Security Game with Non-additive Utilities and Multiple Attacker Resources

1.1 MB·

by  Sinong Wang

Security Game with Non-additive Utilities and Multiple
Attacker Resources
Sinong Wang Ness Shroff
DepartmentofECE DepartmentsofECEandCSE
TheOhioStateUniversity TheOhioStateUniversity
Columbus,OH-43210 Columbus,OH-43210
[email protected] [email protected]
ABSTRACT theoptimaldefensestrategy. Itallowstheanalysttofactordiffer-
7
entialrisksandvaluesintothemodel,incorporategame-theoretic
1 Therehasbeensignificantinterestinstudyingsecuritygamesfor
predictionsofhowtheattackerwouldrespondtothesecuritypol-
0 modelingtheinterplayofattacksanddefensesonvarioussystems
icy,andfinallydetermineanequilibriumstrategythatcannotbeex-
2 involving critical infrastructure, financial system security, politi-
ploitedbyadversariestoobtainahigherpayoff.Inthepastdecade,
calcampaigns,andcivilsafeguarding. However,existingsecurity
n therehasbeenanexplosionofresearchattemptingtoaddressthis
gamemodelstypicallyeitherassumeadditiveutilityfunctions,or
a approach,whichhasledtothedevelopmentofwell-knownmodels
thattheattackercanattackonlyonetarget. Suchassumptionslead
J ofsecuritygames.
to tractable analysis, but miss key inherent dependencies that ex-
0 Theclassicsecuritygameisatwo-playergameplayedbetweena
ist among different targets in current complex networks. In this
3 defenderandanattacker.Theattackerchoosesonetargettoattack;
paper, we generalize the classical security game models to allow
Thedefenderallocates(randomly)limitedresources,subjecttovar-
for non-additive utility functions. We also allow attackers to be
] ious domain constraints, to protect a set of targets. The attacker
T abletoattackmultipletargets.Weexaminesuchageneralsecurity
(defender)willobtainthebenefits(losses)forthosesuccessfullyat-
G gamefromatheoreticalperspectiveandprovideaunifiedview. In
tackedtargetsandlosses(benefits)forthosedefendedtargets. The
particular,weshowthateachsecuritygameisequivalenttoacom-
. goalofthedefenderistochoosearandomstrategysoastoplay
s binatorialoptimizationproblemoverasetsystemε,whichconsists
c of defender’s pure strategy space. The key technique we use is optimallyundersomesolutionconceptssuchasNashequilibrium
[ andstrongStackelbergequilibrium.Thissecuritygamemodeland
basedonthetransformation,projectionofapolytope,andtheelip-

1 soidmethod. Thisworksettlesseveralopenquestionsinsecurity itsgame-theoreticsolutioniscurrentlybeingusedbymanysecu-
rityagenciesincludingUSCoastGuardandFederalAirMarshals
v gamedomainandsignificantlyextendsthestate-of-the-artofboth
Service(FAMS)[1],TransportationSystemAdministration[2]and
4 thepolynomialsolvableandNP-hardclassofthesecuritygame.
eveninthewildlifeprotection[3]; seebookbyTambe[4]foran
4
overview.
6 Keywords
8 1.1 Motivation
Securitygames;Computationalgametheory;Complexity
0
Thereexiststwocommonlimitationsoftheclassicsecuritygame
.
1 1. INTRODUCTION model:first,itdoesnotconsiderthedependencyamongthediffer-
0 enttargets; second, theattackercanattackatmostonetarget. In
Thekeyprobleminmanysecuritydomainsishowtoefficiently
7 particular,thepayofffunctionsforbothplayersareadditive,i.e,the
1 allocatelimitedresourcestoprotecttargetsagainstpotentialthreats.
payoffofagroupoftargetsisthesumofthepayoffsofeachtarget
: Forexample,thegovernmentmayhavealimitedpoliceforcetoop-
v eratecheckpointsandconductrandompatrols.However,theadver- separately. Thisassumptionmeansthatthesecurityagencymea-
Xi sarialaspectinsecuritydomainposesauniquechallengeforallo- surestheimportanceofseveraltargetswithoutconsideringthesyn-
ergyamongthem. Inpractice,theattackercansimultaneouslyat-
catingresources.Anintelligentattackercanobservethedefender’s
r tackmultipletargetsandthereexistssomelinkagestructureamong
a strategyandgatherinformationtoschedulemoreeffectiveattacks.
thosetargetssuchthatattackingonetargetwillinfluencetheother
Therefore,thesimplerandomstrategyof“rollingthedice”maybe
targets.Forinstance,anattackerattemptstodestroytheconnectiv-
exploited by the attacker, which greatly reduces the effectiveness
ityofanetworkandthedefenderaimstoprotectit. Thestrategy
ofthestrategy.
for both players is to choose the nodes of the network. If there
With the development of computational game theory, such re-
aretwonodesthatconstituteabridgeofthisnetwork,successfully
sourceallocationproblemscanbecastingame-theoreticcontexts,
attackingbothofthemwillsplitthenetworkintotwopartsandin-
whichprovidesamoresoundmathematicalapproachtodetermine
curahugedamage,whileattackinganyoneofthemwillhaveno
significanteffect.
Permissiontomakedigitalorhardcopiesofallorpartofthisworkforpersonalor
classroomuseisgrantedwithoutfeeprovidedthatcopiesarenotmadeordistributed
forprofitorcommercialadvantageandthatcopiesbearthisnoticeandthefullcita- EXAMPLE 1. AsshowninFig.1,wehavea20−nodenetwork.
tiononthefirstpage. Copyrightsforcomponentsofthisworkownedbyothersthan Itisclearlythatnodes1,2,3and4arethecriticalbattlefieldsin
ACMmustbehonored.Abstractingwithcreditispermitted.Tocopyotherwise,orre-
thisnetwork. Supposethattheattacker’sanddefender’sstrategies
publish,topostonserversortoredistributetolists,requirespriorspecificpermission
and/[email protected]. are{1},{2},{3},{1,2}or{3,4},where{v}denotestheindexof
the nodes. We adopt the network value function proposed by [5]
as the security measure for different nodes, which calculates the
(cid:13)c 2017ACM.ISBN978-1-4503-2138-9.
importance of a group of nodes via subtracting the value of the
DOI:10.1145/1235
Additive
 Utility
 Function
1 Strategy 1 2 3 4 1,2 3,4
Benefit 39 39 75 75 78 150
3 4
2
Non-­‐additive
 Utility
 Function
Strategy 1 2 3 4 1,2 3,4
Benefit 39 39 75 75 238 142
Figure1:Exampleofsecuritygameina20−nodesnetwork.
network by removing these nodes from the value of the original framework[8]andshowsthatthecompactlyrepresentedsecurity
network1.Forexample,ifweadoptthenetworkvalueasafunction gameispolynomialsolvableinsomecaseswhileNP-hardinother
f({n }) = (cid:80) n2, where n is number of nodes in the ith con- cases. [14] shows that the spatial and temporal security game is
i i i i
nectedcomponent,thevalueoftheoriginalnetworkis202 =400. generallyNP-hard. [15]providesaninterestingresultthat, ifthe
Afterremovingnode3,thenetworkwillbedividedintotwocom- attackerhasasingleresource, thestrongStackelbergequilibrium
ponents:one18−nodenetworkandoneisolatednode,thenetwork is also a Nash equilibrium, which resolves the leader’s dilemma;
valueisreducedto182+12 =325. Thusthebenefitofnode3is iftheattackerhasmultipleresources,thispropertydoesnothold.
equaltothedecrement400−325=75. Similarly,wecangetthe [16]provesthatthegeneralsecuritygamewithcostlyresources3is
benefitsofothernodesasillustratedinthebottomtableofFig1. NP-hardandproposesanapproximationalgorithm.
Intraditionalsecuritygamemodels[6],theyassumethattheben- Theearliermentionedworksstudyingthecomplexityofthese-
efit of strategy {1,2} and {3,4} is equal to 39+39 = 78 and curity game focus on the game with a single attacker resource.
75+75 = 150. Themixedstrategyequilibrium2 underthiscase However, none of these works provide a systematic understand-
isthatdefenderchoosenodes1,2withprobability0.34andnodes ingofcomplexitypropertiesorprovideanefficientalgorithmfor
3,4 with probability 0.66. Instead, if we adopt the true value of thesecuritygamewhenattackerhasmultipleresourcesandutility
nodes{1,2}and{3,4}(asillustratedinredofbottomtable),the functions are non-additive. In [6], the authors extend the classic
equilibriaisthatthedefenderchoosesnodes1,2withprobability securitygamemodeltothescenarioofmultipleattackerresources.
0.63andnodes3,4withprobability0.37. Fromthepointviewof Theydesigna6−statestransitionalgorithmtoexactlycomputethe
thenetwork,thesecondoneprovidesamorereliablestrategy. Nashequilibriuminpolynomialtime. Suchanalgorithmiscom-
plicatedandrestrictedtothecasethatthedefender’sresourceisho-
AscanbeseeninExample1,thetraditionalmodelsthatignore mogenous,i.e.,thedefendercanprotectanysubsetoftargetswith
the inherent synergy effect between the targets are limiting and a cardinality constraint. In the practical scenario such as FAMS,
could lead to catastrophic consequences. A comprehensive anal- thedefender’sresourcesmaybeheterogenousandsolvingsucha
ysisregardingtheeffectofdependenttargetscanbefoundin[7]. scenarioisstillanopenquestioninthesecuritygamedomain.
Inwork[17],theauthorsproposetoinvestigatethesecuritygame
1.2 RelatedWorks
with non-additive utilities. However, they assume that the secu-
Thenatureofresourceallocationinsecuritygamesoftenresults rity game is zero-sum, the defender’s resources are homogenous
in exponentially many pure strategies for the defender, such that andonlyoneonegroupofutilities(benefitfunction). Therecent
thedefender’soptimalmixedstrategyishardtosolve. Inthepast work [18] provides a unified framework of the classical security
decades,therehavebeennumerousalgorithmsdevelopedforvari- gamemodelwithasingleattacker’sresource. Heshowsthatsolv-
ousextensionsoftheclassicalsecuritygamemodeldiscussedear- ingthesecuritygameisequivalenttosolvingalinearoptimization
lier. problemoverasetsystem. Usingtheflexibilityofsuchasetsys-
Onelineofresearchfocusesondesigninganefficientalgorithm tem, their framework can encode most previous security games.
tosolvesuchagame. [8]proposesacompactrepresentationtech- Forexample,ifthesetsystemisauniformmatroid,itrecoversthe
nique,inwhichthesecuritygamecanbeequivalentlyrepresented resultofLAXcheckpointplacementproblem[19]; ifthesetsys-
byapolynomial-sizedmixed-integerlinearprogramming(MILP) temrepresentsacoverageproblem,itrecoversmostresultsin[9],
problem. The issue in [8] is that they only determine the opti- i.e.,thepolynomialsolvabilityof2−weightedcoverageimpliesthe
malsolutionofthecompactgameinsteadoftheoptimaldefender’s polynomialsolvabilityofequilibriumcomputation; ifthesetsys-
mixedstrategy.Tosolvethisproblem,[9]introducestheBirkhoff- temrepresentstheproblemofindependentset,itrecoverstheNP-
von Neumann theorem and show that the defender’s mixed strat- hardnessresultin[2].However,theyleaveopenthequestion:what
egy can be recovered under a specific condition. [10] proposes is the complexity of the security game model when attacker has
adouble-oraclealgorithmtoexactlysolvethesecuritygamewith multipleresourcesandutilityfunctionsarenon-additive.
exponentiallargerepresentation,whichcanberegardedasagen-
1.3 OurResults
eralization of traditional column generation technique in solving
thelarge-scalelinearprogrammingproblem. Therearealsoother Inthispaper,westudytheclassicsecuritygamemodelwhenat-
workssuchastheBayesiansecuritygame[11],thesecuritygame tackerhasmultipleresourcesandutilityfunctionsarenon-additive.
with quantal response [12] and the security game with uncertain Morespecifically,wewonderhowthefollowingquestionsthatare
attackerbehavior[13],etc. wellunderstoodinthecaseofsingleattackerresourceandadditive
Anotherlineofresearchfocusesonexamingthecomplexityof utilityfunctionscanbeaddressedinthisgeneralcase.
thesecuritygame. [9]adoptsthepreviouscompactrepresentation
• Howtocompactlyrepresentthesecuritygamewithmultiple
1Compared with traditional measures such as node degree or be- attackerresourcesandnon-additiveutilityfunctions?
tweenness centrality, the network value provides a more accurate
description of the importance of different nodes. We can use the 3Here “costly resources” means that the defender’s resource are
otherkindsofnetworkvaluesinsteadofthequadraticform. obtainedatsomecostsandthedefenderhasabudgettoallocatehis
2Inthisexample,weadoptthezero-sumsecuritygamemodel. resources.
Table1:SolutionStatusintheSecurityGame
Cases Singleattackerresource Multipleattackerresource
Homogenousresource Heterogenousresource
Zero-sum SSE,NE[8,15,18,20] SSE,NE[6] SSE,NETheorem6,Lemma11
Additiveutilityfunction
Non-zero-sum SSE[20],NE[15,18] NE[6] SSETheorem7,NETheorem8
Zero-sum Sameasabove NE,SSETheorem6,Theorem9
Non-additiveutilityfunction
Non-zero-sum Sameasabove SSETheorem7,Theorem9
• Howtoefficientlysolvesuchacompactlyrepresentedgame? utilityfunctions,andthepreliminariesregardingsomeclassicalre-
sultsincombinatorialoptimization. InSection3, wepresentour
• Whatisthecomplexityofthesecuritygamewhenwecon-
frameworkofpolytopetransformationandprojectiontocompactly
sidernon-additiveutilityfunctionsandallowtheattackersto
representthezero-sumsecuritygame.InSection4,wepresentthe
attackmultipleattackerresources?
reductionbetweenthezero-sumsecuritygameandthecombinato-
To answer these questions, we provide the following contribu- rialoptimizationproblem. InSection5,wefurthergeneralizeour
tions:(1)wefirstproposeapolytopetransformationandprojection frameworktothenon-zero-sumsecuritygameandpresentthemain
framework to equivalently and compactly represent the zero-sum results.Finally,weprovidesnumerousapplicationsofourtheoret-
and non-additive security game with only poly(n) variables; (2) icalframeworkinSection6. WeconcludeourworkinSection7.
WeprovethattheproblemofdeterminingtheNashequilibriumof Duetothespacelimitation,allofourtechnicalproofsareprovided
zero-sumandnon-additivesecuritygameandtheproblemofopti- intheAppendix.
mizingaPseudo-Booleanfunctionoverasetsystemεcanbere-
ducedtoeachotherinpolynomialtime.Themaintechniqueweuse
2. MODELANDPRELIMINARY
istoexploitthegeometricstructureofthelow-dimensionalpoly-
topetoconstructapolynomialtimevertexmappingalgorithm. (3) In the following, we first define the security game with non-
Wethenapplyourframeworktothenon-zero-sumandnon-additive additiveutilityfunctionsandmultipleattackerresourcesasatwo-
securitygame,andfurtherobtainasimilarresultthatdetermining playernormal-formnon-zero-sumgame. Thenwepresentseveral
thestrongStackelbergequilibriumandtheabovecombinatorialop- classicresultsincombinatorialoptimization.
timizationproblemisequivalent.(4)Finally,weexaminetheNash
equilibrium in the non-zero-sum but additive security game. We 2.1 ProblemDescription
provethatdeterminingtheNashequilibriumcanbereducedtothe
The model is similar to the classic security game [8], and the
linearoptimizationoverasetsystemε. Thebasictechniqueisto
onlydifferenceisthatweconsidermultipleattackerresourcesand
add another polytope transformation step in our previous frame-
non-additiveutilityfunctions.
work,andshowthatdeterminingtheNashequilibriumofnon-zero-
Playersandtargets:Thesecuritygamecontainstwoplayers(a
sumgamecanbereducedtoapolynomialdimensionalsaddlepoint defenderandanattacker),andntargets.Weuse[n](cid:44){1,2,...,n}
problem. ThemainresultsandcomparisonissummarizedinTA-
todenotethesetofthesetargets.
BLE1.2.
Strategiesandindexfunction:Thepurestrategyforeachplayer
Theseresultsdemonstratethatthesecuritygamewithnon-additive
is the subset of targets and all the pure strategies for each player
utilityfunctionandmultipleattackerresourceisessentiallyacom-
constitute a collection of subsets of [n]. We assume that the at-
binatorialproblem, andprovidea systematic frameworktotrans- tacker can attack at most c targets, where c > 1 is a constant4.
formthegame-theoreticalproblemtotheproblemofcombinatorial
Theattacker’spurestrategyspaceisauniformmatroidA={A⊆
algorithmdesign. Further, ourresultsnotonlyanswerstheques- [n]||A|≤c}andthenumberofattacker’spurestrategiesisN (cid:44)
tionsproposedinthesecuritygamedomain[6,18],butalsoextends a
|A|.Similarly,weuseD∈2[n]todenotethedefender’spurestrat-
significantlyboththepolynomialsolvableandNP-hardclass:
egy space and N (cid:44) |D|. Note that there exists some resource
d
• The previous result in [18] is dependent on the description allocation constraints in practice and such that D is not always a
lengthofthesetsystem. Onespecialcaseofourresultcan
uniform matroid. For example, if the defender has a budget and
recoverandstrengthenthemainresultsin[18],whichisin-
itsresourceareobtainedatsomecosts,inwhichthecostsarehet-
dependentofthedescriptionlengthofthesetsystem.Italso
erogenous. Inthiscase,thedefender’sfeasiblepurestrategyisall
implies that we also recover and strengthen the results in
thepossiblecombinationsofthetargetswithtotalcostlessthanthe
mostsecuritygamepapers.
budget.
• Thepolynomialtimesolvabilityin[6]correspondingtothat Suppose that the order of the pure strategy of the attacker is
thesetsystemε,isauniformmatroid,whichcanbeeasily givenbyindexfunctionσ(·),whichisaone-onemapping:2[n] →
solvedbysummingthefirstk largestelements. Inthesce- {1,2,··· ,2n}.Then,wedefinethefollowingindexfunctionµ(·)
nario of the heterogenous resource, we can extend several for the pure strategy of the defender as: µ(U) = σ(Uc) for any
polynomialsolvableclasses. U ∈ 2[n]. Forsimplicity,theindexfunctionσ(·)andµ(·)arede-
finedoverallsubsetsof[n]. Thereasonbehindthisdefinitionof
• Solve the security game occurred in the tree network or a
the index function is to simplify the representation of most theo-
sparsenetwork.Inthesecases.althoughtheutilityfunctions reticalresults. Forexample,ifn = 2,A = D = 2{1,2},andthe
are non-additive, we can show such a problem can be re-
orderoftheattacker’spurestrategyisσ({1,2}) = 1,σ({2}) =
ducedtoapolynomialsolvableoraclesuchasasub-modular
2,σ({1})=3andσ({∅})=4,thentheorderfordefender’spure
minimizationproblem.
strategyisµ({∅})=1,µ({1})=2,µ({2})=3andµ({1,2})=
ThedetaileddiscussionscanbeseeninSection6.Therestofthe 4.
paperisorganizedasfollows. InSection2,weintroducethesecu-
ritygamemodelwithmultipleattackerresourcesandnon-additive 4Later,inSection5,wewillrelaxthisconstantassumption.
Successfully Insomeapplicationdomain,thedefendercanbuildfortifications
attacked.
beforetheattackandisthusintheleader’spositionfromthepoint
viewofthegame, andabletomovefirst. Inthiscase, thestrong
Stackelberg equilibrium (SSE) serves as a more appropriate solu-
Successfully tionconcept[21,22],wherethedefendercommitstoamixedstrat-
defended.
egy;theattackerobservesthisstrategyandcomesupwithitsbest
(LimAittetda cBkuedrg et) (resoDurecfee nadlloecr ation responses.Formally,letC(q)=argmaxp∈∆Na Ua(p,q)denote
Select multiple constraint) theattacker’sbestresponsetodefender’smixedstrategyq. Apair
targets to attack and Select a subset of ofmixedstrategies(p∗,q∗)isaSSE,ifandonlyif,
avoid being trapped Interdependent targets to defend.
by defender
Targets q∗ =arg max U (C(q),q)andp∗ =C(q∗).
d
(Non-additive payoffs) q∈∆Nd
Figure2:Securitygamewithnon-additiveutilityfunctionsand Ourgoalistocomputethedefender’sNashequilibriumstrategies
multipleattackerresources. andstrongStackelbergequilibriumstrategiesandwecallitasequi-
libriumcomputationproblem. Thefollowingdefinitionsareof-
Themixed strategyistheprobabilitydistributionoverthepure tenusedinourtheoreticaldevelopment.
strategyspace,whichisemployedwhentheplayerchoosesitsstrat-
egybasedonsomerandomexperiment.Specifically,iftheattacker DEFINITION 1. (Commonutility)Thecommonutilityisdefined
astheMo¨biustransformation[23,24]ofthebenefitandlossfunc-
choosespasitsmixedstrategy,theprobabilitythatthestrategyA
tionB (U)andL (U)forallU ∈2[n],
ischosenisp . Thesetofallthemixedstrategiesofattacker a a
σ(A)
awnhderdeefender can be represented as the simplex ∆Na and ∆Nd, Bac(U)= (cid:88)(−1)|U\V|Ba(V), Lca(U)= (cid:88)(−1)|U\V|La(V).
V⊆U V⊆U
∆ ={p∈RNa| (cid:88) p =1}.
Na + σ(A) Similar definitions hold for defender’s benefit and loss function:
A∈A B (·),L (·)andtheircommonutilities:Bc(·),Lc(·).
d d d d
Similardefinitionholdsfor∆ .
Nd
Payoff Structure: The benefits and losses are represented by DEFINITION 2. (Support set) The support set of the security
utilityfunctionsasfollows. LetsetfunctionB (·) : A → Rand gameisdefinedas
a
B (·):D →Rbetheattacker’sanddefender’sbenefitfunctions,
anddthesetfunctionL (·):A→RandL (·):D→Rbethecor- S ={A∈A|Bac(A)orBdc(A)orLca(A)orLcd(A)(cid:54)=0}. (1)
a d
respondinglossfunctions.Thestandardassumptionisthattheben- andthesupportindexsetσ(S)={σ(A)|A∈S}.
efitisalwayslargerthantheloss: B (A)>L (A)andB (A)>
a a d
Ld(A)forallA∈A. Iftheattackeranddefenderchoosestrategy DEFINITION 3. (Projection operator) The projection operator
A ∈ AandD ∈ D,theattacker’sanddefender’spayoffisgiven π :RN →R|S|is
S
byB (A\D)+L (A∩D)andB (A∩D)+L (A\D),respec-
tivelya5.Asecurityagameiszero-sumdifB (A\D)+d L (A\D)=0 πS((x1,x2,...,xN))=(...,xi,...)i∈σ(S), (2)
a d
andBd(A∩D)+La(A∩D)=0foranyAandD,whichmeans andprojectionofpolytope:Π (∆ )(cid:44){π (x)|x∈∆ }.
S N S N
thatoneplayer’sbenefitisindeedthelossoftheotherplayer.
Bilinear-form:Basedontheabovepayoffstructure,wecande- Thefollowingdefinitionofthesetsystemεisabinaryrepresenta-
finethebenefitmatricesofattackeranddefender: Ba andBd ∈ tionofthedefender’spurestrategyspaceD.
RNa×Nd:∀A∈A,D∈D,
Baσ(A),µ(D) =Ba(A\D),Bdσ(A),µ(D) =Bd(A∩D), 1{iD∈EFDIN}I,T∀IO1N≤4i.≤(Snet,∀SyDste∈mD)T}h.esetsystemε(cid:44){x∈{0,1}n|xi =
andthelossmatrices:LaandLd ∈RNa×Nd:∀A∈A,D∈D, 2.2 Preliminaries
Laσ(A),µ(D) =La(A∩D),Ldσ(A),µ(D) =Ld(A\D), LetH beanon-emptyconvexpolytopeinRn. Givenavector
w∈Rn,onewantstofindasolutiontomax wTx.By“linear
LetMa andMd betheattacker’sanddefender’spayoffmatrices. optimizationoverH”,wemeansolvingtheprxo∈bHlemmax wTx
ItisclearthatMa = Ba+La andMd = Bd+Ld. Thenthe foranyw ∈Rn. AseparationproblemforH isthat,givxe∈nHavec-
expectedpayoffsfortheattackeranddefenderisgivenbyfollowing torx ∈ Rn,decideifx ∈ H,andifnot,findahyperplanewhich
bilinear form, when they play the mixed strategy p ∈ ∆ and
Na separates x from H. The following results are due to Gro¨tschel,
q∈∆ ,by
Nd LovaszandSchrijver[25].
U (p,q)=pTMaq and U (p,q)=pTMdq.
a d THEOREM 1. (Separation and optimization) Let H ∈ Rn be
Solution Concepts: If both players move simultaneously, the aconvexpolytope. Thereisapoly(n)timealgorithmtosolvethe
standardsolutionconceptistheNashequilibrium(NE),inwhich linearoptimizationproblemoverHifandonlyofthereisapoly(n)
nosingleplayercanobtainahigherpayoffbydeviatingunilaterally timealgorithmtosolvetheseparationproblemforH.
fromthisstrategy. Apairofmixedstrategies(p∗,q∗)formsaNE
ifandonlyiftheysatisfythefollowing:∀p∈∆ ,q∈∆ , THEOREM 2. (Separationandconvexdecomposition)LetH ∈
Na Nd Rn be a convex polytope. If there is a poly(n) time algorithm
Ud(p∗,q∗)≥Ud(p∗,q)andUa(p∗,q∗)≥Ua(p,q∗). to solve the separation problem for H, then there is a poly(n)
5A\D isthestandardsetdifference,definedbyA\D = {x|x ∈ time algorithm that, given any x ∈ H, yields (n+1) vertices
A,x∈/ D}andisequaltoA∩Dc,whereDcisthecomplementary v1,...,vn+1 ∈Handconvexcoefficientsλ1,...,λn+1suchthat
setofsubsetD. x=(cid:80)n+1λ vi.
i=1 i
ThefollowingresultisthegeneralizationofthevonNeumann’s Moreover, consideringthefactthatonlythefirstr elementsin
minimax theorem, which provides a condition when we can use vectorp¯ andq¯ andfirstselementsinp¯ andq¯ havethenon-
1 1 2 2
theminimaxormaximinformulationtosolveasaddlepointprob- zerocoefficientsintheaboveoptimizationmodel, wecanfurther
lem[26]. simplifytheaboveoptimizationproblemas
min max p¯TBaq¯ +p¯TLaq¯ , (5)
THEOREM 3. (Sion’s minimax theorem) Let X be be a com- (q¯1,q¯2)∈Hd(p¯1,p¯2)∈Ha 1 r 1 2 s 2
pactconvexsubsetofalineartopologicalspaceandY isaconvex
wheretheH andH isobtainedbyprojectingthepolytope∆a
subsetofalineartopologicalspace. Iff isareal-valuedfunction a d Na
onX×Y with and∆dNd tothosecoordinatesbelongingtothenon-zeroblocks.
Thebasicobservationintheaboveexampleisthatthenumber
1. f(x,·)islowersemicontinuousandquasi-convexony,∀x∈ of variables in the optimization model (5) is equal to the sum of
Xand rankr+sofpayoffmatrices. Basedontherankinequalitythat
therankofamatrixislessthanitsdimension,wehavethatr,s≤
2. f(·,y)isuppersemicontinuousandquasi-concaveonx,∀y∈
min{N ,N }. Since the number of attacker’s pure strategies is
Y, N = Oa(ncd) = poly(n). Therefore,thereexistsatmostpoly(n)
a
thenwehave variablesintheoptimizationmodel(5).
3.2 FormalDescription
minmaxf(x,y)=maxminf(x,y). (3)
y∈Y x∈X x∈X y∈Y
Theaboveillustrativederivationprovidesapossiblepathtocom-
3. THECOMPACTREPRESENTATION pactlyrepresentthegame.However,thereexistsasignificanttech-
nicalchallenge: theelementarymatricesF ,F andtheirinverse
FORZERO-SUMSECURITYGAME 1 2
matrices may have exponential size due to exponential large de-
The Nash equilibrium is equivalent to the strong Stackelberg fender’s pure strategy space. Hence, the key question is whether
equilibrium in the zero-sum game. Therefore, we only focus on wecanfindboththeseelementarymatricesefficiently?Totackle
the computation of Nash equilibrium. Invoking the result in the thisproblem,wefirstshowthatpayoffmatricesBaandLacanbe
vonNeumann’sminimaxtheorem,computingtheNEofzero-sum decomposedastheproductoftheseveralsimplematrices.Thefol-
gamecanbeformulatedasthefollowingminimaxproblem, lowingtechnicallemmaiscriticalinourdecomposition.
min max U (p,q)=pT (Ba+La)q. (4) LEMMA 1. (Utilityandcommonutility)ForallU ∈ 2[n],the
a
q∈∆Ndp∈∆Na benefitandlossfunctionssatisfy
Althoughitcanbecastintoalinearprogramingproblem,suchan B (U)= (cid:88) Bc(V)andL (U)= (cid:88) Lc(V).
optimizationmodelhasΩ(nk)variables,whichisexponentialinn a a a a
V⊆U V⊆U
intheworstcase,i.e.,thedefendercanprotectanysubsetsoftar-
Thelemma1providesapathtorecovertheutilityfunctionsfrom
gets.Thegoalofthissectionistodevelopatechniquetocompactly
the common utility. Indeed, it is named as the zeta transforma-
andequivalentlyrepresentthezero-sumandnon-additivesecurity
tion[24],whichistheinversetransformationoftheMo¨biustrans-
gamewithonlypoly(n)variables.Toconveyourideamoreeasily,
formationgiveninDefinition1.SupposethatBAandLArepresent
webeginwithanexample.
thebenefitandlossmatrixforattackerwhenA = D = 2[n],and
3.1 MotivatingExample Ba,Lacanberegardedasthesub-matrixofBA,LA.Thefollow-
WefirstusegausseliminationonmatricesBa andLa totrans- ingtechnicallemmapresentsthedecomposablepropertyofpayoff
matricesBAandLA.
formthemintorowcanonicalform,whichistoleftandrightmulti-
plysuchmatricesbyelementarymatricesE1,E2 ∈RNa×Na and LEMMA 2. (Decomposition of complete payoff matrix) If the
F1,F2 ∈RNd×Nd. attacker’sanddefender’spurestrategyspaceA = D = 2[n],the
min max pT (Ba+La)q payoffmatricesBA,LA ∈R2n×2n canbedecomposedas,
q∈∆Ndp∈∆Na BA =QDBQT, LA =QDLQTP, (6)
= min max pTE1E−11BaF−11F1q+pTE2E−21LaF−21F2q whereDB,DL ∈R2n×2n arethediagonalmatriceswith
q∈∆Ndp∈∆Na
= min max pTE (cid:20)Bar 0(cid:21)F q+pTE (cid:20)Las 0(cid:21)F q. DBσ(A),σ(A) =Bac(A), DLσ(A),σ(A) =Lca(A),∀A∈2[n].
q∈∆Ndp∈∆Na 1 0 0 1 2 0 0 2 TheQ,P∈R2n×2n arebinarymatrices:∀A,D∈2[n],
where r and s are the rank of matrices Ba, La, and Ba, La are
r s Q =1{Dc ⊆A}, P =1{A=D}.
thecorrespondingnon-zeroblocksoftheirrowcanonicalform. If σ(A),µ(D) σ(A),µ(D)
wedefinetheaffinetransformation: f (p)=(cid:0)pTE (cid:1)T,f (p)= Thenotation1{·}istheindictorfunction.
1 1 2
(cid:0)pTE2(cid:1)T,g1(q)=F1qandg2(q)=F2q.Let6 Thefollowinglemmaconstructstherelationbetweenthematri-
cesBA,LAandthecorrespondingsub-matrices:Ba,La.
∆a ={(f (p),f (p))|p∈∆ },
Na 1 2 Na LEMMA 3. (Completematrixandsub-matrix)Thepayoffma-
∆dNd ={(g1(q),g2(q))|q∈∆Nd}. tricesBa,La ∈RNa×Nd canbeexpressedas
wecanobtainthefollowingequivalentoptimizationproblem, Ba =SBAR, ,La =SLAR, (7)
min max p¯T (cid:20)Bar 0(cid:21)q¯ +p¯T (cid:20)Las 0(cid:21)q¯ . wherematrixS∈RNa×2n andR∈R2n×Nd aretheblockmatri-
(q¯1,q¯2)∈∆dNd(p¯1,p¯2)∈∆aNa 1 0 0 1 2 0 0 2 ces,definedas,
6Thenotation(·,·)denotestheconcatenationoperatorofvector. S =1{A=Uc},∀A∈A,U ∈2[n],
σ(A),µ(U)
R =1{D=Uc},∀D∈D,U ∈2[n]. Sincethesizeofthesupportset|S| ≤ N andN = poly(n),
σ(U),µ(D) a a
we arrive at a compact representation of zero-sum security game
Intuitively, the matrices S and R in Lemma 3 plays the role
withonlypoly(n)variables. Notethatintheabovecompactrep-
of extracting the rows and columns of the matrices BA and LA,
resentation framework, the affine transformation f and f is the
1 2
whoseindicesbelongtothefeasiblepurestrategiesofattackerand
same as in our compact representation. The following theorem
defender. For example, suppose that the defender’s pure strategy
guaranteesthecorrectnessofourcompactrepresentation.
space is a uniform matroid, i.e., D = {D ⊆ [n]||D| ≤ k}, if
theindexfunctionsatisfiesσ(U1) ≤ σ(U2),|U1| ≥ |U2|, which THEOREM 5. (Correctnessofcompactrepresentation)(p∗,q∗)
meansthattheindexofattacker’s(defender’s)purestrategyisin- is a Nash equilibrium of zero-sum security game if and only if
creasing(decreasing)withthedecreasingofthecardinalityofeach (πS(f(p∗)),(πS(g1(q∗)),πS(g2(q∗)))istheoptimalsolutionof
strategy,thenthepayoffmatricesBaandLacomesfromthebot- compactminimaxproblem(10).
tomleftofthematricesBAandLA. TheblockmatricesSandR
canberepresentedas REMARK 1. Basedthespecificformof(10),thecomplexityof
obtainingtheabovecompactrepresentation(10)isdependenton
(cid:20) (cid:21)
S=(cid:2)0Na×(2n−Na) INa×Na(cid:3),R= 0(2InN−dN×dN)×dNd , tphoelyctoopmepsleHxiat,yHodf.oTbhtaeinnionng-zmeraotreilceemseDn(cid:101)tbs,oDf(cid:101)mlaatrnidcetshDe(cid:101)pbraonjedcDt(cid:101)edl
whereIistheidentitymatrix. arethecommonutilities,whichcanbecalculatedintimeO(nc)by
CombiningtheresultsofLemma2andLemma3,wehavethe Definition1. ThecomplexityofrepresentingHaandHdisdepen-
followingdecompositionofthepayoffmatrixMa. dentontheircorrespondingdescriptionlength.
3.3 LinearProgrammingApproach
THEOREM 4. (Decompositionofthepayoffmatrix)Thepayoff
matrixMa =Ba+Lacanbedecomposedas Thecompactminimaxproblemhasalinearobjectivefunction.If
thepolytopeH andH isconvex,suchaproblemcanbecastinto
Ma =E(DbJ+DlK), (8) a d
a linear programming approach. The following technical lemma
whereDb,Dl ∈RNa×Na arethediagonalmatriceswith showsthattheabovetransformedandprojectedpolytopeisconvex.
Db =Bc(A), Dl =Lc(A),∀A∈A.
σ(A),σ(A) a σ(A),σ(A) a
TheE∈RNa×Na andJ,K∈RNa×Ndarebinarymatrices: LEMMA 4. ThepolytopesHaandHdisconvex.
Based on Lemma 4, we can formulate the minimax problem by
E =1{U ⊆A},∀A,U ∈A;J =1{A⊆Dc},
σ(A),σ(U) σ(A),µ(D) followingequivalentlinearprogrammingmodel,
K =1{A⊆D},∀A∈A,D∈D.
σ(A),µ(D) CompactLinearProgramming
AscanbeseeninTheorem4,wedecomposetheoriginalexpo- min u (11)
nentiallargepayoffmatrixMa intothesummationandtheprod-
uct of several simple matrices including binary matrices E,J,K s.t. vT(D(cid:101)bq¯1+D(cid:101)lq¯2)≤u,∀v∈Ia,
and two polynomial-sized diagonal matrices Db and Dl. More- (q¯1,q¯2)∈Hd,
over, suchadecompositionhasaclosed-formexpressionandthe whereI denotesthesetofverticesoftheconvexpolytopeH .
a a
elementsinthosesimplematricescanbeimplicitlyrepresented. Inthesequel,werefertothecompactproblemastheabovelin-
Based on the above decomposition results, we can let elemen- earprogrammingproblem.Basedonourcompactrepresentation,a
tarymatricesE1 = E2 = E,F1 = JandF2 = K,andthecor- naturalquestionthatarisesiswhethercanweefficientlysolvesuch
respondingaffinetransformationf(p) = ETpandg1(q) = Jq, alinearprogrammingproblemandimplementtheoptimalsolution
g2(q) = Kqtoyieldtwopolytopes: ∆aNa = {f(p)|p ∈ ∆Na} bythedefender’smixedstrategy? Wewillanswerthisquestionin
and∆dNd = {(g1(q),g2(q))|q ∈ ∆Nd}. Thenwecanrepresent thenextsection.
theminimaxproblem(4)as
min max p¯T(Dbq¯ +Dlq¯ ), (9) 4. SOLVING THE ZERO-SUM SECURITY
1 2
(q¯1,q¯2)∈∆dNdp¯∈∆aNa GAME IS A COMBINATORIAL PROB-
Based on the definition of our support set S and matrices Db, LEM
Dl,onlythevariableswithindicesbelongingtoσ(S)hasnon-zero Inthissection, wewillbuildtheconnectionbetweentheequi-
coefficients. Therefore,wecaneliminatethosevariableswithzero libriumcomputationinthezero-sumsecuritygameandthefollow-
coefficientsin(9)andprojectthepolytopes∆a and∆d intothe ingdefenderoracleproblem(DOP)viaourcompactrepresentation
Na Nd
coordinateswithindicesbelongingtoσ(S).Thefurthersimplified framework. Forsimplicity, weuseI todenotesetofverticesof
d
modelcanbeexpressedas theconvexpolytopeH .
d
CompactMinimaxProblem DEFINITION 5. (Defenderoracleproblem)Foranygivenvec-
min maxp¯T(D(cid:101)bq¯1+D(cid:101)lq¯2), (10) torw∈R2|S|,determine,
(q¯1,q¯2)∈Hdp¯∈Ha x∗ =argminwTx. (12)
iwshoebrtea7inHedab=yeΠxtSra(c∆tinaNga)th,eHndon=-zeΠroSc(o∆ludNmdn),smanadtrrioxwD(cid:101)sbofamndatDr(cid:101)ixl Themainresultofthissectionxi∈sItdhefollowingtheorem.
DbandDl.
THEOREM 6. (NEcomputationanddenfenderoracleproblem)
7Notethateachvectorin∆d isconsistsoftwopartsg (q)and Thereisapoly(n)timealgorithmtocomputethedefender’sNash
Nd 1
g (q).Herethecorrepondinglow-dimensionalpointis(π (g (q), equilibrium(strongStackelbergequilibrium),ifandonlyifthereis
2 S 1
π (g (q)). apoly(n)timealgorithmtocomputethedefenderoracleproblem.
S 2
To obtain above reduction, we adopt the following path: we where(cid:107)·(cid:107)isthematrix’sspectrumnorm. Thenwetake(q¯ ,q¯ )
1 2
firstshowhowthecompactproblemandthedefenderoracleprob- andu astheinputoftheseparationoracleforthecompactprob-
0
lemcanbereducedtoeachotherinpoly(n)time;thenweexploit lem. Iftheoutputisyes,wealsogetacertificatethat(q¯ ,q¯ ) ∈
1 2
the geometric structure of polytope H and H to construct two H ;ifnot,outputa(2|S|+1)−dimensionalseparatinghyperplane:
a d d
poly(n)timevertexmappingalgorithmstoobtainthereductionbe-
tweentheequilibriumcomputationandthecompactproblem. aT1q¯1+aT2q¯2+bu0 >aT1q1+aT2q2+bu,
4.1 ReductionbetweenCompactProblemand ∀(q1,q2),usatisfyconstraintsof(11). Duetothespecificchoice
DOP ofu0,wecanshowthatsuchaT1q¯1+aT2q¯2alsoformsaseparating
hyperplaneforall(q ,q )∈H .Theintuitionbehindthisstepis
1 2 d
Thelinearprogramingproblem(11)haspoly(n)numberofvari-
thatwhenwechoosealargeenoughu ,thefeasibleregiondefined
0
ablesandpossiblyexponentiallymanyconstraintsduetotheirreg-
bytheconstraintsinLP(11)willdegeneratetothepolytopeH .
d
ularityofthepolytopeH . Therefore,wecanapplytheellipsoid
d Based on Lemma 5 and Lemma 6, we arrive at the reduction
method to solve such an LP, given a poly(n) time separation or-
betweenthecompactproblemandthedefenderoracleproblem.
acle. Specifically, the separation oracle of such LP is defined as
following. 4.2 Reduction between Equilibrium Compu-
tationandCompactProblem
DEFINITION 6. (SeparationoracleforLP(11))Foranygiven
(q¯ ,q¯ )andu¯,either(q¯ ,q¯ )andu¯satisfyalltheconstraintsof To obtain the reduction between the equilibrium computation
1 2 1 2
(11)orfindsahyperplane: andthecompactproblem,thereexisttwoissues:first,howtotrans-
formtheinputinstanceofeachproblemtotheotheroneinpoly(n)
aTq¯ +aTq¯ +bu¯>aTq +aTq +bu,
1 1 2 2 1 1 2 2 time;second,howtomaptheoptimalsolutionofeachproblemto
∀(q ,q ),usatisfyconstraintsof(11) theotherinpoly(n)time.Sincetheinputoftheequilibriumcompu-
1 2
tationproblemaretheuitlityfunctions{B (U)}and{L (U)}and
a a
Similarly,theseparationoracleofthedefenderoracleproblemis theinputofcompactproblemarethecommonutilities{Bc(U)}
a
definedasfollowing. and{Lc(U)}(alltheelementsofmatricesDbandDlarethecom-
a
monutilities),suchtransformationcanbecompletedinO(2cnc)=
DEFINITION 7. (SeparationoracleforDOP(12))Foranygiven poly(n)timebasedonDefinition1andLemma1.
(q¯ ,q¯ ), either (q¯ ,q¯ ) belongs to H , or finds a hyperplane:
1 2 1 2 d Toresolvethesecondissue,wefirstconsiderhowtomaptheop-
∀(q ,q )∈H suchthat
1 2 d timalsolutionofcompactproblemtothedefender’soptimalmixed
aTq¯ +aTq¯ >aTq +aTq . strategies. Based on Theorem 2, we obtain that if the separation
1 1 2 2 1 1 2 2
problemofLP(11)canbesolvedinpoly(n)time,wecandecom-
Based on Theorem 1, both our compact problem (11) and de- pose any feasible point x into a convex combination of at most
fenderoracleproblemisequivalenttotheircorrespondingsepara- (2|S|+1) vertices of the polytope defined by those constraints.
tionoracleproblem. Toobtaintheequivalencebetweenthecom- Note that this is precisely the DOP required for above reduction.
pactproblemandthedefenderoracleproblem,itremainstoshow Applyingthisresulttotheoptimalsolution(q∗,q∗)oftheLP(11),
1 2
theequivalencebetweentheabovetwoseparationoracles. wecangetaconvexdecompositionthat
Wefirstexamineonedirection: theseparationoracleofsuchan
2|S|+1
LPcanbereducedtotheseparationoracleofDOP.Specifically,the (q∗,q∗)= (cid:88) λ (vi,vi), (13)
separationoracleofsuchanLPcanbereducedtothefollowingtwo 1 2 i 1 2
parts: givenany(q¯ ,q¯ )andu¯,(1)membershipproblem: decide i=1
1 2
iwnhge(tqh¯e1r,q(¯q¯21),,qu¯¯2f)ro∈mHHdd.;I(f2n)oint,egqeunaelriatytecoanhsytrpaeirnptlapnroebthleamt:sedpeacriadte- wstrhaetreegy(vc1ia,nvb2ie)re∈gaIrdd.edTahseabcaosincvefaxcctoimstbhiantattihoendoeffietnsdpeurr’sesmtriaxteed-
whetheralltheinequalityconstraintshold.Ifnot,findoneviolating gies,eachofwhichcorrespondstoavertexofsimplex∆ .Ifwe
constraint.Wehavethefollowingresultfortheseproblems. canmapthevertices(vi,vi)backtothevertices(pureNsdtrategy)
1 2
oftheoriginalgame,denotedbyh((vi,vi)),themixedstrategies
LEMMA 5. The membership problem and the inequality con- 1 2
ofthedefendercanbeexpressedas
straintproblemofLP(11)canbereducedtotheseparationoracle
ofthedefenderoracleprobleminpoly(n)time. 2|S|+1
q∗ = (cid:88) λ h((vi,vi)). (14)
ThebasicideaintheproofofLemma5istoshowthatthenum- i 1 2
berofverticesofH ispoly(n)andeachvertexhasaclose-form i=1
a
expression.Thenwecanimplicitlychecktheinequalityconstraint Thus,thekeyliesinhowtocomputeh((vi,vi))inpoly(n)time.
1 2
problem in poly(n) time, and the membership problem is indeed To tackle this problem, we need to investigate the geometric
theseparationproblemfortheDOP.Thereversedirectionisguar- structureofpolytopeH .First,consideringanarbitrarydefender’s
d
anteedbythefollowinglemma. purestrategyD ∈ D, thecorrespondingvertexin∆ isaunit
Nd
vector eD ∈ RNd with only one non-zero element eD = 1.
µ(D)
LEMMA 6. Theseparationoracleofthedefenderoracleprob- Basedonthedefinitionofthetransformationg (q)andg (q),the
1 2
lemcanbereducedtotheseparationoracleofLP(11)inpoly(n) correspondingpointofpolytopeH is
d
time.
(g (eD),g (eD))=(JeD,KeD)=(J ,K ), (15)
1 2 µ(D) µ(D)
TheideaoftheproofofLemma6isthat,foranyinput(q¯ ,q¯ )of
1 2
separationoracleforDOP,wecanchooseaspecificu0 ∈ Rsuch whereJµ(D)andKµ(D)istheµ(D)thcolumnofmatrixJandK.
that ThenthecorrespondingpointvDoftheprojectedpolytopeHdis
u0 =|S|((cid:107)D(cid:101)b(cid:107)+(cid:107)D(cid:101)l(cid:107)) vD =(cid:0)πS(Jµ(D)),πS(Kµ(D))(cid:1), (16)
Algorithm1VertexMappingfromVertextoPureStrategy Algorithm2VertexMappingfromPureStrategytoVertex
Input: Vertex(v ,v )∈Id Input: Defender’sPureStrategyD
1 2
Output: Defender’spurestrategyD. Output: VertexvD ∈Id
T =∅; foreachV ∈Ado
foreachi∈[n]do ifV ⊆Dc thenv1D,σ(V) =1;else v1D,σ(V) =0.
Examineeachcoordinateofvertex: ifV ⊆D thenvD =1;else vD =0.
2,σ(V) 2,σ(V)
ifv (cid:54)=0 thenT =T ∪{i}; endfor
1,σ({i})
endfor OutputvertexvD =(vD,vD).
1 2
D=Tc;
5. NON-ZERO-SUMSECURITYGAME
which is the sub-vector of J and K . The problem is
µ(D) µ(D) In this section, we assume that the security game is non-zero-
that the vertex in the high-dimensional polytope may not project
sum, inwhichthebenefitB (U)(B (U))ofattacker(defender)
a d
toavertexofitslow-dimensionalimage. However,thefollowing
maynotequaltothethelossL (U)(L (U))ofthedefender(at-
d a
lemmawillprovideapositiveresult.
tacker). Weconsiderthecomputationoftwomostlyadoptedcon-
LEMMA 7. (Geometric structure of Hd) For any support set cepts including the strong Stackelberg equilibrium (SSE) and the
[n] ∈ S ∈ A, the vertices of the polytope H are the columns Nash equilibrium (NE). For the SSE, we prove analogous equiv-
d
(cid:20)J(cid:21) alence theorem as the zero-sum case. For the NE, we relax the
of the sub-matrix of K , which is formed by extracting the row assumptionthattheattacker’sresourcelimitcisconstanttoanar-
bitrary number, but assuming the additive utility function. Then
whoseindexbelongstoσ(S).
weproveourequivalencetheoremfortheproblemofequilibrium
Sincewehaveaclosed-formexpressionofthematrixJandK, computation.
wecanconstructavertexmappingalgorithmfromlow-dimensional
5.1 StrongStackelbergEquilibrium
vertextothedefender’spurestrategy. Theefficiencyandthecor-
rectnessofAlgorithm1isjustifiedbyfollowinglemma. Themainresultofthissubsectionisgivenbythefollowingthe-
orem.
LEMMA 8. (Correctnessofvertexmappingalgorithm)Thever-
texmappingalgorithm1runsinO(n)timeandmapseachvertex THEOREM 7. (SSE and DOP) There is a poly(n) time algo-
ofH toauniquepurestrategy. rithmtocomputethedefender’sstrongStackelbergequilibrium,if
d
andonlyifthereisapoly(n)timealgorithmtocomputethede-
Note that our vertex mapping algorithm only examines n instead fenderoracleproblem(12).
ofallthecoordinatesofeachvertexofH torecoveradefender’s
d
The “only if” direction follows straightforwardly from Theo-
pure strategy. The reason behind this result is that there exists a
rem 6. Considering the fact that the set of Nash equilibrium is
one-one correspondence between each pure strategy and those n
equivalenttothesetofstrongStackelbergequilibriuminthezero-
coordinatesofeachvertexofpolytopeH . Intuitively,thosenco-
d
sum game, and the zero-sum game is a special case of the non-
ordinatesofeachvertexofH isbinaryandthereforethereexists
d
possibly2npossibilities,eachofwhichcorrespondstoapurestrat- zero-sumgame,suchadirectioncanbeobtainedbyconverse. For
the“if”direction,weneedthefollowingexistingtechnicallemma,
egy.
whichshowsthattheSSEofthegamecanbecomputedbymultiple
Theotherdirectionfollowsfromthefollowingargument. Sup-
linearprogrammingapproach.
posethattheproblemofequilibriumcomputationissolvedinpoly
(n)timeandtheoptimaldefender’smixedstrategyisdenotedby
LEMMA 10. (MultiplelinearprogrammingofSSE[28])Solv-
q∗. Invokingaknownresultingametheory(Theorem4in[27]),
ingthestrongStackelbergequilibriumcanbeformulatedasamul-
thenumberofnon-zeroprobabilityoftheNashequilibriumisless
tiplelinearprogrammingproblem.
thantherankofthepayoffmatrix. Sincetherankofpayoffmatrix
MaisO(nc),thenumberofnon-zerocoordinatesinq∗isatmost MultipleLP
O(nc)=poly(n)andq∗canbeexpressedas max (eA)TMdq (18)
q∗ =po(cid:88)ly(n)λ ei. (17) s.t. (eA)TMaq≥(eA(cid:48))TMaq,∀A(cid:48) ∈A,
i q∈∆ ,
Nd
i=1
Solving above linear programming for all A ∈ A, then picking
Therefore,wecandeterminetheoptimalsolutionofthecompact
the optimal defender’s mixed strategy of the LP with the largest
problem in poly(n) time by constructing the following poly(n)
timevertexmappingalgorithmfromapurestrategyei toavertex objectivevalue.
ofHd. The intuition behind this result is that, in linear programming
Theintuitionbehindthisresultissimilartothepreviousvertex problem(foreachA∈A),thedefenderoptimizeshismixedstrat-
mappingalgorithmandthecorrectnessofAlgorithm2isguaran- egyundertheconstraintthattheattacker’sbestresponseisA.Once
teedbythefollowinglemma. we have solved theseLPs for all the attacker’s best response, we
comparealltheoptimalmixedstrategiesandchoosethebestone,
LEMMA 9. (Correctness of vertex mapping algorithm) Vertex
mappingalgorithm2runsinO(nc)timeandmapseachdefender’s whichisalsotheoptimalsolutionoverall(withoutconstraintson
whichisthebestresponse).
purestrategyDtoauniquevertexofH .
d
Sincetheattacker’sresourcelimitcisconstant,thereexistspoly
BasedonLemma5,Lemma6,Lemma8andLemma9,wear- (n)numberofLPsintheaboveapproach,buttherestillexistsex-
rivedthedesiredresultinTheorem6. ponentialnumberofvariablesineachLP.Fortunately,wecanstill
followourtransformationandprojectionapproachintheprevious Asimilarresultholdsfordefender’sutilityfunctionsB (·),L (·)
d d
zero-sum scenario. Specifically, based on the same argument of and their common utilities Bc(·), Lc(·). Note that the common
d d
Theorem4,wecandecomposebothattacker’sanddefender’spay- utilityfunctionisindeedtheoriginalutilityfunctionwhenU isa
offmatricesas singletonset. Then,accordingtothedefinitionofmatrixDb and
a
Dl, wecanobservethatalthoughtheyareexponentiallargema-
Ma =E(DbJ+DlK) and Md =E(DbJ+DlK), (19) a
a a d d trices,thereexistsonlynnon-zeroelementsinthemaindiagonal.
andchoosethesametransformationsuchthatf(p) = ETpand Therefore,thesupportsetS =[n]basedonthedefinition,andwe
g (q) = Jq,g (q) = Kq. Thenweeliminatethevariableswith candefinethefollowingn−dimensionalpolytope.
1 2
zero-coefficientsandprojectthepolytopeintocorrespondinglow-
H(cid:48) =Π (∆a ), H(cid:48) ={π (Jq),∀q∈∆ }. (23)
dimensional image. Therefore, we can formulate above multiple a [n] Na d [n] Nd
linear programming problem as following compact problem with
whichprojectsourtransformedpolytopestothecoordinateswhose
onlypoly(n)variables.
indicesbelongtothesingletonset. Notethat,inthiscase,weonly
CompactMultipleLP use the vector Jq instead both of Jq and Kq to form the trans-
formed and projected polytope. The reason is that there exists a
max vT(D(cid:101)bdq¯1+D(cid:101)ldq¯2) (20) linearcouplingbetweenπ[n](Jq)andπ[n](Kq). Theformalde-
s.t. vT(D(cid:101)baq¯1+D(cid:101)ldq¯2)≥v¯T(D(cid:101)baq¯1+D(cid:101)ldq¯2),∀v¯ ∈Ia, scriptioncanbeseeninLemma15. Thefollowingresultprovides
(q¯ ,q¯ )∈H , acompactrepresentationofthenon-zero-sumsecuritygame.
1 2 d
wbyheerxetrtahcetimngatrthixeD(cid:101)nobdna-znedrothecooltuhmernsdiaangdontahlemraotwrisceosfamreaotrbixtaiDnebd. gamLeE)MTMhAe s1tr2a.te(gCyomprpoaficletr(eppr∗e,sqe∗n)taitsioanoNfansohne-zqeurioli-bsruimumseocufrtihtye
d
Clearly, such a compact problem can be reduced to the defender non-zero-sumsecuritygameifandonlyif
oracleproblem(12)andtheoptimalsolutionofcompactproblem U(cid:48)(a∗,t∗)≥U(cid:48)(a,t∗),∀a∈H(cid:48),
canbemappedtoadefender’soptimalmixedstrategybasedona a a a (24)
similarargumentwithLemma7andLemma8.Thus,wearriveour Ud(cid:48)(a∗,t∗)≥Ud(cid:48)(a∗,t),∀t∈Hd(cid:48).
desiredresultsinTheorem7.
where8
5.2 NashEquilibrium
n
ItiswellknownthatcomputingtheNashequilibriumofatwo- Ua(cid:48)(a,t)=(cid:88)ai[tiLa(i)+(1−ti)Ba(i)],
player normal form game is PPAD-hard [29, 30]. In the security i=1
gtiammeei,nwtehecacnapseotoenfttihaellysicnogmlepauttteacthkeerNraesshouerqcueili[b1r8i]u,mbeincapuosley(wne) Ud(cid:48)(a,t)=(cid:88)n ai[tiBd(i)+(1−ti)Ld(i)].
cantransformthenon-zero-sumgameintoanequivalentzero-sum i=1
game[15].However,whentheattackerhasmultipleresources,the AscanbeseeninLemma12,thestrategyprofile(p∗,q∗)isa
solvabilityisstillanopenproblem. Inthissubsection,weassume
Nashequilibriuminnon-zero-sumsecuritygameifandonlyifits
thattheattacker’sresourcelimitcisanarbitrarynumberinsteadof lowdimensionalimage(a∗,t∗)isalsoanequilibriumpoint.Based
aconstant,andalltheutilityfunctionsareadditive.Themainresult
onthedefinitionofmatrixEandJ,wehave
ofthissubsectionisgivenbythefollowingtheorem.
(cid:88) (cid:88)
a = p , t = q ,
THEOREM 8. (Non-zero-sumNEcomputationandDOP)There i σ(A) i µ(D)
isapoly(n)timealgorithmtocomputethedefender’sNashequi- A∈A:i∈A D∈D:i∈D
librium,ifandonlyifthereisapoly(n)timealgorithmtocompute whichcanberegardedasthemarginalprobabilitythattargetiis
thedefenderoracleproblem:foranygivenw∈Rn attackedanddefendedamongthestrategyprofile(p,q).
argminwTx. (21)
x∈Id REMARK 2. Lemma 12 provides a compact representation of
thenon-zero-sumsecuritygame,andtheobjectivefunctionU(cid:48) and
The“onlyif”directioncanbeobtainedbytheconverseandThe- a
U(cid:48) arequitesimilartothetheoneusedin[8]. However,thepre-
orem6,thusweonlyfocusonthehowtocomputethedefender’s d
vious result is limited to the assumption that the defender’s pure
Nash equilibrium in poly(n) time given a poly(n) time defender
strategyspaceisauniformmatroid. However, ourresultismore
oracle. Inthiscase,thereexiststwokeychallenges: first,notonly
generalanddoesnotdependentonthestructureofD.
isthedefender’spurestrategyspaceexponentiallarge,butalsothe
attacker’spurestrategyspace;second,theNashequilibriuminnon-
Although we have a polynomial-sized representation, it is still
zero-sumgamemaynotbethecorrespondingminimaxequilibrium
a non-zero-sum “game” and its optimal solution may not corre-
whentheattackerhasmultipleresources[15],thuswecannotapply
spondtoitsminimaxsolution. Fortunately,inspiredbythetrans-
thelinearprogrammingapproachtostraightforwardlysolvesucha
formationintroducedin[15],wecanexploitthespecificstructure
problem.
oftheaboveproblemtotransformthecompactlyrepresentedprob-
Thefollowinglemmaexhibitsacrucialpropertyofcommonutil-
lemintoan−dimensionalsaddlepointproblem.
itywhenalltheutilityfunctionisadditivesuchthatallthecommon
utilityisequaltozeroexceptthosedefinedonthesingletonset.
LEMMA 13. (Saddlepointproblem)The(a∗,t∗)satisfycondi-
LEMMA 11. (Additiveutilityfunction)Iftheattacker’sutility tion(24)ifthe(h(a∗),t∗)isasaddlepointoffollowingproblem,
(cid:80)
functions are additive such that B (U) = B ({i}) and
La(U)=(cid:80)i∈ULa({i})forallU a∈2[n],thentih∈eUcoraresponding Ua(cid:48)(h(a∗),t∗)≥Ua(cid:48)(h(a),t∗),∀a∈Ha(cid:48) (25)
commonutilitiessatisfy U(cid:48)(h(a∗),t∗)≤U(cid:48)(h(a∗),t),∀t∈H(cid:48),
a a d
Bac(U)=Lca(U)=0, if|U|>1. (22) 8Forsimplicity,wewriteaσ({i})andtµ({i})asaiandti.
wheretheone-to-onetransformfunctionh : Rn → Rnisdefined REMARK 3. The reduction from the equilibrium computation
asfollows: tothedefenderoracleproblemdoesnotrequiretheassumptionthat
theattacker’spurestrategyspaceAisauniformmatroid.Indeed,if
B ({i})−L ({i})
h (a)= d d a , (26) Aencodesapolynomialsolvableproblemsuchasuniformmatroid,
i B ({i})−L ({i}) i
a a bipartite matching and 2−weighted cover, we can obtain a same
result in Theorem 8; otherwise, the polynomial solvability of the
Based on Lemma 13, we can first determine a saddle point of
equilibriumcomputationwilldependsonbothAandD.
functionU(cid:48)(a,t).DuetothefactthatfunctionU(cid:48)(h(a),t)islin-
a a
earinaandtandthepolytopeH(cid:48) andH(cid:48) isconvex(Lemma4),
a d 6. CONSEQUENCESANDAPPLICATIONS
wecanapplytheSion’sminimaxtheoremandsolvesuchproblem
bytheminimaxproblem, Inthissection,wefurtherinvestigatethedefenderoracleprob-
lemandprovidesvariousinterestingapplicationsofourtheoretical
min maxUa(cid:48)(h(a),t), (27) framework.
t∈Hd(cid:48) a∈Ha(cid:48)
6.1 WhatistheDefenderOracleProblem
whichcanbefurthertackledbythefollowinglinearprogramming
approach. Through a series of reductions, we determine that the security
gamewithnon-additiveutilityfunctionsandmultipleattackerre-
min u (28) sourcesisessentiallyadefenderoracleproblemdefinedonalow-
s.t. (cid:80)n h (v)[t L (i)+(1−t )B (i)]≤u,∀v∈I(cid:48), dimensionalpolytopeHd.However,thecomplicatedformofpoly-
i i a i a a topeH stillpreventsusfromuncoveringtheeffectofourassump-
i=1 d
t∈H(cid:48). tions. Thepromisingobservationthoughisthatsinceoracleprob-
d
lemattainsitsmaximuminsomeverticesofH ,weonlyneedto
d
Similarly,suchanLPcanbereducedtothemembershipproblem examinethegeometricstructureofitsvertices.
andinequalityconstraintproblem. Themembershipproblemcan First,consideringthefactthateachvertexvofpolytopeH con-
d
bereducedtothedefenderoracleproblem(12),andtheonlydiffer- sistsoftwopartssuchthatv = (v ,v ),andv ,v comesfrom
1 2 1 2
enceisthatHd(cid:48) isan−dimensionalpolytopeinthiscase.Tosolve twodifferenttransformationsandsameprojectionoftheonevertex
theinequalityconstraintproblem, wefirstexaminethegeometric ofpolytope∆ .Therefore,thereexistsalinearcouplingbetween
structureofpolytopeHa(cid:48). thesetwopartNsdv1 andv2. Thefollowinglemmajustifiesthisin-
tuitionandshowthatthecoordinatesofv andv whoseindices
LEMMA 14. (Geometric structure of Ha(cid:48)) The polytope Ha(cid:48) is belongtothosesingletonsetsexhibitsaco1mpleme2ntaryrelation.
theintersectionofan−dimensionalcubeandn−dimensionalhy-
perplane, LEMMA 15. (GeometricstructureofHd)Foranyvertex(v1,v2)
inthepolytopeH ,itscoordinatessatisfy
n d
Ha(cid:48) ={x∈Rn|(cid:88)xi ≤c,xi ≥0,∀1≤i≤n}. (29) v1,σ({i})+v2,σ({i}) =1,∀i∈[n]. (32)
i=1
Basedonthisresult,Theorem4andLemma7,wecanshowthat
Sinceanylinearprogramachievesoptimalityatsomevertexofits
theDOPisindeedacombinatorialoptimizationproblemoveraset
feasibleregion,basedonLemma14,theinequalityconstraintprob-
system.
lemcanbefurtherreducedtothefollowingpolynomial-sizedlinear
programmingproblem. THEOREM 9. (DOPiscombinatorialoptimization)Thedefender
oracleproblemis, foranyvectorw ∈ R|S|, maximizeapseudo-
n
(cid:88) booleanfunctionoverasetsystemε.
max h (v)[t L (i)+(1−t )B (i)] (30)
i i a i a
  
i=1
(cid:88) (cid:89)
s.t. (cid:80)n vi ≤c,vi ≥0,∀1≤i≤n. mx∈aεx wσ(V) xi, (33)
i=1 V∈S {i}∈V
whichcanbesolvedinthepoly(n)timebytheinteriorpointmethod. Clearly,thecomplexityoftheDOPisnotonlydependentonthe
Iftheoptimalvalueoftheabovelinearprogrammingislessthanu, setsystem,butalsodependentonthesupportsetS,whichdescribes
thenalltheinequalityconstraintsaresatisfied;iftheoptimalvalue thedegreeoftheabovepseudo-booleanfunction. Forexample,If
islargerthanu, weoutputthepointv attainsthisoptimalvalue, theattackercanattackatmosttwotargetsandtheutilityfunctions
whichcorrespondstoaviolatingconstraint. arenon-additive,thesupportsetS ={A∈2[n]||A|≤2}. Inthis
Now the remaining is to determine how to map such a saddle case,theDOPisageneralconstrainedbinaryquadraticprogram-
point back to the optimal defender’s mixed strategy. Since our mingproblem,whichisNP-hard. InSection5,weassumethatall
transformationislinearandtheinversefunctionisgivenby the utility functions are additive. In this case, Lemma 11 shows
thatthesupportsetS = [n]andthedefenderoracleproblemwill
h−1(a)= Ba({i})−La({i})a , (31) degeneratetothefollowinglinearoptimizationproblem:
i B ({i})−L ({i}) i
d d maxwTx. (34)
wecanfirstdecomposesuchsaddlepointintotheconvexcombi- x∈ε
nationoftheverticesgivenapoly(n)timeseparationoracle;then Inthiscase,thecomplexityofsuchanoracleproblemisonlyde-
usetheinversefunctiontomapeachvertexbacktothevertexof pendent on the complexity of set system ε. For example, if the
polytopeH(cid:48). ThemappingfromthethevertexofpolytopeH(cid:48) to defender can attack at most k targets, the set system ε is a uni-
d d
adefender’spurestrategyfollowstheAlgorithm1. formmatroidandsolvingtheDOPonlyrequiressummingfirstk
CombiningtheresultofLemma12andLemma13,wearrived largestelementsofw. Ifthedefender’sresourcesareobtainedat
thedesiredresultinTheorem8. some costs and there exists a resource budget, the set system in