1

ObjectiveValueIsAlwaysNewcombizable

ArifAhmedandJackSpencer

Words:5885

Words,includingfootnotesandappendices:9060

Thisessayarguesthatevidentialdecisiontheory(EDT)isincompatiblewithoptionshaving

objectivevalues.

After somescene-setting (§§1-2),weconsider threearguments forour thesis: the

argument fromNewcomb’sProblem(§§3-4), theargument fromExpectationism(§§5-6),

andtheargumentfromNewcombizability(§7).Thefirsttwoargumentsfailforinstructive

reasons.But the third succeeds.EDT is inconsistentwithoptionshavingobjectivevalues

becauseobjectivevalueisalwaysNewcombizable.

§1/ObjectiveandSubjective

Webeginbysupposing:

TwoOughts.Thereisbothanobjectiveoughtandasubjectiveought.

2

OnecandenyTwoOughts—Mackie(1977),forinstance,rejectstheobjectiveought,and

Moore(1903)andThomson(2008)rejectthesubjectiveought.1Butcaseslikethefollowing

convincemanyphilosopherstoacceptit:2

Miners.Tenminersaretrapped inshaftAorshaftB,but theagentdoesnotknow

which.Floodwatersthreatentofloodtheshafts.Theagenthasenoughsandbagsto

blockoneshaft,butnotboth.Iftheagentblocksoneshaft,allthewaterwillgointo

theother,killinganyminersinside.Iftheagentblocksneithershaft,bothshaftswill

fillhalfwaywithwater,andjustoneminer,thelowestintheshaft,willbekilled.(In

fact,theminersaretrappedinshaftA.)

With regard to Miners, one might ask: what ought the agent to do? And is the agent’s

uncertaintyrelevant towhatsheought todo?IfTwoOughtsis true, thesequestions lack

univocalanswers.

Intheobjectivesense,theagent’suncertaintyisirrelevant.Howeverconfidentsheis

thattheminersareinshaftB,sheobjectivelyoughttoblockshaftA.Inthesubjectivesense,

theuncertaintyisparamount.Iftheagentis5%confidentthattheminersareinshaftAand

95%confidentthattheyareinshaftB,thenshesubjectivelyoughttoblockshaftB.If,aswe

supposehenceforward,theagentis50%confidentthattheminersareinshaftAand50%

1 Also seeKolodnyandMacFarlane2010,which argues for onlyoneought that is neitherpurely

objectivenorpurelysubjective. 2Parfit(unpublished);cf.Jackson1991andRegan1980.ForadifferentmotivationofTwoOughts,

seeOddieandMenzies1992.

3

confidentthattheyareinshaftB,shesubjectivelyoughttoblockneithershaft—despite

knowingforcertainthatblockingneithershaftisnotwhatsheobjectivelyoughttodo.

Given the distinction between objective and subjective ought, we can saywhat it

wouldbeforoptionstohaveobjectiveandsubjectivevalues.

Saying that options have objective values is equivalent to saying that some

consequentializabletheoryoftheobjectiveoughtistrue.Inotherwords,it’stosaythatthere

is a numerically representable propertyof options such that,whenever an agent faces a

decision,theoptionsthatareobjectivelypermissibleforhertochooseareexactlytheoptions

thatmaximizetheproperty.

Somephilosophersconjecturethatevery(plausible)theoryoftheobjectiveoughtis

consequentializable.3Althoughwewillnotassumethat thisconjecture istrue, if it is,our

argumentsestablishanevenmoresurprisingmetaethicalthesis—thatEDTisincompatible

withevery(plausible)theoryoftheobjectiveought.

Saying that options have subjective value is equivalent to saying that some

consequentializabletheoryofthesubjectiveoughtistrue.Likeobjectivevalues,subjective

values are numerically representable properties of options. But unlike objective values,

subjectivevaluesarehadonlyrelativetoacredencefunction.Sotoclaimthatoptionshave

subjective values is to claim that there is some numerically representable property of

optionsrelativetoacredence functionsuchthat,wheneveranagent facesadecision, the

optionsthataresubjectivelypermissiblefortheagenttochooseareexactlytheoptionsthat

maximizethepropertyrelativetotheagent’scredencefunction.

3Seee.g.Dreier1993;2011,Louise2004,Portmore2007.

4

TounderstandwhyEDTisinconsistentwithoptionshavingobjectivevalues,itwill

behelpfultocontrastitwithitschiefrival,causaldecisiontheory(CDT).

§2/EDTandCDT

EDTandCDTarebothconsequentializabletheoriesofthesubjectiveought.Buttheydisagree

overwhatsubjectivevalueis.

According to EDT, subjective value is evidential expected value. To characterize

evidentialexpectedvalue,weneedsomefamiliarformalism.

Let𝑊 = {𝑤%,𝑤', … ,𝑤)}bethesetofpossibleworlds.Weassumeforsimplicitythat

𝑊isfinite.Andtoeasetheformalism,weignorethedistinctionbetweenapossibleworld

anditssingleton.

Let𝐴 = {𝑎%, 𝑎', … , 𝑎-}bethesetoftheagent’soptions,eachofwhichisaproposition

(i.e. a setofpossibleworlds) that theagent canmake truebydeciding.4Weassume that

optionsarealwayspair-wiseexclusiveandjointlyexhaustive.

Let 𝐶 be the agent’s credence function, a probability function mapping each

proposition to a real number in the unit interval and thereby representing the agent’s

confidencethatthepropositionistrue.Let𝐶(𝑝)betheagent’scredencein𝑝.Andintheusual

way,let𝐶(𝑝|𝑞) = 𝐶(𝑝𝑞)/𝐶(𝑞)betheagent’sconditionalcredencein𝑝given𝑞.

Finally,let𝑢betheworld-valuationfunctionmappingeachpossibleworldtoareal

numberandtherebyrepresentingthevalueofthatworld.Valuecomesinmanyflavors,and

4FollowingJeffrey1983:83-4.

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forourpurposesanyflavorwilldo.Ifweareinterestedinmorality,wemaytake𝑢(𝑤)to

representhowmorallygoodwis.Ifweareinterestedinprudentialvalue,wemaytake𝑢(𝑤)

torepresenthowprudentiallygood(i.e.desirable)w is;andsoon.OurclaimthatEDTis

incompatiblewithoptionshavingobjectivevaluesholdsforanyflavorofvaluethatmight

belongtoindividualworlds.

With this formalism in place we can characterize evidential expected value. The

evidentialexpectedvalueof𝑎,written𝑉(𝑎),is:

EvidentialExpectedValue:𝑉(𝑎) = ∑ 𝐶(𝑤|𝑎)𝑢(𝑤)8 .5

According to CDT, subjective value is causal expected value. The simplest way to

characterizecausalexpectedvalue,which,forconvenience,weadopthere,involvesahighly

discerning causal similaritymetric.6 Thatmeans: ameasure of distance betweenpossible

worlds(asimilaritymetric)suchthat: (i) it ishighlydiscerning: foranyoption𝑎andany

world𝑤9 ,some𝑎-world𝑤:is,bythelightsofthismeasure,uniquelyclosestto𝑤97;(ii)itis

5Forsimplicity,thischaracterizationofevidentialexpectedvalueinvolvescredencesinindividual

worlds, thus departing from the technical assumption in standard axiomatizations of EDT that

preferenceisdefinedoveranatomlessfield(Jeffrey1983:146).Thisisharmless,sincenothingthat

wesaydependsontherepresentationtheoremforEDTthatrequiresatomlessnessforitsproof;in

anycase,wecoulddowithoutatomsaslongasvalueisnot‘gunky’(asHájek(2013:438)putsit).6Fordiscussionofothercharacterizations,seee.g.Ahmed2014,ch.2,Joyce1999ch.5,Lewis1981.7Nonaturalsimilaritymetricisquitethisdiscerning:fordiscussionseeLewis1973andJoyce1999

sect.5.4.Amorerealisticproposalisthat,foranyoption𝑎andworld𝑤9 ,thereissomesetof𝑎-worlds,

{𝑤:, 𝑤;, … ,𝑤)}thatare,bythelightsofthecausalsimilaritymetric,moresimilarto𝑤9 thanareany

other𝑎-worlds.Nothingexcepteaseofexpositionisgoingtoturnonthis.

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causal:itreckonstheclosest𝑎-worldto𝑤9 tomatch𝑤9 onallparticularmattersoffactto

which𝑎iscausallyirrelevant.

Giventhishighlydiscerningcausalsimilaritymetric,wecangivetruth-conditionsfor

nonbacktracking counterfactual conditionals. If ⟨𝑎 ⇒ 𝑤:⟩ is the nonbacktracking

counterfactual—thatworld𝑤:wouldhavebeenactualhadoption𝑎beenchosen—then

⟨𝑎 ⇒ 𝑤:⟩istrueatsomepossibleworld𝑤9 ifandonlyif𝑤:isthe𝑎-worldthatis,bythelights

ofthehighlydiscerningcausalsimilaritymetric,uniquelymostsimilarto𝑤9 .

The causal expected value of option𝑎, written𝑈(𝑎), is a function of the agent’s

credencesinthesenonbacktrackingcounterfactuals:

CausalExpectedValue:𝑈(𝑎) = ∑ 𝐶(⟨𝑎 ⇒ 𝑤⟩)𝑢(𝑤)8 .

§3/TheArgumentfromNewcomb’sProblem

ThefirstpurportedargumentforourthesisistheargumentfromNewcomb’sProblem.EDT

and CDT often agree on which option maximizes subjective value, but not always. One

famouscruxis:

Newcomb’sProblem.Thereisatransparentboxandanopaquebox.Theagenthastwo

options:she can takeonly theopaquebox (she can ‘one-box’), or takebothboxes

(‘two-box’).The transparentbox contains$1,000.Theopaquebox containseither

$1,000,000or$0,dependingonapredictionmadeyesterdaybyareliablepredictor.

Ifthepredictorpredictedthattheagentwouldtwo-box,thentheopaqueboxcontains

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$0. If the predictor predicted that the agentwould one-box, then the opaque box

contains$1,000,000.Theagentknowsallofthis.

The agent reasonably takes her choice to be strong evidence about whether the

opaqueboxcontains$1,000,000or$0.Equatingunitsofvalueanddollars,the𝑉-scoreof

one-boxingisthereforeslightlyunder1,000,000,andthatoftwo-boxingslightlyover1,000.8

Buttwo-boxinguniquelymaximizes𝑈.Interpretingthefollowingcounterfactualsin

termsofacausalsimilaritymetric,theagent’scredencethatshewouldhavereceived$1,000

hadshetwo-boxedexactlyequalshercredencethatshewouldhavereceived$0hadsheone-

boxed, and her credence that she would have received $1,001,000 had she two-boxed

exactlyequalshercredencethatshewouldhavereceived$1,000,000hadsheone-boxed.

The𝑈-scoreof two-boxing is thereforeexactly1,000greater than thatofone-boxing,no

matterwhatthe𝑈-scoreofone-boxingis.

TheargumentfromNewcomb’sProblemreferstoNewcomb’sProblemandhastwo

premises:

Newcomb Knowledge: If options have objective values then: an agent facing

Newcomb’sProblemknowsforcertainthattwo-boxinguniquelymaximizesobjective

value.

8E.g.iftheagentexpectsthepredictortobeaccurateon99%oftrialswhicheveroptionischosen,

then the V-score of one-boxing is (0)(0.01) + (0.99)(1,000,000) = 990,000; and that of two-

boxingis(1,000)(0.99) +(0.01)(1,001,000) = 11,000.

8

Bridge:Ifoptionshaveobjectivevaluesthen:ifanagent’soptionsare𝑎%, 𝑎',… , 𝑎- ,

andtheagentknowsforcertainthatoption𝑎9 uniquelymaximizesobjectivevalue,

thenoption𝑎9 uniquelymaximizessubjectivevalue.

NewcombKnowledgeismotivatedbytheclaimthatobjectivevalueiscausalvalue.

Thecausalvalueofanoptionisthevalueoftheworldthatwouldhavebeenactualhadthe

agentchosentheoption.Thatis:if⟨𝑎 ⇒ 𝑤:⟩istrueatworld𝑤9 ,thenthecausalvalueof𝑎at

𝑤9 is𝑢(𝑤:).Causalvalueisafamiliarconceptionofobjectivevalue.Itfeatures,forexample,

inMoore’sdefenseofconsequentialism,wherehewrites:

Inordertoshowthatanyaction[maximizesobjectivevalue],itisnecessarytoknow

both what are the other conditions, which will, conjointly with it, determine its

effects;toknowexactlywhattheeffectsoftheseconditionswillbe;andtoknowall

theeventswhichwillbe inanywayaffectedbyouractionsthroughoutan infinite

future.Wemusthaveall thiscausalknowledge….Andnotonlythis:wemustalso

possessallthisknowledgewithregardtotheeffectsofeverypossiblealternative;and

mustthenbeabletoseebycomparisonthatthetotalvalueduetotheexistenceofthe

actioninquestionwillbegreaterthanthatwhichwouldbeproducedbyanyofthese

alternatives.(1903:149)

ItisuncontroversialthatanagentfacingNewcomb’sProblemknowsforcertainthat

two-boxinguniquelymaximizescausalvalue.So,ifobjectivevalueiscausalvalue,Newcomb

Knowledgeistrue.

9

Bridgeismotivatedbymetaethicalconsiderations.Thesubjectiveoughtarisesfrom

theagent’s subjectiveuncertainty about theobjectiveought.When anagent isuncertain

what she objectively ought to do, there can be a discrepancy between what the agent

subjectively and objectively ought to do. Indeed, asMiners illustrates, when an agent is

uncertainaboutsheobjectivelyoughttodo,anagentcanknowforcertainthattheydiffer.

Butifanagentknowsforcertainwhatsheobjectivelyoughttodo,thentheobjectiveand

subjectiveoughtmust coincide.9TwoOughts thusentails the following conditional: if an

agentknowsforcertainthatsheobjectivelyoughttochooseoption𝑎9 ,shesubjectivelyought

tochooseoption𝑎9 .10Andgiventhatoptionshavebothobjectiveandsubjectivevalues,this

conditionalisequivalenttotheconsequentofBridge.11

9Zimmermandefendsthestrongerclaimthat‘[a]nagent[subjectively]oughttoperformanactifand

only ifhebelieves that it is [objectively] thebest option thathehas’ (2008:5).WethinkMiners

refutestheleft-to-rightdirectionofthisclaim;buttheright-to-leftdirectionisplausibleandentails

Bridge.

10OneapparentchallengetoBridgeinvolvesupwardmonotonicity.TheagentinMinersobjectively

oughttoblockshaftA.Iftheobjectiveoughtisupwardmonotonic,thentheagentinMinerswill(if

rational)knowforcertainthatsheobjectivelyoughttoblocksomeshaftorother.Butitisnotthe

casethattheagentsubjectivelyoughttoblocksomeshaftorother.Inresponseonemightreasonably

denyupwardmonotonicity;butinanycase,theexampledoesnotthreatenBridgeasstated,because

Bridgeonlycoversanagent’soptions,andsinceoptionsarepair-wiseexclusive,blockingsomeshaft

orothercannotbeanoptionifblockingshaftAisanoption.11Bridgemustbedistinguishedfromnearbyprinciples.

Onenearbyprincipleconcernspermissibility.Itsays:ifanagentknowsforcertainthatan

option is objectively permissible, then the option is subjectively permissible. This principle is

plausible, but less plausible than Bridge. Opaque sweetening cases, discussed inHare 2010, put

pressureonthispermissibilityprinciple,butputnopressureonBridge.

10

ItmightappearthattheargumentfromNewcomb’sProblemestablishesourthesis

— that Nozick, way back in 1969, showed that EDT is inconsistentwith options having

objectivevalues.ButtheargumentfromNewcomb’sProblemcanberesisted.

§4/ManyConceptionsofObjectiveValue

Bridgeisundeniable,butNewcombKnowledgeisnot.AnagentfacingNewcomb’sProblem

knowsforcertainthattwo-boxinguniquelymaximizescausalvalue.ButwhyshouldEDT’ists

grantthatobjectivevalueiscausalvalue?Afterall,therearemanyconceptionsofobjective

Another nearby principle concerns inevitable knowledge. It says: if an agent knows for

certainthatshewouldbeobjectivelyrequiredtochooseaparticularoptionifsheknewthatpwas

trueandalsoknowsforcertainthatshewouldbeobjectivelyrequiredtochoosethatsameoptionif

sheknewthatpwasfalse,thentheagentissubjectivelyrequiredtochoosetheoption.Thisinevitable

knowledge principle is clearly false. It might be the case that an agent, who is uncertain about

whetherp,objectivelyoughttopayasmallsumofmoneytocometolearnthetruth-valueofp,even

thoughtheagentknowsforcertainthatshewouldbeobjectivelyrequirednottopaythesmallsum

ifsheknewthatpwastrueorknewthatpwasfalse.

ThedifferencebetweenBridgeandtheprincipleconcerninginevitableknowledgeisrelevant

tothediscussioninHare2016.Haremakestwoclaims:(1)thatkillingonetosavefiveissubjectively

permissiblewhenitisunknownwhichofthesixisbeingsacrificedtosavetheotherfive,and(2)that

killingonetosavefiveissubjectivelyimpermissibleiftheidentityoftheonebeingsacrificedtosave

thefivewasknown.Itisnotobvioustousthatbothclaimsaretrue.Buteveniftheyare,theypose

nothreattoBridge.Weclaimthatif(1)istrue,thenitisalsoobjectivelypermissibletokilloneto

savefivewhentheidentityoftheonebeingsacrificedtosavethefiveisunknown.Andweclaimthat

if(2)istrue,thenitisalsoobjectivelyimpermissibletokillonetosavefivewhentheidentityofthe

onebeingsacrificed tosave the five isknown.Hare’sexamplemightbeacounterexample to the

inevitableknowledgeprinciple,butitisnotacounterexampletoBridge.

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value(whereby‘conceptionofobjectivevalue’wemeanaproposalaboutwhatobjective

valueis).

Anaturalconceptionofobjectivevaluehastwocomponents.Thefirstisasimilarity

metric.Theinputtoasimilaritymetricissomeoption𝑎andsomeworld𝑤9 ,andtheoutput

issomesetF𝑤:,𝑤;,… , 𝑤)Gof𝑎-worldsthatare,onthatmetric,themostsimilara-worldsto

𝑤9 .Thesecondcomponentisamethodofaveraging.Theobjectivevalueofoption𝑎atworld

𝑤9 issomeweightedaverageof𝑢H𝑤:I, 𝑢(𝑤;), … , 𝑢(𝑤)),thevaluesofthe𝑎-worldsthatare,

bythelightsofthesimilaritymetric,mostsimilarto𝑤9 .Ifasimilaritymetricalwaysoutputs

asetcontainingonlyonepossibleworld,wecandisregardthemethodofaveraging,forthe

objectivevalueofanoption𝑎atworld𝑤9 isthenjustthevalueofthe𝑎-worldthatis,bythe

lightsofthesimilaritymetric,uniquelymostsimilarto𝑤9 .Causalvalueisonememberofthis

familyofnaturalconceptionsofobjectivevalue,buttherearemanyothers.Andmanyare

uniquelymaximizedbyone-boxinginNewcomb’sProblem.12

Forinstance,considerwhatHorgansays,inhisdefenseofone-boxing:

I shall assume that there is indeed a standard resolution of the vagueness of the

similarityrelationamongworlds,andthatLewis’saccountofitisessentiallycorrect.

Returning to Newcomb’s problem, it is clear that [the] premises…of the one-box

12Thefullclassofobjectivevalueconceptsextendswellbeyondthisnaturalfamily.Initsbroadest

sense, an objective value concept is any function taking each pair of a possible world and a

propositiontoarealnumber.Althoughwefinditmorenaturalandintuitivetofocusontherestricted

family that themain textdescribes,ourresultextends toanyobjectivevalueconceptwithin this

broaderclass.WeclaimthatEDTrulesoutidentifyingobjectivevaluewithanyfunctionwithinthat

broaderclass.

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argument cannot…be true under the standard resolution. For the beingmade his

predictionaboutmychoice,andhaseitherputthe$1millioninthe[opaquebox]or

not,wellbeforeIchoose.Thus,hisactual-worldpredictionandtheactual-worldstate

of[theopaquebox]remainintactintheclosestworldinwhichItakebothboxes,and

alsointheclosestworldinwhichItake[theopaquebox]only.

[… The] intuitive plausibility of the one-box argument rests upon a

nonstandardresolution,onethatseemsquiteappropriate in thiscontext. Itdiffers

fromthestandardresolutiontotheextentthatitgivestopprioritytomaintainingthe

being’spredictivecorrectnessinthenearestpossibleworldwhereItakebothboxes,

and also in the nearest world where I take [only the opaque box]. Under this

backtrackingresolution…theclosestworldinwhichItakebothboxesisoneinwhich

thebeingcorrectlypredictedthisandputnothingin[theopaquebox],andtheclosest

worldinwhichItakeonly[theopaquebox]isoneinwhichhecorrectlypredictedthis

andput$1millionin[it].(1981:336)

AlthoughHorgandoesnotgivedetails,wecanspelloutasimilaritymetriconpossible

worldsthatsatisfieshisdesiderata.OnewaytodoitwouldbetograftHorgan’s‘toppriority’

onto something like Lewis’s (1979) lexicographic criteria for measuring the relative

similarityofworlds,insistingthatwhatcountsmostforclosenessofapossibleworld𝑤in

this context is whether the predictor’s accuracy at𝑤 with regard to the agent’s choice

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matchesthepredictor’sactualaccuracyonthatquestion.13WethenusetheHorganmetric

todefineaconceptionofobjectivevalue,whichwemightcall:

Horganvalue:if𝑤:isthe𝑎-worldthatismostsimilarto𝑤9 bytheHorganmetric,

thentheHorganvalueofaat𝑤9 is𝑢(𝑤9).

13Moreformally,defineameasureoftherelativesimilarityofworlds𝑤and𝑤′toafixedworld𝑤9

byappealtofivepartialordersonworlds:

1) 𝑤 >% 𝑤′ iff: the Newcomb predictor’s correctness on this occasion at𝑤 matches the

predictor’saccuracyat𝑤9 ,butthepredictor’scorrectnessat𝑤′doesnot.

2) 𝑤 >' 𝑤′iff:therearebig,widespreadanddiverseviolationsofthelawsof𝑤9 at𝑤′and

notat𝑤.

3) 𝑤 >L 𝑤′iff:thespatio-temporalregionthroughoutwhichperfectmatchoverparticular

factswith𝑤9 prevailsislargerat𝑤thanat𝑤′.

4) 𝑤 >M 𝑤′iff:therearesmall,localized,simpleviolationsofthelawsof𝑤9 at𝑤′andnotat

𝑤.

5) 𝑤 >N 𝑤′iff:𝑤achievesapproximatesimilarityto𝑤9 overmattersofparticularfactand

𝑤′doesnot.

Say that 𝑤: is Horgan-closer to 𝑤9 than is 𝑤; if and only if: either (i) F𝑛P𝑤; >- 𝑤:G = ∅ and

F𝑛P𝑤: >- 𝑤;G ≠ ∅;or(ii)minF𝑛P𝑤; >- 𝑤:G > minF𝑛P𝑤: >- 𝑤;G.

Criteria2)-5)arefromLewis(1979:47-8),minusLewis’shedgingover5).Takenjointlyas

ananalysisofthe‘standardresolution’ofclosenessinnaturallanguage,2)-5)areimplausible(see

e.g.McDermott1999forcriticism).Ofcoursewearenotmakinganyclaimsaboutnaturallanguage

counterfactualsbutratherusingLewis’scriteriatoconstructakindofobjectivevaluethatone-boxing

maximizes.

14

Often,theHorganvalueofanoptionisequaltoitscausalvalue.Forinstance,equating

livesandunitsof value, inMiners, the causalvalueofblocking shaftA (B) is10 (0), and

likewisetheHorganvalueofblockingshaftA(B)is10(0).

But in Newcomb’s Problem, the two diverge. The causal metric holds fixed the

contentsoftheopaqueboxbutnotthepredictor’scorrectness.Whatevermoneyisinthe

opaqueboxatthetwo-boxing-worldthatiscausallyclosesttotheactualworldisalsointhe

opaqueboxattheone-boxing-worldthatiscausallyclosest,sothecausalvalueoftwo-boxing

exceedsthatofone-boxingby1,000.TheHorgansimilaritymetricholdsfixedthepredictor’s

correctnessbutnotthecontentsoftheopaquebox.So,ifthepredictorisactuallycorrect,the

two-boxing-worldthatisHorgan-closestisonewheretheopaqueboxcontains$0,andthe

one-boxing-worldthatisHorgan-closestisonewheretheopaqueboxcontains$1,000,000.

Ifthepredictorisactuallyincorrect,thesituationisreversed—thetwo-boxing-worldthat

isHorgan-closest isonewhere theopaquebox contains$1,000,000, and theone-boxing-

worldthat isHorgan-closest isonewheretheopaqueboxcontains$0. Ineithercase, the

Horganvalueofsomeoptiondivergesfromitscausalvalue.

If objective value is Horgan value, Newcomb Knowledge is false. An agent facing

Newcomb’sProblemdoesnotknowforcertainthattwo-boxinguniquelymaximizesHorgan

value. After all, she is confident that the predictor is accurate on thisoccasion, so she is

confident thatone-boxinguniquelymaximizesHorganvalue. Indeed, if objectivevalue is

Horganvalueandthepredictorisknowntobesufficientlyreliable,theagentmightknowfor

certainthatone-boxinguniquelymaximizesobjectivevalue.AndHorganvalueisnotunique

15

in this respect. Uncountably many conceptions of objective value can be known to be

uniquelymaximizedbyone-boxinginNewcomb’sProblem.14

The argument from Newcomb’s Problem is therefore too quick. The argument

establishessomething.SinceBridgeistrue,theargumentestablishesthatEDTisinconsistent

with any conception of objective value that validates Newcomb Knowledge. But most

conceptionsofobjectivevalueinvalidateNewcombKnowledge.Withoutsomeindependent

argument that the true conceptionofobjectivevaluevalidatesNewcombKnowledge, the

argumentfromNewcomb’sProblemfailstoestablishourthesis.

§5/TheArgumentfromExpectationism

ThesecondpurportedargumentforourthesisistheargumentfromExpectationism.

Expectationismisathesisabouthowobjectivevalueandsubjectivevaluerelate.It

saysthatsubjectivevalueisexpectedobjectivevalue.Let⟨𝑂(𝑎) = 𝑣⟩bethepropositionthat

(onsomearbitrarilychosenscale)theobjectivevalueofanoption𝑎 is𝑣 ∈ ℝ.Then,more

formally,wehave:

14Thusconsider theviewonwhichtheobjectivevalueofanoption𝑎 isgivenbytheconditional

chancesatsometimebeforethedecision.Supposethepredictionisat𝑡% ,andconsidertheconditional

chances taggedtosomeearlier𝑡X .Wecan formulateasimilaritymetric thatmakestheclosesta-

worldsto𝑤9 bethoseinthesetofworlds{𝑤:, 𝑤;, … ,𝑤)}thathavepositivechanceattime𝑡Xat𝑤9 ,

conditionalon𝑎.Tocalculatetheobjectivevalue,averagethevaluesoftheoutputtedworldsbytheir

respectivechancesconditionalon𝑎.Thus,accordingtotheproposal,theobjectivevalueof𝑎atworld

𝑤9 equals∑ 𝐶ℎZ[,\](𝑤|𝑎)𝑢(𝑤)8 .Ifthepredictorhasa99%chanceofcorrectnessbackat𝑡X ,thenthe

objectivevalueofone-boxingis(0)(0.01) +(1,000,000)(0.99) = 990,000,andtheobjectivevalue

oftwo-boxingis(1,000)(0.99) +(0.01)(1,001,000) = 11,000.

16

Expectationism:Foranycredencefunction𝐶,thesubjectivevalueofarelativeto𝐶

is∑ 𝑣𝐶(⟨𝑂(𝑎) = 𝑣⟩)^ .

Expectationism is widely accepted. Often it’s just assumed,15 but there are some

explicitdefensesofit.Forexample,onemightdefendExpectationismbyclaimingthatthe

subjectivevalueofanoptionshouldbetheagent’sbestestimateofitsobjectivevalue,and

thenarguingthattheagent’sbestestimateofaquantityistheirexpectationofit.16

Inthedisputebetweenone-boxersandtwo-boxers,Expectationismisneutral.Ifwe

combineExpectationismwiththeclaimthatobjectivevalueiscausalvalue,wegetCDT,since

thecausalexpectedvalueofanoptionistheagent’sexpectationofthecausalvalueofthe

option.17 But we can combine Expectationism with conceptions of objective value that

invalidateNewcombKnowledge.Forexample,ifwecombineExpectationismwiththeclaim

thatobjectivevalueisHorganvalue,thenwegetHorgandecisiontheory(HDT)—theview

thatthesubjectivevalueofanoptionistheagent’sexpectationoftheHorganvalueofthe

option.Whereastwo-boxinguniquelymaximizesexpectedcausalvalue(i.e.causalexpected

value),one-boxinguniquelymaximizesexpectedHorganvalue.

15Seee.g.Parfit1984:25.16ForadefenseofExpectationismalongtheselines,seee.g.OddieandMenzies1994andPettigrew

2015.17Proof:Let𝑂Z(𝑎)bethecausalvalueofoption𝑎atworld𝑤.Theexpectationofthecausalvalueof

𝑎relativetocredencefunctionCis∑ 𝐶(𝑤)𝑂Z(𝑎)8 .LetWibetheworldsatwhich⟨𝑎 ⇒ 𝑤9⟩istrue.

Then ∑ 𝐶(𝑤)𝑂Z(𝑎)8 = ∑ 𝐶(𝑤)𝑢(𝑤%) +⋯+ ∑ 𝐶(𝑤)𝑢(𝑤-)8̀8a = 𝐶(⟨𝑎 ⇒ 𝑤%⟩)𝑢(𝑤%) + ⋯+

𝐶(⟨𝑎 ⇒ 𝑤-⟩)𝑢(𝑤-) =∑ 𝐶(⟨𝑎 ⇒ 𝑤⟩)𝑢(𝑤) = 𝑈(𝑎)8 .

17

ButExpectationismishostiletoEDT,aswenowargue.

Everyremotelyplausibleconceptionofobjectivevaluemustallowanagenttoregard

aasevidenceaboutwhattheobjectivevalueofais.Takethesimplestcase,inwhichanagent

isuncertainwhethertheobjectivevalueofoption𝑎 is𝑣%or𝑣'.Everyremotelyplausible

conceptionofobjectivevaluemustvalidate:

Relevance:Itispossiblethatanagent’scredencesbesuchthat,forsome𝑣% ≠ 𝑣':

i. 𝐶(⟨𝑂(𝑎) = 𝑣%⟩ ∨ ⟨𝑂(𝑎) = 𝑣'⟩) = 1,

ii. 𝐶(⟨𝑂(𝑎) = 𝑣%⟩) = 𝑥 < 1,and

iii. 𝐶(⟨𝑂(𝑎) = 𝑣%⟩|𝑎) = 𝑦 ≠ 𝑥.

Relevancemustholdbecausethefactthatanagentregards𝑎asevidenceastowhether𝑝

cannotprecludetheobjectivevalueof𝑎fromdependingonwhether𝑝.

Tosee thismore concretely, consideravariationonMiners.On thisvariation, the

agentrememberstheminers’tellingherwhichshafttheywouldbeworkingin,butcannot

consciouslyrecallwhich.Asbefore,sheis50%confidentthattheminersareinshaftAand

50%confidentthattheyareinshaftB.Butshe(reasonably)thinksthatthereisanonzero

chancethatherunconsciousmemorywillinfluenceherchoiceifshechoosestoblockoneof

the shafts. Therefore, her confidence that the miners are in shaft A, conditional on her

blockingshaftA,is(say)52%,upfrom50%.

Sinceclauses(i)and(ii)ofRelevanceclearlyhold in thiscase,anyonewhodenies

RelevancewouldhavetoholdthattheobjectivevalueofblockingshaftAdoesnotdepend

onwheretheminersare.Theywouldhavetoholdthattheobjectivevalueofblockingshaft

18

AataworldatwhichtheminersareinshaftAisequaltotheobjectivevalueofblockingshaft

AataworldatwhichtheminersareinshaftB.Butthat’sabsurd.Onanyremotelyplausible

conceptionofobjectivevalue,theobjectivevalueofblockingshaftAdependsonwherethe

minersactuallyare,eveniftheagentregardsblockingshaftAasevidenceaboutwherethe

minersare.Thus,Relevancemusthold.18

Nowwecanprove:

Result#1.Relevance,Expectationism,andEDTarejointlyinconsistent.

The(verysimple)proofisinAppendixA.Informally,theideaisthatExpectationismmakes

thesubjectivevalueofanoptiondependonlyontheagent’sunconditionalcredencesinits

havingthisorthatobjectivevalue,whereasEDTmakesitturnonitsexpectedobjectivevalue

conditionalonitsperformance.RelevancethereforeimpliesthatEDTandExpectationismcan

disagree about the subjective value of an option. For instance, in the variant onMiners,

18Insayingthis,weareeffectivelysettingaside‘indexical’valueconceptsofthesortthatHájekand

Pettit(2004)discussinconnectionwithLewis’s(1988,1996)argumentsagainst‘DesireasBelief’

(DAB).Indexicalvaluesdependnotonlyonthestateofthemind-independentworldbutalsoonthe

beliefs of the agent herself.We agreewithHájek and Pettit that indexical value concepts evade

Lewis’sarguments.TheyalsoviolateRelevance.(Forinstance,iftheindexicalvalueofanoption𝑎is

just𝑉(𝑎),then,solongasconditionalizationcanchangetheevidentialexpectedvalueofthetautology

(seeBradleyandStefánsson2016:699-702),Relevanceasappliedtoindexicalvaluefails.)Butwe

denythatindexicalvalueisobjective.Thenotionofobjectivevaluethatinterestsusissuchthat,if

optionshaveobjectivevalues, theobjectivevalueof (say)blockingshaftA inMinersdependson

wheretheminersactuallyare,regardlessofwhattheagentthinks.(FormoreonDABseen.23.)

19

Expectationism implies that the subjective value of blocking shaft A is 5, whereas EDT

reckonsitat5.2.

TheargumentfromExpectationismexploitsResult#1.ItsaysthatEDTisinconsistent

withoptionshavingobjectivevaluebecause,ifoptionshaveobjectivevalues,Relevanceand

Expectationismarebothtrue.

The argument from Newcomb’s Problemwas too narrow. It showed that EDT is

inconsistentwithanyconceptionofobjectivevaluethatvalidatesNewcombKnowledge,but

was silentabout conceptionsofobjective thatdonotvalidateNewcombKnowledge.The

argumentfromExpectationism,bycontrast,purportstoestablishthatEDTisinconsistent

withevery(remotelyplausible)conceptionofobjectivevalue,eventhose,likeHorganvalue,

that invalidate Newcomb Knowledge. It might seem, then, that the argument from

Expectationismestablishesourthesis—thataproperunderstandingofthemathematical

relationshipbetweensubjectivevalueandobjectivevaluerevealsthatEDTisinconsistent

withoptionshavingobjectivevalues.ButtheargumentfromExpectationismcanberesisted.

§6/ResistingExpectationism

EveryremotelyplausibleconceptionofobjectivevaluevalidatesRelevance,sotheargument

fromExpectationismestablishesthatEDTandExpectationismareinconsistent.Evidential

expectedvalueisnotexpectedobjectivevalue.CDT’istsandHDT’istsagreethatsubjective

valueisexpectedobjectivevalueanddisagreeaboutwhatobjectivevalueis,butEDT’istsdo

notshareinthisagreement.NoconceptionofobjectivevaluestandstoEDTascausalvalue

standstoCDT.

20

ButExpectationismcanbequestioned.Ifoptionshavebothobjectiveandsubjective

values,thentheremustbesomewell-behaved,intimaterelationshipbetweenthesubjective

valueofanoptionrelativetoanagent’scredencefunctionandtheagent’shypothesesabout

itsobjectivevalue.Butthisrelationshipneedn’tbeexpectation.

Aminimalnecessaryconditionfortherelationshipbetweenobjectiveandsubjective

valuebeingwell-behavedandintimateisthetruthofthefollowingprinciple:

Certain Reflection: For any credence function 𝐶, if 𝐶(⟨𝑂(𝑎) = 𝑣⟩) = 1, then the

subjectivevalueofoption𝑎relativeto𝐶equals𝑣.

ButCertainReflectionisstrictlyweakerthanExpectationism,aswecanseebyconsidering

someviewsthatverifyCertainReflectionwhilefalsifyingExpectationism.

FirstconsiderMaximin—theviewthatthesubjectivevalueof𝑎relativetoanagent’s

credencefunctionistheleast𝑣suchthattheagentassignsnonzerocredenceto⟨𝑂(𝑎) = 𝑣⟩.

MaximinsatisfiesCertainReflection,butitfalsifiesExpectationism.

Next,considerRisk-adjustedExpectationism—theviewthatthesubjectivevalueof𝑎

isnotthestraightexpectationofobjectivevaluebutratheraweightedsumofitspossible

objectivevaluesthatattachesmoreimportancetosomeofthesepossibilitiesthantoothers

dependingontheirrankandnotonlyontheirprobability: for instance, itweightsworse

possibilities more heavily than better ones, other things being equal.19 On most such

weightings, the view that subjective value is risk-adjusted expectation ofobjective value

19Buchak2013ch.2.

21

falsifiesExpectationism,butagainitsatisfiesCertainReflection,becauseifyouarecertainof

whattheobjectivevalueofanoptionis,thentherearenoalternativehypothesesaboutthis

towhichyoucangivemoreorlessweightdependingontheirrank.20

Athirdalternative,whichisamenabletoEDT,takessubjectivevaluetobeconditional

expectedobjectivevalue.Wecallthis:

ConditionalExpectationism:Foranycredencefunction𝐶,thesubjectivevalueofa

relativeto𝐶is∑ 𝑣𝐶(⟨𝑂(𝑎) = 𝑣⟩|𝑎)^ .21

ConditionalExpectationismentailsCertainReflection. AndgivenCertainReflection,EDT

entailsConditionalExpectationism.22

20Wecanrealizethisideaformallybymeansofadistortioni.e.anon-decreasingfunction𝑟: [0,1] →

[0,1] such that 𝑟(0) = 0, 𝑟(1) = 1. For any option𝑎, arrange its epistemically possible objective

valuesinincreasingorder𝑥%, 𝑥', … , 𝑥-.ThecorrespondingversionofRisk-adjustedExpectationism

(RE) says that relative to a credence function 𝐶 the subjective value of 𝑎 is 𝜎(𝑎) = 𝑥% +

∑ (𝑥9l% − 𝑥9)𝑟H𝐶(⟨𝑂(𝑎) ≥ 𝑥9l%⟩)I-o%9p% .If𝑟istheidentityfunctioni.e.𝑟(𝑥) = 𝑥thensubjectivevalue

coincides with expected objective value; but if 𝑟 is convex to the 𝑥-axis, e.g. if 𝑟(𝑥) = 𝑥', then

subjective value weights worse possibilities more heavily than expectationism demands. But

trivially,itsatisfiesCertainReflection.21Oddie1994:460.AlsoseeBroome’s(1991)discussionofConditionalExpectationism(whichhe

somewhatmisleadinglycallsDesire-as-Expectation).22Proof:Let𝑤9 beanyworldatwhich𝑎and⟨𝑂(𝑎) = 𝑣⟩arebothtrue,andlet𝐶concentrateallofits

credenceon𝑤9 .Then,byCertainReflection,thesubjectivevalueof𝑎relativeto𝐶is𝑣.And,byEDT,

thesubjectivevalueof𝑎relativeto𝐶is∑ 𝐶(𝑤|𝑎)𝑢(𝑤8 ) = 𝑢(𝑤9).So𝑢(𝑤9) = 𝑣.ThismeansthatEDT

entailsConditional Expectationism, since,∑ 𝐶(𝑤|𝑎)𝑢(𝑤) = ∑ 𝐶(𝑤|𝑎)𝑢(𝑤) +⋯ =Z∈⟨q(r)p^a⟩∩rZ∈8

∑ 𝐶(𝑤|𝑎)𝑣% + ⋯ = ∑ 𝑣𝐶(⟨𝑂(𝑎) = 𝑣⟩|𝑎)^Z∈⟨q(r)p^a⟩ .

22

There is something intuitiveaboutExpectationism,which says that the subjective

valueofanoptionshouldbetheagent’sestimateofitsobjectivevalue,andsomeonewho

acceptsConditionalExpectationismmustrejectExpectationism,since,givenRelevance,the

two claims cannot both be true. But there is also something intuitive about Conditional

Expectationism,which says that the subjective value of an option should be the agent’s

estimateofitsobjectivevalueinworldswhereitisrealized.Thetheoreticalcostofrejecting

ExpectationisminfavorofConditionalExpectationismthusseemstouslow.Andthereisno

argumentfromConditionalExpectationism.WhereasRelevance,Expectationism,andEDT

arejointlyinconsistent,Relevance,ConditionalExpectationism,andEDTareconsistent.

TheassumptionthatEDT’istsmustacceptExpectationismisunjustified.(Evenamong

opponentsofEDT,Expectationismisnotcommonground.)So,withoutsomeindependent

argumentthatEDT’istsmustacceptExpectationism,theargumentfromExpectationismfails

toestablishourthesis.23

23WeshouldbrieflyrelatethepresentdiscussiontoLewis’s(1986,1988)argumentagainsttheanti-

Humean‘DesireasBelief’(DAB)thesis,towhichtheargumentfromExpectationismbearsanobvious

resemblance.ThebasicideabehindLewis’sargumentisthatifevidentialexpectedvalue𝑉measures

theagent’sdesire for the truthofaproposition, thenDABsays that𝑉(𝐴) = 𝐶H�̇�I,where �̇� isthe

propositionthatitisgoodthat𝐴.(Lewis’sproofinvolvesanungradednotionofgoodnessbutcould

easily be extended to cover a graded notion, as the argument fromExpectationism does.) Given

Lewis’sassumptionthat𝑉(𝐴|𝐴) = 𝑉(𝐴) (whichhasbeenquestioned:seeBradleyandStefánsson

2016:699-702)itfollowsfromDABthat𝐶H�̇�P𝐴I = 𝐶H�̇�I;butthisisinconsistentwiththeanalogue

of Relevance that Lewis implicitly assumes (1996: 309). Oneway for the anti-Humean to resist

Lewis’sargument(Price1989:122)wouldbetoreformulatetheanti-Humeanthesisas‘Desireas

ConditionalBelief’(DACB),whichsaysthat𝑉(𝐴) = 𝐶H�̇�P𝐴I.ThisthesisgivesnotractiontoLewis’s

argument,forjustthesamereasonthatConditionalExpectationismgivesnonetotheargumentfrom

Expectationism. In response, Lewis argues (1996: 310-11) that DACB is a version of desire by

23

§7/TheArgumentfromNewcombizability

ThethirdpurportedargumentforourthesisistheargumentfromNewcombizability.

ItstartsfromaweakeningofBridge.Aswesaid,Bridgeseemsundeniable.Ifoptions

haveobjectivevaluesthen:ifanagent’soptionsare𝑎%, 𝑎', … , 𝑎-,andtheagentknowsfor

certain that option 𝑎9 uniquely maximizes objective value, then 𝑎9 uniquely maximizes

subjective value. But the subjective values of options relative to a credence function

supervene on the agent’s credences and theworld valuations.We therefore canweaken

Bridge, appealing to any of the agent’s certainties, and not just those that constitute

knowledge.Theresultis:

Dominance. If options have objective values then: if an agent’s options are

𝑎%, 𝑎', … , 𝑎- , and the agent is certain that option𝑎9 uniquelymaximizes objective

value,then𝑎9 uniquelymaximizessubjectivevalue.

necessity: it is committed to the existence of aproposition𝐺 which the agent desires to be true

whateverhercredences.WearenotclearwhythispointhasanyforceagainstDACB.Maybetheidea

isthattheanti-Humeanthesisisadescriptivepsychologicalthesisaboutaperson’sdesires,andthat

itissimplyfalseasamatteroffactthatthereisanythingthateveryonedesires(cf.Lewis1996:304-

5). However this may be, we are clear enough that no analogous point threatens Conditional

Expectationism. After all, the latter is a normative thesis, because of the connection between

subjectivevalueandwhatonesubjectivelyoughttodo.Evenifthereisnothingthateveryonedoes

value,theremightbesomethingthateveryoneshouldvalue.Inshort,wethink:(a)thatthesame

objectionarisesagainstbothLewis’sargumentandtheargumentfromExpectationism;and(b)that

eveniftheformersurvivesit,thelatterdoesnot.

24

DominanceislikeCertainReflection.CertainReflectionensuresthatsubjectivevalue

conformstoobjectivevaluenumerically—itsaysthat,ifanagentiscertainthattheobjective

valueofanoptionequals𝑣, then,relativeto theagent’scredence function, thesubjective

value of the option also equals𝑣. Dominance ensures that subjective value conforms to

objective valueordinally— it says that, if an agent is certain that some optionuniquely

maximizesobjectivevalue, then,relativeto theagent’scredence function, theoptionalso

uniquelymaximizessubjectivevalue.

Dominance is entailed bymany views about how objective and subjective values

relate, including Minimax, Risk-adjusted Expectationism, CDT, HDT, and any form of

Expectationism.24ButDominanceishostiletoEDT.ThereisageneralizationofRelevance—

24ProofthatExpectationismimpliesdominance:let𝑂Z(𝑎)betheobjectivevalueofoption𝑎atworld

w.If𝐶(⟨𝑂(𝑎) > 𝑂(𝑏)⟩) = 1andExpectationismistrue,then,relativeto𝐶,thedifferencebetweenthe

subjectivevalueofoption𝑎andthesubjectivevalueofoption𝑏equals∑ 𝐶(𝑤)(𝑂Z(𝑎) − 𝑂Z(𝑏))8 .

Sincetheagentiscertainthattheobjectivevalueof𝑎exceedsthatof𝑏,atanyworld𝑤towhich𝐶

assignsnonzerocredence,𝑂Z(𝑎) − 𝑂Z(𝑏)ispositive.Hence,∑ 𝐶(𝑤)(𝑂Z(𝑎) − 𝑂Z(𝑏))8 ispositive,

whichentailsthat,relativeto𝐶,thesubjectivevalueofoption𝑎exceedsthesubjectivevalueof𝑏.

Proofthatmaximinimpliesdominance:let𝑎 = min{𝑥|𝐶(⟨𝑂(𝑎) = 𝑥⟩) > 0}.If𝑎 ≤ 𝑏then𝐶H⟨𝑂(𝑎) ≤

𝑏⟩I > 0, so 𝐶(⟨𝑂(𝑎) ≤ 𝑂(𝑏)⟩ > 0. Contrapositively, 𝐶(⟨𝑂(𝑎) > 𝑂(𝑏)⟩) = 1 implies 𝑎 > 𝑏, so the

maximinsubjectivevalueof𝑎exceeds thatof𝑏.For theproof thatRisk-adjustedExpectationism

implies dominance, see Buchak 2013: 245-6. Note that this proof assumes that the distortion

function𝑟isstrictlyincreasing.Butevenifweassumeonlythat𝑟isnon-decreasingwecanstillprove

thefollowingweakeneddominanceprinciple:ifanagent’soptionsare𝑎%, 𝑎',… , 𝑎-,andtheagentis

certainthatoption𝑎9uniquelymaximizesobjectivevalue,thennootheroptionuniquelymaximizes

subjective value. The argument from Newcombizability still goes through on this weakening of

Dominance.

25

a principle that is strictly stronger than Relevance, but no less obviously true — that

contradictstheconjunctionofEDTandDominance.

Wewillbuilduptotherelevantprinciple in twostages.Tostart,supposethat the

agentiscertainthattheobjectivevalueofoption𝑎%exceedstheobjectivevalueofoption𝑎'

bysomedefinitepositivemargin𝑧,butisuncertainwhethertheobjectivevaluesof𝑎%and

𝑎'equal𝑣%and𝑣' = 𝑣% − 𝑧,respectively,orinsteadequal𝑣Land𝑣M = 𝑣L − 𝑧,respectively,

where𝑣% − 𝑣L > 𝑧.Anyremotelyplausibleconceptionofobjectivevaluemustvalidate:

BaselineRelevance.Itispossiblefortheagent’scredencestobesuchthat:

i. 𝐶(⟨𝑂(𝑎%) = 𝑣%⟩ ∨ ⟨𝑂(𝑎%) = 𝑣L⟩) = 1,

ii. 𝐶(⟨𝑂(𝑎%) = 𝑣%⟩|𝑎%) = 𝑥 < 1,

iii. 𝐶(⟨𝑂(𝑎%) = 𝑂(𝑎') + 𝑧⟩) = 1,and

iv. 𝐶(⟨𝑂(𝑎%) = 𝑣%⟩|𝑎') = 𝑦 ≠ 𝑥.

The rationale behind Baseline Relevance is the same as that behind Relevance. Baseline

Relevanceholdsbecauseanyremotelyplausibleconceptionofobjectivevaluemustallowan

agenttoregard𝑎%asevidenceaboutwhattheobjectivevalueof𝑎% is,eveniftheagentis

certainthattheobjectivevalueof𝑎%exceedstheobjectivevalueofsomeotheroption𝑎'by

somemargin𝑧.

IfBaselineRelevanceholdsofeveryremotelyplausibleconceptionofobjectivevalue,

thensotoodoesavariantinwhichclause(iv)isstrongerthanabareinequality.Usingthe

samenotation,theprincipleis:

26

Newcombizability.Itispossiblefortheagent’scredencestosatisfy:

i. 𝐶(⟨𝑂(𝑎%) = 𝑣%⟩ ∨ ⟨𝑂(𝑎%) = 𝑣L⟩) = 1,

ii. 𝐶(⟨𝑂(𝑎%) = 𝑣%⟩|𝑎%) = 𝑥 < 1,

iii. 𝐶(⟨𝑂(𝑎%) = 𝑂(𝑎') + 𝑧⟩) = 1,and

iv. 𝐶(⟨𝑂(𝑎%) = 𝑣%⟩|𝑎') = 𝑦 > 𝑥 + y^ao^z

.

Asfaraswecansee,almostanyconceptionofobjectivevalueisNewcombizable.25

Indeed,thereseemstobeageneralrecipeforNewcombizing.Let𝑂beanyconceptionof

objectivevalue.Ifthereareoptions𝑎%and𝑎',thentherearepropositions𝑆% =def. ⟨𝑂(𝑎%) =

𝑣% ∧ 𝑂(𝑎') = 𝑣% − 𝑧⟩ and 𝑆' =def. ⟨𝑂(𝑎%) = 𝑣L ∧ 𝑂(𝑎') = 𝑣L − 𝑧⟩. We can therefore

constructadecisionprobleminwhichthepayoffsareasfollows:

𝑺𝟏 𝑺𝟐

𝒂𝟏 𝑣% 𝑣L

𝒂𝟐 𝑣' = 𝑣% − 𝑧 𝑣M = 𝑣L − 𝑧

Table1

25Thereasonforthequalification‘almost’isjustthis.(i)Weareassumingthattheobjectivevalue

conceptimpliesthatobjectivevaluetakesontheappropriatevaluesforsomeoptions𝑎%and𝑎'in

somepossibleworlds.Thus,forexample,wedisregardanyconceptionofobjectivevalueonwhich

every option has the same objective value at every world. (ii)We are assuming that any given

distributionofobjectivevaluesofoptionsacrosspossibleworldsisconsistentwithanydistribution

ofcredencesacrossthoseworlds.Thisdoesnotseemcontentioustous,sincetheobjectivevalueof

options are at least sometimes totally insensitive to changes in the agent’s credences. See the

correspondingdiscussionofRelevanceatn.18.

27

Weconstructacredencefunction𝐶ontheatomsF𝑎9𝑆:G9,:p%,'asfollows.Let𝑥 =%'�1 − y

^ao^z�.

Let 𝑦 = %M�3 + y

^ao^z�. Choose an arbitrary 𝑘, 0 < 𝑘 < 1. Let 𝐶(𝑎%𝑠%) = 𝑥𝑘, 𝐶(𝑎%𝑠') =

(1 − 𝑥)𝑘,𝐶(𝑎'𝑠%) = 𝑦(1 − 𝑘)and𝐶(𝑎'𝑠') = (1 − 𝑦)(1 − 𝑘).Thenitiseasytocheckthat𝐶

satisfiesallofclauses(i)-(iv)intheNewcombizabilitycondition.Ifoneneedsaback-story,

imaginethatthemechanismthattypicallycausesonetochoose𝑎%alsoandindependently

tendstopromoteastate𝑆'inwhichbothoptionspossesslessofwhicheverkindofobjective

valueisatissue.26

26Asaconcreteillustrationofthisgeneralresult,notethatHorganvalueisNewcombizable.Suppose

that youhave a choicebetween takingbox1 andbox2, bothboxesbeingopaque.Takingbox1

releases2unitsofwelfare;takingbox2doesnothing.ButaNewcombianpredictorhaswrittendown

whatheexpectsyourchoicetobe;andifyoumanagetooutwitthepredictor,yougetabonusof8

unitsofwelfare.Moreover,youknowthatthepredictorissomewhatbetteratpredictingpeoplewho

choosebox1(successrateof0.625)thanatpredictingpeoplewhochoosebox2(successrateof

0.1875).Thisneednotbebecauseyouareantecedentlyandrobustlyconfidentthatthepredictionis

thatyouwillchoosebox1;itcouldbethatactivationofthepartofyourbrainthatinclinesyouto

choosebox2interfereswiththepredictor’sbrain-scanningdevice.(Forasimilarexampleseethe

‘Semi-Frustrater’probleminSpencerandWellsforthcoming.)

ThiscasesatisfiesthefourclausesofNewcombizability.(i)youarecertainthattheHorgan

valueof(𝑎%)takingbox1iseither𝑣% = 10(ifthepredictorisactuallyinaccurate)or𝑣L = 2(ifthe

predictorisactuallyaccurate).Tospelloutwhy:on theHorganresolutionofcounterfactuals, the

closest𝑎%-worldtoactuality, call it𝑤%, isonewhere thepredictor isaccurate ifandonly ifhe is

actuallyaccurate.Soifthepredictorisactuallyinaccuratethen𝑢(𝑤%) = 10;otherwise𝑢(𝑤%) = 2.So

certainly𝑂(𝑎%) = 10 ∨ 𝑂(𝑎%) = 2.(ii)YourconfidencethattheHorganvalueoftakingbox1is10,

giventhatyoutakebox1, is𝑥 = 0.375.(iii)YouarecertainthattheHorganvalueoftakingbox1

exceedstheHorganvalueof(𝑎')takingbox2by𝑧 = 2 units,becauseifthepredictorisactually

accuratethentheformeris2andthelatteriszero,andifthepredictorisinaccuratethentheformer

is10andthelatteris8.(iv)YourconfidencethattheHorganvalueoftakingbox1is10,giventhat

28

If every remotely plausible conception of objective value is Newcombizable, then

DominanceandEDTcannotbothbetrue,because,asweproveinAppendixB:

Result#2.Newcombizability,Dominance,andEDTarejointlyinconsistent.

Theintuitiveideabehindthisisasfollows.Whateverobjectivevalueis,wecanconstructa

casewhereanoption𝑎%hasmoreofitthananotheroption𝑎'ateverypossibleworld;but

thedominatedoption,𝑎',isverygoodevidencethatbothoptionshavehighobjectivevalue.

Dominancethereforedemandsthatthesubjectivevalueof𝑎%exceedsthatof𝑎';butEDT

impliestheopposite.

The argument fromNewcombizability follows fromResult #2. It says that EDT is

inconsistentwithoptionshavingobjectivevaluesbecause,ifoptionshaveobjectivevalues,

NewcombizabilityandDominancearetrue.

The argument from Expectationism can be resisted by retreating from

Expectationism to Conditional Expectationism, but no analogous ‘conditionalizing’

maneuverhelpsagainsttheargumentfromNewcombizability.Toseethis,considerafew

proposals.

TheobviousfirstproposalistorejectDominanceinfavorof:

you take box 2, is 𝑦 = 0.8125 > 𝑥 + y

^ao^z= 0.625. So the case represents a Newcombization of

Horganvalue.

29

ConditionalDominance1:Ifoptionshaveobjectivevaluesthen:ifanagent’soptions

are𝑎%, 𝑎',… , 𝑎-,andtheagentiscertainthatoption𝑎9 uniquelymaximizesobjective

valuegiventhat𝑎9 isrealized,thenoption𝑎9 uniquelymaximizessubjectivevalue.27

Butthisgetsusnofurther,becauseConditionalDominance1entailsDominance.Ifanagent

iscertainthat𝑎9 uniquelymaximizesobjectivevalue,thentheagentisalsocertainthat𝑎9

uniquely maximizes objective value given that 𝑎9 is realized. So, Newcombizability,

ConditionalDominance1,andEDTarejointlyinconsistent.

AsecondproposalmightbetorejectDominanceinfavorof:

ConditionalDominance2:Ifoptionshaveobjectivevaluesthen:ifanagent’soptions

are 𝑎%, 𝑎',… , 𝑎-, and the agent is certain that: the objective value of option 𝑎9

conditional on its realization exceeds the objective value of any other option 𝑎;

conditionalonitsrealization,thenoption𝑎9 uniquelymaximizessubjectivevalue.28

Conditional Dominance 2 is not inconsistent with the conjunction of EDT and

Newcombizability,butthat’sbecauseitsaysnothingmeaningfulatall.Theobjectivevalueof

anoptionisnothadrelativetoacredencefunction,sothereisnosuchthingastheobjective

valueofanoptionconditionalonitsrealization(orconditionalonanything).

27Formally,wecanwritetheconsequentofConditionalDominance1:𝐶(⋀ ⟨𝑂(𝑎9) > 𝑂(𝑎;)⟩;�9 |𝑎9) =

1 → ⋀ 𝜎(𝑎9) > 𝜎(𝑎;);�9 .28 Formally, we might try writing the consequent of Conditional Dominance 2 as follows:

𝐶(⋀ ⟨𝑂(𝑎9|𝑎9) > 𝑂(𝑎;|𝑎;)⟩;�9 ) = 1 → ⋀ 𝜎(𝑎9) > 𝜎(𝑎;);�9 .

30

Onecouldtrytogive‘conditionalobjectivevalue’anobjectivemeaning,perhapsby

explicatingitasacounterfactualconditionalrelativetosomesimilaritymetric.Theobjective

valueofanoptionconditionalonitsrealizationthenwouldbetheobjectivevaluethatthe

optionwouldhavewereittoberealized.Thiswouldgiveus:

ConditionalDominance3:Ifoptionshaveobjectivevaluesthen:ifanagent’soptions

are𝑎%, 𝑎',… , 𝑎-,andtheagentiscertainthat:theobjectivevalueofoption𝑎9 wereit

toberealizedexceedstheobjectivevalueofanyotheroption𝑎;wereittoberealized,

thenoption𝑎9 uniquelymaximizessubjectivevalue.29

Butthisjustrearrangesthedeck-chairs.If𝑂isaconceptionofobjectivevalue,thenthe𝑂-

valueofanoptionwere it toberealized is justanother conceptionofobjectivevalue,and

ConditionalDominance3isjustDominanceassertedaboutit.Thisalternativeconceptionof

objectivevalueis,likeanyremotelyplausibleconception,Newcombizable,sonoprogressis

made.Newcombizability,ConditionalDominance3,andEDTarejointlyinconsistent.

ThereisnowayaroundDominance,justasthereisnowayaroundCertainReflection.

To reject either is to reject thewhole idea of objective values—objective properties of

options to which subjective values conform. Anyone who rejects Certain Reflection or

Dominance rejects the idea that differences between objective and subjective values are

alwaysmereartifactsoftheagent’suncertaintyaboutobjectivevalues.

29 Formally, we might write the consequent of Conditional Dominance 3 as: 𝐶H∀𝑛⋀ : (𝑎9 ⇒;�9

⟨𝑂(𝑎9) = 𝑛⟩) → (𝑎; ⇒ ⟨𝑂(𝑎;) < 𝑛⟩)I = 1 → ⋀ 𝜎(𝑎9) > 𝜎(𝑎;);�9 , where ⇒ is the selected

counterfactualoperator.

31

WhattheargumentfromNewcombizabilityrevealsisthereal,metaethicallessonof

Newcomb’sProblem.InthefiftyyearssinceNozickintroducedtheproblem,therehasbeen

much ado about causation. Over and again, opponents of EDT make the same causal

observations:theagentfacingNewcomb’sProblemhasnocontroloverthecontentsofthe

opaquebox;theamountofmoneycontainedintheopaqueboxatthecausallyclosestone-

boxingworld is likewise contained in the opaque box in the causally closest two-boxing

world; the agent is in a position to know for certain that the causal valueof two-boxing

exceedsthecausalvalueofone-boxing,andsoon.Theseobservationssuggestthatthelesson

ofNewcomb’sProblemhassomethingessentiallytodowithcausation.

Butitdoesn’t.Causallanguageisusedbecauseitispresupposedthatobjectivevalue

iscausalvalue,butthemetaethicallessonofNewcomb’sProblemconcernstherelationship

betweenobjectivevalueandsubjectivevalue,onanyplausibleconceptionofobjectivevalue.

Takeanyremotelyplausibleconceptionofobjectivevalue,beitcausalorwhollynoncausal.

ThatconceptionofobjectivevaluewillbeNewcombizable.TherewillbeacasewhereEDT

recommends an option that the agent knows for certain to be objectivelyworse on that

conceptionthantheonlyalternative.TheEDT’ist’sclaimthatsubjectivevalueisevidential

expected value thuswill be inconsistentwith the claim that that conception of objective

valueistrue.

Maximin, Risk-adjusted Expectationism, and the various form of Expectationism,

including CDT and HDT, are consistent with options having objective values. Indeed,

arguably these theories are defensible only if options have objective values. But EDT is

metaethicallyverydifferent.EDTisnotconsistentwithoptionshavingobjectivevalues.

32

§8/Conclusion

Pastthispoint,theauthorsofthispaperpartways.WeagreethatEDTisinconsistentwith

optionshavingobjectivevalue,butwedon’tagreeaboutwhattomakeofthatfact.Oneusof

hasantecedentcommitmentstooptionshavingobjectivevalues,soisinclinedtotakethe

inconsistency to amount to a metaethical refutation of EDT. The other has antecedent

commitments toEDT,so is inclined totake the inconsistency toamount toametaethical

refutationoftheclaimthatoptionshaveobjectivevalues.Obviously,wecannotsettlehere

whethertorejectEDTortheclaimthatoptionshaveobjectivevalue.Butoneofthemhasgot

togo.

33

AppendixA

HereweproveResult#1:thatRelevance,Expectationism,andEDTarejointlyinconsistent.

Start from a possible case thatwitnesses the truth of Relevance. Then, according to the

agent’s credences, 𝐶(⟨𝑂(𝑎) = 𝑣%⟩ ∨ ⟨𝑂(𝑎) = 𝑣'⟩) = 1, 𝐶(⟨𝑂(𝑎) = 𝑣%⟩) = 𝑥 < 1, and

𝐶(⟨𝑂(𝑎) = 𝑣%⟩|𝑎) = 𝑦 ≠ 𝑥.Theagent’sexpectationoftheobjectivevalueofoption𝑎is:

∑ 𝐶(⟨𝑂(𝑎) = 𝑣⟩)𝑣� = 𝑥𝑣% + (1 − 𝑥)𝑣' = 𝑥(𝑣% − 𝑣') +𝑣'.

TheV-valueof𝑎is:

∑ 𝐶(⟨𝑂(𝑎) = 𝑣⟩|𝑎)𝑣� = 𝑦𝑣% + (1 − 𝑦)𝑣' = 𝑦(𝑣% −𝑣') + 𝑣'.

IfExpectationismistrue,thesubjectivevalueof𝑎relativetoCisequaltotheexpectationof

theobjectivevalueof𝑎relativeto𝐶.Hence:

𝑥(𝑣% − 𝑣') +𝑣' = 𝑦(𝑣% −𝑣') + 𝑣'.

But 𝑣% ≠ 𝑣' implies that 𝑥(𝑣% − 𝑣') +𝑣' = 𝑦(𝑣% −𝑣') + 𝑣' only if 𝑥 = 𝑦, and 𝑥 ≠ 𝑦.

Therefore,EDT,Relevance,andExpectationismcannotallbetrue.QED.

Note thatExpectationism,aswehave characterized it, is a thesis aboutnumerical

values.Itsaysthatthenumericalrepresentationofthesubjectivevalueofanoptionrelative

toa credence function (on somearbitrarily chosen scale)mustbeara certain arithmetic

34

relationtothenumericalrepresentationoftheobjectivevalueoftheoption(onthatsame

scale).Butonemightbeconcernedwithamoregeneral,purelynormativethesis:namely,

NormativeExpectationism.Agentsalwayssubjectivelyoughttomaximizeexpected

objectivevalue.

TheproofabovedoesnotestablishthatEDT,Relevance,andNormativeExpectationismare

inconsistent, but the inconsistency between these three claims can now be proven very

simply. Just imagine a case which, relative to some arbitrary determination of a scale,

satisfiesRelevance, and supposewithout lossof generality that𝑣% > 𝑣' and𝑥 > 𝑦. Then

imagineachoicebetweenoptions𝑎and𝑏,where𝑎isasdescribedaboveand𝑏isalottery

withchance �l�'ofrealizinganoutcomewithobjectivevalue𝑣%andchance1 − �

�l�'�of

realizinganoutcomewithobjectivevalue𝑣'.NormativeExpectationism implies that the

agentsubjectivelyoughttorealize𝑎,butEDTimpliesthattheagentsubjectivelyoughtto

realize𝑏.

AppendixB

Here we prove Result #2: that Newcombizability, Dominance, and EDT are jointly

inconsistent.StartfromacasethatwitnessesthetruthofNewcombizability.Then,according

totheagent’scredences:𝐶(⟨𝑂(𝑎%) = 𝑣%⟩ ∨ ⟨𝑂(𝑎%) = 𝑣L⟩) = 1,𝐶(⟨𝑂(𝑎%) = 𝑣%⟩|𝑎%) = 𝑥 < 1,

𝐶(⟨𝑂(𝑎%) = 𝑂(𝑎') + 𝑧⟩) = 1, and 𝐶(⟨𝑂(𝑎%) = 𝑣%⟩|𝑎') = 𝑦 > 𝑥 + y^ao^z

. Then since EDT

entailsConditionalExpectationism(seen.22),theV-valueofoption𝑎%equals:

35

∑ 𝐶(⟨𝑂(𝑎%) = 𝑣⟩|𝑎%)𝑣� = 𝑥𝑣% + (1 − 𝑥)𝑣L = 𝑥(𝑣% − 𝑣L) + 𝑣L.

TheV-valueofoption𝑎'equals:

∑ 𝐶(⟨𝑂(𝑎') = 𝑣⟩|𝑎')𝑣� = 𝑦𝑣' + (1 − 𝑦)𝑣M = 𝑦(𝑣% − 𝑧) + (1 − 𝑦)(𝑣L − 𝑧) =

𝑦(𝑣% − 𝑣L) + 𝑣L − 𝑧.

Andsince𝑦 > 𝑥 + y^ao^z

,

𝑦(𝑣% − 𝑣L) + 𝑣L − 𝑧 > �𝑥 + y^ao^z

� (𝑣% − 𝑣L) + 𝑣L − 𝑧 = 𝑥(𝑣% − 𝑣L) + 𝑣L.

Thus,ifEDTistrue,thesubjectivevalueof𝑎'relativetoCexceedsthesubjectivevalueof𝑎%

relativetoC.

Buttheagentiscertainthattheobjectivevalueof𝑎%exceedstheobjectivevalueof

𝑎'.So,ifDominanceistrue,thesubjectivevalueof𝑎%relativetoCexceedsthatof𝑎'relative

toC.Therefore,Newcombizability,EDT,andDominancecannotallbetrue.QED.

36

References

Ahmed,A.2014.Evidence,DecisionandCausality.Cambridge:CambridgeUniversityPress.

Buchak,L.2013.RiskandRationality.Oxford:OxfordUniversityPress.

Bradley,R.andH.O.Stefánsson.2016.Desire,ExpectationandInvariance.Mind125:691-

725.

Broome,J.1991.Desire,BeliefandExpectation.Mind100:265-7.

Dreier,J.1993.StructuresofNormativeTheories.Monist76:22-40.

———. 2011. In Defense of Consequentializing. In Timmons, M. (ed.),Oxford Studies in

NormativeEthics,Vol.1.Oxford:OxfordUniversityPress:97-119.

Hájek,A.2015.OnthePluralityofLewis’sTrivialityResults. InLoewer,B.andJ.Schaffer

(ed.),ACompaniontoDavidLewis.Oxford:Blackwell:425-45.

———andP.Pettit.2004.DesireBeyondBelief.AustralasianJournalofPhilosophy82:77-

92.

Hare,C.2010.TaketheSugar.Analysis70:237-47.

———.2016.ShouldWeWishWelltoAll?PhilosophicalReview125:451-72.

Horgan,T.1981.CounterfactualsandNewcomb’sProblem.JournalofPhilosophy78:331-56.

Jackson, F. 1991. Decision-Theoretic Consequentialism and the Nearest and Dearest

Objection.Ethics101:461-88.

Jeffrey,R.C.1983.TheLogicofDecision.Seconded.Chicago:UniversityofChicagoPress.

Joyce,J.1999.TheFoundationsofCausalDecisionTheory.Cambridge:CambridgeUniversity

Press.

Kolodny,N.andJ.MacFarlane.2010.IfsandOughts.JournalofPhilosophy107:115-43.

Lewis,D.K.1973.Counterfactuals.Oxford:Blackwell.

37

———.1979.CounterfactualDependenceandTime’sArrow.Noȗs13:455-76.Reprintedin

hisPhilosophicalPapersVolumeII(1986).Oxford:OxfordUniversityPress.

———.1981.CausalDecisionTheory.AustralasianJournalofPhilosophy59:5-30.

———.1988.DesireasBelief.Mind97:323-32.

———.1996.DesireasBeliefII.Mind105:303-13.

Louise,J.2004.RelativityofValueandtheConsequentialistUmbrella.PhilosophicalQuarterly

54:518-36.

McDermott,M.1999.CounterfactualsandAccessPoints.Mind108:291-334.

Mackie,J.L.1977.Ethics:InventingRightandWrong.London:PenguinBooks.

Moore,G.E.1903.PrincipiaEthica.Cambridge:CambridgeUniversityPress.

Oddie,G.1994.Harmony,Purity,Truth.Mind103:451-72.

———andP.Menzies.1992.AnObjectivist’sGuidetoSubjectiveValueEthics102:512-33.

Parfit,D.1984.ReasonsandPersons.Oxford:OxfordUniversityPress.

———.Unpublished.WhatWeTogetherDo.

http://individual.utoronto.ca/stafforini/parfit/parfit_-_what_we_together_do.pdf

Pettigrew, R. 2015. Risk, Rationality, and Expected Utility Theory Canadian Journal of

Philosophy45:796-826.

Portmore,D.W.2007.ConsequentializingMoralTheories.PacificPhilosophicalQuarterly88:

39-73.

Price,H.1989.DefendingDesire-as-Belief.Mind98:119-27.

Regan,D.1980.UtilitarianismandCooperation.Oxford:OxfordUniversityPress.

Spencer, J. and I. Wells. Forthcoming. Why Take Both Boxes? Philosophy and

PhenomenologicalResearch.

38

Thomson,J.J.2008.Normativity.Peru,Illinois:OpenCourt.

Zimmerman,M.2008.LivingWithUncertainty.Cambridge:CambridgeUniversityPress.