilde{\rho }^{*})<1. Then, we can write U_d(\tilde{\rho }^{*}, \widetilde{\sigma }_a^*(\tilde{\rho }^{*})) as fol... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.004749203100800514,
0.05676155909895897,
-0.03295832499861717,
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0.00570667302235961,
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0.03265315294265747,
0.0024718742351979017,
0.028090868145227432,
-0.029326805844902992,
-0.0... | |
5c445fc1ef61eddf98337b80148431e99e2d19f7 | subsection | 75 | 129 | Proofs of Section | Since \epsilon is sufficiently small and \widetilde{\sigma }_a(\emptyset , \tilde{\rho }^{*})<1, we obtain that U_d(\tilde{\rho }^{\prime }, \widetilde{\sigma }_a(\tilde{\rho }^{\prime }))> U_d(\tilde{\rho }^{*}, \widetilde{\sigma }_a(\tilde{\rho }^{*})). Therefore, \tilde{\rho }^{*} cannot be a SPE. We can conclude th... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.004561296198517084,
0.03179178014397621,
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0.012349061667919159,
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0.0163382887840271,
0.003466737922281027,
0.05049461871385574,
-0.02861870266497135,
-0.02120... | |
8d85627b5fc9632fff500c666d39bca25d01a29c | subsection | 76 | 129 | Proofs of Section | Hence, the utility of the defender given \tilde{\rho }^{\prime } increases by \epsilon p_d compared to that given \tilde{\rho }, because the expected usage cost \mathbb {E}_{\sigma }[C] does not change, but the expected defense cost decreases by \epsilon p_d. Thus, such \tilde{\rho } cannot be the defender's equilibriu... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.004784594289958477,
0.040108177810907364,
-0.041420698165893555,
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-0.00733714085072279,
-0.04969262704253197,
-0... | |
4872244b2b669fdbc3063d587b161f378bf32ff8 | subsection | 77 | 129 | Proofs of Section | Note that functions \lbrace p_d^{ij}\rbrace _{j=1}^{i} are defined on the range \left[0, ~\frac{\sum _{k=1}^{i}E_{(k)}}{\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}}\right]. Since \lbrace C_{(k)}\rbrace _{k=1}^{i} satisfies (), we have:\frac{\sum _{k=1}^{i}E_{(k)}}{\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\e... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.04838959500193596,
0.04744376987218857,
-0.04796244949102402,
-0.02607119083404541,
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0.01673498935997486,
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-0.009336202405393124,
-0.011159202083945274,
-0.... | |
c3f1e3bb48af228e7d36f28e30b47234b4e6b054 | subsection | 78 | 129 | Proofs of Section | We want to argue that \hat{j}\ne i:&&p_a\cdot \left(\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}\right)&=\left(C_{(i)}-C_{\emptyset }-\epsilon \right) \cdot \left(\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}\right)\\
&& &= E_{(i)}+\sum _{k=1}^{i-1}\frac{\left(C_{(i)}-C_{\emptyset }\right) E_{(k)}}{C... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.01208468433469534,
0.028838451951742172,
-0.02847224846482277,
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0.009200839325785637,
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0.01917985826730728,
-0.015044821426272392,
-0... | |
0b5bb3988b973859fda6c64584c500d02ebe5fe5 | subsection | 79 | 129 | Proofs of Section | Since \epsilon is a sufficiently small positive number, we have:&&\sum _{k=\hat{j}+1}^{i}E_{(k)} &\le \left(\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}\right) \cdot \left(C_{(i)}-C_{\emptyset }-\epsilon \right)\\
&& &=E_{(i)}+\sum _{k=1}^{i-1}\frac{\left(C_{(i)}-C_{\emptyset }\right) E_{(k)}}{C_{(k)}-C_{\emp... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.02747628092765808,
0.03524164855480194,
-0.043876614421606064,
-0.029200222343206406,
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0.0416187085211277,
0.009405400604009628,
0.01304398663341999,
0.016873273998498917,
-0.0194363035261631,
0.009847828187048435,
-0.011930289678275585,
-0.0... | |
64e4d6981638e4236cc243704f73469067d783ee | subsection | 80 | 129 | Proofs of Section | Then,\lim _{p_a\rightarrow \left(C_{(i)}-C_{\emptyset }\right)^{-}}\widetilde{p}_d(p_a)&=\lim _{\epsilon \rightarrow 0} p_d^{i\hat{j}}(C_{(i)}-C_{\emptyset }-\epsilon )\\
&\stackrel{(\ref {cdij})}{=}\frac{C_{(\hat{j})}-C_{\emptyset }}{\left(C_{(\hat{j})}-C_{\emptyset }\right) \cdot \left(\sum _{k=1}^{\hat{j}-1} \frac{E... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.04590009152889252,
0.05490310117602348,
-0.03555426001548767,
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0.01890632137656212,
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0.016495345160365105,
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-0.01951669529080391,
-... | |
68b565855b1c50bf1a53083a5512b1fc7e0d3b1c | subsection | 81 | 129 | Proofs of Section | From (REF ), we can check that p_d^{KK}(0)=\left(\sum _{k=1}^{K} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}\right)^{-1}=\bar{p_d}(0). If p_a approaches C_{(1)}-C_{\emptyset }, then \widetilde{p}_d(p_a)=p_d^{11}(p_a), and we have:\lim _{p_a\rightarrow C_{(1)}-C_{\emptyset }}\widetilde{p}_d(p_a)\stackrel{(\ref {cdij})}{=}\li... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.035558491945266724,
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0.030339177697896957,
-0.0063562714494764805,
0.009355088695883751,
-0.006993424613028765,... | |
10d4630e6ecf607010fb9a403c53e8668244f8fe | subsection | 82 | 129 | Proofs of Section | Lemma REF characterizes SPE in sets \left\lbrace \Lambda ^i\right\rbrace _{i=0}^{K}, and Lemma REF characterizes SPE in sets \left\lbrace \Lambda ^i_j\right\rbrace _{i=1, j=1}^{i=K, j=i}.Lemma 6
In \widetilde{\Gamma }, for any \left(p_a, p_d\right) in the set \Lambda ^i, where i=0, \dots , K:If i=0, then SPE is as giv... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.029568839818239212,
0.037929896265268326,
-0.03268134966492653,
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0.0478166900575161,
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0.04226299747824669,
-0.040004901587963104,
0.007918... | |
8c5b24f5db5490be2c90af122dce24180f4d691a | subsection | 83 | 129 | Proofs of Section | The utility of the defender can be written as:
U_d(\tilde{\rho }, \widetilde{\sigma }_a^*(\tilde{\rho }))=-\lambda -\left(\sum _{e\in \mathcal {E}} \tilde{\rho }_e\right)\cdot p_d.
We now consider \tilde{\rho }^{\prime } as follows:
\tilde{\rho }_e^{\prime }&=\tilde{\rho }_e+\frac{\epsilon }{C_{e}-C_{\emptyset }}, &... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.007108509074896574,
0.03968249633908272,
-0.011103468015789986,
-0.0008446839055977762,
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0.019459685310721397,
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0.010096142999827862,
-0.04053719714283943,
-... | |
9b22463c9a1ece32203844ba4854d674d9db4b44 | subsection | 84 | 129 | Proofs of Section | Therefore, the defender's utility can be written as:
&U_d(\tilde{\rho }^{\prime }, \widetilde{\sigma }_a(\tilde{\rho }^{\prime }))=-\lambda +\epsilon -\left(\sum _{e\in \mathcal {E}} \tilde{\rho }^{\prime }_e\right)\cdot p_d=-\lambda +\epsilon -\left(\sum _{e\in \mathcal {E}} \tilde{\rho }_e\right)\cdot p_d-\sum _{e\i... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.014046627096831799,
0.046817004680633545,
-0.027894876897335052,
-0.027742279693484306,
-0.01303185150027275,
-0.0336020328104496,
0.028871502727270126,
0.03470074012875557,
0.04245270788669586,
0.039309192448854446,
-0.005020467564463615,
0.03946179151535034,
-0.04635921120643616,
-0.0... | |
05152d08afc100517058c9f49e03e8d6fee3e5af | subsection | 85 | 129 | Proofs of Section | From Lemma REF , we know that the defender can either secure all vulnerable facilities e\in \cup _{k=1}^{i} \bar{\mathcal {E}}_{(k)} with the threshold effort \widehat{\rho }_e defined in (REF ), or leave at least one vulnerable facility secured less than the threshold effort. We discuss the two cases separately:If any... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.011255450546741486,
0.028050871565937996,
-0.019031250849366188,
0.009225654415786266,
-0.00019112804147880524,
-0.06709011644124985,
0.03006540797650814,
0.036414243280887604,
0.03348401188850403,
0.013056322932243347,
-0.02524273283779621,
0.02904287911951542,
-0.03244622051715851,
-0... | |
6b62c03dd98af5a8c9c1bb0d2ee3998da7c71b2f | subsection | 86 | 129 | Proofs of Section | Then, \tilde{\rho }^\dagger can be written as:
&\tilde{\rho }^\dagger \in \underset{\tilde{\rho }\in \widetilde{P}}{\mathrm {argmax}}~ U_d(\tilde{\rho }, \widetilde{\sigma }_a^*(\tilde{\rho }))=\underset{\tilde{\rho }\in \widetilde{P}}{\mathrm {argmax}} \left(-\mathbb {E}_{\left(\tilde{\rho }, \widetilde{\sigma }_a^*(... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.00466283829882741,
0.03287644311785698,
-0.007249912247061729,
-0.03541009873151779,
0.003044963115826249,
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0.03681429103016853,
0.02324550785124302,
0.04319421201944351,
-0.023703396320343018,
-0.026496... | |
6f74e11247fe412a9fd240a2944979898990d237 | subsection | 87 | 129 | Proofs of Section | Therefore, \sum _{e\in \mathcal {E}} \widetilde{\sigma }_a^*(e, \tilde{\rho })=1, and (REF ) can be re-expressed as:
\tilde{\rho }^\dagger &\in \underset{\tilde{\rho }\in \widetilde{P}}{\mathrm {argmax}} \left(-\mathbb {E}_{\left(\tilde{\rho }, \widetilde{\sigma }_a^*(\tilde{\rho })\right)}[C]-\left(\sum _{e\in \mathc... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.03609263151884079,
0.026375385001301765,
-0.029975494369864464,
-0.06141544133424759,
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0.02298883907496929,
0.053879618644714355,
0.008001092821359634,
0.0615374781191349,
-0.04405558854341507,
-0.0191... | |
8c77d2c1744ede398bed756d61f00daf38ea33aa | subsection | 88 | 129 | Proofs of Section | Hence, \tilde{\rho }^\dagger can be re-expressed as:
\tilde{\rho }^\dagger &\stackrel{(\ref {zero_sum_again})}{=} \underset{\tilde{\rho }\in \widetilde{P}}{\mathrm {argmax}} \left(-\max _{\widetilde{\sigma }_a\in \Delta (S_a)} \left(\mathbb {E}_{\left(\tilde{\rho }, \widetilde{\sigma }_a\right)}[C]- \left(\sum _{e\in ... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.010580293834209442,
0.03071107715368271,
-0.026179933920502663,
-0.03649324178695679,
-0.007433667313307524,
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0.04525040090084076,
-0.007887544110417366,
0.044090915471315384,
-0.032770685851573944,
-0... | |
bc6f415c4c56826b8abfb3c0ca4754f2369c2fc6 | subsection | 89 | 129 | Proofs of Section | The defender's utility in this case is:
U_d(\tilde{\rho }^\dagger , \widetilde{\sigma }_a^*(\tilde{\rho }^\dagger ))=-C_{(j)}-\left(\sum _{k=1}^{j-1} \frac{\left(C_{(k)}-C_{(j)}\right) \cdot E_{(k)}}{C_{(k)}-C_{\emptyset }}\right) \cdot p_d.Finally, by comparing U_d in (REF ) and (REF ), we can check that if p_d>p_d^{... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.0266977958381176,
0.04796449467539787,
-0.04884933680295944,
-0.016873005777597427,
-0.012189450673758984,
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0.01867320016026497,
-0.012929360382258892,
0.032617077231407166,
-0.016995053738355637,
-0.02... | |
87deb511265427c7b76d738a0771b4a256daa36e | subsection | 90 | 129 | Proofs of Section | For any C_{(i+1)}-C_{\emptyset }\le p_a<C_{(i)}-C_{\emptyset }, there is a unique \hat{j}\in \lbrace 1, \dots , i\rbrace such that \frac{\sum _{k=\hat{j}+1}^{i}E_{(k)}}{\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}}\le p_a<\frac{\sum _{k=\hat{j}}^{i}E_{(k)}}{\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset ... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.01296753715723753,
0.013982056640088558,
-0.04689671844244003,
-0.022227885201573372,
0.007074935827404261,
-0.05458570644259453,
0.022899145260453224,
0.02221262827515602,
0.01755957119166851,
0.02167867124080658,
-0.01445499062538147,
0.01078594010323286,
-0.019680146127939224,
-0.006... | |
6557bbbfabe450581829f23eaab59d2ac54b1bc0 | subsection | 91 | 129 | Proofs of Section | We now show that in \Lambda ^i_j, p_d<p_d^{ij}(p_a):
p_d^{ij}(p_a)\stackrel{(\ref {cdij})}{>}p_d^{ij}\left(\frac{\sum _{k=j}^{i}E_{(k)}}{\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}}\right)=\left(\sum _{k=1}^{j-1} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}\right)^{-1}\stackrel{(\ref {partition})}{>}p_d.
Hence, ... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.038763731718063354,
0.027516145259141922,
-0.04260958358645439,
-0.04788999632000923,
0.002672637114301324,
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0.054757583886384964,
0.04175494983792305,
0.02258674055337906,
0.02727196365594864,
-0.016344863921403885,
0.017840472981333733,
-0.019381865859031677,
-0.... | |
d98ad04dc1b6850513c5c7abb742f76a11e8fd47 | subsection | 92 | 129 | Proofs of Section | Therefore, we can re-express \widetilde{\Lambda }^1 as follows:
\widetilde{\Lambda }^1&\stackrel{(\ref {regimej_constraint_1})}{=} \left\lbrace \left(p_a, p_d\right) \left|p_a< \widetilde{p}_d^{-1}(p_d), ~ p_d> \left(\frac{E_{(1)}}{C_{(1)}-C_{\emptyset }}\right)^{-1} \right.\right\rbrace \\
&=\left\lbrace \left(p_a, p... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.01766798086464405,
0.06987854093313217,
-0.022580837830901146,
-0.028531193733215332,
-0.0009440468857064843,
-0.05013556405901909,
0.048030052334070206,
0.0453447625041008,
0.017210260033607483,
0.03078927844762802,
-0.025556016713380814,
0.0031830589286983013,
-0.006514877080917358,
-... | |
e3f1d3c3d62a9e2df3433eca15a9008c56106f9e | subsection | 93 | 129 | Proofs of Section | Analogous to (REF ), we re-express the set \widetilde{\Lambda }_j as follows:
\widetilde{\Lambda }_j&\stackrel{(\ref {regimej_constraint})}{=} \left\lbrace \left(p_a, p_d\right) \left|p_a< \widetilde{p}_d^{-1}(p_d), ~ \left(\sum _{k=1}^{j} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}\right)^{-1}\le p_d< \left(\sum _{k=1}^{j... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.03578881546854973,
0.05369848012924194,
-0.02869512513279915,
-0.023142214864492416,
-0.008794651366770267,
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0.04027386009693146,
0.0245304424315691,
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0.03313440456986427,
-0.024667739868164062,
0.015034048818051815,
-0.007864081300795078,
0.... | |
64addb2552b54e607afa2fdc6ef70f11004af24c | subsection | 94 | 129 | Proofs of Section | Consider any cost parameters \left(p_a, p_d\right) in the set \Lambda ^i_j\cap \lbrace \left(p_a, p_d\right)|p_d> \widetilde{p}_d(p_a)\rbrace , from (REF ), we can find \hat{j} such that \frac{\sum _{k=\hat{j}+1}^{i}E_{(k)}}{\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}}\le p_a<\frac{\sum _{k=\hat{j}}^{i}E_{(k... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.024256104603409767,
0.021723706275224686,
-0.04387456551194191,
-0.026193542405962944,
-0.01197549793869257,
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0.01312728226184845,
0.018199706450104713,
0.017482701689004898,
0.030007395893335342,
-0.010213498026132584,
0.018321748822927475,
-0.038779258728027344,
0... | |
592599fcf9a4962f824cb496dce055a5aeaab6a7 | subsection | 95 | 129 | Proofs of Section | If \hat{j}<j, then since p_a\ge \frac{\sum _{k=\hat{j}+1}^{i}E_{(k)}}{\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}}, from (REF ), we have \widetilde{p}_d(p_a)=p_d^{i\hat{j}}(p_a)\ge \left(\sum _{k=1}^{\hat{j}} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}\right)^{-1} \ge \left(\sum _{k=1}^{j-1} \frac{E_{(k)}}{C_{(k)... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.041112251579761505,
0.03390921652317047,
-0.032077934592962265,
-0.04685026407241821,
0.020327216014266014,
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0.044652730226516724,
0.041142772883176804,
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0.036778222769498825,
-0.025576887652277946,
0.029285231605172157,
-0.04065443202853203,
-0.... | |
42f623721b9ab664dc9286dd35f47bb15ce3d67d | subsection | 96 | 129 | Proofs of Section | Now we check that \sigma _d in (REF ) indeed induces \rho . Consider any e\in \mathcal {E} such that \rho _e=0. Then, since e\notin \left\lbrace \mathcal {E}|\rho _e\ge \rho _{(i)}\right\rbrace for any i=1, \dots , m, and e \notin \emptyset , for any s_d\ni e, we must have \sigma _d(s_d)=0. Thus, \sum _{s_d\ni e} \sigm... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.0006573903956450522,
0.04756474867463112,
-0.006452818401157856,
-0.038625381886959076,
-0.03533032536506653,
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0.02285182476043701,
0.05650411173701286,
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0.011448794975876808,
-0.0022539100609719753,
0.024194255471229553,
-0.030006369575858116,
... | |
87e2379a1d473371f852948b939c4c5fce107002 | subsection | 97 | 129 | Proofs of Section | Therefore, any s_d such that s_d\nsubseteq \bar{\mathcal {E}} is a strictly dominated strategy. Hence, in \Gamma , any equilibrium strategy of the defender satisfies \sigma _d^*(s_d)=0. From (REF ), we know that \rho _e^{*}=0 for any e\in \mathcal {E}\setminus \bar{\mathcal {E}}.We denote the set of defender's pure str... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.002237871056422591,
0.05219791457056999,
-0.03855319693684578,
-0.031807150691747665,
-0.022252794355154037,
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0.0344323068857193,
0.004117072559893131,
0.04246040806174278,
-0.02596159465610981,
-0.0112... | |
e8325a12ea6e88d26dbaeaea19069c1f220a7a8d | subsection | 98 | 129 | Proofs of Section | Thus, \Gamma ^0 and \Gamma are strategically equivalent, i.e. they have the same set of equilibrium strategy profiles. Using the interchangeability property of equilibria in zero-sum games, we directly
obtain that for any \sigma _d^*\in \Sigma ^{*}_d and any \sigma _a^*\in \Sigma ^{*}_a, (\sigma _d^*, \sigma _a^*) is a... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.01770450547337532,
-0.005784489680081606,
-0.05302194505929947,
-0.03571426123380661,
-0.012942604720592499,
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0.049603141844272614,
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0.03150181099772453,
0.018925506621599197,
0.02576310932636261,
-0.021245406940579414,
-0.0... | |
529bb107c61e3e86a8cc03c39adac3968ec65c1e | subsection | 99 | 129 | Proofs of Section | \quad &\sum _{e\in \bar{\mathcal {E}}} \sigma _a(e)+ \sigma _a(\emptyset )=1, \\
&\sigma _a(\emptyset ) \ge 0, \quad \sigma _a(e) \ge 0, \quad \forall e\in \bar{\mathcal {E}}.Given any s_d\in S_d, we can express the objective fucntion in (REF ) as follows:&\sum _{e\in \bar{\mathcal {E}}} \left(C(s_d, e)+|s_d| p_d-p_a\r... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.022745629772543907,
0.021265866234898567,
-0.05967867374420166,
-0.04387420043349266,
-0.006060161627829075,
-0.016994386911392212,
0.012722907587885857,
0.014477265067398548,
0.027978193014860153,
0.05104418098926544,
-0.009725243784487247,
0.04869486764073372,
-0.006639862433075905,
-... | |
b9eeb6a59fe87d0bc9d12e715be136fa09d9b835 | subsection | 100 | 129 | Proofs of Section | We first argue that the defender's best response is in (REF ). For edge e\in \mathcal {E} such that \sigma _a(e)<\frac{p_d}{C_{e}-C_{\emptyset }}, we have \left(C_{\emptyset }-C_{e}\right)\sigma _a(e)+p_d>0. Since \rho \in BR(\sigma _a) maximizes U_d(\sigma _d, \sigma _a) as given in (REF ), \rho _e must be 0. Addition... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.01955655962228775,
0.04164540395140648,
-0.020792191848158836,
-0.01459877286106348,
-0.006441308185458183,
-0.0582120381295681,
0.010655426420271397,
0.024941477924585342,
0.0472896508872509,
0.02510927990078926,
0.00520567549392581,
0.03407905995845795,
-0.02520080842077732,
-0.0246668... | |
94f3721fef1c1941010401523faf5a256dc59778 | subsection | 101 | 129 | Proofs of Section | We obtain:
V(\sigma _a^{^{\prime }})-V(\sigma _a)\stackrel{}{=}\epsilon \left(C_{\widehat{e}}-C_{\emptyset }\right)>0
The last inequality holds from () and \widehat{e}\in \bar{\mathcal {E}}. Therefore, \sigma _a cannot be an attacker's equilibrium strategy.
If there does not exist such \bar{e} as defined in case (a)... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.047884054481983185,
0.02932860143482685,
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0.05734489485621452,
-0.0056383549235761166,
0.03405902162194252,
0.02691761404275894,
-0.023667357861995697,
-0.... | |
1c796e9167d6789fcdf9d7d4b41febbead1e991f | subsection | 102 | 129 | Proofs of Section | There is no vulnerable facility, and thus \sigma _a^*(\emptyset )=1.
i=1, \dots , K:
Since p_d satisfies (REF ) or (REF ), we obtain:
\sum _{e\in \cup _{k=1}^{i} \bar{\mathcal {E}}_{(k)}}\frac{p_d}{C_{e}-C_{\emptyset }}=\sum _{k=1}^{i} \frac{p_d\cdot E_{(k)}}{C_{(k)}-C_{\emptyset }} < 1
Therefore, the set of feasibl... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.008148585446178913,
0.017228001728653908,
-0.05575830116868019,
-0.014313433319330215,
-0.005741395987570286,
-0.039430610835552216,
0.0403461828827858,
0.014000613242387772,
0.04226887971162796,
0.013771720230579376,
0.01805201731622219,
0.025956450030207634,
-0.015175596810877323,
-0.... | |
1698973878e349e97c69416c3f8cc7622677f04a | subsection | 103 | 129 | Proofs of Section | By definition of Nash equilibrium, the probability vector \rho ^{*} is induced by an equilibrium strategy if and only if it satisfies the following two conditions:\rho ^{*} is a best response to any \sigma _a^*\in \Sigma ^{*}_a.
Any attacker's equilibrium strategy is a best response to \rho ^{*}, i.e. the attacker has... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.0129086310043931,
-0.01863054186105728,
-0.015990832820534706,
0.00913979858160019,
-0.02648863196372986,
-0.026366565376520157,
0.04308980330824852,
0.022948676720261574,
0.049650926142930984,
0.04598890617489815,
-0.015128732658922672,
0.007518590893596411,
-0.01829485595226288,
-0.028... | |
35243bf71d12d7c939c57f9ad93efaff912b2a18 | subsection | 104 | 129 | Proofs of Section | Therefore, \rho ^{*} in (REF )-() satisfies both conditions (1) and (2). \rho ^{*} is the unique equilibrium strategy.
Type II regimes \Lambda _j:
If j=0:
Consider an attacker's strategy \sigma _a such that:
\sigma _a(e)&=\frac{1}{E_{(1)}}, \quad \forall e\in \bar{\mathcal {E}}_{(1)}, \\
\sigma _a(e)&=0, \quad \for... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.009736181236803532,
0.008584016002714634,
-0.026171710342168808,
0.0069587756879627705,
-0.005451804026961327,
-0.034580230712890625,
0.03644200786948204,
0.04361443221569061,
0.02171565219759941,
0.04962705820798874,
0.011491135694086552,
0.017946315929293633,
-0.030032610520720482,
-0... | |
747b13ac2f9a98b5af091a1f52f558fe7ddcc7e4 | subsection | 105 | 129 | Proofs of Section | Therefore, for any e\in \cup _{k=1}^{j-1}\bar{\mathcal {E}}_{(k)}, \rho ^{*} satisfies:
&& &&U_a(\rho ^{*}, e)&=C_{(j)}-p_a, \\ &&\stackrel{(\ref {Ua_rewrite})}{\Rightarrow } \quad &&\rho _e^{*}\left(C_{\emptyset }-p_a\right)+(1-\rho _e^{*}) \left(C_{(k)}-p_a\right)&=C_{(j)}-p_a, \\&&\stackrel{\text{ }}{\Rightarrow } ... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.01712198741734028,
0.046818412840366364,
-0.04257606714963913,
-0.006237622816115618,
-0.010270141065120697,
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0.018525930121541023,
0.02751420997083187,
0.007355435285717249,
0.04993150010704994,
-0.0021555088460445404,
0.008850938640534878,
-0.008980650454759598,
-0... | |
cd46851c8288e06e7730865dd4548cf966461605 | subsection | 106 | 129 | Proofs of Section | Hence, the set of best response strategies of the attacker is \Delta (\bar{\mathcal {E}}^{*}\cup \lbrace \emptyset \rbrace ), where \bar{\mathcal {E}}^{*} is the set defined in (REF ).Otherwise, if there exists a facility e\in \lbrace \mathcal {E}| C_{e}-p_a>C_{\emptyset }\rbrace such that \tilde{\rho }_e<\widehat{\rho... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.005161856301128864,
0.03064303658902645,
-0.04266829043626785,
-0.016908608376979828,
0.003944833297282457,
-0.050695303827524185,
0.012567002326250076,
0.04529309272766113,
0.03918890282511711,
0.028369225561618805,
-0.01127749215811491,
0.06000419333577156,
-0.023134881630539894,
0.00... | |
7163b90e97021579a89d2035a425bedcbf4109a8 | subsection | 107 | 129 | Proofs of Section | For the sake of contradiction, we assume that in SPE, there exists a facility e\in \bar{\mathcal {E}}^{*} such that \widetilde{\sigma }_a^*(e, \tilde{\rho }^{*})>0, i.e. \widetilde{\sigma }_a^*(\emptyset , \tilde{\rho }^{*})<1. Then, we can write U_d(\tilde{\rho }^{*}, \widetilde{\sigma }_a^*(\tilde{\rho }^{*})) as fol... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.004749203100800514,
0.05676155909895897,
-0.03295832499861717,
-0.028350261971354485,
0.00570667302235961,
-0.052489183843135834,
0.014259052462875843,
0.04690457880496979,
0.047575950622558594,
0.03265315294265747,
0.0024718742351979017,
0.028090868145227432,
-0.029326805844902992,
-0.0... | |
1ae9c9eee6294aed4d6ef818041a51ea9e964fdc | subsection | 108 | 129 | Proofs of Section | Since \epsilon is sufficiently small and \widetilde{\sigma }_a(\emptyset , \tilde{\rho }^{*})<1, we obtain that U_d(\tilde{\rho }^{\prime }, \widetilde{\sigma }_a(\tilde{\rho }^{\prime }))> U_d(\tilde{\rho }^{*}, \widetilde{\sigma }_a(\tilde{\rho }^{*})). Therefore, \tilde{\rho }^{*} cannot be a SPE. We can conclude th... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.004561296198517084,
0.03179178014397621,
-0.03453771024942398,
0.012349061667919159,
-0.0017333689611405134,
-0.07194338738918304,
0.02942722663283348,
0.03420209512114525,
0.020197847858071327,
0.0163382887840271,
0.003466737922281027,
0.05049461871385574,
-0.02861870266497135,
-0.02120... | |
0d87eadafa03236bc9bc59a7ccda92beb94e03ea | subsection | 109 | 129 | Proofs of Section | Hence, the utility of the defender given \tilde{\rho }^{\prime } increases by \epsilon p_d compared to that given \tilde{\rho }, because the expected usage cost \mathbb {E}_{\sigma }[C] does not change, but the expected defense cost decreases by \epsilon p_d. Thus, such \tilde{\rho } cannot be the defender's equilibriu... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.004784594289958477,
0.040108177810907364,
-0.041420698165893555,
-0.009279212914407253,
-0.01551368460059166,
-0.04200064763426781,
0.03436971828341484,
0.028326019644737244,
0.028692303225398064,
0.011797420680522919,
-0.010904601775109768,
-0.00733714085072279,
-0.04969262704253197,
-0... | |
539529f981f4968bfc5f5f21281280d93a1a7c13 | subsection | 110 | 129 | Proofs of Section | Note that functions \lbrace p_d^{ij}\rbrace _{j=1}^{i} are defined on the range \left[0, ~\frac{\sum _{k=1}^{i}E_{(k)}}{\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}}\right]. Since \lbrace C_{(k)}\rbrace _{k=1}^{i} satisfies (), we have:\frac{\sum _{k=1}^{i}E_{(k)}}{\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\e... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.04838959500193596,
0.04744376987218857,
-0.04796244949102402,
-0.02607119083404541,
-0.008695482276380062,
-0.038077060133218765,
0.02254723198711872,
0.01381361298263073,
0.005911402404308319,
0.01673498935997486,
-0.015140817500650883,
-0.009336202405393124,
-0.011159202083945274,
-0.... | |
094af91598fa71785d8e0b569036b08d783942d3 | subsection | 111 | 129 | Proofs of Section | We want to argue that \hat{j}\ne i:&&p_a\cdot \left(\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}\right)&=\left(C_{(i)}-C_{\emptyset }-\epsilon \right) \cdot \left(\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}\right)\\
&& &= E_{(i)}+\sum _{k=1}^{i-1}\frac{\left(C_{(i)}-C_{\emptyset }\right) E_{(k)}}{C... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.01208468433469534,
0.028838451951742172,
-0.02847224846482277,
-0.018630554899573326,
-0.00906351301819086,
-0.057646386325359344,
0.010207896120846272,
0.016646957024931908,
0.006755674257874489,
0.009200839325785637,
-0.006790005601942539,
0.01917985826730728,
-0.015044821426272392,
-0... | |
d583d32dd0ffc38cf65d11343eeca8a2293e503d | subsection | 112 | 129 | Proofs of Section | Since \epsilon is a sufficiently small positive number, we have:&&\sum _{k=\hat{j}+1}^{i}E_{(k)} &\le \left(\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}\right) \cdot \left(C_{(i)}-C_{\emptyset }-\epsilon \right)\\
&& &=E_{(i)}+\sum _{k=1}^{i-1}\frac{\left(C_{(i)}-C_{\emptyset }\right) E_{(k)}}{C_{(k)}-C_{\emp... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.02747628092765808,
0.03524164855480194,
-0.043876614421606064,
-0.029200222343206406,
-0.0009325306164100766,
-0.05504409968852997,
0.0416187085211277,
0.009405400604009628,
0.01304398663341999,
0.016873273998498917,
-0.0194363035261631,
0.009847828187048435,
-0.011930289678275585,
-0.0... | |
072ff92b2528782ccf389e8818c3126d59a30f1a | subsection | 113 | 129 | Proofs of Section | Then,\lim _{p_a\rightarrow \left(C_{(i)}-C_{\emptyset }\right)^{-}}\widetilde{p}_d(p_a)&=\lim _{\epsilon \rightarrow 0} p_d^{i\hat{j}}(C_{(i)}-C_{\emptyset }-\epsilon )\\
&\stackrel{(\ref {cdij})}{=}\frac{C_{(\hat{j})}-C_{\emptyset }}{\left(C_{(\hat{j})}-C_{\emptyset }\right) \cdot \left(\sum _{k=1}^{\hat{j}-1} \frac{E... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.04590009152889252,
0.05490310117602348,
-0.03555426001548767,
-0.022415969520807266,
-0.025956137105822563,
-0.036591894924640656,
0.048005882650613785,
0.01890632137656212,
0.028382370248436928,
0.016495345160365105,
-0.020584849640727043,
-0.006832369137555361,
-0.01951669529080391,
-... | |
9d2e6c2560066a67f2bd2668bba7459f5cd7e2b8 | subsection | 114 | 129 | Proofs of Section | From (REF ), we can check that p_d^{KK}(0)=\left(\sum _{k=1}^{K} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}\right)^{-1}=\bar{p_d}(0). If p_a approaches C_{(1)}-C_{\emptyset }, then \widetilde{p}_d(p_a)=p_d^{11}(p_a), and we have:\lim _{p_a\rightarrow C_{(1)}-C_{\emptyset }}\widetilde{p}_d(p_a)\stackrel{(\ref {cdij})}{=}\li... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.035558491945266724,
0.017077233642339706,
-0.061899248510599136,
-0.015581640414893627,
-0.016008952632546425,
-0.04456257447600365,
0.07172743231058121,
0.0010244051227346063,
0.018939094617962837,
0.030339177697896957,
-0.0063562714494764805,
0.009355088695883751,
-0.006993424613028765,... | |
d5031a00ccd7b34bcde6c7d0c827183077cf98e6 | subsection | 115 | 129 | Proofs of Section | Lemma REF characterizes SPE in sets \left\lbrace \Lambda ^i\right\rbrace _{i=0}^{K}, and Lemma REF characterizes SPE in sets \left\lbrace \Lambda ^i_j\right\rbrace _{i=1, j=1}^{i=K, j=i}.Lemma 6
In \widetilde{\Gamma }, for any \left(p_a, p_d\right) in the set \Lambda ^i, where i=0, \dots , K:If i=0, then SPE is as giv... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.029568839818239212,
0.037929896265268326,
-0.03268134966492653,
-0.02049068734049797,
0.02078057825565338,
-0.0664612352848053,
0.00808642152696848,
0.04049313813447952,
0.014700504019856453,
0.0478166900575161,
-0.005721524823457003,
0.04226299747824669,
-0.040004901587963104,
0.007918... | |
4461a38401513e762e2a51e843ed7fca38541103 | subsection | 116 | 129 | Proofs of Section | The utility of the defender can be written as:
U_d(\tilde{\rho }, \widetilde{\sigma }_a^*(\tilde{\rho }))=-\lambda -\left(\sum _{e\in \mathcal {E}} \tilde{\rho }_e\right)\cdot p_d.
We now consider \tilde{\rho }^{\prime } as follows:
\tilde{\rho }_e^{\prime }&=\tilde{\rho }_e+\frac{\epsilon }{C_{e}-C_{\emptyset }}, &... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.007108509074896574,
0.03968249633908272,
-0.011103468015789986,
-0.0008446839055977762,
-0.008997242897748947,
-0.06959699839353561,
0.020405961200594902,
0.04313182085752487,
0.05039677023887634,
0.019459685310721397,
-0.015430386178195477,
0.010096142999827862,
-0.04053719714283943,
-... | |
7ab5294183c8638e674435be09de343f2716a206 | subsection | 117 | 129 | Proofs of Section | Therefore, the defender's utility can be written as:
&U_d(\tilde{\rho }^{\prime }, \widetilde{\sigma }_a(\tilde{\rho }^{\prime }))=-\lambda +\epsilon -\left(\sum _{e\in \mathcal {E}} \tilde{\rho }^{\prime }_e\right)\cdot p_d=-\lambda +\epsilon -\left(\sum _{e\in \mathcal {E}} \tilde{\rho }_e\right)\cdot p_d-\sum _{e\i... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.014046627096831799,
0.046817004680633545,
-0.027894876897335052,
-0.027742279693484306,
-0.01303185150027275,
-0.0336020328104496,
0.028871502727270126,
0.03470074012875557,
0.04245270788669586,
0.039309192448854446,
-0.005020467564463615,
0.03946179151535034,
-0.04635921120643616,
-0.0... | |
f79294541443fe8a10a0c337b7a9f20cb52fed4c | subsection | 118 | 129 | Proofs of Section | From Lemma REF , we know that the defender can either secure all vulnerable facilities e\in \cup _{k=1}^{i} \bar{\mathcal {E}}_{(k)} with the threshold effort \widehat{\rho }_e defined in (REF ), or leave at least one vulnerable facility secured less than the threshold effort. We discuss the two cases separately:If any... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.011255450546741486,
0.028050871565937996,
-0.019031250849366188,
0.009225654415786266,
-0.00019112804147880524,
-0.06709011644124985,
0.03006540797650814,
0.036414243280887604,
0.03348401188850403,
0.013056322932243347,
-0.02524273283779621,
0.02904287911951542,
-0.03244622051715851,
-0... | |
cd4d249ffe6120c4dbfa59f293406d43f469681c | subsection | 119 | 129 | Proofs of Section | Then, \tilde{\rho }^\dagger can be written as:
&\tilde{\rho }^\dagger \in \underset{\tilde{\rho }\in \widetilde{P}}{\mathrm {argmax}}~ U_d(\tilde{\rho }, \widetilde{\sigma }_a^*(\tilde{\rho }))=\underset{\tilde{\rho }\in \widetilde{P}}{\mathrm {argmax}} \left(-\mathbb {E}_{\left(\tilde{\rho }, \widetilde{\sigma }_a^*(... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.00466283829882741,
0.03287644311785698,
-0.007249912247061729,
-0.03541009873151779,
0.003044963115826249,
-0.04636891186237335,
0.017460841685533524,
0.08516738563776016,
0.02729019522666931,
0.03681429103016853,
0.02324550785124302,
0.04319421201944351,
-0.023703396320343018,
-0.026496... | |
718da6dd967c230bebbc914940349698c98c344c | subsection | 120 | 129 | Proofs of Section | Therefore, \sum _{e\in \mathcal {E}} \widetilde{\sigma }_a^*(e, \tilde{\rho })=1, and (REF ) can be re-expressed as:
\tilde{\rho }^\dagger &\in \underset{\tilde{\rho }\in \widetilde{P}}{\mathrm {argmax}} \left(-\mathbb {E}_{\left(\tilde{\rho }, \widetilde{\sigma }_a^*(\tilde{\rho })\right)}[C]-\left(\sum _{e\in \mathc... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.03609263151884079,
0.026375385001301765,
-0.029975494369864464,
-0.06141544133424759,
-0.012859716080129147,
-0.01717679761350155,
0.020639615133404732,
0.05482541024684906,
0.02298883907496929,
0.053879618644714355,
0.008001092821359634,
0.0615374781191349,
-0.04405558854341507,
-0.0191... | |
74925962f949b33556190d15eed4ea87daac2bd5 | subsection | 121 | 129 | Proofs of Section | Hence, \tilde{\rho }^\dagger can be re-expressed as:
\tilde{\rho }^\dagger &\stackrel{(\ref {zero_sum_again})}{=} \underset{\tilde{\rho }\in \widetilde{P}}{\mathrm {argmax}} \left(-\max _{\widetilde{\sigma }_a\in \Delta (S_a)} \left(\mathbb {E}_{\left(\tilde{\rho }, \widetilde{\sigma }_a\right)}[C]- \left(\sum _{e\in ... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
"Saurabh Amin"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.010580293834209442,
0.03071107715368271,
-0.026179933920502663,
-0.03649324178695679,
-0.007433667313307524,
-0.018689054995775223,
0.03609657660126686,
0.06163574382662773,
0.024242375046014786,
0.04525040090084076,
-0.007887544110417366,
0.044090915471315384,
-0.032770685851573944,
-0... | |
351fd9c71e00643fed8631c6f81ae9953570b67b | subsection | 122 | 129 | Proofs of Section | The defender's utility in this case is:
U_d(\tilde{\rho }^\dagger , \widetilde{\sigma }_a^*(\tilde{\rho }^\dagger ))=-C_{(j)}-\left(\sum _{k=1}^{j-1} \frac{\left(C_{(k)}-C_{(j)}\right) \cdot E_{(k)}}{C_{(k)}-C_{\emptyset }}\right) \cdot p_d.Finally, by comparing U_d in (REF ) and (REF ), we can check that if p_d>p_d^{... | {
"cite_spans": []
} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
"Manxi Wu",
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0bf57b52c75fb24e738b8af205b7114c90ce1d10 | subsection | 123 | 129 | Proofs of Section | For any C_{(i+1)}-C_{\emptyset }\le p_a<C_{(i)}-C_{\emptyset }, there is a unique \hat{j}\in \lbrace 1, \dots , i\rbrace such that \frac{\sum _{k=\hat{j}+1}^{i}E_{(k)}}{\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}}\le p_a<\frac{\sum _{k=\hat{j}}^{i}E_{(k)}}{\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset ... | {
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fdd5fad49358db72b653b9c9a9d0ff434d16d57e | subsection | 124 | 129 | Proofs of Section | We now show that in \Lambda ^i_j, p_d<p_d^{ij}(p_a):
p_d^{ij}(p_a)\stackrel{(\ref {cdij})}{>}p_d^{ij}\left(\frac{\sum _{k=j}^{i}E_{(k)}}{\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}}\right)=\left(\sum _{k=1}^{j-1} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}\right)^{-1}\stackrel{(\ref {partition})}{>}p_d.
Hence, ... | {
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"Manxi Wu",
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332f064f09a89e4599affc2bf1f9f1067583b568 | subsection | 125 | 129 | Proofs of Section | Therefore, we can re-express \widetilde{\Lambda }^1 as follows:
\widetilde{\Lambda }^1&\stackrel{(\ref {regimej_constraint_1})}{=} \left\lbrace \left(p_a, p_d\right) \left|p_a< \widetilde{p}_d^{-1}(p_d), ~ p_d> \left(\frac{E_{(1)}}{C_{(1)}-C_{\emptyset }}\right)^{-1} \right.\right\rbrace \\
&=\left\lbrace \left(p_a, p... | {
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} | 1804.00391 | Securing Infrastructure Facilities: When does proactive defense help? | [
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900edd61fff056fa230bb6b142e9468f151f36d6 | subsection | 126 | 129 | Proofs of Section | Analogous to (REF ), we re-express the set \widetilde{\Lambda }_j as follows:
\widetilde{\Lambda }_j&\stackrel{(\ref {regimej_constraint})}{=} \left\lbrace \left(p_a, p_d\right) \left|p_a< \widetilde{p}_d^{-1}(p_d), ~ \left(\sum _{k=1}^{j} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}\right)^{-1}\le p_d< \left(\sum _{k=1}^{j... | {
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b56ad12269ad1b49cad9e95258b7be3dd4d3701a | subsection | 127 | 129 | Proofs of Section | Consider any cost parameters \left(p_a, p_d\right) in the set \Lambda ^i_j\cap \lbrace \left(p_a, p_d\right)|p_d> \widetilde{p}_d(p_a)\rbrace , from (REF ), we can find \hat{j} such that \frac{\sum _{k=\hat{j}+1}^{i}E_{(k)}}{\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}}\le p_a<\frac{\sum _{k=\hat{j}}^{i}E_{(k... | {
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4776d61ab4c1d336e51c870e18f23c23d106d843 | subsection | 128 | 129 | Proofs of Section | If \hat{j}<j, then since p_a\ge \frac{\sum _{k=\hat{j}+1}^{i}E_{(k)}}{\sum _{k=1}^{i} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}}, from (REF ), we have \widetilde{p}_d(p_a)=p_d^{i\hat{j}}(p_a)\ge \left(\sum _{k=1}^{\hat{j}} \frac{E_{(k)}}{C_{(k)}-C_{\emptyset }}\right)^{-1} \ge \left(\sum _{k=1}^{j-1} \frac{E_{(k)}}{C_{(k)... | {
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c2d295c2c586b7c27cf4062f180ba56869bd9253 | abstract | 0 | 16 | Abstract | In mathematics, many notations have been invented for the concise
representation of mathematical formulae. Tensor index notation is one of such
notations and has been playing a crucial role in describing formulae in
mathematical physics. This paper shows a programming language that can deal
with symbolical tensor indic... | {
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Index Notation into Programming Languages | [
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0123874513357cd0acb31aca764ac30ab0cbab65 | subsection | 1 | 16 | Introduction | Tensor index notation invented by Ricci and Levi-Civita has been playing a crucial role to develop differential geometry and the wide range of theoretical physics including the general theory of relativity.
This is because this notation makes the description of tensor calculus intuitive.Intuitive representation is imp... | {
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2041dbf85bb89ae974cdb02d4cfd574cda5e2f7e | subsection | 2 | 16 | Introduction | For existing work using this method, there are Maxima , , , a computer algebra system that introduces index notation through the extension library itensor, and Ahalander's work , which implements index notation on C++.
These studies introduce index notation by implementing two special functions “+” and “\cdot ” that su... | {
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... | 1804.03140 | Symbolical Index Reduction and Completion Rules for Importing Tensor
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e367dbe10c258cce11519a0ecfc3dbf61f9486d7 | subsection | 3 | 16 | Result | This paper proposes a method that enables us to apply arbitrary user-defined functions to tensor arguments using index notation without requiring an additional description to enable each function to handle tensors.
It is achieved by introducing two types of parameters, scalar parameters and tensor parameters.
First, we... | {
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9be07607f17304c3d1aba1c6d8fd88000825b39e | subsection | 4 | 16 | Scalar and Tensor Parameters | The basic contribution of this paper is that it introduces two types of parameters, scalar and tensor parameters.
Scalar and tensor parameters are used to define two types of functions, scalar functions and tensor functions, respectively.
Scalar functions are functions that are defined for scalar arguments.
For example... | {
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} | 1804.03140 | Symbolical Index Reduction and Completion Rules for Importing Tensor
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61b714f1e42e5c7e697ac1712b5d75fa2a9bcd8e | subsection | 5 | 16 | Scalar and Tensor Parameters | In particular, the reason that the loop structure by the Sum expression in the Wolfram language does not appear in our expression to express \Gamma ^{m}_{\;jk} \Gamma ^{i}_{\;ml} - \Gamma ^{m}_{\;jl} \Gamma ^{i}_{\;mk} is that the “.” function can handle Einstein summation notation.The part that we would like the reade... | {
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ab34eee8c4ab7e50b063e032b091b2a2bb7ee0f7 | subsection | 6 | 16 | Reduction Rules for Tensors with Indices | In this section, we discuss the index reduction rules that are compatible with the idea of scalar and tensor parameters.
Tensors are combined by the unified way by scalar and tensor functions as we explained in the previous section.
Therefore, we consider the reduction rules just for a single tensor.First, to access th... | {
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ecf39a6f6700f370dcc28a291f0df7ca4d71005c | subsection | 7 | 16 | Reduction Rules for Tensors with Indices | However, in that case, the summarized indices become a supersubscript, which is represented by “~_”.[|[|11 12 13|] [|21 22 23|] [|31 32 33|]|]~i_i;[|11 22 33|]~_iEven when three or more indices of the same symbol appear that contain both supersubscripts and subscripts, our system converts it to the tensor composed of d... | {
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61ea93d47aa527b7687de7a34d4bbf40ef70a287 | subsection | 8 | 16 | Implementation of Scalar and Tensor Parameters | In this section, we explain how to implement scalar and tensor parameters.As with ordinary parameters, when a tensor parameter obtains a tensor as an argument, the function treats the tensor as it is.
It means the implementation of tensor parameters is same with the ordinary parameters.In contrast, when a scalar parame... | {
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bb669daf6de1e38a93738b7082f27cac55af5c02 | subsection | 9 | 16 | Inverted Scalar Arguments | The “∂/∂” function appearing in Figure REF is a scalar function.https://github.com/egison/egison/blob/master/lib/math/analysis/derivative.egi
However, “∂/∂” is not a normal scalar function.
“∂/∂” is a scalar function that inverts indices of the tensor given as its second argument.
For example, the program “(∂/∂ Γ~i_j_k... | {
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Index Notation into Programming Languages | [
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d14e51c33f9255d9e9110f971f19e6d6d85f5cbf | subsection | 10 | 16 | The | The with-symbols expression is syntax for generating new local symbols, such as the Module http://reference.wolfram.com/language/ref/Module.html expression in the Wolfram language.
One-character symbols that are often used as indices of tensors such as “i”, “j”, and “k” are often used in another part of a program.
Gene... | {
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05653294b4d98312e31ff884bda3bc285bbbb923 | subsection | 11 | 16 | Index Completion Rules for Tensors with Omitted Indices | By designing the index completion rules for omitted indices properly, we can extend our method explained so far to express a calculation handling the differential forms . | {
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bd41782633097156937e4f32f2327d518fb8e199 | subsection | 12 | 16 | Differential Forms | In mathematics, we treat a n-th order tensor as a k-form when only (n-k) indices are appended to it.
For example, we treat a third order tensor “ω~i_j” as a matrix-valued 1-form.
We import the same convention into programming.
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e6e5307a17e2c801f20d28d466414f158484ca96 | subsection | 13 | 16 | Differential Forms | We can see the sample programs that use the functions defined above in Egison Mathematics Notebookhttps://www.egison.org/math/. | {
"cite_spans": []
} | 1804.03140 | Symbolical Index Reduction and Completion Rules for Importing Tensor
Index Notation into Programming Languages | [
"Satoshi Egi"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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