Download 3. Thomas Lidbetter, Using a best response orac..

Usingabestresponseoracletosolvesearchgamesongraphs
ThomasLidbetter*andLisaHellerstein**
*RutgersBusinessSchool
**NewYorkUniversity
GRASTA2017
BroadcharacterizationofSearchGamesforanimmobileHider
• Zero-sumgamesbetweenaSearcherandaHider
• Strategyset𝐻 forHideristypicallyasetof𝑛 locations
• Strategyset𝑆 forSearchercouldbemuchlarger,eg.allorderingsof
n : = {1, … , 𝑛},orasubsetthereof(couldberestrictedbysomegraph
structure)
• PayoffissomecosttoSearcheroffindingHiderorprobabilityoffinding
him
• Mainproblem:findoptimal(minimax)strategiesandvalueofthegame
• Ideallyclosedformsolutions,butifnot,isthereanefficientalgorithm
(polynomialin𝑛)forfindingsolutions?
• LPapproachnotefficient
Example:searchingforballsinboxes
• 𝐻 = 𝑛 = setofboxes,𝑆 = allpermutationson[𝑛]
• Costs𝑐/ , … , 𝑐0 ofsearchingboxes
• Payoffiscostofallboxessearcheduntilballfound(searchcost)
• Eg.
$3
$7
$3
$2
$6
Example:searchingforballsinboxes
• 𝐻 = 𝑛 = setofboxes,𝑆 = allpermutationson[𝑛]
• Costs𝑐/ , … , 𝑐0 ofsearchingboxes
• Payoffiscostofallboxessearcheduntilballfound(searchcost)
• Eg.
$3
$7
$3
$2
$6
Example:searchingforballsinboxes
• 𝐻 = 𝑛 = setofboxes,𝑆 = allpermutationson[𝑛]
• Costs𝑐/ , … , 𝑐0 ofsearchingboxes
• Payoffiscostofallboxessearcheduntilballfound(searchcost)
• Eg.
$3
$7
$3
$2
$6
Example:searchingforballsinboxes
• 𝐻 = 𝑛 = setofboxes,𝑆 = allpermutationson[𝑛]
• Costs𝑐/ , … , 𝑐0 ofsearchingboxes
• Payoffiscostofallboxessearcheduntilballfound(searchcost)
• Eg.
$3
$7
$3
$2
$6
Example:searchingforballsinboxes
• 𝐻 = 𝑛 = setofboxes,𝑆 = allpermutationson[𝑛]
• Costs𝑐/ , … , 𝑐0 ofsearchingboxes
• Payoffiscostofallboxessearcheduntilballfound(searchcost)
• Eg.searchcost= 7 + 6 + 3 = 16
$3
$7
$3
$2
$6
• ThreedifferentclosedformsolutionsfoundbyAlpern &L(2013),
L(2013)and(implicitly)by Condon,Deshpande,Hellerstein &Wu(2009)
Example:searchingforballsinboxes
• Wecanconsiderthebestresponseproblem:foragivenmixed
(randomized)strategyoftheHider,whatisthebestresponseofthe
Searcher?
• Eg.
$3
2/15
$7
5/15
$3
3/15
$2
1/15
$6
4/15
• Answer:openboxesindecreasingorderofprobabilitytocost
(Blackwell,1964)
Example:searchingforballsinboxes
• Wecanconsiderthebestresponseproblem:foragivenmixed
(randomized)strategyoftheHider,whatisthebestresponseofthe
Searcher?
• Eg.
$3
2/15
$7
5/15
$3
3/15
$2
1/15
$6
4/15
• Answer:openboxesindecreasingorderofprobabilitytocost
(Blackwell,1964;Smith,1956)
• Generalquestion:canweusesolution(orapproximatesolution)tobest
responseproblemtosolveagame?
Maintheorems(Hellerstein &L,2017)
• Considerazero-sumgamewithvalue𝑉 wherePlayerI(maximizer)has
𝑛 purestrategiesandPlayerII(minimizer)hasanarbitrarynumberof
purestrategies.
• Supposewehaveanbestresponseoraclethatfindsan𝛼-approximation
tothebestresponseproblem(𝛼 ≥ 1)intimepolynomialinn.
• Thentherearepolynomialtime*algorithmsthatfindamixedstrategy
forPlayerIthatguaranteesanexpectedpayoffofatleast𝛼𝑉 anda
mixedstrategyforPlayerIIthatguaranteesanexpectedpayoffofat
most𝑉/𝛼.(Callthesestrategies𝛼-optimal.)
Approach1
• WritegameasLPanduseellipsoidapproachwiththebestresponse
oracletosimulatetheseparationoracle
• Sameapproach(butnotforgames)usedbyJansen(2003),Carr and
Vempala (2002),Friggstad andSwamy (2012)andFeldmanetal.(2012)
• Reliesonellipsoidalgorithmsoispolynomialinnotonly𝑛 butlog𝑀,
where𝑀 = largestpossiblepayoffingame.
Approach2
• Multiplicativeweightsupdatemethod
• More“combinatorial”algorithmandruntimedoesnotdependon𝑀
• However,algorithmactuallyoutputs𝛼(1 + 𝜀)-optimalstrategies,and
algorithmispolynomialinnotjust𝑛 but1/𝜀 and𝛼
• SimilarapproachasFreundandSchapire (1999),buttheydonot
consider𝛼-approximations,andtheygivestrategiesthatarewithinan
additivedistanceof𝜀 from𝑉
Example1:Expandingsearchgame(Alpern &L,2013)
Anexpandingsearchofanedge-weighted,connectedgraphwithroot
vertex𝑂 isasequenceofedges,startingat𝑂,eachoneofwhichis
incidenttoapreviouslychosenedge.
Hiderstrategy:avertex
2
3
1
1
Searcherstrategy:anexpanding
search
Payoff:costoffindingHider
2
3
𝑂
Example1:Expandingsearchgame(Alpern &L,2013)
• Bestresponseproblemsolvedfortrees(Alpern &L,2013;Sidney,1975)
• NP-hardforgeneralgraphs;unknownwhetherapproximatesolutions
canbefound
Example2:Expandingsearchgameonadirectedacyclicgraph(DAG)
• Bestresponseproblemiswell-knownNP-hardschedulingproblem
• Many2-approximationsknown:
•
•
•
•
•
•
•
Ambuhl &Mastrolili (2009)
Chekuri &Motwani (1999)
Chudak &Hochbaum (1999)
Hall,Schultz&Shmoys (1997)
Margot,Queyranne &Wang(2003)
Pisaruk (2003)
Schulz(1996)
Example3:Expandingsearchratio(Angelopoulos,Dürr,L,2016)
• Likeexpandingsearchgame,butpayoffisdividedbyshortestpath
distanceofvertexfromroot
• Bestresponseproblemissameasforexpandingsearchgame
• Sothisgamecanbesolvedpreciselyfortreesanda2-approximationcan
befoundforDAGs
Example4:Pursuit-evasiongame(Gal,Cassas,2014)
• Hiderchoosesoneof𝑛 locations
• Searchercansearch𝑘 ofthem,andiftheycontainHider’slocation𝑖,a
pursuitensuesandheisfoundwithprobability𝑝G
• Moregeneralversion:locationshavecosts𝑐G andsearchercansearcha
subsetoftotalcost≤ 𝑘
• BestresponseproblemisKnapsackProblem,whichhasaFully
PolynomialTimeApproximationScheme
Example5:VonStengel&Werchner (1997)
• Hiderchoosesavertexofanunweightedgraph
• Searcherchoosesaconnectedsubgraphwith𝑘 edges
• SearcherwinsifhissubgraphcontainstheHider’svertex
WinforSearcher
• Bestresponseproblemisweighted𝑘-CARDTREE(Fischetti etal,1994)
• NP-hardandnoapproximationsknown