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DoubleDummyHandEvaluator

ZhenyangZhangAdvisor:DavidAldous

1.IntroductionThegoalofthisstudyistoanalyzemethodsthatalreadyexistandcomeupwithanewwayofthinkingabouttrick-takingvaluesofdifferenthands.Ratherthanmakinganyconclusionsonhandevaluation,Iwouldliketobemorecautious,andconsidermymodela“baby”versionofadoubledummyhandevaluator.Ihopethatthiswillbringupmorethoughtsabouthandevaluationmodelinginthefuture.IamusingdoubledummydatafromMattGinsberg’sdoubledummylibrary.Onemayarguethatdoubledummytricksarenot“realistic”sincesometimesresultsarenotachievableintherealworldplays.Forexample,doubledummytrickswillalwaysrunasuitmissingQ,whileinrealityplayersmayguesstheQonthewrongsideandloseonetrickasaresult.However,therearestillalotofadvantagesofusingit.First,therearealotofhandsinGinsberg’slibrary,717102handsintotal.Andinsomesensedoubledummytricksaremore“pure”.Wedonothavetotakeintoaccountifplayersmadeamistakeorusedsomeat-tableinformation(likehandgestures,facialexpressions,etc.)duringtheplaysothattheywonmoreorfewertricksthanexpected.Also,expertplayershaveinclinationtobidgames,whichmeansthatminorsuitcontractswillbelessplayedandtheresultwillbesomehowbiased,whiledoubledummyresultswillnotbeaffected.2.ObservationsonexistingmethodsBeforeweactuallystepintoourmodelofevaluatinghands,wewishtoexamineafewexistingmethodsofevaluatinghands.Thefourmethodsarehighcardpoints,distributionalpoints,controlsandlosers.Theyareamongthemostpopularmethodsofevaluatingahandthathavebeenusedbyprofessionalplayers.Butthroughthebelowanalysiswecanseethatthesearenotidealwaysoftellinghowgoodyourhandis.-HighCardPointsItisaclassicandthemostpopularmethodofevaluatingone’shandthateveryonewilllearnatthebeginningofplayingbridge.Aceiscountedas4points,King3,Queen2andJack1.Intotalthereare40pointsandahandofmorethan10pointsisconsideredgood.

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Hereisahistogramofhighcardpointsof1000randomsampleddeals.Wecanseethatthedistributionisroughlynormalandthusarobustmethod.However,boththeplotandcorrelationbetweenHCPandtricksimplythatthisisnotsoprecise.Andthisisexactlywhatmanybridgeexpertshaveargued:HCPdoesnottakehandshapeintoaccount;QueensandJacksareoverestimatedsincetheyarenotsopowerfulintherealplaysincetheyarelikelytobecoveredbyAcesandKings.

Histogram for hcp

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-DistributionpointsArevisedversionofHCPisdistributionpoint:thatistoadddistributionalvaluestooriginalhighcardpointsofahand.Adoubletonisworth1pointmore,singleton3andvoid5.SinceitbasicallyinheritsfromHCP,wecanseethatthedistributionisstillnormal-like.Thecorrelationofdistributionpointandnumberoftrickstowinisaround0.29,whichislargerthanbefore.ThereforeitisbetterthanHCPinsomesensewhilestillnotgoodenough.

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hcp vs tricks

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Histogram for distp

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ControlsSincetherearecontroversialideasonvaluesofQueenandJacks,expertsreevaluatetheirhandsusingcontrolswhentheyarethinkingaboutslamcontracts.AceiscountedastwocontrolsandKingasone.Butfromthehistogramwecanseethatmosthandshave0-2controlsandareconsideredabadhand.BecauseitismorelikeababyversionofHCPnotconsideringQueensandJacks,thecorrelationbetweencontrolsandtrickswonarearound0.08whichimpliesitisnotagoodindicatorof

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trickwinningpower.

Histogram for control

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-LosersThisisthelastmethodwearegoingtolookat.Losersaresomehowmorecomplicatedtocount.Foreachsuit,justlookatthethreelargestcardsinthesuitarecounthowmanyofAceKingandQueenismissing.Forexample,AQhasoneloser,KQ432hasoneloser,andQ82hastwolosers.Themorelosersonehave,theworsehishandis.Thecorrelationisaround-0.2,adecentone,butstillnotgoodenough.

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controls vs tricks

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Histogram for losers

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3.HandEvaluatorModelAftershowingthatexistingmethodsarenotidealweaponstouse,wenowwanttocomeupwithsomethingmore.Firstwewanttodecidewhichfactorswearelookingatinthemodel.Generallywehavefouraspectsthatweareconcerningaboutahandwhenwearebidding:notrumpcontractsandtrumpcontracts,anddefendinganddeclaringineachcase.Intheprogram,wearegoingtouseabbreviation,“NTOffense”,“NTDefense”,“TrumpOffense”and“TrumpDefense”torepresentthesefouraspects.Weassumethatthevalueofahandisthelinearsumofthevaluesofeachsuit,andthuswehavethefollowingequation,wheretisthenumberoftricksyouexpecttogetinacertainsuit.

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V (h) = t(s)s∈h∑

Weallknowthatalongersuitismorelikelytowinmoretrickssincewehavepotentialchancetorunthesuitandwintherestofthesmallcardsinthesuit.Thuswewishtonormalizeandreducetheeffectofthissuit.Wewishtotaketheexpectedwinningtricksofthesuitinthispattern(thatis,suitsthathavethesamehighcards),minustheexpectedwinningtricksamongallthesuitsofthesamelength.Theequationisstatedbelow:

t(s) = ave(s)− ave(length(s))

Basedonthesetwoequations,weareabletoanalyzetheexpectedtrickswecanwingiven a hand.Wewrote the code in python andwith an input, the hand analysisfunctionwouldprintoutexpectedtrickstowininfouroccasionsrespectively.

Forexample,Ijustrandomlypickedahandfrommyhandhistoryandwishtoknowhowgooditis.ThusIinputthehandinterminalandruntheprogram.

Wecanseefromtheoutputthatweareexpectedtowin7tricksinNTcontracts,and7.8tricksintrumpcontractsifwearedeclarer.Andweareexpectedtowin8.8tricksinNTcontractsand9tricksintrumpcontractsifwearedefending.Thisissuggestingthatourhandisbetterthanaverage,soweshoulddefinitelyopenthebidding.However,itisabetterhandindefense,andhasalotofpotentialindefendingopponents’contract.Sowearealsohappytodefendiftheopponentsstarttointerfereinthebidding.

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4.StrengthandWeaknessoftheModel:Thismodelusesthewholesetofdataoutofover700000hands,andthusthenumericalanalysisisindeedreliable.Fromtheaboveanalysisweareabletomakedecisionsinbiddingmoreconfidently.However,wehavenotconsideredpartner’shandinthemodel,whichmeansthatweeliminatehalfoftheinformation.Furthermore,duringthebidding,asweknowmoreandmoreaboutpartner’shand,weshouldbeabletodynamicallyconditionontheknownfacttoreevaluateourhand’svalue,butunfortunatelywehavenotfoundgoodwaystogetthatinvolved.Finally,inthemodel,weassumethelinearityofhonorsindifferentsuitssothatwecansumthemtogethertogetthefinalresult.However,intherealworld,atrumpsuitwillplayalargerrolethanothersuit,butwecannotdecidethetrumpsuitjustlookingatourhandsincepartner’shandisunclear.Alsowejustignorethecomplexcorrelationbetweenhonorsinasuitandthusitmakesthemodelinaccurate.Inconclusion,Iwillrecommendthismodelmoreasareferenceforreconsideringyourhandvaluethananaccuratetrickwinningpowerreportofasinglehand.5.ReferenceMattGinsberg’spaperonLawoftotaltricks:http://bocosan.tripod.com/ginsberg/total.HTMLExplanationsofbinaryfileofdoubledummylibrary:http://bocosan.tripod.com/ginsberg/library_notes.HTML