EpistemicTimeBiasinNewcomb’sProblem*MelissaFusco

CausaldecisiontheoristslikeDavidLewisholdthat,whileanagentshouldchooseactsbyusinghercurrent,ratherthananticipated,credencesovercausaldependencyhypotheses,sheshouldalsoupdatebyconditionalizationonherownactassheperformsit(Lewis1981a,1976).Thispackageleadstoanunflatteringformoftime-bias,whichcanbehighlightedbyconsideringiteratedNewcombproblems.Afterpresentingthepuzzle,IdiscussaCDTer’sbestresponse.

1. IntroductionSupposeyouwilleitherexperienceanhourofpainonTuesdayandanhourofpleasureonThursday,oranhourofpleasureonTuesdayandanhourofpainonThursday.Neitherpossibilityseemsclearlybetterthantheother—unless,thatis,todayisWednesday.Forafixedmenuofpleasuresandpains,time-biasedagentspreferdistributionswhereinmorepainsareinthepast,andmorepleasuresareinthefuture.

Figure1:BasicTimeBias.

ThetwopossibilitiesinFigure1illustratethisbasiccommitment.

AclassicdiscussionoftimebiascanbefoundinChapter8ofParfit’sReasonsandPersons(Parfit1984).1Parfitimaginesthathewakesupinthehospital,andistoldbyanursethateitherhehasjustsurvivedaverylongandpainfuloperation(whichhedoesnotremember,sincepatientsundergoinghissurgeryareregularly

*WarmthankstoArifAhmed,LaraBuchak,AlexKocurek,andJohnMacFarlane(viapandoc)fortheirhelponthispaper.

1Parfitdistinguishestwoformsofbias,futurebias—beingmoreconcernedaboutone’sfuturewelfarethanforone’spastwelfare—andnearbias—beingmoreconcernedforthenearfuturethanforthedistantfuture.Futurebiasisthoughttobethemorerationallydefensiblethannearbias(seeop.cit.,pg.158ff.)Ifocusonlyonfuturebiashere.

givenpost-operativemedicationwhichinducesamnesia),orhehasyettoundergoashorterversionofanoperationofthesametype.WhileParfit’scharactergenerallypreferstosufferless,ratherthanmore,henowfindsthathewouldmuchprefertolearnthathehasalreadyhadthelonger,earlieroperation.Hispreferencesseemtohavebeenreversedbythemerepassageoftime.Noristhisareversalbetweenoutcomeswhichareintuitivelyequal.Rather,Parfit’scharacternowprefersapossibilityinwhichhislifecontainsmoretotalhoursofsuffering.

Figure2:Parfit’sSurgeries.

InthispaperIinvestigateaformoftimebiaswhichisentailedbyacommonformofcausaldecisiontheory.Itcanbeseeninthecausalist’sresponsetoNewcomb’sproblem.

Figure3:NewcombTimeBias.

AstandardcausaldecisiontheoristinNewcomb’sproblem—anagentwhopreferstotwo-box,giventhechance—willnonethelessprefertolearnthatNewcombproblemswhichinvolveher,butoverwhichshecurrentlyhasnoagency,areonesinwhichsheone-boxed.WhenParfittianamnesiaisonthescene,timebiaseasilytakesroot:learningaboutone’spastactionsisaparadigmofbeinginvestedinadecisionoverwhichonedoesnotcurrentlyexerciseagency.2Henceanagent,locatedat𝑁𝑂𝑊,whoseesthepastasfixedbutthefutureasherstochoose,willprefertheupperstateinthefiguretothelowerone.Aswewillsee,theCDTer’stimebiasis,if2ArifAhmedpointsouttomethatanact’sbeinginthepastisnotessentialtothedifferencebetweenchoosingandlearning.TherelevanttypeofCDTerwillalsoprefertolearnthatshewillone-boxinthefuture.Thisistrue,butsuch“learnings"arepuzzlingbecauseofthedifficultymanypeople(myselfincluded)willhavereconcilingsuchforeknowledgewithfreeagency.Forthisreason,Iwillfocusinthisessayonlearningaboutthepast.

anything,strangerthanthetimebiasinvestigatedbyParfit,becauseitistraceabletoanormative,ratherthanaffective,source.

Onepuzzleaboutthissetupwillimmediatelypresentitself.Isaythat,inFigures1-3,time-biasedagentswouldpaytoswitchfromthelowerpossibilityintotheupper.But—itwillbeobjected—anagentlocatedat𝑁𝑂𝑊cannotreallymakesuchaswitch,sincethatwouldinvolvechangingherpast.Thisistrue,butIshowinsection3thatthebasicstructureofthetime-biasedpreferencesillustratedinFigure3can,withsuitablemanipulation,beturnedintoactionablepreferenceswhichoperationalizethebasicdisputebetweenthebiasedandunbiasedperspectives.

Nonetheless,thisisaconfessionalessay:ImaintaintheCDTer’scommitmentsinbothcases,despitethe(literal)costs.Damagecontrolcommencesinsection4,whereIarguethatEvidentialDecisionTheoryisalsocommittedtoaformoftimebias.Thequestion,then,iswhetherthetwoformsofbiashaveanequalclaimtobefeaturesofrationalbehavior.

2. EvidenceandDecisionInthissectionIsketchthesharedbackgroundofcausalandevidentialdecisiontheorywithaneyetotwoissues,screening-offandconditionalization,whichbearontheargumenttocome.

Iwillassumethatinalldecisionproblems,agentshavethegoalofmaximizingexpectedutility.Theycometotheseproblemsequippedwithavaluefunction𝑉𝑎𝑙(⋅)onpossibleworlds,whichcanbeliftedontopropositionscalledoutcomes;3followingstandardidealization,Iwilloccasionallyspeakasiftheunitsintherangeofthevaluefunctionareequivalenttodollars.Anagent’sdecisionproblematatime𝑡canberepresentedasatriple⟨𝒜, 𝒮, 𝐶𝑟1⟩,where

• 𝒜 = {𝑎5, … , 𝑎7}isasetofactsavailabletotheagentat𝑡,

• 𝒮 = {𝑠5, … , 𝑠:}isasetofadmissiblestatesofnature,whereasetofstatesisadmissibleinadecisionproblemonlyifeachact-stateconjunction(𝑎; ∧ 𝑠=)isassignedadeterminateimage𝑉𝑎𝑙(𝑎; ∧ 𝑠=)underthevaluefunction,and

• 𝐶𝑟1(⋅)istheagent’stime-𝑡credencefunction.4

3Theliftingofthevaluefunctioncanbeachievedasfollows.Where𝑣𝑎𝑙(⋅)isaworldlyvaluefunctionin𝑊 → ℝ,(i)𝑉𝑎𝑙({𝑤}) = 𝑣𝑎𝑙(𝑤),and(ii)𝑉𝑎𝑙(𝑝 ∪ 𝑞)isdefinedonlyif𝑉𝑎𝑙(𝑝) = 𝑉𝑎𝑙(𝑞),inwhichcase𝑉𝑎𝑙(𝑝 ∪ 𝑞) = 𝑉𝑎𝑙(𝑝).

4Inwhatfollows,Itreatthemembersof𝒜,𝒮,andthedomainof𝐶𝑟1(⋅)alikeaspropositions.ThisdiffersfromSavage’soriginalpictureoftherelevantprimitives,onwhichactsarefunctionsfromstatestooutcomes.SeeJoyce(1999Ch.2)fordiscussionofthisshift.

Atafirstpass,theagent’scredencefunction𝐶𝑟1(⋅)willbeconceivedofinBayesianterms.Iassumethatitisaprobabilityfunction,andthattheagentisdisposedtorespondtonewinformationbyconditioningthisfunctiononthestrongestpropositionofwhichsheisinformed.I’llalsoassumethatherexpectationsaregenerallyfrequentist,inthesensethatwhentheyconcerneventsinthepresentorfuture,theywill,ceterisparibus,conformtoobservedpastfrequenciesoverevent-classesofthesametype.

Againstthisbackground,wecanintroduceSavage(1972)’sdecisiontheory.Itappliesthegeneralstatisticalnotionofexpectationtothevalueofanact𝑎 ∈ 𝒜.5

Equation1(SavageExpectedUtility).𝑆𝐸𝑈1(𝑎) = 𝐶; 𝑟1(𝑠;)𝑉𝑎𝑙(𝑎 ∧ 𝑠;).Savage’snorm:maximizeexpectedutilityat𝑡bychoosinganact𝑎 ∈ 𝒜suchthat𝑆𝐸𝑈1(𝑎)ismaximal:thatis,suchthat𝑆𝐸𝑈1(𝑎) ≥ 𝑆𝐸𝑈1(𝑎′),forany𝑎′ ∈ 𝒜.

EDTandCDTcanthenbeframedastwodifferentwaysofcashingoutafinalconditionSavageputonwell-formulateddecisionproblems:therequirementthattheset𝒮ofstatesintheproblemare,attime𝑡,independentoftheagents’acts.

EDTconstruesact-stateindependenceintermsofprobabilisticconfirmation.Ingeneral,theEDTerholdsthatrationalityrequirescalculatingexpectedutilitiesintermsofact-conditionedprobabilities𝐶𝑟1(𝑠|𝑎):

Equation2(EvidentialExpectedUtility).𝐸𝐸𝑈1(𝑎) = 𝐶; 𝑟1(𝑠;|𝑎)𝑉𝑎𝑙(𝑎 ∧ 𝑠;).ConditionalProbabilitynorm:maximizeexpectedutilitybychoosinganact𝑎 ∈ 𝒜suchthat𝐸𝐸𝑈1(𝑎) ≥ 𝐸𝐸𝑈1(𝑎′),forany𝑎′ ∈ 𝒜.

ThisdoesjusticetooneattractiveglossonSavage’sindependencerequirementon𝒮:itentailsthatSavage’sequationwillapplytoanyprobleminwhich𝐶𝑟1(𝑠|𝑎)isequalto𝐶𝑟1(𝑠)—inotherwords,toanyprobleminwhichtheagenttakes𝑎tobeprobabilisticallyindependentof𝑠.

CDT,ontheotherhand,favorsaninterpretationofact-stateindependencethatiscausal.SherejectstheEDTer’sglossbecauseprobabilisticindependence—alackofprobabilistic(dis)confirmation—isneithernecessary,norsufficient,forcausalindependence.6Inthispaper,IwilluseLewis(1981a)’s“dependencyhypothesis”characterizationoftheCDT.Thistakesasprimitiveapartition𝒦ofdependencyhypotheses,which,arestipulatedtobecausallyindependentofthe

5Where𝐹isafunction,thegenericnotionofexpectedvaluesaysthat𝐸[𝐹] = 𝑓;; 𝑃𝑟(𝐹 = 𝑓;).Here,ourprobabilityfunctionisthesubjectivecredencefunction𝐶𝑟1(⋅),andthefunction𝐹is𝑉𝑎𝑙(⋅)acrossact-statepairs,wheretheactisheldfixed.

6Fornon-necessity,wecanconsideranycase(suchasNewcomb)inwhichanact𝑎probabilisticallyconfirmssomestate𝑠whosetruth-valuewasfixedbefore𝑎occurred.Fornon-sufficiency,wecanappealtoexamplessuchasHesslow(1976)’swell-knownthrombosiscase:whilebirthcontrolpillscausethrombosis,theeffectisstatisticallymaskedbecausebirthcontrolpillscounteractpregnancy,whichitselfcausesthrombosis.

agent’savailableacts,aswellassufficienttofixanoutcomevalueforeach(𝑎 ∧ 𝑘;)conjunction.7

Equation3(CausalExpectedUtility).𝐶𝐸𝑈1(𝑎=) = 𝐶; 𝑟1(𝑘;)𝑉𝑎𝑙(𝑎= ∧ 𝑘;),where𝒦 = {𝑘5. . . 𝑘7}isapartitionofdependencyhypotheses.CausalSupportnorm:maximizeexpectedutilitybychoosinganact𝑎 ∈ 𝒜suchthat𝐶𝐸𝑈1(𝑎) ≥ 𝐶𝐸𝑈1(𝑎′),forany𝑎′ ∈ 𝒜.

ThiswayofstatingCDTleavesopenthemodel-theoreticimplementationofwhatisspecialaboutthe𝑘’s—leavesopen,thatis,howdoxasticstatesmightbemodeledsoastoreflectanagent’sconvictionthatthe𝑘’sarecausallyindependentofactsin𝒜.Itisclear,however,thatthesemodelingrequirementswillexceedthoseofaustereBayesianism.8Notethat,sinceEEUispartition-invariant,theevidentialdecisiontheoristcanalsocalculateexpectedutilityusingthe𝒦-partition.9

WearenowinapositiontodefineNCProblems,decisionproblemswiththegeneralstructureofNewcomb’spuzzle.Formypurposes,anNCProblemwillbeaproblemwithtwoacts𝒜 = {𝑎∗, ¬𝑎∗}andtwocausaldependencyhypotheses𝒦 ={𝑘5, 𝑘V},inwhichadominanceargumentacross𝒦givestheagentaone-utileincentivetochoose𝑎∗;nonetheless,because¬𝑎∗indicatesamuchmorevaluableoutcome(whichI’llrefertogenericallyas“thebigprize”),theconditionalprobabilitynormfavoredbytheEDTerrecommends¬𝑎∗.HencefromtheCDTer’spointofview,thedistinctivethingtosayaboutNCproblemsisthatmaximizingexpectedutilitywillentailgettingsome“badnewsontheside"(Joyce2007pg.542)—newsthatstatisticallydisconfirmsone’sgettingthebigprize.

Hereisanexamplewhichwewillrevisitthroughout.

MoodCandles.Lightingaromaticmoodcandles(=𝐿)slightlyincreaseshappiness.Bobisdecidingwhethertolightamoodcandle.Healsofindshimselfunabletorememberwhetherhesuffersfromdepression(=𝐷).Inhiscurrentstateofignoranceandindecision,hispossibleoutcomes,andtheirvalues,are:

NotDepressed(¬𝐷) Depressed(𝐷)Don’tlight(¬𝐿) 9 0

Light(𝐿) 10 1

7Foraprobabilisticversionofthesameidea—whichdoesnotrequirethatdependencyhypothesesentailoutcomes,butmerelythattheyfixtheirchances—see§10ofLewis(1981a).

8CandidatesincludethedirectedacyclicgraphsofPearl(2000),andtheworldlydistancemetricsofLewis(1973)’streatmentofnaturallanguagecounterfactuals.

9Evidentialexpectedutilityispartitioninvariantinthesensethattheexpectedutilityassignedtoallacts𝑎 ∈ 𝒜w.r.t.acredalstate𝐶𝑟1(⋅)willbethesameinanyproblems⟨𝒜, 𝒮5, 𝐶𝑡1(⋅)⟩and⟨𝒜, 𝒮V, 𝐶𝑡1(⋅)⟩solongas⋃𝒮5 = ⋃𝒮V.SeeJoyce(1999pg.176ff.),andreferencestherein,fordiscussionofthisfeature.

There’sacatch,however.Knowledgeofthetherapeuticvalueofmoodcandlesisrare.Onlypeoplewhoaredepressedtendtohaveit,sinceonlytheyareregularlyupdatedbytheirdoctorsaboutmoodtherapies.HenceBob’scredencesaresuchthat𝐶𝑟1(𝐷|𝐿) > 𝐶𝑟1(𝐷):whilelightingmoodcandlesmodestlyconducestohappiness,itisastrongindicatorofunhappiness.

Aflatfooted(perhapsnaive;seebelow)calculationbytheEDTer’sconditionalprobabilitynormwillrecommendthatBobrefrainfromlightingamoodcandle,solongas𝐿raisesthestatisticalprobabilityof𝐷bymorethan1/9.10GiventhatBobalsobelievesthatlightingmoodcandlescannotcausedepression,however,itisunclearthatthisistherightrecommendation.

2.1 ScreeningOffTheformaldescriptionofanNCproblemisclearlycoherent,andMoodCandlesprimafaciefitsthebill.Somephilosophershaveargued,however,thattherearenoNCproblemsinreallife—atleast,notforrationalagents.Thisisacommonfactoroftwovenerablethreadsintheliterature:theso-called“Ramseythesis"((Ahmed2014Ch.8),(Ramsey1990)),andElleryEell’s“tickledefense"ofEDT(Eells1982).Thesharedideaisthataslongasaresponsibleagenttakesherfullpre-decisionevidenceintoaccount,andisnotirrationallyinfluencedbyfactorsoutsideofhercontrol,shewillbeabletoaccessevidence𝑍thatscreensoffthestatisticalsupportthatheractwouldapparentlyprovideforanystate𝑘:11

Screeningoff.𝑍screensoff𝑘from𝑎relativetoaprobabilityfunction𝐶𝑟1(⋅)justincase𝐶𝑟1(𝑘|𝑎 ∧ 𝑍) = 𝐶𝑟1(𝑘|𝑍).

10Calculation:

𝐸𝐸𝑈1(¬𝐿) > 𝐸𝐸𝑈1(𝐿)iff

𝐶;

𝑟1(𝑘;|¬𝐿)𝑉𝑎𝑙(𝑘; ∧ ¬𝐿) > 𝐶;

𝑟1(𝑘;|𝐿)𝑉𝑎𝑙(𝑘; ∧ 𝐿)iff

> [𝐶𝑟1(¬𝐷|𝐿)×10 + 𝐶𝑟1(𝐷|𝐿)×1]iff> [𝐶𝑟1(¬𝐷|𝐿)×10 + 𝐶𝑟1(𝐷|𝐿)]

Since𝐶𝑟c1 (⋅)=𝐶𝑟1(⋅ |𝜙)isitselfaprobabilityfunction,thisisequivalentto

[𝐶𝑟¬e1 (¬𝐷)×9] > [𝐶𝑟g1(¬𝐷)×10 + (1 − 𝐶𝑟e1(¬𝐷))]

Letting𝑛 = 𝐶𝑟¬e1 (¬𝐷)and𝑚 = 𝐶𝑟e1(¬𝐷),thisobtainsjustincase

9𝑛 > 10𝑚 + (1 − 𝑚)iff𝑛 > 𝑚 + 1/9.

11Eells’sdiscussionemphasizesthefirstidea—screeningevidenceisalwaysavailablepre-decision—whileRamseyemphasizesthesecond—thatanagentnotirrationallyinfluencedbyfactorsoutsideofhercontrolinfactcannotlearnanythingfromheracts.BeyondEells,theissueisdiscussedinAhmed(2014Ch.8)andBriggs(2010).

Ascreeningproblem.Anagent’sdecisionproblem⟨𝒜,𝒦, 𝐶𝑟1(⋅)⟩isascreeningproblemjustincasetheagent’sevidentialpositionat𝑡makesaccessiblesome𝑍whichscreensoffall𝑎 ∈ 𝒜fromall𝑘 ∈ 𝒦,relativeto𝐶𝑟1(⋅).

Whensomesuch𝑍isavailablepre-decision,theadditionalevidentialvalueofactuallyacting—actuallymaking𝑎insteadof¬𝑎thecase,orviceversa—isnothing.𝐶𝑟1(𝑘) = 𝐶𝑟l1(𝑘) = 𝐶𝑟l1(𝑘|𝑎)forall𝑎and𝑘,sothecharacteristicdecisionrulesofEDTandCDTwillalwaysendorsethesameact.

MoodCandlesisdesignedtosuggestscreening-off.Therearetwoparticularlyappealingcandidatesfortheproposition𝑍whichwouldscreenof𝐿from𝐷.Thefirstwilllikelyhaveoccurredtothereaderupabove:itcomesfromBob’sabilitytoreflectonwhathealreadybelievesabouttheproblemheisfacing.InMoodCandles,Bobbelievesthatlightingupgiveshimanextrautile,andthatitiscausallyindependentfromwhetherhegetsthebigprize.(Asnotedabove,exactlyhowhisdoxasticstatereflectsthisconvictionaboutcausalindependenceisaquestionIamcurrentlyleavingunanswered,buthedoesbelieveit.)ItfollowsthatBobalreadyhasevidencethatheisamemberofacertainclass:theclassofpeoplewhoareawarethatlightingmoodcandlesisefficaciousforhappiness.Thisisstatisticalevidencefor𝐷—bythedescriptionoftheproblem,onlydepressedpeopletendtobethatinformedabout(small,but)effectivewaysofelevatingone’smood.Thefirstcandidate“screener”proposition,then,supportsahighcredencein𝐷.Onthisinterpretation,Bob’smeremiddlingcredencein𝐷looksratherlikeabitofepistemicnegligence.12

Asecondcandidateisalsointrospective,butpullsintheotherdirection.ItfocusesonBob’sintrospectiveaccesstohowhefeels,ratherthanwhathebelieves.𝐷,afterall,isthepropositionthatBobisdepressedat𝑡.Surelyagentshavesomeprivilegedaccesstowhethertheyaredepressed,whichcomesfromconcentratingonhowtheyfeel.Bythedescriptionoftheproblem,though,Bobdoesn’tfeelstronglydepressed.13Reflectiononthissupportsalowcredencein𝐷.Onthisinterpretation,Bob’smeremiddlingcredencein𝐷mightstillberemiss—butbecauseitshouldbelowerthanitis,ratherthanhigher.

Ofcourse,thereisnotreallyan“either-or”inthiscase:Bobshouldbeepistemicallyresponsibleintakingintoaccountbothwhathe(already)believesandhowhe(already)feels.Bobmaybeinsuchasingularevidentialsituationvis-à-vis

12IntheNewcombliterature,thisthoughtfindsanallyinargumentthatifanagentcanreflectthather“decisionalgorithm”recommendstwo-boxing,shecanalreadyconcludethatthepredictorleftnomoneyintheopaquebox.Foraversionthisargument,seeYudkowsky(2010pg.26ff.).

13Iconstruethistobeunderlyingthefactthat𝐶𝑟1(𝐷)isn’thigh,intheoriginaldescriptionofthecase.

histotalevidencethatitisjustdifficulttosaywhatajustifiedcredencein𝐷wouldbe.

Thisiswhythesuggestionthatreal-lifeNCproblemsareallscreeningproblemshashadaequivocaleffectontheliterature.Briggsweighsinonthethesisasfollows:

WhetherthetickledefenserulesoutallcasesofconflictbetweenEDT[and]CDT...[is]controversial.Perhapsagentscanbereasonableenoughtolooktodecisiontheoryforadvicewithoutbeingasrationalandself-awareas[screening-off]requires...[However,]itseemsclearthatagreatmanysuitableagentswillhaveinformationthatrenderstheiractionsevidentiallyirrelevanttothedependencyhypotheses.Totheextentthatsituations[tothecontrary]arerare,thetaskofdecidingamongvariouspossibledecisiontheoriesisrenderedmuchlessurgent.(Briggs2010,pg.28,emphasisadded)

Twonondenominationalmorals,Ithink,emergefromconsiderationsofthescreening-offdebate.Thefirstisthatscreening-offargumentsinNCProblemswillalwayssupportthedominantoption–thatis,theactprimafaciefavoredbytheCDT,ratherthantheEDTer.14Thesecondisthatanagent’spre-decisionreflectiononherfullinformation—anactivityIwillcallepistemicentrenchment—mustbeconsideredbyboththeoriesinanyputativeNCproblem,eveniffullscreening-offiscontroversial.Thisoftenleadstosubtledisputesaboutwhatintrospectioncandeliver,fromtheagent’spointofview.

ForCDTers,entrenchmentservesasareminderthatcausalistsdonotactuallyignoretheirevidence.InNCproblems,dominancereasoningcanobscurethefactthatevenaCDT-rationalagentmusthavesomeparticularcredenceovereachdependencyhypothesisin𝒦beforesheacts.Decisionprocedureaside,shewillnotmakethebestdecisionsshecouldmakeifthesecredencesfailtoreflecthertotalevidence.15

14Why?Ina2×2NCProblem,let𝒜 = {𝑎∗, ¬𝑎∗}and𝒦 = {𝑘5, 𝑘V}.Byscreeningoff,thereissome𝑍whichtheagentcanupdateonbeforeactingsuchthat∀𝑘: 𝐶𝑟l1(𝑘|𝑎) = 𝐶𝑟l1(𝑘).Itfollowsfromthisthat𝐶𝑟l1(𝑘|𝑎∗) = 𝐶𝑟l1(𝑘|¬𝑎∗)forall𝑘 ∈ 𝒦.Callingthisnumber𝑛,wehave𝐸𝐸𝑈(𝑎∗) = 𝑛𝑉𝑎𝑙(𝑎∗ ∧𝑘5) + (1 − 𝑛)𝑉𝑎𝑙(𝑎∗ ∧ 𝑘V),and𝐸𝐸𝑈(¬𝑎∗) = 𝑛𝑉𝑎𝑙(¬𝑎∗ ∧ 𝑘5) + (1 − 𝑛)𝑉𝑎𝑙(¬𝑎∗ ∧ 𝑘V).Bydominance,𝑉𝑎𝑙(𝑎∗ ∧ 𝑘5) > 𝑉𝑎𝑙(¬𝑎∗ ∧ 𝑘5)and𝑉𝑎𝑙(𝑎∗ ∧ 𝑘V) > 𝑉𝑎𝑙(¬𝑎∗ ∧ 𝑘V).Hence𝐸𝐸𝑈(𝑎∗) > 𝐸𝐸𝑈(¬𝑎∗).ThisisnotanembarrassmenttotheEDTer’stheory,sincescreeningoffmakesgoingfordominantoptionscompatiblewiththattheory.

15ForasimilarobservationabouthowdominanceargumentscanobscureaCDTer’sevidentialduties,seeJoyce(2012pg.239).HereItakethetraditionalviewthatepistemicnormssuchasthePrincipleofTotalEvidencearesuigenerisprinciplesofepistemicrationality,whosejustificationispriortothejustificationofanypracticalmaximstheyinform(see,forexample,(HareandHedden2016pg.615)).Forargumentthatinvertsthisorderofpriority—offeringapracticalvindicationofthePrincipleofTotalEvidence—seeGood(1967);foranargumentthatEDT,incontrasttoCDT,hastroubleunderwritingGood’sTheorem,seeSkyrms(1990).

EDTers,fortheirpart,needepistemicentrenchmenttocombattheobjectionthattheirviewrecommendspatentlyabsurdactsthatwouldapparentlyresultfromgivingtoomuchweighttogeneralstatisticalcorrelations.ConsiderPearl’scomplaint:16

[AccordingtoEDT,][p]atientsshouldavoidgoingtothedoctortoreducetheprobabilitythatoneisseriouslyill;workersshouldneverhurrytowork,toreducetheprobabilityofhavingoverslept...remedialactionsshouldbebanishedlesttheyincreasetheprobabilitythataremedyisindeedneeded.(Pearl2000pg.108-9)

Thereasonablereplyhere,onbehalfoftheevidentialist,appealstoanagent’sneedtoincorporateherfullinformationintohercredalstatebeforesheacts.FocusonPearl’sexampleofhurryingandlateness(whichwecancallHurryingtoClass).GrantPearl’sassumption,thathurrying(e.g.toa10amclass)onanygivendayiscorrelatedwithlatenessonthatday:

HurryingtoClass.Myoutcomematrixandcredencesareasfollows:

NotLate(¬𝐿) Late(𝐿)

hurry(𝐻) 9 0

don’thurry(¬𝐻) 10 1

𝐶𝑟1(late|hurry) > 𝐶𝑟1(late)

Whydoesn’ttheEDTer’snormrecommendthatIavoidhurryingtoday,rightnow,asIheadtocampus?

Inthesimplestcase,Ihaveatimekeepingdevicewithme,andeitherdo,oreasilycould,glanceatit,toseethatitis(e.g.)9:46am.Relativetoanyspecifictime—afortiori,relativeto9:46am—thecorrelationbetweenlatenessandhurryingisreversed:

𝐶𝑟1(late|hurry ∧ 9: 46am) < 𝐶𝑟1(late|9: 46am)

Hencerelativetomyfullinformation,hurryingisnotcontraindicatedbymyact-conditionalizedcredences.

WhathappensifIdonothaveawatch?Itseemsthatthesamereasoningoughttohold—afterall,forebearingtohurrywhenoneisunsureofthetimeseemslikeanevenworseideathanforebearingtohurrywhenonedoesknowthetime.

Hereisawaytogeneralizethestrategy.Theargumentsketchedaboveclearlydidnotdependonthetime’sbeingexactly9:46.Rather,itdependedonthefactthattheagent(here,me)waslocatedatsomeparticulartime,whichshecouldthinkaboutinadirectway.Supposethatagentsgenerallyhavethepowertopick16ItakethequotefromPearlviaAhmed(2014pg.82).

thecurrenttimesrigidlybymeansofsomedemonstrativetoken—say,𝜇.17RelativetomyfullinformationinHurryingtoClass,Iaccept

𝐶𝑟1(late|hurry ∧ thetimeis𝜇) < 𝐶𝑟1(late|thetimeis𝜇)

Justinvirtue,then,ofthepatentlyintrospectablefactthatthetimeis𝜇,anEDTagentinaproblemlikeHurryingtoClasscanconcludethatthereisnoin-respect-of-latenessreasonnottohurry.Iwillreturntothisargumentin§4.

2.2 ConditionalizationandLewis’sPackageUpabove,IdescribedcredencebysayingthatagentsarepresumedtohaveBayesiandispositions—includingadispositiontoupdatebyconditionalizingontheanynewevidencetheyreceive.Thispresentation,however,wasneutralwithrespectthenormativestatusofthattendency.Bayesianism’sadditional,normativeclaimisthatconditionalprobabilitiesprovideadiachronicnormofbeliefrevision—thatanagentoughttorevisehercredenceinlightofevidence𝐸bymovingfrom𝐶𝑟1(⋅)to𝐶𝑟1~(⋅) = 𝐶𝑟1(⋅ |𝐸).IwillcallthisnormConditionalization(withacapital‘C’.)

Conditionalizationentails,asaspecialcase:

(CNC)InanNCproblem,oneoughttoupdatebyconditionalizingonone’schosenact𝑎.

(CNC)appearstodovetailwellwiththeCDTer’sconciliatorypositiononnewsvalues.Relativetoafixedcredalstate,propositionslike𝐿—amoodcandleislit—or𝐻—theagenthurries—carryastablenewsvaluebothbefore,andafter,thatpropositionisanavailableact.TheCDTpositionissimplythat,whene.g.𝐿isanavailableoption,itshouldbechosen(orpassedup)accordingtoitscausalexpectedutilityratherthanitsnewsvalue.

Lewishimselfseemstohaveembraced(CNC).18Theconjunctionof(CNC)andtheCDTer’sexpectedutilityequationcompriseajointCDTerpositionwhichwecancall“Lewis’spackage"(LP):

(LP)Atalltimes𝑡,oneoughtto:(i)assignexpectedutilitytoanavailableact𝑎usingone’stime-𝑡credencesacrossthedependencyhypotheses𝒦,selectingthe𝐶𝐸𝑈-maximalact;

17SeeMoss(2012)’s“Dr.Demonstrative”example(§1)forasimilarmaneuver,inthecaseofself-locatingbeliefs.

18InLewis(1976pg.302),hewritesthat“therearegoodreasonswhythechangeofbeliefthatresultsfromcomingtoknowanitemofnewevidenceshouldtakeplacebyconditionalizingonwhatwaslearned."AfootnotefromthequotedsentencerefersthereadertoTeller(1973)’sDutchBookargumentinfavorofConditionalization.LewisadvanceshisownDutchBookforConditionalizationinLewis(1999).

(ii)whenmakingpropositionsin𝒜truebyone’sactions,plantoevolve𝐶𝑟1(⋅)accordingtoConditionalization.

Conceptually,thereissomethingoddabout(LP).IfanagentwithLewis’sformofCDTisself-aware,shecananticipatewhathercredenceinthevariousdependencyhypotheses𝑘 ∈ 𝒦willbeatanyfuturetime𝑡~whenshehasperformedagivenavailableact𝑎.19ByConditionalization,thesefuturecredencesarejust

{𝐶𝑟1(𝑘|𝑎): 𝑘 ∈ 𝒦}

ByEquation2,thesearethesamenumberstheEDTerwillcurrentlyapplytocalculatetheexpectedvalueof𝑎.Infavoring𝐶𝐸𝑈over𝐸𝐸𝑈asaguidetoaction,then,theLewisianCDTerdefactoholdsthatone’scurrentcredencesinthedependencyhypothesesoverrulethecredencesoneplanstoadopt.Thisseemsintensionwiththeideathatthereissomethingmoreinformedaboutthemoreopinionatedcredencefunction𝐶𝑟1(⋅ |𝑎),invirtueofwhichitistheoneanagentshouldplan(followingConditionalization)totransitiontouponperforming𝑎.

Thistensioncanbeseeninthejuxtapositionofnormsgoverningactionandbeliefchange.FortheLewisianCDTer,alackofcausalefficacymakesitthecasethatoneshould,inchoosinganact,beunmoved,sotospeak,bythefactthattheactprovidesstatisticalevidenceforadesirablestate.Toreturntoourfirstexample,inlightofitslackofefficacyinpromoting𝐷,Bobshouldbeunmoved,inhisdecision-making,bythefactthat𝐿providesgoodstatisticalevidencefor𝐷.ButbecausetheLewisianCDTeralsothinksConditionalizationapplies,hebelievesthatifBobdoesbringabout𝐿,hemust,inaway,be“moved"byitsevidentialforceafterall:Bobisrationallyrequiredtoincreasehisconfidencein𝐷uponprocessingthefactthathemade𝐿true.20

3. TheNewcombHospitalCanthisberight?Thepresentsectionwillbedevotedtosketchinganoddconsequenceof(LP),drawingonananalogywithParfit’samnesiacases.Thesettingisahospitalfundedbyaneccentricbillionaire,whereamnesiacpatients—besetbyvarioushandicaps—gettofacedailyNewcombpuzzles.

SupposeyouwakeupintheNewcombhospitaltoday,knowingthatyoufacedNewcomb’sproblemyesterday,butunabletorememberwhatyoudid.Youarecompletelyevidentiallyindifferentonthematter,assigningaprobability1/2to

19Andhaslearnednothingmore:𝑎is,at𝑡~,hertotalevidence.

20Again,Lewisseemstohavebeencomfortablewiththisresult.AddressingNewcomb,hewriteswithanairofresignationthatatwo-boxermustexpectone-boxerstocomeoutahead:“[w]e[two-boxers]...didnotpleadsurprise.Weknewwhattoexpect."(Lewis1981bpg.378).SeeByrneandHájek(1997)formorediscussionofLewis’spackageview.

thepropositionthatyouone-boxedandaprobability1/2tothepropositionthatyoutwo-boxed.Thenewsthatyoudidtheformerwouldbeanear-to-perfectindicationthatyoumade$1,000yesterday,andthenewsthatyoudidthelatterwouldbeanear-to-perfectindicationthatyoumade$1𝑀yesterday.Henceinyourstateoftotalignoranceyouestimateyourwinningsat$500,500.ALewisianrationalamnesiac,whileatwo-boxer,muchprefersthenewsthatsheone-boxedinthepast.

Atradereflectingthispreferencecanbeengineered.

NewcombPast.Fry(aCDTer)awakens,bedriddenandwithamnesia,intheNewcombHospital.TodayisWednesday,andFryisinbed336.Bender,Fry’shospitalroommate,inbed335.WhileFrycan’trememberwhetherheone-boxedonTuesday,Bendertruthfullytellshimthathe,Bender,one-boxedonTuesday.Fry’swinningsfromyesterdayareinalockboxmarked“336-Tuesday”.Bender’sareinalockboxmarked“335-Tuesday”.For$𝛥,BenderofferstoswitchlockboxkeyswithFry(callthistrade𝜎).

Whatisthevalue𝜎?BecauseFryconditionalizesonhisevidence,whenBendertellshimthatheone-boxedyesterday,Frybecomes90%confidentthatBender’slockbox,336-Tuesday,has$1Minit.Heestimatesthevalueofhisownlockboxatamere$500,500.So𝜎,thedealBenderisoffering,isattractiveunlesstheamountofmoneyhedemandsforthetrade,𝛥,isgreaterthan$399,500.

TheattractivenessofthetradeinNewcombPastreflectsthefactthat,onLewis’spicture,itisveryfinethingtohavethepastofaone-boxer.Ofcourse,itisalsoafinethingtobeatwo-boxer,sincetwo-boxingmaximizescausalexpectedutility.Wecaneasilyengineerabetwhoseappealreflectsthispreferenceaswell,byinducingsomeuncertaintyoverwhichactisbeingperformed:

NewcombPresent.It’sstillWednesdayinthehospitalwithdailyNewcombrounds.Butnowit’stimeforFrytochoosetoday’smove.Itworkslikethis:thepredictorhasalreadydepositedeither$1Mornothinginopaquelockbox“Wednesday-336",dependingonwhethershepredictedthatFrywould,today,chooseact2𝐵(whichdumpsanextrathousandintoWednesday-336).Likewise,shehasalreadydepositedeither$1MornothinginBender’slockbox,“Wednesday-335",dependingonwhethershepredictedBenderwouldperform2𝐵(whichdumpsanextrathousandintoWednesday-335).

Unfortunately,Fryhasaninjurythatmakesspeechimpossible,andduetoamix-up,thenurseincorrectlybelievesheisamonolingualDingbatsspeaker.Thenursegiveshimtheday’sNewcombmenuwithtwooptionsprintedentirelyinDingbats:hecaneitherchoose“@”or“#".Hedoesnotknowwhichonecorrespondsto2𝐵.Infrustration,Fryrandomlycirclesthefirstoption,“@”.

Bender’sformisinEnglish—hecaneitherchoose“1𝐵"or“2𝐵"—anditisstillblank.For$𝛥,BenderofferstoswitchformswithFry(callthistrade𝜏).

HowshouldFryevaluate𝜏?ThevalueofthetradedependsonwhathewoulddowithBender’sformifhegotit.Butthatpartiseasy:fromtheCausalPointofView,atwo-boxforminaNewcombproblemisalwaysworth$1000morethanaone-boxform(itensuresthattheextrathousandisdumpedintoone’sbox).SoaCDTerlikeFry,whochooses𝜏andgetsablankformwillcertainlygoontopickthetwo-boxoption.Indeed,thetradeBenderisofferingisappealingtohimsolongas𝛥 < $500.

Argument:let𝛽betheamountofmoneyalreadyinthebox.AsintheoriginalNewcombpuzzle,becausetheidentityof𝛽iscausallyindependentofanythingFrydoesnow,wecanframetheproblemfromtheCDTer’spointofviewviaanargumentthatabstractsfromthevalueof𝛽:

𝑝(@ = 1𝐵) ¬𝑝(@ = 2𝐵)

𝜏 𝛽+(1000-𝛥) 𝛽+(1000-𝛥)¬𝜏 𝛽+0 𝛽+1000

IfFryiscompletelyindifferentontheidentityof@,then𝐶𝑟(𝑝)=.5.Hence

𝐶𝐸𝑈1(𝜏) = .5(𝛽 + (1000 − 𝛥)) + (1 − .5)(𝛽 + (1000 − 𝛥))= 𝛽 + (1000 − 𝛥)

𝐶𝐸𝑈1(¬𝜏) = .5(𝛽 + 0) + (1 − .5)(𝛽 + 1000)= 𝛽 + 500

Hence𝐶𝐸𝑈1(𝜏) > 𝐶𝐸𝑈1(¬𝜏)iff𝛥 < $500.(Forapresentationoftheargumentwithoutthevariable𝛽,seeAppendix.)

3.1 PuttingitalltogetherOurlaststeptogettingthepreferencesreflectedinFigure3istocombinethebettingbehavioroftheCDTersinNewcombPastandNewcombPresent.Thefollowingsetup,whilebaroque,willdothejob:

NewcombHospitalFusion.Asbefore,it’sWednesday.Thelockboxes,Fry’samnesia,andmutualknowledgethatBenderone-boxedyesterdayareasinNewcombPast.AsinNewcombPresent,FryhasbeengivenaDingbatsformfortoday’sroundandhasrandomlymarked“@"onit,whileBenderwasgivenaforminEnglish.BenderoffersFrya“combodeal":for$𝑛,hewillswitchlockboxkeysandformswithFry(callthistrade𝜔).

BeforeFrymakeshisdecision,helearnsoneadditionalpieceofinformationfromtheDingbats-languagechartbyhisbed:yesterday,theacthechosewas#.

HowwillFryweighthecombodeal?Thecausalexpectedutilityofchoosing“@"onthisroundisequaltothecausalexpectedutilityofchoosing“#"givenhiscurrentcredences,sinceeachacthasa50%chanceofbeingtheactofone-boxinganda50%chanceofbeingtheactoftwo-boxing.Becauseheknowshechose#yesterdayandhascurrentlymarked@onhisform,heknows,evenbeforehetrades,thatheisoneofthetwosituationsinFigure3.Foranadditional$𝛥—bychoosing𝜔—hecanguaranteetheoutcomeaLewisianCDTerconfidentlydesires:thathehasaone-box-pastandtwo-box-futureasopposedtothereverse.

NewcombTimeBias:Figure3,repeated.

AsinNewcombPresent,thepartofthecombodealthatensuresatwo-boxingpresentisappealing.Evenmoreappealingisthethepartthatmimicsbeingabletoensureaone-boxingpast—thatis,Fry’sbeingabletotradehisTuesdayboxforBender’s.SinceBenderone-boxedyesterday,hisbox,box335-Tuesday,is90%certaintocontainamilliondollars.Indeed,aslongas𝛥 <$400,500,Frywillchoosetotrade.

Argument:let𝛽betheamountofmoneythatisalreadyinFry’sWednesdaybox—thatis,theboxhewillgettoday.Nomatterwhat,FryisnottradingawayhisWednesdaybox(rather,ifhetakes𝜔heswitchesTuesdayboxesandWednesdayformswithBender.)Sothevalueof𝛽—thoughitisactdependent—is,again,causallyindependentfromwhatFrychoosestodo.

IfFrychooses𝜔,heswitchesformswithBender,soheinsulateshisownpayofffromtheidentityof@;hewillget𝛽 + (1000 − 𝛥)todayinanycase.Fry’spayoffswillthenfurtherdependonwhetherBender’sTuesdayboxcontains$1𝑀ornothing(letthesebepropositions5𝑇1and5𝑇0,respectively).

IfFrydeclinesthecombodeal,optingfor¬𝜔,hekeepsthecontentsofhisownTuesdaybox,andhispayoffswillthendependonwhetherhisTuesdayboxcontains$1𝑀ornothing(letthesebepropositions6𝑇1and6𝑇0,respectively).Moreover,histotalpayoffdoesnowdependontheidentityof@—hewillgetanextrathousandjustincase@= 2𝐵(row2,below).

I. 5𝑇1, 𝑝, 6𝑇1 II. 5𝑇1, 𝑝, 6𝑇0 III. 5𝑇1,¬𝑝, 6𝑇1 IV. 5𝑇1,¬𝑝, 6𝑇0𝜔 1𝑀 + (1000 − 𝛥)

+ 𝛽1𝑀 + (1000 − 𝛥)

+ 𝛽1𝑀 + (1000 − 𝛥)

+ 𝛽1𝑀 + (1000 − 𝛥)

+ 𝛽¬𝜔 1𝑀 + 0 + 𝛽 0𝑀 + 0 + 𝛽 1𝑀 + 1000 + 𝛽 0𝑀 + 1000 + 𝛽 V. 5𝑇0, 𝑝, 6𝑇1 VI. 5𝑇0, 𝑝, 6𝑇0 VII. 5𝑇0,¬𝑝, 6𝑇1 VIII. 5𝑇0,¬𝑝, 6𝑇0𝜔 0𝑀 + (1000 − 𝛥)

+ 𝛽0𝑀 + (1000 − 𝛥)

+ 𝛽0𝑀 + (1000 − 𝛥)

+ 𝛽0𝑀 + (1000 − 𝛥)

+ 𝛽¬𝜔 1𝑀 + 0 + 𝛽 0𝑀 + 0 + 𝛽 1𝑀 + 1000 + 𝛽 0𝑀 + 1000 + 𝛽

Utilities.

Whatoftheappropriatecausalistcredences?KnowingBenderone-boxedyesterday,Fryis90%confidentin5𝑇1,thepropositionthatthepredictorput$1𝑀inBender’sboxonTuesday.Hencehehascredence90%acrosscolumnsI-IVandcredence10%acrosscolumnsV-VIII.Withineachofthesepossibilities,heis90%confidentthepredictorpredictedhisownmove(#)correctlyyesterday,soheexpectsacorrelationof.9between¬𝑝(whichentailsthat#isone-boxing)and6𝑇1,thepropositionthatthepredictorput$1𝑀in336-Tuesday.Finally,hisoverallconfidencein𝑝is.5.Thatyields:

I. 5𝑇1, 𝑝, 6𝑇1 II. 5𝑇1, 𝑝, 6𝑇0 III. 5𝑇1,¬𝑝, 6𝑇1 IV. 5𝑇1,¬𝑝, 6𝑇0𝜔 0.045 0.405 0.405 0.045¬𝜔 0.045 0.405 0.405 0.045 V. 5𝑇0, 𝑝, 6𝑇1 VI. 5𝑇0, 𝑝, 6𝑇0 VII. 5𝑇0,¬𝑝, 6𝑇1 VIII. 5𝑇0,¬𝑝, 6𝑇0𝜔 0.005 0.045 0.045 0.005¬𝜔 0.005 0.045 0.045 0.005

Credences.

Whatoftheappropriatecausalistcredences?KnowingBenderone-boxedyesterday,Fryis90%confidentin5𝑇1,thepropositionthatthepredictorput$1𝑀inBender’sboxonTuesday.Hencehehascredence90%acrosscolumnsI-IVandcredence10%acrosscolumnsV-VIII.Withineachofthesepossibilities,heis90%confidentthepredictorpredictedhisownmove(#)correctlyyesterday,soheexpectsacorrelationof.9between¬𝑝(whichentailsthat#isone-boxing)and6𝑇1,thepropositionthatthepredictorput$1𝑀in336-Tuesday.Finally,hisoverallconfidencein𝑝is.5.Thatyields:

Acalculationshowsthat𝐶𝐸𝑈(𝜔)=900,000+(1000-𝛥)overthebaseline𝛽,while𝐶𝐸𝑈(¬𝜔)=$500,500inexcessofthesamebaseline.HenceFrywilltakethedealaslongas𝛥 <$400,500.

3.2 KarmaForetoldFromtheLewisianpointofview,time-biasedtradingattheNewcombHospitalseemstobemotivated.Ofcoursethecausaldecisiontheoristwantsthepastofaone-boxer—thisisanexcellentindicatorthatsheisalreadyamillionaire!Andofcourseshewantstotwo-boxtoday—thatguaranteesthatshewalksawaywith$1,000morethanshewouldhavegottenotherwise!21

21IhereusethecounterfactuallocutionLewisfavoredindescribingcausalistreasoning;see,forexample,(Lewis1981b).

Thesesentimentsarearticulatedfromthepointofviewofaparticulartime—forexample,thetimelabeled“𝑁𝑂𝑊"inFigure3.Butatthepointintimemarked“after"—whenbothroundslieinthepast—takingthecombodealwillseem,bytheCDTer’sownlights,likebadnews.Moretothepoint,Ifear,itlookslikeanirrationaldecision.SupposeFrywakesupamnesiaconThursday,andistoldthatBenderofferedhimthecombodealonWednesday.OnWednesday,Frycouldhavekepthis@formforfree,ensuringthattheactionheperformedonWednesdaywasdifferentinkindthantheactionheperformedonTuesday—hence,thathewasinoneofthescenariosinFigure3,withoutknowingwhich.Now,forFrytolearnwhetherhetookBender’scombodealistolearneitherthat:

(a)Inthelasttwodays,hetwo-boxedonceandone-boxedonce,insomeunknownorder,or

(b)Inthelasttwodays,heone-boxedonceandtwo-boxedonce,andwillinglygaveBender$400,000toensurethatthefirsthappenedbeforethesecond.

ThoughIamaCDTer,theideaoflearningthatIsecured(b)over(a)insuchasituationmakesmeuncomfortable.Itishardtojustifytheclaimthatonewayoforderingmyactsintimeisworth(morethan)$400,000morethantheother.22

4. CDT’sresponseTheoddityinNewcombHospitalFusionarisesbecausethetiebetweenone-boxingandbeingamillionaireisinvisibletoFry’sdeliberationsatthemomentofchoice—onlytoreappearwithavengeancelater,whenheassesseswhathebelievesisinyesterday’slockboxes.ThismakesitimpossibletostablyassumethatFrydoes,ordoesn’t,takethefactthatanagentisaone-boxertobegoodreasontoincreaseone’sconfidencethatthatagentisamillionaire.

Inmostordinarysituations,theredoesseemtobeastablefact.InMoodCandles,forexample,thecausalist’sintuitionisthatBobshouldlightacandleeventhoughcandle-lightingis—atleast,inthegeneralpopulation—correlatedwith

22Itisworthnotingthat,whilethepost-factonewsvalueof(a)exceedsthatof(b),NewcombHospitalFusionisnotaDutchBook:choosing(b)over(a)doesnotguaranteethatyouarepoorerineverypossibleworld(thoughitdoesmakeitlikely).Aworld𝑤willmaketheagentbetteroff,where

• 335-Tuesdaycontains$0in𝑤,

• 336-Tuesdaycontains$1millionin𝑤,and

• @isone-boxing.

ForadiachronicNewcombproblemforthetwo-boxerthatdoesmakeastandardCDTerpoorerineverypossibleworld,andtheextraassumptionsthisinvolves,seeAhmed’s“NewcombInsurance"cases(Ahmed2017,2014pg.202ff.).

depression.Italsoseemslike,ifhedoeslightup,hefailstoacquireagoodreasontoincreasehisconfidencethatheisdepressed.Atleast,thatiswhatIsuggestedabove:thereissomethingoddaboutthinking,withLewis,thatifBobdoeslightup,hewouldtherebyacquiregoodevidenceforthinkinghewasdepressedafterall.Heknowswhyhe’scontemplatinglightingup—heisinpursuitoftheextrautile,whichtheactwillsecurehim—andit’snotbecauseheisdepressed.

4.1 TimeBiasandEDTAsindicatedintheintroduction,myfirstdefensivemaneuveristoarguethatEDTisalsosusecptibletotimebias.ItwillbeuptothereadertodeterminewhethertheEDTer’sformofbiasisasunflatteringasFry’s.

RecallthatinthePearl-inspiredexampleHurryingtoClass(§2.1),hurryingonanygivendaywasgenerallycorrelatedwithlatenessonthatday:

𝐶𝑟1(late|hurry) > 𝐶𝑟1(late)

Nonetheless,relativetoaspecifictime—like9:46am—thatcorrelationfailed(infact,wasreversed),sothatrelativetotheagent’sfullavailableinformationhurryingwasnotafterallcontraindicated,byeitherdecisiontheory.Forcaseswheretheagentlackedawristwatchorothertimekeepingdevice,Iextendedthescreening-offargumentbyappealtointrospection—totheagent’sabilityto“lockon”tohercurrenttimebymeansofsomedemonstrative𝜇,anduse𝑍=thetimeis𝜇toscreenoffthesupportlenttolatenessbyhurrying.

Thisability,whichextendsonlytothepresentmoment,givesrisetotime-bias.SupposeanEDTer,whohasjustwokenat𝑡unabletorememberwhatshehasdoneforthelastmonth(theintervaloftime“𝑇�”),findsherselfinaniteratedversionofHurryingtoClass.Ifshecaresabouthertrackrecordofarrivingontimetoclass,shemaywell:

(i)prefertohurryat𝑡,and

(ii)prefertolearnthatshedidn’thurryatanypasttime𝑡′ ∈ 𝑇�.

TheEDTerinHurryingtoClassshouldhaveattitude(i)forthereasonsexplainedabove:overturningPearl’sargumentdependsonherleveragingher(difficult-to-express)knowledgethatitisnowthecurrenttime.Butthispowerislimitedto𝑡:withregardtoherpast,itisrationaltocontinuetomaintainattitude(ii),despitetakingupattitude(i).Thoughsomethingsimilartothethoughtinvolving𝜇wastrueinthepast—afterall,foranymorningintheagent’spastitwasthattimethenasmuchasitisthistimenow—thepropositionexpressedbytheformerthoughtdoesnothavetheforce,at𝑡,thatthelatterthoughtdoes.EDT’sappealtoscreeningoffinmanyputativeNCproblemswillcommittheviewtotimebiasincaseswhere,asinNewcombHospital,thoseNCproblemsareiterated.

Figure4:ASharedStructure(toppreferredtobottom).

Itisworththinkingaboutwhythisisso,andwhatthismeansforLewis’sviewsonupdating.AtissueinLewis’sPackageiswhetheranagent—forexample,inanNCProblem—shouldplantoconditionalizeontheactsheperforms.Onewayofthinkingaboutthisistoaskwhether,inanNCproblem,yourattitudetowardstheacts𝑎′ ∈ 𝒜youarecurrentlycontemplatingshouldbethesameasthepreferencesyouwouldhaveifyouwokeuptomorrowwithamnesiaandwereabouttobetoldwhich𝑎′ ∈ 𝒜youperformed.

Butintrospection’spowertocreateanddiscoverpropositionsconcerningthetime𝑡ofchoiceissharplydifferentacrossthetwosituations.InHurryingtoClass,Icanthinktomyself,thetimeis𝜇.Oddly,atthetimeofchoice,Iseemtogainsomeknowledgebythisexercise—knowledgethataffectsmyrationalcalculations.Later,post-amnesia,Iwillnothavethispower.Itispatentlyuninformativetobetold,“youreflectedthenthatthetimewas𝜈,whereby𝜈youmeant:thetimeitinfactwas.”

AcomparisonbetweenthissituationandaCDTer’sreasoninginNCProblemsistelling.InanNCproblem,Icantellmyself,“Iwillnowchoosethedominantoptionbecauseofthe(small)benefitIgetfromit”.Iamintrospectingmymotives,whichgivemeareasontoperform𝑎∗withoutgivingmeanextrareasontothinkthatIwon’tgetthebigprize.IfIamsubsequentlysubjectedtoamnesia,laterIwillonlybeabletolearnthatIeitherperformedthedominantoptionorIdidn’t.AnyextraknowledgeIhadaccesstoattheearliertimewillbewipedout:itfeelsuninformativetobetold“youperformedthedominantoptionbecauseitwasthedominantoption”.Whilenotvacuous,thisthoughtseems,again,obvious—likethethoughtthatifyouhurried,youhurriedatsomeparticulartime.

5. Conclusion

IbeganbyarguingthatCDTistime-biased,andgoingontoshowthatthisleadstooddchoicesintheNewcombHospital.Inresponsetothisbias,IleveledatuquoqueagainstEDT.Time-biasmaybeapervasivefeatureofdecision-making,notaweaknessinanyparticulartheoryofit.

ItwouldbebesttoclosebyreturningtoNewcombitself.EDTersmayfeelthat,inshiftingthefocustoanagent’stheact-priorviewpoint,andtheepistemicreasonsthatcanbearticulatedfromit,Ihavequietlysidesteppedtherealbiteoftheoriginalproblem.Forinthatpuzzle,whetheritispossible,beforeyouact,tocometoanevidentiallysupportedcredenceinthe𝒦-partition{Predictorpredictedyou’done-box,Predictorpredictedyou’dtwo-box}(={𝑃1,𝑃2})ispreciselywhatisatissue.Thereisawidespreadtendencyintheliteraturetoassume

(P)InNewcomb’spuzzle,theonlyevidenceavailablewithrespecttodistributingcredenceoverthepartition{𝑃1,𝑃2}ishowtheplayerhim/herselfreasonsabouttheexpectedutilityofone-boxing.

Thiswouldapparentlyrenderfullreflection,andanyattemptatscreeningoff,impossible.Ahmed,forexample,pressesthepointthat“[a]nyonefacingNewcomb’sproblemhasnoevidencethatrelevantlydistinguisheshimnowfromanyoneelsewhomthestatisticalgeneralization[𝑃𝑟(𝑃1|1𝐵)and𝑃𝑟(𝑃2|2𝐵)arehigh]covers;thatis,allotherpersonswhoeverfacethisproblem"(Ahmed2014pg.191).Thissuggests(P)’sepistemologicalmoral—thatthereisjustnoway,inNewcomb,toeventrytotakeyourfullevidenceintoaccountbeforeyouact,sincenothingcountsasrelevantevidenceotherthantheactitself.

Butwhile(P)mightcharacterizeNewcomb’sProblem,itisn’tsupported,eitherbydeductiveorabductiveconsiderations,bythebasicmechanicsofanNCproblem—wherethelatterisdescribedsimplyintermsofcausaldependencyhypotheses,credences,andoutcomestates,andwhereitisthelatterthatusefullydiagnosesthesplitbetweencausalandevidentialdecisiontheory.

6. Appendix:NewcombPresent:LongargumentAsintheoriginalNewcombproblem,Fry’sestimatehowmuchmoneyisalreadyinhisownbox—thequantitycalled“𝛽”inthemaintext—isact-dependent.Itdepends,inpart,onhowlikelyhethinksitisthathewilltakethetrade.Fryknowsthatifhechoosestotrade,thenhewillcertainlytwo-box.Thepredictorwillhavepredictedthiswithherusualaccuracy,sothat𝐶𝑟1(𝑃1|trade) = .1and𝐶𝑟1(𝑃2|trade) = .9.However,ifFrydeclinestotrade,stickingwiththewildcardmove@,thentheprobabilitythatthepredictorpredicted𝑃1isequaltotheprobabilitythatthepredictorpredicted𝑃2(thehighcorrelationsof𝑃1with1𝐵and𝑃2with2𝐵arenowbalancedbythefactthatFryhasa50%chanceofdoing1𝐵anda50%chanceofdoing2𝐵.)

𝒦 = { 𝑃1 ∧ 𝑝 , 𝑃1 ∧ ¬𝑝 , 𝑃2 ∧ 𝑝 , 𝑃2 ∧ ¬𝑝 }isadependencyhypothesispartition.

𝑃1 ∧ 𝑝 𝑃1 ∧ ¬𝑝 𝑃2 ∧ 𝑝 𝑃2 ∧ ¬𝑝𝜏 1𝑀 + (1000 − 𝛥) 1𝑀 + (1000 − 𝛥) 0𝑀 + (1000 − 𝛥) 0𝑀 + (1000 − 𝛥)¬𝜏 1𝑀 + 0 1𝑀 + 1000 0𝑀 + 0 0𝑀 + 1000

Utilities.

Wehave:

𝐶𝑟(𝑘5) = 𝐶𝑟(𝑃1 ∧ 𝑝) = 𝐶𝑟(𝑝)𝐶𝑟�(𝑃1)= 𝐶𝑟(𝑝)[𝐶𝑟�(𝑃1|𝜏)𝐶𝑟�(𝜏) + 𝐶𝑟�(𝑃1|¬𝜏)𝐶𝑟�(¬𝜏)]= 𝐶𝑟(𝑝)[.1𝐶𝑟�(𝜏) + .9𝐶𝑟�(¬𝜏)]= .5[.1𝐶𝑟�(𝜏) + .9𝐶𝑟�(¬𝜏)]= .5[.1𝐶𝑟(𝜏) + .9𝐶𝑟(¬𝜏)]

𝐶𝑟(𝑘V) = 𝐶𝑟(𝑃1 ∧ ¬𝑝)= 𝐶𝑟(¬𝑝)𝐶𝑟¬�(𝑃1)= 𝐶𝑟(¬𝑝)[𝐶𝑟¬�(𝑃1|𝜏)𝐶𝑟¬�(𝜏) + 𝐶𝑟¬�(𝑃1|¬𝜏)𝐶𝑟¬�(¬𝜏)]= 𝐶𝑟(𝑝)[.1𝐶𝑟¬�(𝜏) + .1𝐶𝑟¬�(¬𝜏)]= .5[.1𝐶𝑟(𝜏) + .1𝐶𝑟(¬𝜏)]= .5(.1) = .05

𝐶𝑟(𝑘�) = 𝐶𝑟(𝑃2 ∧ 𝑝) = 𝐶𝑟(𝑝)𝐶𝑟�(𝑃2)= 𝐶𝑟(𝑝)[𝐶𝑟�(𝑃2|𝜏)𝐶𝑟�(𝜏) + 𝐶𝑟�(𝑃2|¬𝜏)𝐶𝑟�(¬𝜏)]= 𝐶𝑟(𝑝)[.9𝐶𝑟�(𝜏) + .1𝐶𝑟�(¬𝜏)]= .5[.9𝐶𝑟�(𝜏) + .1𝐶𝑟�(¬𝜏)]= .5[.9𝐶𝑟(𝜏) + .1𝐶𝑟(¬𝜏)]

𝐶𝑟(𝑘�) = 𝐶𝑟(𝑃2 ∧ ¬𝑝)= 𝐶𝑟(¬𝑝)𝐶𝑟¬�(𝑃2)= 𝐶𝑟(¬𝑝)[𝐶𝑟¬�(𝑃2|𝜏)𝐶𝑟¬�(𝜏) + 𝐶𝑟¬�(𝑃2|¬𝜏)𝐶𝑟¬�(¬𝜏)]= 𝐶𝑟(𝑝)[.9𝐶𝑟¬�(𝜏) + .9𝐶𝑟¬�(¬𝜏)]= .5[.9𝐶𝑟(𝜏) + .9𝐶𝑟(¬𝜏)]= .5(.9) = .45

Forshorthand,westipulatethat𝔱 = 𝐶𝑟(𝜏).HenceFry’scredencesareasfollows:

𝑘1 𝑘2 𝑘3 𝑘4. 45− .4𝔱

. 05 . 4𝔱+ .05

. 45

Credences,takeI.

Letting𝑧 = .4𝔱,thisisequivalentto:

𝑘1 𝑘2 𝑘3 𝑘4. 45 − 𝑧 . 05 𝑧 + .05 . 45

Credences,takeII.

Calculatingcausalexpectedutilities:

𝐶𝐸𝑈1(𝜏) = (.45 − 𝑧)(1𝑀 + (1000 − 𝛥)) + .05(1𝑀 + (1000 − 𝛥))+(𝑧 + .05)(0𝑀 + (1000 − 𝛥)) + .45(0𝑀 + (1000 − 𝛥))= (.5 − 𝑧)(1𝑀) + (1000 − 𝛥)

𝐶𝐸𝑈1(¬𝜏) = (.45 − 𝑧)(1𝑀 + 0) + .05(1𝑀 + 1000) + (𝑧 + .05)(0𝑀 + 0) + .45(0𝑀 + 1000)= (.45 − 𝑧)(1𝑀) + (.05)(1𝑀) + (.05)(1000) + (.45)(1000)= (.45 − 𝑧 + .05)(1𝑀) + (.05 + .45)(1000)= (.5 − 𝑧)(1𝑀) + 500

Onceagain,wehavesomethingoftheform:𝐶𝐸𝑈1(𝜏) = 𝑋 + (1000 − 𝛥),while𝐶𝐸𝑈1(¬𝜏) = 𝑋 + 500.Hence𝐶𝐸𝑈1(𝜏) > 𝐶𝐸𝑈1(¬𝜏)iff𝛥 < $500.QED.

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