Fitness Beats Truth in the Evolution of Perception

FitnessBeatsTruthintheEvolutionofPerception1

2

ChetanPrakasha*,KyleD.Stephensb,DonaldD.Hoffmanb,3

ManishSinghc,ChrisFieldsd4

aDepartmentofMathematics,CaliforniaStateUniversity,SanBernardino,CA92407;5

bDepartmentofCognitiveSciences,UniversityofCalifornia,Irvine6

cDepartmentofPsychologyandCenterforCognitiveScience,RutgersUniversity,NewBrunswick,7

d243WestSpainStreet,Sonoma,CA954768

9

10

Abstract11

Doesnaturalselectionfavorveridicalperceptions—thosewhichaccurately,thoughperhaps12

notexhaustively,depictobjectivereality?Prominentvisionscientistsandevolutionary13

theoristsclaimthatitdoes.Hereweformalizethisclaimusingthetoolsofevolutionary14

gametheoryandBayesiandecisiontheory.Wethenpresentandprovea"Fitness-Beats-15

Truth(FBT)Theorem"whichshowsthattheclaimisfalse.Wefindthatincreasingthe16

complexityofobjectivereality,orperceptualsystems,orthetemporaldynamicsoffitness17

functions,increasestheselectionpressuresagainstveridicalperceptions.Weillustratethe18

FBTTheoremwithaspecificexampleinwhichveridicalperceptionminimizesexpected19

fitnesspayoffs.WeconcludethattheFBTTheoremsupportsthe"interfacetheoryof20

perception,"whichproposesthatoursenseshaveevolvedtohideobjectiverealityand21

guideadaptivebehavior.Italsosupportstheassertionofsomeproponentsofembodied22

*[email protected];+1(909)537-5390

cognitionthat“representingtheanimal-independentworldisnotwhataction-oriented23

representationsaresupposedtodo;theyaresupposedtoguideaction”(Chemero,2009).24

25

Keywords:ReplicatorDynamics;InterfaceTheoryofPerception;EvolutionaryGame26

Theory;UniversalDarwinism;Sensation27

28

29

1.Introduction30

Itisstandardintheperceptualandcognitivesciencestoassumethatmoreaccurate31

perceptionsarefitterperceptionsand,therefore,thatnaturalselectiondrivesperceptionto32

beincreasinglyveridical,i.e.toreflecttheobjectiveworldinanincreasinglyaccurate33

manner.Thisassumptionformsthejustificationfortheprevalentviewthathuman34

perceptionis,forthemostpart,veridical.Forexample,inhisclassicbookVision,Marr35

(1982)arguedthat:36

“We...verydefinitelydocomputeexplicitpropertiesoftherealvisiblesurfacesout37

there,andoneinterestingaspectoftheevolutionofvisualsystemsisthegradual38

movementtowardthedifficulttaskofrepresentingprogressivelymoreobjective39

aspectsofthevisualworld”.(p.340)40

Similarly,inhisbookVisionScience,Palmer(1999)statesthat:41

“Evolutionarilyspeaking,visualperceptionisusefulonlyifitisreasonably42

accurate...Indeed,visionisusefulpreciselybecauseitissoaccurate.Byandlarge,43

whatyouseeiswhatyouget.Whenthisistrue,wehavewhatiscalledveridical44

perception...perceptionthatisconsistentwiththeactualstateofaffairsinthe45

environment.Thisisalmostalwaysthecasewithvision.”46

Indiscussingperceptionwithinanevolutionarycontext,GeislerandDiehl(2003)similarly47

assumethat:48

“Ingeneral,(perceptual)estimatesthatarenearerthetruthhavegreaterutility49

thanthosethatarewideoffthemark.”50

Intheirmorerecentbookonhumanandmachinevision,Pizloetal.(2014)gosofarasto51

saythat:52

“…veridicalityisanessentialcharacteristicofperceptionandcognition.Itis53

absolutelyessential.Perceptionandcognitionwithoutveridicalitywouldbelike54

physicswithouttheconservationlaws.”(p.227,emphasistheirs.)55

Ifhumanperceptionisinfactveridical,itfollowsthattheobjectiveworldsharesthe56

attributesofourperceptualexperience.Ourperceivedworldisthree-dimensional,andis57

inhabitedbyobjectsofvariousshapes,colors,andmotions.Perceptualandcognitive58

scientiststhustypicallyassumethattheobjectiveworldissoinhabited.Inotherwords,59

theyassumethatthevocabularyofourperceptualrepresentationsisthecorrectvocabulary60

fordescribingtheobjectiveworldand,moreover,thatthespecificattributesweperceive61

typicallyreflecttheactualattributesoftheobjectiveworld.Theseassumptionsare62

embodiedwithinthestandardBayesianframeworkforvisualperception,whichwe63

considerinthenextsection.64

Someproponentsofembodiedcognitionrejecttheclaimthatperceptionisnormally65

veridical.Forinstance,Chemero(2009)arguesthat“…perceptualsystemsevolvedtoguide66

behavior.Neitherhumansnorbeetleshaveaction-orientedrepresentationsthatrepresent67

theanimal-independentworldexactlycorrectly.Indeed,representingtheanimal-68

independentworldisnotwhataction-orientedrepresentationsaresupposedtodo;they69

aresupposedtoguideaction.Sothesetofhumanaffordances,thatis,action-oriented70

representeds,isjustastightlygearedtohumanneedsandsensorimotorcapacitiesasthose71

ofothertypesofanimal.Thisleavesuswithamultiplicityofconflictingsensorimotor72

systems,eachofwhichisappropriateforguidingtheadaptivebehaviorofanimalswhose73

systemstheyare.”TheFBTTheorem,whichwepresentbelow,supportsChemero’sclaim.It74

issupported,inturn,byspecificexamplesofnon-veridicalperceptions,suchasthose75

discussedbyLoomis(2004)andKoenderinket.Al.(2010).76

77

2.ThestandardBayesianframeworkforvisualperception78

Thestandardapproachtovisualperceptiontreatsitasaproblemofinverseoptics:The79

“objectiveworld”—takentobe3Dscenesconsistingofobjects,surfaces,andlightsources—80

projects2Dimagesontotheretinas.Givenaretinalimage,thevisualsystem’sgoalistoinfer81

the3Dscenethatismostlikelytohaveprojectedit(e.g.Adelson&Pentland,1996;Feldman,82

2013;Knill&Richards,1996;Mamassian,Landy,&Maloney,2002;Shepard,1994;Yuille&83

Bulthoff,1996).Sincea2Dimagedoesnotuniquelyspecifya3Dscene,theonlywaytoinfer84

a3Dsceneistobringadditionalassumptionsor“biases”tobearontheproblem—basedon85

priorexperience(whetherphylogeneticorontogenetic).Forexample,ininferring3Dshape86

fromimageshading,thevisualsystemappearstomaketheassumptionthatthelightsource87

ismorelikelytobeoverhead(e.g.Kleffner&Ramachandran,1992).Similarly,ininferring88

3Dshapefrom2Dcontours,itappearstousetheassumptionthat3Dobjectsaremaximally89

compactandsymmetric(e.g.Lietal.,2013).90

Formally,givenanimage𝑥",thevisualsystemaimstofindthe“best”(generallytakento91

mean“mostprobable”)sceneinterpretationintheworld.Inprobabilisticterms,itmust92

comparetheposteriorprobabilityℙ 𝑤 𝑥" ofvarioussceneinterpretations𝑤,giventhe93

image𝑥".ByBayes’Rule,theposteriorprobabilityisgivenby:94

ℙ 𝑤 𝑥" = ℙ 𝑥" 𝑤 ∙ℙ(()ℙ(*+)

95

Sincethedenominatortermℙ(𝑥")doesnotdependon𝑤,itplaysnoessentialrolein96

comparingtherelativeposteriorprobabilitiesofdifferentscenesinterpretationsw.The97

posteriorprobabilityisthusproportionaltotheproductoftwoterms:Thefirstisthe98

likelihoodℙ 𝑥" 𝑤 ofanycandidatesceneinterpretationw;thisistheprobabilitythatthe99

candidatescenewcouldhaveprojected(orgenerated)thegivenimage𝑥".Becauseany2D100

imageistypicallyconsistentwithmanydifferent3Dscenes,thelikelihoodwilloftenbe101

equallyhighforanumberofcandidatescenes.Thesecondtermisthepriorprobability102

ℙ(𝑤)ofasceneinterpretation;thisistheprobabilitythatthesystemimplicitlyassignsto103

differentcandidatescenes,evenpriortoobservinganyimage.Forexample,thevisual104

systemmayimplicitlyassignhigherpriorprobabilitiestosceneswherethelightsourceis105

overhead,ortoscenesthatcontaincompactobjectswithcertainsymmetries.Thus,when106

multiplesceneshaveequallyhighlikelihoods(i.e.areequallyconsistentwiththeimage),107

thepriorcanserveasadisambiguatingfactor.108

ApplicationofBayes’Ruleyieldsaprobabilitydistributiononthespaceofcandidate109

scenes—theposteriordistribution.Astandardwaytopickasingle“best”interpretation110

fromthisdistributionistochoosetheworldscenethathasthemaximalposterior111

probability—onethat,statisticallyspeaking,hasthehighestprobabilityofbeingthe112

“correct”one,giventheimage𝑥".Thisisthemaximum-a-posterioriorMAPestimate.More113

generally,thestrategyoneadoptsforpickingthe“best”answerfromtheposterior114

distributiondependsonthechoiceofaloss(orgain)function,whichdescribesthe115

consequencesofmaking“errors,”i.e.pickinganinterpretationthatdeviatesfromthe“true”116

(butunknown)worldstatebyvaryingextents.TheMAPstrategyfollowsunderaDirac-117

deltalossfunction—nolossforthe“correct”answer(or“nearlycorrect”withinsome118

tolerance),andequallossforeverythingelse.Otherlossfunctions(suchasthesquared-119

errorloss)yieldotherchoicestrategies(suchasthemeanoftheposteriordistribution;see120

e.g.Mamassianetal.,2002).ButwefocusontheMAPestimateherebecause,inawell-121

definedsense,ityieldsthehighestprobabilityofpickingthe“true”sceneinterpretation122

withinthisframework.123

ThisstandardBayesianapproachembodiesthe“veridicality”or“truth”approachtovisual124

perception.Bythiswedonotmean,ofcourse,thattheBayesianobserveralwaysgetsthe125

“correct”interpretation.Giventheinductivenatureoftheproblem,thatwouldbea126

mathematicalimpossibility.Itisneverthelesstruethat:127

(i) ThespaceofhypothesesorinterpretationsfromwhichtheBayesianobserver128

choosesisassumedtocorrespondtotheobjectiveworld.Thatis,thevocabulary129

ofperceptualexperiencesisassumedtotherightvocabularyfordescribing130

objectivereality.131

(ii) Giventhissetup,theMAPstrategymaximizes(statisticallyspeaking)the132

probabilityofpickingthe“true”worldstate.133

134

3.EvolutionandFitness135

TheBayesianframework,summarizedabove,focusesonestimatingtheworldstatethathas136

thehighestprobabilityofbeingthe“true”one,givensomesensoryinputs.Thisestimation137

involvesnonotionofevolutionaryfitness.2Inordertobringevolutionandfitnessintothe138

picture,wethinkoforganismsasgatheringfitnesspointsastheyinteractwiththeir139

environment.ThuseachelementwoftheworldWhasassociatedwithitafitnessvalue.In140

general,however,thefitnessvaluedependsnotonlyontheworld,butalsoontheorganism141

oinquestion(e.g.,lionvs.rabbit),itsstates(e.g.,hungryvs.satiated),andtheactionclassa142

inquestion(e.g.,feedingvs.mating).Givensuchafitnesslandscape,naturalselectionfavors143

perceptionsandchoicesthatyieldmorefitnesspoints.144

Wemaythusdefineaglobalfitnessfunctionasa(non-negative)real-valuedfunctionf(w,o,145

s,a)ofthesefourvariables.However,oncewefixanorganism,itsstateandagivenaction146

class,i.e.,oncewefixo,sanda,aspecificfitnessfunctionissimplya(non-negative)real-147

valuedfunction𝑓:𝑊 → [0,∞)definedontheworldW.148

Inordertocomparethefitnessofdifferentperceptualand/orchoicestrategies,onepits149

themagainstoneanotherinanevolutionaryresourcegame(forsimulationsexemplifying150

theresultsofthispaper,see,e.g.,Mark,Marion,&Hoffman,2010;Marion,2013;andMark,151

2013).Inatypicalgame,twoorganismsemployingdifferentstrategiescompetefor152

availableterritories,eachwithacertainnumberofresources.Thefirstplayerobservesthe153

availableterritories,chooseswhatitestimatestobeitsoptimalone,andreceivesthefitness154

payoffforthatterritory.Thesecondplayerthenchoosesitsoptimalterritoryfromthe155

remainingavailableones.Thetwoorganismsthustaketurninpickingterritories,seeking156

tomaximizetheirfitnesspayoffs.157

Inthiscase,thequantityofresourcesinanygiventerritoryistherelevantworldattribute.158

Thatis,Wishereinterpretedasdepictingdifferentquantitiesofsomerelevantresource.1592Asnotedabove,Bayesianapproachesofteninvolvealoss(orgain)function.However,thisisquitedistinctfromafitnessfunction,asdefinedbelow.Specifically,lossfunctionsarefunctionsoftwovariablesl(x,x*),wherex*isthe“true”worldstate,andxisahypotheticalestimatearrivedatbytheobserver.Afitnessfunctionis,however,notafunctionoftheobserver’sestimatex.

WecanthenconsideraperceptualmapP :W → X ,whereXisthesetofpossiblesensory160

states,togetherwithanorderingonit:Ppicksoutthe“best”elementofXinasenserelevant161

totheperceptualstrategy.Onemay,forinstance,imagineasimpleorganismwhose162

perceptualsystemhasonlyasmallnumberofdistinctsensorystates.Itsperceptualmap163

wouldthenbesomewayofmappingvariousquantitiesoftheresourcetothesmallsetof164

availablesensorystates.Asanexample,Figure1showstwopossibleperceptualmappings,165

i.e.twowaysofmappingthequantityofresources(here,rangingfrom0through100)to166

fouravailablesensorycategories(heredepictedherebythefourcolorsR,Y,G,B).167

168

Figure1.AsimpleexampleshowingtwodifferentperceptualmappingsP :W → X from169

worldstates,W=[1,100]tosensorystatesX={R,Y,G,B}.170

Inaddition,thereisafitnessfunctiononW,𝑓:𝑊 → [0,∞),whichassignsanon-negative171

fitnessvaluetoeachresourcequantity.Onecanimaginefitnessfunctionsthatare172

monotonic(e.g.fitnessmayincreaselinearlyorlogarithmicallywiththenumberof173

resources),orhighlynon-monotonic(e.g.fitnessmaypeakforacertainnumberof174

resources,anddecreaseineitherdirection).Non-monotonicfitnessfunctions(suchasthe175

oneshowninFigure2)areinfactquitecommon:toolittlewaterandonediesofthirst,too176

muchwaterandonedrowns.Similarargumentsapplytothelevelofsalt,ortothe177

proportionofoxygenandindeedanynumberofotherresources.Indeed,giventhe178

ubiquitousneedfororganismstomainhomoeostasis,oneexpectsnon-monotonicfitness179

functionstobeprevalent.(Moreover,fromapurelymathematicalpointofview,thesetof180

monotonicfitnessfunctionsisanextremelysmallsubsetofthesetofallfunctionsona181

givendomain.Thatistosay,thereare“manymore”non-monotonicfunctionsthan182

monotonicones;hencearandomsamplingoffitnessfunctionsismuchmorelikelytoyielda183

non-monotonicone.)184

185

Figure2.Anexampleofanon-monotonicfitnessfunction𝑓:𝑊 → [0,∞).Fitnessismaximal186

foranintermediatevalueoftheresourcequantityanddecreasesineitherdirection.Given187

theubiquitousneedfororganismstomainhomoeostasis,oneexpectsthatsuchfitness188

functionsarequitecommon.189

190

4.Comparingperceptualstrategies:“Truth”vs.“Fitness-only”191

Inthecontextoftheseevolutionarygames,inwhichperceptualstrategiescompetefor192

resourceacquisition,wetakeasfixedandknowntotheorganism:thespecificfitness193

function,itsprior(inaparticularstateandforaparticularactionclass)anditsperceptual194

map(seeFigure3).Onanygiventrial,theorganismobservesanumberofavailable195

territoriesthroughitssensorystates,sayx1,x2,…,xn.Itsgoalistopickoneofthese196

territories,seekingtomaximizeitsfitnesspayoff.Onecannowconsidertwopossible197

resourcestrategies:198

The“Truth”strategy:Foreachofthensensorystates,theorganismestimatestheworld199

stateorterritory-theBayesianMAPestimate-thathasthehighestprobabilityofbeingthe200

“true”one,giventhatsensorystate.Itthencomparesthefitnessvaluesforthoseestimated201

worldstates.Finally,itmakesitschoiceofterritorybasedonthesensorystatexithatyields202

thehighestfitness.ItschoiceisthusmediatedthroughMAPestimateoftheworldstate.203

The“Fitness-only”strategy:Inthisstrategy,theorganismmakesnoattempttoestimate204

the“true”worldstatecorrespondingtoeachsensorystate.Ratheritdirectlycomputesthe205

expectedfitnesspayoffthatwouldresultfromeachpossiblechoiceofxi.Foragivensensory206

statexi,thereisaposteriorprobabilitydistribution(given,aswiththeTruthstrategy,by207

Bayes’formula)onthepossibleworldstates,aswellasafitnessvaluecorrespondingto208

eachworldstate.Theorganismweightsthesefitnessvaluesbytheposteriorprobability209

distribution,inordertocomputetheexpectedfitnessthatwouldresultfromthechoicexi.210

Anditpickstheonewiththehighestexpectedfitness.211

212

Figure3.Theframeworkwithinwhichwedefinethetworesourcestrategies.Weassumea213

fixedperceptualmapP :W → X aswellasafixedfitnessfunction𝑓:𝑊 → [0,∞).Givena214

choiceofavailableterritoriessensedthroughthesensorystates,sayx1,x2,…,xn,the215

organism’sgoalistopickoneofthese,seekingtomaximizeitsfitnesspayoff.216

217

5.TheoremsfromEvolutionaryGameTheory218

Inanevolutionarygamebetween the twostrategies, sayA andB,thepayoffmatrix is as219

follows:220

𝑎𝑔𝑎𝑖𝑛𝑠𝑡𝐴 𝑎𝑔𝑎𝑖𝑛𝑠𝑡𝐵𝐴𝑝𝑙𝑎𝑦𝑠 𝑎 𝑏𝐵𝑝𝑙𝑎𝑦𝑠 𝑐 𝑑

221

Herea,b,c, andd denote thevariouspayoffs to the rowplayerwhenplayingagainst the222

column player. E.g., b is the payoff to Awhen playing B.We will refer to three main223

theoremsfromevolutionarygametheoryrelevanttoouranalysis,asfollows.224

We first consider games with infinite populations. These are investigated bymeans of a225

deterministic differential equation, called the replicator equation, where time is the226

independentvariable and the relativepopulation sizes𝑥C, 𝑥Dare thedependentvariables,227

with𝑥C + 𝑥D = 1(TaylorandJonker,1978,HofbauerandSigmund,1990,Nowak2006).In228

thiscontext,therearefourgenericbehaviorsinthelongrun:229

Theorem1. (Nowak 2006) Inagamewithan infinitepopulationof twotypes,AandB,of230

players,either231

(i) Adominates B (in thesense thatanon-zeroproportionofAplayerswilleventually232

take over the whole population), if𝑎 ≥ 𝑐 and b≥ 𝑑 (with at least one of the233

inequalitiesbeingstrict);234

(ii) BdominatesA,if𝑎 ≤ 𝑐andb≤ 𝑑(withatleastoneoftheinequalitiesbeingstrict);235

(iii) AandBcoexist,if𝑎 ≤ 𝑐andb≥ 𝑑(withatleastoneoftheinequalitiesbeingstrict),236

atastableequilibriumgivenby𝑥C∗ =JKL

JMNKOKL(and𝑥D∗ = 1 − 𝑥C∗);237

(iv) Thesystemisbistable,if𝑎 ≥ 𝑐andb≤ 𝑑(withatleastoneoftheinequalitiesbeing238

strict)andwilltendtowardseitherallAorallBfromanunstableequilibriumatthe239

samevalueof𝑥C∗ asabove.240

A fifth,non-genericpossibility is that𝑎 = 𝑐andb= 𝑑, inwhichcasewehave thatAandB241

areneutralvariantsofoneanother:anymixtureofthemisstable.242

Games with a finite population size N can be analyzed via a stochastic, as against243

deterministic, approach. The dynamics are described by a birth-death process, called the244

Moranprocess(Moran1958).Theresultsaremorenuancedthanintheinfinitepopulation245

sized case: there are now eight possible equilibrium behaviors, and they are population246

dependent,notjustpayoffdependent.247

Let𝜌CD denote the fixation probability of a single A individual in a population of N-1 B248

individuals replacing (i.e., taking over completely) that population. Similarly, let Let𝜌DC249

denotethefixationprobabilityofasingleBindividualinapopulationofN-1ofAindividuals250

replacing(i.e.,takingovercompletely)thatpopulation.Intheabsenceofanyselection,we251

havethesituationofneutraldrift,wheretheprobabilityofeitherof theseevents is justRS.252

WesaythatselectionfavorsAreplacingBif𝜌CD >RSandthatselectionfavorsBreplacingAif253

𝜌DC >RS.254

By analyzing the probabilities of a single individual of each type interacting with an255

individualofeithertype,orofdyingoff,wecanusethepayoffmatrixabovetocomputethe256

fitness𝐹V ,when there are i entities of typeA, and the fitness𝐺V of (theN-i individuals) of257

type B. If we setℎV = 𝐹V − 𝐺V (𝑖 = 1, . . . , 𝑁), we can see thatℎR > 0implies that selection258

favorsAinvadingB,whileℎSKR > 0impliesthatselectionfavorsBinvadingA.Therearenow259

sixteen possibilities, depending upon whether selection favors A replacing B or not; B260

replacing Aor not; whether selection favors A invading Bor not; and whether selection261

favorsBinvadingAornot.Ofthese,eightareruledoutbyatheoremofTaylor,Fudenberg,262

SasakiandNowak(2004).Afulldescriptionisprovidedinthatpaper,alongwithanumber263

of theorems detailing the possibilities in terms of the payoff values and population size.264

TheirTheorem6, interpretedbelowasourTheorem2, ismostrelevanttoouranalysisof265

evolutionary resource games: it gives conditions underwhich selection is independent of266

population size and is reproduced below. Interestingly, for finite populations the267

relationshipbetweenpayoffsbandcbecomesrelevant:268

Theorem2. Inagamewithafinitepopulationoftwotypesofplayers,AandB,if𝑏 > 𝑐, 𝑎 >269

𝑐and𝑏 > 𝑑,wehaveforallN,ℎV > 0∀𝑖and𝜌CD >RS> 𝜌DC:selectionfavorsA.270

Finally,wealsoconsider,withinlargefinitepopulations,thelimitofweakselection.Inorder271

tomodelthestrengthofselection,anewparameterwisintroduced.Thisparameter,lying272

between0and1,isameasureofthestrengthofselection:wewritethefitnessofAnowas273

𝑓V = 1 − 𝑤 + 𝑤𝐹V and the fitness of B now as𝑔V = 1 − 𝑤 + 𝑤𝐺V . When𝑤 = 0, there is no274

selection:thefitnessesareequalandwehaveneutraldrift.When𝑤 = 1,wehaveselection275

atfullstrength.AnanalysisofthedynamicsoftheMoranprocessunderweakselection(i.e.,276

inthelimitas𝑤 → 0),reveals(followingNowak2006,equation7.11)that:277

Theorem3.Inagamewithafinitepopulationoftwotypesofplayers,AandB,andwithweak278

selection, 𝑎 − 𝑐 + 2 𝑏 − 𝑑 > ] OKN K(JKL)S

implies that𝜌CD >RS. Thus, if𝑎 > 𝑐 and𝑏 > 𝑑 ,279

forlargeenoughN,selectionfavorsA.3280

281

6.EvolutionaryResourceGames282

Foroursituationoftworesourcestrategies,wemaydefinethepayoffmatrixasfollows:283

a:toFitness-Onlywhenplayingagainst

Fitness-Only

b:toFitness-Onlywhenplayingagainst

Truth

c:toTruthwhenplayingagainstFitness-Only d:toTruthwhenplayingagainstTruth

284

In a game with a very large (effectively infinite) number of players, the Fitness-Only285

resource strategy dominates the Truth strategy (in the sense that Fitness-Only will286

eventually driveTruthto extinction) if the payoffs toFitness-Only as first player always287

exceedthoseofTruthasfirstplayer,regardlessofwhothesecondplayeris,i.e.if and288

andatleastoneoftheseisastrictinequality.Ifneitheroftheseinequalitiesisstrict,289

thenattheleastFitness-OnlywillneverbedominatedbyTruth.290

3ThevalueofNatwhichthishappensdependsuponthepayoffmatrix,butcanbearbitrarilylargeoverthesetofallpayoffmatricessatisfying𝑎 > 𝑐and𝑏 > 𝑑.

!a ≥c

!b ≥d

OurmainclaiminthispaperisthattheTruthstrategy—attemptingtoinfertothe“true”291

stateoftheworldthatismostlikelycorrespondtoagivensensorystate—confersno292

evolutionaryadvantagetoanorganism.Inthenextsection,westateandproveatheorem—293

the“FitnessBeatsTruth"theorem—whichstatesthatFitness-Onlywillneverbedominated294

byTruth.Indeed,theTruthstrategywillgenerallyresultinalowerexpected-fitnesspayoff295

thantheFitness-Onlystrategy,andisthuslikelytogoextinctinanyevolutionary296

competitionagainsttheFitness-Onlystrategy.(ThestatementoftheFBTtheorem297

articulatestheprecisewayinwhichthisistrue.)Webegin,first,withanumericalexample298

thatexemplifiesthis.299

6.1NumericalExampleofFitnessBeatingTruth300

Wegiveasimpleexampletopavethewayfortheideastofollow.Supposetherearethree301

states of the world,𝑊 = {𝑤R, 𝑤], 𝑤_}and two possible sensory stimulations,𝑋 = {𝑥R, 𝑥]}.302

Each world state can give rise to a sensory stimulation according to the information303

contained in Table 1. The first two columns give the likelihood values,ℙ 𝑥 𝑤 ,for each304

sensorystimulation,givenaparticularworldstate;forinstance,ℙ(𝑥R|𝑤]) = 3/4.Thethird305

column gives the prior probabilities of the world states. The fourth column shows the306

fitnessassociatedwitheachworldstate. Ifwe thinkof theworldstatesas threedifferent307

kindsoffoodthatanorganismmighteat,thenthesevaluescorrespondtothefitnessbenefit308

anorganismwouldgetbyeatingoneofthefoods.Withthisanalogy,𝑤Rcorrespondstoan309

extremely healthful food,while𝑤]and𝑤_correspond tomoderately healthful foods,with310

𝑤]beingmorehealthfulthan𝑤_(seeTable1).Thissetupisthebackdropforasimplegame311

where observers are presented with two sensory stimulations and forced to choose312

betweenthem.313

Likelihood:𝒙𝟏given𝒘𝒋

ℙ(𝑥R 𝑤j

Likelihood:𝒙𝟐given𝒘𝒋

ℙ(𝑥] 𝑤j

Prior

ℙ 𝑤j

Fitness

𝑓 𝑤j

𝑤R 1/4 3/4 1/7 20

𝑤] 3/4 1/4 3/7 4

𝑤_ 1/4 3/4 3/7 3

Table1:Likelihoodfunctions,priorsandfitnessforoursimpleexamplewheretheTruth314

observerminimizesexpectedfitness,whileFitness-onlyobservermaximizesit.315

Using Bayes’ theorem we have calculated (see Appendix) that for𝑥Rthe Truth (i.e. the316

maximum-a-posteriori) estimate is𝑤], and that for𝑥]thisestimate is𝑤_.Thus, if aTruth317

observer isofferedachoicebetweentwofoodstoeat,onethatgives itstimulation𝑥Rand318

onethatgivesitstimulation𝑥],itwillperceivethatithasbeenofferedachoicebetweenthe319

foods𝑤]and𝑤_. Assuming that it has been shaped by natural selection to choose, when320

possible, the foodwith greater fitness, itwill alwaysprefer𝑤]. So,whenoffered a choice321

between𝑥Rand𝑥],theTruthobserverwillalwayschoose𝑥R,withanexpectedutilityof5.322

NowsupposeaFitness-Onlyobserverisgiventhesamechoice.TheFitness-Onlyobserver323

isnotatallconcernedwithwhich“veridical”foodthesesignalsmostlikelycorrespondto,324

buthasbeenshapedbynaturalselectiontoonlycareaboutwhichstimulusyieldsahigher325

expected fitness.We have calculated (see Appendix) that the expected fitness of sensory326

stimulation𝑥Ris 5 and the expected utility of stimulation𝑥]is 6.6. Thus, when offered a327

choice between𝑥Rand𝑥], the Fitness-Only observer will always, maximizing expected328

fitness,choose𝑥].329

Theimplicationsoftheseresultsareclear.ConsiderapopulationofTruthobservers330

competingforresourcesagainstapopulationofFitness-Onlyobservers,bothoccupyingthe331

nichedescribedbyTable1.Since,inthiscase,theTruthobserver’schoiceminimizes332

expectedutilityandtheFitness-Onlyobserver’schoicemaximizesexpectedutility,the333

Fitness-OnlypopulationwillbeexpectedtodrivethepopulationofTruthobserversto334

extinction.Seeingtruthcanminimizefitness;therebyleadingtoextinction.Thisconclusion335

isapartfromconsiderationsoftheextraenergyrequiredtokeeptrackoftruth(seeMark,336

MarionandHoffman2010fordiscussiononenergyresources).337

338

7.MathematicalBackgroundfortheMainTheorem339

Weassumethatthereisafixedpreliminarymap,p,whichassociatestoeachworldstate340

𝑤 ∈ 𝑊asensorystate𝑥 ∈ 𝑋.AndweassumeafitnessmaponW(recallFigure3).This341

placestheTruthstrategyandtheFitness-onlystrategyonacommonfootingwherethey342

canbesetindirectcompetitionagainsteachotherwithinthecontextofanevolutionary343

resourcegame.344

Webeginwithsomemathematicaldefinitionsandassumptionsregardingthesespacesand345

maps.346

Itwillsufficeforabasicunderstandingofthedevelopmentinwhatfollows,tothink347

ofWas a finite set (as in the example in 6.1).4In general, we take theworldW to be a348

compactregularBorelspacewhosecollectionofmeasurableeventsisa𝜎-algebra,denoted349

ℬ.5Weassume that< 𝑊,ℬ >comesequippedwithanaprioriprobabilitymeasure𝜇on350

4inwhichcasealltheintegralsignsbelowcanbereplacedbysummations.5Anexampleisaclosedrectangleinsomek-dimensionalEuclideanspace,suchastheunitinterval[0,1]inonedimension,ortheunitsquareintwo.

ℬ.Wewillconsideronlythoseprobabilitymeasures𝜇thatareabsolutelycontinuouswith351

respect to the Borel measure onℬ. That is, if we writed𝑤for the uniform, or Borel,352

probability measure on W, then the a priori measure satisfies𝜇 𝑑𝑤 = 𝑔 𝑤 d𝑤.Here353

𝑔:𝑊 → ℝMis some non-negative measurable function, called the density of𝜇,satisfying354

𝑔(𝑤)d𝑤 = 1.Wewilltakeanysuchdensitytobecontinuous,sothat italwaysachieves355

itsmaximumonthecompactsetW.Thisconstitutesthestructureoftheworld:astructure356

thatappliestomostbiologicalandperceptualsituations.357

We assume that a given species interacts with its world, employing a perceptual358

mapping that “observes” theworldvia ameasurablemap𝑝:𝑊 → 𝑋.We refer to this as a359

pureperceptualmap because it involves no dispersion: eachworld state can yield only a360

singlesensorystatex.WeassumethatthesetofperceptualstatesXisafiniteset,withthe361

standarddiscrete𝜎-algebra𝒳,i.e.,itspowerset(sothatallsubsetsofXaremeasurable).In362

thegeneralcase,theperceptualmapmayhavedispersion(ornoise),andismathematically363

expressed as aMarkovian kernel𝑝:𝑊×𝒳 → 0,1 .That is, for every elementw inW, the364

kernelpassignsaprobabilitydistributiononX(henceitassignsaprobabilityvaluetoeach365

measurablesubsetofX).BecauseX is finiteandallof itssubsetsaremeasurable,herethe366

kernelmaybeviewedsimplyasassigning,foreveryelementwinW,aprobabilityvalueto367

eachelementofX.368

7.1GeneralPerceptualMappingsandBayesianInference369

We use the letterℙto indicate any relevant probability. Bayesian inference consists in a370

computation of the conditional probability measureℙ(d𝑤|𝑥)on the world, given a371

particular perception𝑥in X. The likelihood function is the probabilityℙ(𝑥|𝑤)that a372

particular world state𝑤could have given rise to the observed sensory state𝑥.Then the373

conditionalprobabilitydistributionℙ(𝑑𝑤|𝑥)istheaposteriorprobabilitydistributionina374

(partially)continuousversionofBayesformula:375

ℙ(d𝑤|𝑥) =ℙ(𝑥|𝑤)ℙ d𝑤

ℙ 𝑥.376

Since𝜇,theprioronW,hasadensity𝑔withrespecttotheBorelmeasured𝑤,wecanrecast377

thisformulaintermsof𝑔:indeed,ℙ(d𝑤|𝑥)alsohasaconditionaldensity,𝑔(𝑤|𝑥),with378

respecttotheBorelmeasure6andweobtain379

𝑔(𝑤|𝑥) =ℙ(𝑥|𝑤)𝑔 𝑤ℙ(𝑥|𝑤′)𝑔 𝑤′

.380

We now define amaximum a posteriori estimate for𝑥in X to be any𝑤* at which this381

conditional density is maximized:𝑔 𝑤* 𝑥) =max 𝑔 𝑤 𝑥 |𝑤 ∈ 𝑊}. At least one such382

maximumwillexist,since𝑔isboundedandpiecewisecontinuous;however,therecouldbe383

multiplesuchestimatesforeach𝑥.384

Foragivensensorystate𝑥, theonlyworldstatesthatcouldhavegivenriseto it lie inthe385

fiberover𝑥,i.e., theset𝑝KR 𝑥 ⊂ 𝑊.So, foragiven𝑥,themapping𝑤 → ℙ(𝑥|𝑤)takes the386

value1onthefiber,andiszeroeverywhereelse.Thismappingmaythusbeviewedasthe387

indicatorfunctionofthisfiber.Wedenotethisindicatorfunctionby1wxy * 𝑤 .388

Forapuremappingtheconditionaldensityisjust389

𝑔(𝑤|𝑥) =𝑔 𝑤 ⋅ 1wxy * 𝑤

𝜇 𝑝KR 𝑥,390

where𝜇 𝑝KR 𝑥 istheapriorimeasureofthefiber.391

6Thatis,ℙ(d𝑤|𝑥) = 𝑔(𝑤 𝑥 d𝑤.

In this special case of a pure mapping that has given rise to the perception𝑥,we can392

diagram the fiber overxon which this average fitness is computed.This is the shaded393

regioninfigure3below.394

395

Figure4.Theexpectedfitnessof𝑥istheaverage,usingtheposteriorprobability,overthe396

fiber𝑝KR 𝑥 .397

7.2ExpectedFitness398

Givenafitnessfunction𝑓:𝑊 → [0,∞)thatassignsanon-negativefitnessvaluetoeach399

worldstate,theexpectedfitnessofaperception𝑥is400

𝐹 𝑥 = 𝑓 𝑤 ℙ(d𝑤|𝑥) = 𝑓 𝑤 𝑔(𝑤|𝑥)d𝑤 .401

7.3TwoPerceptualStrategies.402

Wemaybuildourtwoperceptualstrategies𝑃}, 𝑃~ ,called“Truth”and“Fitness-Only”403

respectively,ascompositionsofa“sensory”map𝑝:𝑊 → 𝑋thatrecognizesterritoriesand404

!

W p-1(x)

!.!X! X

“ordering”maps𝑑}, 𝑑~: 𝑋 → 𝑋,where𝑃} = 𝑑} ∘ 𝑝and𝑃~ = 𝑑~ ∘ 𝑝.Thatis,themap𝑑} re-405

namestheelementsofXbyre-orderingthem,sothatthebestone,intermsofitsBayesian406

MAPestimate,isnowthefirst,𝑥R,thesecondbestis𝑥]etc.Themap𝑑~ ,ontheotherhand,407

re-orderstheelementsofXsothatthebestone,intermsofitsexpectedfitnessestimate,is408

𝑥R,thesecondbestis𝑥]etc.Theorganismpicks𝑥Rifitcan,𝑥]otherwise.409

Wecannowassertourmaintheorem,invariouscontextsofevolutionarygames:with410

infinitepopulations,finitepopulationswithfullselection,andsufficientlylargefinite411

populationswithweakselection.412

413

8.Results414

8.1The“FitnessBeatsTruth”Theorem415

Thefollowingtheoremappliestoinfinitepopulations,ortolargefinitepopulations416

includingthosewithweakselection:417

Theorem4:Overallpossiblefitnessfunctionsandapriorimeasures,theprobabilitythatthe418

Fitness-onlyperceptualstrategystrictlydominatestheTruthstrategyisatleast( 𝑋 −419

3)/( 𝑋 − 1),where 𝑋 𝑖𝑠𝑡ℎ𝑒𝑠𝑖𝑧𝑒𝑜𝑓𝑡ℎ𝑒𝑝𝑒𝑟𝑐𝑒𝑝𝑡𝑢𝑎𝑙𝑠𝑝𝑎𝑐𝑒.Asthissizeincreases,this420

probabilitybecomesarbitrarilycloseto1:inthelimit,Fitness-onlywillgenericallystrictly421

dominateTruth,sodrivingthelattertoextinction.422

Proof:Foranygiven𝑥,theBayesianMAPestimateisaworldpoint𝑤* (itisthe𝑤*suchthat423

𝑔 𝑤* 𝑥) =max 𝑔(𝑤|𝑥 𝑤 ∈ 𝑊 ). This point has fitness𝑓 𝑤* ;let𝑥� be that𝑥forwhich424

thecorresponding𝑓 𝑤* ismaximized.Thenthis𝑥� is,ifavailable,ischosenbyTruthand425

𝐹(𝑥�),itsexpectedfitness,isthepayofftoTruth.426

On the other hand, the fitness payoff to the Fitness-only strategy is, by definition the427

maximumexpectedfitness𝐹(𝑥�)overallfibers,soclearly,𝐹(𝑥�) ≤ 𝐹(𝑥�).428

Asdefined earlier, our evolutionary gamehas as payoffs,a:toFitness-onlywhenplaying429

against Fitness-only; b: to Fitness-only when playing against Truth; c: to Truth when430

playingagainstFitness-only;d:toTruthwhenplayingagainstTruth.431

Weneedtoestimatetheprobability that and Weassumethat ifbothstrategies432

arethesame,theneachhasanevenchanceofpickingitsbestterritoryfirst.Thusif,inany433

givenplay of the game, two competing strategies both take aparticular territory as their434

mostfavoredone,theneachstrategyhasanevenchanceofpickingthatterritoryandthen435

theotherstrategypicksitsnext-bestchoiceofterritory.436

If Fitness-only meets Fitness-only, then each has an even chance of choosing its best437

territory, say𝑥�; thesecond tochoose thenchooses its secondbest territory, say𝑥��. Since438

eachplayerhasanequalchanceofbeingfirst,wehave439

𝑎 = 𝐹 𝑥� + 𝐹 𝑥�� /2.440

IfTruthmeetsFitness-only,itschoicewillbe𝑥� ,aslongasthisvaluediffersfrom𝑥� .Inthis441

instance, we have𝑎 > 𝑐. If, however,𝑥� = 𝑥�,half the timeTruth will choose𝑥� and the442

otherhalf𝑥�� , where𝑥�� isthesecondbestoftheoptimalterritoriesfor𝑻𝒓𝒖𝒕𝒉.Hence443

𝑐 =𝐹(𝑥�),ifdifferentbestterritories𝐹(𝑥�) + 𝐹 𝑥��

2,ifsamebestterritories

444

andsince𝐹 𝑥� ≤ 𝐹 𝑥� and𝐹(𝑥�� ) ≤ 𝐹 𝑥�� weget 445

WhathappenswhenFitness-onlymeetsTruth?IfFitness-onlygoesfirst,thepayoffwillbe446

𝑏 = 𝐹(𝑥�).ThesameistrueifTruthgoesfirstandthetwobestterritoriesaredifferent.If,447

!a ≥c !!b ≥d.

!!a ≥c.

however,thetwobestterritoriesarethesame,thenthepayofftoFitness-onlyisitssecond-448

bestoutcome:449

𝑏 =𝐹(𝑥�),ifdifferentbestterritories𝐹 𝑥�� ,ifsamebestterritories

450

Finally,whenTruthmeetsTruth,wehavethat451

𝑑 =𝐹 𝑥� + 𝐹 𝑥��

2.452

Soitisclearthat𝑏 ≥ 𝑑,aslongasthetwobestterritoriesaredifferent.Iftheyarethesame,453

this may or may not be true: it depends on the relative size of the average d and𝐹 𝑥�� 454

(which,inthisinstance,alsoliesinbetween𝐹 𝑥�� and𝐹(𝑥�) = 𝐹 𝑥� ).455

Now,apriori,thereisnocanonicalrelationbetweenthefunctionsfandg,bothofwhichcan456

beprettymucharbitrary(infact,fneednotevenbecontinuousanywhere,andcouldhave457

big jumps as well as bands of similar value separated from each other in W). Also,458

generically themaximum foreachstrategywillbeuniqueandalso theexpected fitnesses459

forthedifferentterritorieswillallbedistinct.460

Thus,generically,𝐹(𝑥�)and𝐹 𝑥�� willbedifferentfromandindeedstrictlylessthan𝐹(𝑥�)461

(and also𝐹 𝑥�� < 𝐹 𝑥�� ). The only impediment to the domination of Fitness-Only can462

come fromthesituationwhere thebest territories forbothstrategiesare thesame.LetX463

havesize 𝑋 = 𝑛.Thereare𝑛waysthetwostrategiescanoutputthesameterritory,outof464

the𝑛! [2! 𝑛 − 2 !]waysofpairingterritories.Thus,acrossallpossibilitiesfor fandg, the465

probabilitythatrandomlychosenfitnessandapriorimeasureswouldresultinchoosingthe466

sameterritoryforbothstrategies,i.e.,that𝐹 𝑥� = 𝐹(𝑥�),willhappenwithaprobabilityof467

𝑛𝑛!2! 𝑛 − 2 !

=2

𝑛 − 1468

Finally, theprobabilityofthetwofibersbeingdifferent isthecomplement:1 − ] KR

=  K_ KR

.469

470

8.1DynamicFitnessFunctions471

Apossibleobjectiontotheapplicabilityof thistheoremisthat itseemstoassumeastatic472

fitnessfunction,whereasrealisticscenariosmayinvolvechanging,orevenrapidlychanging,473

fitnessfunctions.However,aclosescrutinyoftheproofofthetheoremrevealsthatatany474

moment, the fitness functionatthattime being the same for both strategies, the relative475

payoffsremaininthesamegenericrelationasatanyothermoment.Hencethetheoremalso476

appliestodynamicallychangingfitnessfunctions.477

478

9.Discussion479

AswenotedintheIntroduction,itisstandardintheliteraturetoassumethatmoreaccurate480

perceptionsarefitterperceptionsandthat,therefore,naturalselectiondrivesperceptionto481

increasingveridicality—i.e.tocorrespondincreasinglytothe“true”stateoftheobjective482

world.Thisassumptioninformstheprevalentviewthathumanperceptionis,forthemost483

part,veridical.484

Ourmainmessageinthispaperhasbeenthat,contrarytothisprevalentview,attemptingto485

estimatethe“true”stateoftheworldcorrespondingtoagivenasensorystate,confersno486

evolutionarybenefitwhatsoever.Ratherastrategythatsimplyseekstomaximizeexpected-487

fitnesspayoff,withnoattempttoestimatethe“true”worldstate,doesconsistentlybetter488

(intheprecisesensearticulatedinthestatementofthe“FitnessBeatsTruth”Theorem).489

Indeed,this“Fitness-only”strategydoesnotestimateanysingleworldstate;itsimply490

!

averagesoverallpossibleworldstatestocomputetheexpected-fitnesspayoff491

correspondingtoanygivensensorystate(thisisanalogoustoamodel-averagingstrategyin492

modelselection).Andyet,asthetheoremshows,inanevolutionarycompetition,this493

strategyislikelytodrivethe“truth”strategytoextinction.494

Atfirstglance,thisexpected-fitnessstrategy,basedonaveragingoverallpossibleworld495

states,mayseemimplausible:Afterall,inourownperceptualexperience,weperceive496

thingstobeoneparticularway;wecertainlydon’texperienceasuperpositionor“smear”497

resultingfromaveragingovervariouswaysthattheworldcouldbe.Whilethisis498

undoubtedlytrue,oneshouldnotethatthisisafactaboutperceptualexperience,and499

providesnosupportwhatsoeverforastrategythatinvolvesestimatingthe“true”stateof500

theworld.Inwhatfollows,wesketchoutamorecompleteanswertotheseeming501

implausibilityofaveraging,basedonourInterfaceTheoryofPerception(Hoffman,Singh,&502

Prakash,2015).503

Forthepurposeofthecurrentanalysis,itwasessentialtoplacethetwostrategiestobe504

compared—“Truth”and“Fitness-only”—withinacommonframeworkinvolvingBayesian505

inferencefromthespaceofsensorystates,X,totheworld,W(recallFigure3).Thisallowed506

ustoplacethetwostrategiesonthesamefooting,sotheycouldcompetedirectlyagainst507

eachother.However,thisresultstronglysupportsourbeliefthattheveryideaofperception508

asprobabilisticinferencetostatesoftheobjectiveworldismisguided.Perceptionisindeed509

fruitfullymodeledasprobabilisticinference,buttheinferencehappensinaspaceof510

perceptualrepresentations,andnotinanobjectiveworld.511

Theseideasarepartoflargertheory,theInterfaceTheoryofPerception,thatwehave512

describedindetailelsewhere(Hoffman,2009;Hoffman&Prakash,2014;Hoffman&Singh,513

2012;Hoffman,Singh,&Prakash,2015;seealsoKoenderink,2011;2013;2014;von514

Uexkull,1934).Forthepurposesofthecurrentdiscussion,thekeypointisthatthestandard515

Bayesianframeworkforvisualperceptionconflatestheinterpretationspace(orthespace516

ofperceptualhypothesesfromwhichthevisualsystemmuchchoose)withtheobjective517

world.Thisisamistake;itisessentiallytheassumptionthatthelanguageofourperceptual518

representationisthecorrectlanguagefordescribingobjectivereality—ratherthansimplya519

species-specificinterfacethathasbeenshapedbynaturalselection.InourITPframework,520

theprobabilisticinferencethatresultsinperceptualexperiencetakesplaceinaspaceof521

perceptualrepresentations,say,X1,thatmayhavenoisomorphicorevenhomomorphic522

relationwhatsoevertoW.TheextendedframeworkofthisComputationalEvolutionary523

PerceptionissketchedinFigure5(seeHoffman&Singh,2012;Hoffman,Singh,&Prakash,524

2015;Singh&Hoffman,2013).525

526

Figure5.TheframeworkofComputationalEvolutionaryPerceptioninwhichperceptual527

inferencestakeplaceinaspaceofrepresentationsX1thatisnotisomorphicor528

homomorphictoW.ThemorecomplexrepresentationalformatofX1evolvesbecauseit529

permitsahigher-capacitychannelP1 :W → X1 forexpectedfitness,therebyallowingthe530

organismtochooseandactmoreeffectivelyintheenvironment(i.e.inwaysthatresultin531

higherexpected-fitnesspayoffs).532

533

Thus,thereasonwegenerallyperceiveasingleinterpretationisbecausetheprobabilistic534

inferenceintheperceptualspaceX1generallyresultsinauniqueinterpretation.Butthe535

perceptualspaceX1isnottheobjectiveworld,norisithomomorphictoit.Itissimplya536

representationalformatthathasbeencraftedbynaturalselectioninordertosupportmore537

effectiveinteractionswiththeenvironment(inthesenseofresultinginhigherexpected-538

fitnesspayoff).Inotherwords,amorecomplexorhigher-dimensionalrepresentational539

format(e.g.involving3Drepresentationsin𝑋R,inplaceof2Drepresentationsin𝑋")evolves540

becauseitpermitsahigher-capacitychannelP1 :W → X1 forexpectedfitness(seeFigure541

5).Butthisdoesnotinanywayentailthatthisrepresentationalformatsomehowmore542

closely“resembles”theobjectiveworld.Evolutioncanfashionperceptualsystemsthatare,543

inthissense,ignorantoftheobjectiveworldbecausenaturalselectiondependsonlyon544

fitnessandnotonseeingthe“truth.”545

Theseconsiderationsstronglyunderminethestandardassumptionsthatseeingmore546

veridicallyenhancesfitness,andthatthereforeonecanexpectthathumanperceptionis547

largelyveridical.Ashumanobservers,wearepronetoimputingstructuretotheobjective548

worldthatisproperlypartofourownperceptualexperience.Forexample,ourperceived549

worldisthree-dimensionalandpopulatedwithobjectsofvariousshapes,colors,and550

motions,andsowetendtoconcludethattheobjectiveworldisaswell.Butif,astheFitness-551

beats-TruthTheoremshows,evolutionarypressuresdonotpushperceptioninthedirection552

ofbeingincreasinglyreflectiveofobjectivereality,thensuchimputationshavenological553

basiswhatsoever.7554

Acknowledgments5557SeealsotheInventionofSpace-TimeTheoreminHoffman,Singh,&Prakash(2015).

Wethank,FedericoFagginforenlighteningdiscussions.Thisworkhasbeenpartially556

fundedbytheFedericoandElviaFagginFoundation.557

558

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33. vonUexkull, J.(1934).Astrollthroughtheworldsofanimalsandmen:Apicturebook629of invisible worlds. Reprinted in Instinctive behavior: Development of a modern630concept,C.H.Schiller(1957).NewYork:HallmarkPress.631

34. Yuille, A., & Bulthoff, H. (1996). Bayesian decision theory and psychophysics. In:632Perception as Bayesian inference, D. Knill and W. Richards, Eds., New York:633CambridgeUniversityPress.634

635

Appendix:CalculationsforthenumericalexampleinTable1.636

InthisappendixweperformtheBayesianandexpectedfitnesscalculationsusingthedata637

giveninTable1.638

TocomputetheTruthestimates,wefirstneedtheprobabilityofeachstimulationℙ(𝑥R)and639

ℙ(𝑥]).Thesecanbecomputedbymarginalizingoverthepriorsintheworldasfollows:640

ℙ(𝑥R) = 𝑝(𝑥R 𝑤R 𝜇 𝑤R + p 𝑥R 𝑤] 𝜇 𝑤] + p 𝑥R 𝑤_ 𝜇 𝑤_ = R¡. R¢+ _

¡. _¢+ R

¡. _¢= R_

]£641

ℙ(𝑥]) = 𝑝(𝑥] 𝑤R 𝜇 𝑤R + p 𝑥] 𝑤] 𝜇 𝑤] + p 𝑥] 𝑤_ 𝜇 𝑤_ = _¡. R¢+ R

¡. _¢+ _

¡. _¢= R¤

]£642

ByBayes’Theorem,theposteriorprobabilitiesoftheworldstates,given𝑥R,are643

𝑝(𝑤R 𝑥R = 𝑝 𝑥R 𝑤R .𝜇 𝑤Rℙ 𝑥R

=14.17/1328

=113644

𝑝(𝑤] 𝑥R = 𝑝 𝑥R 𝑤] .𝜇 𝑤]ℙ 𝑥R

=34.37/1328

=913645

𝑝(𝑤_ 𝑥R = 𝑝 𝑥R 𝑤_ .𝜇 𝑤_ℙ 𝑥R

=14.37/1328

=313646

Thusthemaximumaposteriori,orTruthestimateforstimulus𝑥Ris𝑤].647

Posteriorprobabilitiesoftheworldstates,given𝑠],are:648

𝑝(𝑤R 𝑥] = 𝑝 𝑥] 𝑤R .𝜇 𝑤Rℙ 𝑥]

=34.17/1528

=15649

𝑝(𝑤] 𝑥] = 𝑝 𝑥] 𝑤] .𝜇 2ℙ 𝑥]

=14.37/1528

=15650

𝑝(𝑤_ 𝑥] = 𝑝 𝑥] 𝑤_ .𝜇 𝑤_ℙ 𝑥]

=34.37/1528

=35651

Thusthemaximumaposteriori,orTruthestimateforstimulus𝑥]is𝑤_.652

Finally, the expected-fitness values of the different sensory stimulations𝑥Rand𝑥]are,653

respectively:654

𝐹 𝑥R = 𝑝(𝑤R 𝑥R 𝑓 𝑤R + 𝑝(𝑤] 𝑥R 𝑓 𝑤] + 𝑝(𝑤_ 𝑥R 𝑓 𝑤_ =113. 20 +

913. 4 +

313. 3 = 5;655

𝐹 𝑥] = 𝑝(𝑤R 𝑥] 𝑓 𝑤R + 𝑝(𝑤] 𝑥] 𝑓 𝑤] + 𝑝(𝑤_ 𝑥] 𝑓 𝑤_ = R¤. 20 + R

¤. 4 + _

¤. 3 = 6.6.656

Thus𝑥]hasalargerexpectedfitnessthan𝑥R.657

658

659

660

661

Highlights662

• Wemakerigorousmathematicaldefinitionsoftwoperceptualstrategiesemployable663byagivenspecies,foragivenactionclassandwithinagivenenvironment:Truth,664basedonBayesianestimationofassumedobjectivepropertiesoftheworld,and665Fitness,tunedtoanarbitraryfitnessfunction;666

• UndertheassumptionofuniversalDarwinism(Dennett,1995)wesubjectthetwo667strategiestoanevolutionarygameanalysis;668

• WeconcludethattheFitnesswillgenerallydriveTruthtoextinction,forgeneric669fitnessfunctionsandpriors;670

• ThelikelihoodofFitnessdominatingTruthexceeds1/2assoonasthesensoriumhas671morethanfiveelements,andrisesmonotonicallyto1asthesizeofthesensorium672growstowardsinfinity;673

• Thistheoremholdsinthepresenceofchangingfitnessfunctionsandforlargefinite674populations.675

676