A Model of Spontaneous Remission from Addiction

DOI: 10.4018/IJABE.2019010102



A Model of Spontaneous

Remission From Addiction

Chiara Mocenni, Department of Information Engineering and Mathematics, University of Siena, Siena, Italy Giuseppe Montefrancesco, Prevention of Pathological Addictions, ASL Sudest, Siena, Italy

Silvia Tiezzi, Department of Economics and Statistics, University of Siena, Siena, Italy

ABSTRACT

This article develops a formal model of spontaneous recovery from pathological addiction.Itregardsaddictionasaprogressivesusceptibilitytostochasticenvironmental cues and introduce a cognitive appraisal process in individual decision making dependingonpastaddictionexperiencesandontheirfutureexpectedconsequences. Thisprocessaffectsconsumptionchoicesintwoways.Therewardfromusedecreases withage.Atthesametime,cognitiveincentivesemergethatreducetheprobability ofmakingmistakes.Inadditiontomodelingtheroleofcue-triggeredmistakesin individualdecisionmaking,theanalysishighlightstheroleofotherfactorssuchas subjectiveself-evaluationandcognitivecontrol.Theimplicationsforsocialpolicy andforthetreatmentofdrugandalcoholdependencearediscussed. KEywoRdS

1. INTRodUCTIoN

Addictionisdefinedastheconsequenceofrepeateduseofpsychoactivedrugs.It ischaracterizedbyalossofcontroloverdrugseekingwithharmfuleffectsonthe individualandahighprobabilityofrelapseevenmonthsoryearsaftercessationof drugtaking(Volkow&Fowler2000;Fertigetal.2004;Koobetal.2004).Themain problemistounderstandhowtheindividual,substanceandenvironment-relatedfactors involvedcantriggerthestart,sustainrecurrenceorgeneraterelapse. Economistshavedevelopedtheoriestomodeladdiction,theirintereststemming from the social costs and externalities generated by the consumption of addictive substances. These theories can be loosely classified as generalizations of the rational addiction model (Becker & Murphy 1988). Generalizations allow for the presenceofrandomcuesthatincreasethemarginalutilityofconsumption(Laibson 2001);“projectionbias”(Loewensteinetal2003);present-biasedpreferencesand sophisticatedornaiveexpectations(GruberandKoszegi2001);“temptation”(Gul& Pesendorfer2001)wherepreferencesaredefinedbothoverchosenactionsandover actionsnotchosen.BernheimandRangel(2004)inanattempttoharmonizeeconomic theorywithevidencefrompsychology,theneuroscienceandclinicalpractice,regard addictionasaprogressivesusceptibilitytostochasticenvironmentalcuesthatcan triggermistakenusage1_{,thusexplainingtherelationshipbetweenbehaviorandthe} characteristicsoftheuser,ofthesubstanceandoftheenvironment.Neuroscienceand clinicalpracticehaveshownthataddictivesubstancessystematicallyinterferewiththe properoperationofaprocessusedbythebraintoforecastneartermhedonicrewards and lead to strong impulses to consume that may interfere with higher cognitive control.Thereforeconsumptionchoicesaresometimesdrivenbyarationaldecision makingprocess,sometimesbystrongimpulsesleadingtomistakes,i.e.divergences betweenpreferencesandchoices.

These theories explain several patterns of addictive behavior, but one aspect leftunexplainedisspontaneousremissionalsoknownasnaturalrecovery.Although addictionisdefinedasachronicandpersistentdiseasebythescientificcommunity (seee.g.theAmericanPsychologicalAssociation’sDiagnosticandStatisticalManual ofMentalDisorders,knownastheDSM-V),recentstudieshavecalledintoquestion whetherthisisanaccuraterepresentation(Slutzke2006;Breidenbach&Tse2016). Clinicalpracticeshowsthatnaturalrecoverycharacterizesasubstantialfractionof individualswithahistoryofpathologicaladdictionandthatthisisnotaninfrequent patternofbehaviorinlongtermaddicts.Howeverthereasonsforitarestilltobe understood. Thispaperofferstwocontributions.First,ittriestosolvetheinterestingpuzzleof naturalrecoverybyidentifyingsomeofitsdeterminantsanditsdynamics.Second,it highlightstheroleofcognitiveprocessestoexplainnaturalrecoveryeveninindividuals withanimportantaddictionhistory. BuildingontheworkofMocennietal.(2011),weextendBernheimandRangel (2004)addictiontheorybyintroducinga“cognitiveappraisal”functiondependingon

futureexpectedlossesaswellasonthepastaddictionhistory.Suchprocessleadstoa reductionoftherewardfromuseasthedecisionmakergrowsolder,anditincreases cognitiveincentivescompetingwiththeHedonicForecastingMechanismthusreducing the probability of making mistakes. A similar mechanism, based on the struggle betweentheimpulsiveandreflectivesystems,isproposedbyBechara(2005)inhis neurocognitivetheoryofdecisionmaking.Wealsoexploretheroleofotherfactors, suchaslearningandindividualheterogeneity.Ourmodeliswellsuitedtoexplain thefollowingdynamicsofquittingbehavior:(i)naturalrecoveryoccurringascold turkeyquittingwithoutanexogenousshock;(ii)gradualquittingafteraperiodof decreasingconsumption;and(iii)quittingoccurringafteraseriesoffailedattempts. Performanceanalysisofthisextendedmodeliscarriedout. Ourresultposesahighvalueonpolicymeasuresincreasingcognitivecontrol suchaseducation,creationofcountercuesandpoliciesthathelptheaccumulationof socialcapital,butitdoesnotruleouttheeffectivenessofmoreconventionalpolicy measures,suchasregulationortaxationoflegaladdictivesubstances. Theremainderofthepaperisstructuredasfollows.Sections2and3providea narrativeofaddictionandofnaturalrecovery.Section4containstheformalanalysis. Section 5 concludes. Appendix A reports the results of the stability analysis of equilibria.AppendixBdevelopstheoptimizationmethodandalgorithm.

2. THE NEURoSCIENCE oF AddICTIVE BEHAVIoR

Drugs stimulate the nigrostriatal and corticolimbic dopaminergic systems (Wise 2004)thusincreasingdopamineconcentrationattarget-cells’receptorlevels.These cerebralsystemshaveevolvednottoentertainaddictivesubstances,buttoensurethe survivaloftheindividualbycontrollingbasicfunctionssuchasmatingorsearching forfoodandwater.Whenthesesystemsareengagedbyaddictivesubstances(Fertig etal.2004;Nestler2003)thedopaminereleasethatoccursinthenucleusaccumbens causesspecificemotionalstates(e.g.euphoria)thatarepowerfuldriversofbehavior. Addictivesubstancesproducehigherdopamineconcentrationsthannaturalrewards. Thus,thereisanincentivetorepeatexperienceswiththosesubstances(Kelley& Berridge2002;Bechara2005;Kalivas&Volkow2002).Moreover,whileinthecase ofnaturalrewardsthedevelopmentofhabitsovertimereducesthequalityandquantity ofpleasure,thisdoesnothappenwithaddictivesubstancesastheyactivateeachtime thesamehedonicresponse(Berridge&Robinson2003). Chronicsubstanceabuseinducesprofoundalterationsofthecerebralmechanisms justmentionedforcingtheusertomakecompulsorychoices.Bypowerfullyactivating dopamine transmission, drugs reinforce the associated learning process ending upbyconstrainingtheindividual’sbehavioralchoices(Berke&Hyman2009).In otherwords,drugsseemtoaffectthebasicHedonicForecastingMechanism2_(HFM

henceforth), a simple and fast system for learning correlations between current conditions,decisionsandshorttermrewards.Thereisagrowingconsensusinthe neuroscienceaccordingtowhichaddictionresultsfromtheimpactaddictivesubstances

haveontheHFM.Withrepeateduseofasubstance,thecuesassociatedwithpast consumptioncausetheHFMtoforecastexaggeratedpleasureresponses,creatinga disproportionateimpulsetouse.Thepleasurefollowinguse,theexcessiveandrapid hedonicexpectationinducedbytheHFM,theprogressivefailingofthefrontalcortex tocounterbalancewithrationalchoicesthemorealluringofferofdrugs,allportraya processthatinvariablyregeneratesitselfandseemstohavenoend(Kelley&Berridge 2002;Berridge2004). Althoughdrugaddictionseemstoleadtojustonepossibleresult,forstillunclear reasonsoftenthepatientstopsparticipatingintheineluctabledynamicsofhercaseand ceasestohavethiscompulsionforthedrug.Onecouldsaythatthemultifactoriality sustainingdrugaddictionsometimesceasestoofferthoseprofitsandconveniences considereduptillthenasindispensable.Whenthishappenswithoutprofessionalhelp naturalrecoveryoccurs.

3. NATURAL RECoVERy

Natural recovery is more common than suggested by conventional wisdom and characterizesthewholespectrumofdrugssuchasalcohol(Cunninghamet al.2005; Bischofet al.2003;Weisneret al.2003;Matzgeret al.2005;GrellaandStein2013; Breidenbach&Tse2016),marijuana(Copersinoet al.2006),heroin(Waldorfand Biernacki1979),bingeeating,smoking,sexandgambling(Nathan2003).Longitudinal studieshaveshownevidenceofnaturalrecoveryeveninpathologicalgambling(Slutske 2006)questioningthedefinitionofaddictionasgivenbytheDSM-V. Awell-establishedstrandofliteraturehasfocusedonnaturalrecoveryandseveral reviewsofstudiesexist(Carballoetal2007;Klingemannetal2009;Smart2007; Sobelletal2000;Watson&Sher1998;Klingemannetal.2009),butfindingsin thesereviewshaveyettobesystematizedintoawellformulatedconceptualization explainingwhydecisionstochangeoccur(Klingemannetal.2009). Clinicalandexperimentalresearchhasstudiednaturalrecoveryfromsubstance abuse since the mid-1970s (Vaillant 1982) focusing on triggering mechanisms, maintenancefactorsandontryingtoidentifycommonreasonsforchangeinsubstance use (Prochaska et al. 1992). According to Matzger et al. (2005) factors leading a persontomovefromproblematictonon-problematicalcoholuse,forexample,can beheterogeneous.Inthisstudytwogroupsofproblemdrinkingadults,whoreported drinkinglessattheoneyearfollowup,wereidentifiedinNorthernCalifornia.The firstgroupcamefromaprobabilitysampleinthegeneralpopulation;thesecondwas originatedthroughasurveyofconsecutiveadmissionstopublicandprivatealcohol anddrugproblems.Alogitmodelwasusedtoassessthedeterminantsofsustained remission from problem drinking. The two most frequently endorsed reasons for drinkinglesswerethesameinthetwogroups:acost-benefitanalysisofdrinking,and majorlifechanges.Drinkingcausinghealthproblemswasalsoanimportantreason forquitting.Medicalpersonnelandfamilyinterventionswerefoundtobeunrelatedto improvementsintreateddrinkers.Carballoetal.(2007)recognizecognitiveappraisal

andself-evaluationasacentralaspectofself-changeinotheraddictivebehaviors.In Cunninghametal.(2005)cognitiveappraisalandlifechangesarethemainreasonsfor quitting.Cognitiveappraisalisdescribedasaprocessofself-appraisalofthecostsand benefitsofquitting.Inthelifechangesmotivationthepatients’lifestylesarelinkedto successfulattemptstoquit.Respondentsexperiencingthegreatestreductionintheir negativelifeeventspretopostquitattemptswerehypothesizedtobemostlikelyto havesuccessfullyreducedorquittheiraddiction.Nathan(2003)inhisstudyofnatural recoveryfrompathologicalgambling,alsoarguesthatself-changershavelesssevere drinkinghistoriesandfewersymptomsofdependence.Finally,Breidenbach&Tse (2016)alsoreportednaturalrecoveryfromalcoholanddrugaddictionamongEnglish speakingHongKongresidentsasanevolutionaryprocessoccurringoveralongperiod oftimewithintervalsofcognitiveappraisalandquitattempts. Despitethisempiricalevidence,thereareveryfewmodelsofdecisionmaking describing pathways to natural recovery. The Becker and Murphy (1988) model generatescoldturkeyquittingthroughexogenousshocksorstressfulevents.Suranovic etal.(1999)extendBeckerandMurphytogeneratecold-turkeyquittingofcigarettes’ smokingwithoutrelyingonexogenousshocks.Themotivationtoquitisbasedinstead onchangesintheaddict’sperspectiveasshegrowsolder.Inaddition,thismodel showsthatsomeindividualsmayquitaddictionbygraduallyreducingconsumption overtime.Theseresultsareobtainedbyexplicitlytakingintoaccountthewithdrawal effects(quittingcosts)experiencedwhenuserstrytoquitandbyexplicitrecognition thatthenegativehealtheffectsofaddictiongenerallyappearlateinanindividual’slife. Bothmodelspresupposestandardinter-temporaldecisionmakingimplyingacomplete alignmentofchoicesandtimeconsistentpreferences,therebydenyingthepossibility ofmistakes.Insightsfrompsychologyandtheneurosciencehaveledtonewtheories ofaddictiontryingtobridgethegapbetweenneuroscienceanddecisionmakingand depictingaddictionasaprogressivesusceptibilitytostochasticenvironmentalcues triggeringmistakenusage(Loewenstein1999;Bernheim&Rangel2004).Thesenew theories,however,donotexplicitlymodelpathwaystonaturalrecovery3_. Klingemannetal.(2009)highlightedthefollowingaspotentialdriversofnatural recovery: • Someindividualsconfrontedwithaddictioncanmakeinformeddecisionsand developresolutionstrategies;

• An individual’s capacity to terminate chronic substance misuse is very much afunctionoftheresourcesshehasdevelopedandmaintainedoverthecourse of her life. These resources consist of personal attributes, physical and socio-environmentalstructures,culturaldispositionsandrelatedlifecircumstances; • Consequent-drivenreasons(e.g.particularlifeevents)forrecoverycomparedto drifting-outreasons(e.g.rolechanges,growingolder)occursignificantlymore frequentlyamongthosewithmoresevereaddictionproblems; • Muchself-changeresearchhashighlightedtheimportanceofcognitivedecision processes(e.g.balancing)asacentralcharacteristicofindividualchange;

• Acoreconceptofself-changeresearchistoregardchangeasaprocessthat,in mostcases,occursovertime. Inordertoincludethesefactorsinanaddictiontheory,weextendBernheimand Rangel(2004)modelbyexplicitlyconsideringlearning,self-appraisalandcognitive control.AsshowninthenextSection,naturalrecoveryisonepossibleoutcomeof theextendedmodel. Figure1outlinesaschematicrepresentationofourmodel.Consumptionchoices resultfromthecombinationofexogenouscues,innatepropensity,age,lifestyleand sensitizationtoaddictivegoods.ThereinforcingmechanismleadingtheDMtobecome anaddictcanbeaffectedbycognitivefactorsandbytheevaluationofthepastand futurelossesduetocompulsiveconsumptionofaddictivegoods.

4. A ModEL oF NATURAL RECoVERy

BernheimandRangel’s(2004)theoryisbasedontheassumptionthatsubstances’ consumptionisoftenacue-triggeredmistake.Howeveraddictscandevelopagrowing awarenessleadingtoattemptstocontroltheprocess. Drawingfromthistheory,weconsideradecisionmaker(DMhenceforth)whocan operateineithercoldorhotdecision-makingmode(Loewenstein1996and1999).In hotmode,shealwaysconsumesthesubstanceirrespectiveofhertruepreferences.In coldmodeallalternativesandconsequencesareconsidered,includingthelikelihood

Figure 1. Scheme of the mathematical model. External and innate factors influencing the DM. Sequence of decisions taken by the DM (central path). Hot and cold modes of decision leading to compulsive and non compulsive behaviors (left and right cycles). Natural recovery (triangle) may be activated by increasing costs (loss function) and cognitive factors producing switches in the DM’s behavior (dotted line).

ofenteringthehotmodeinthefuture.Metcalfe&Mischel(1999)adoptedsuchhot/ coldframeworktoexplainthedynamicsofwillpower.Theyalsoreviewedprevious studiesandfieldsofinvestigationinpsychologyinwhichahot/coolframeworkhad beenadopted(Figure2).

Timeisdiscrete,indexedbythenonnegativeintegers,t∈

{

0 1 2, , ,…

}

.Ineachtime periodttheDMfirstselectsalifestyleafromtheset

{

E A R, ,

}

.(e.g.goingtoabar orstayingathomewatchingTV).IflifestyleE,“exposure”,ischosenthereisahigh

probability that the DM will encounter a large number of substance-related cues. ActivityA,“avoidance”,entailsfewersubstance-relatedcuesandmayalsoreduce

sensitivitytothem.ActivityR,“rehabilitation”,impliesacommitmenttoclinical

treatmentthecostofwhichisr_s,anditmayfurtherreduceexposureandsensitivity

tosubstance-relatedcues.TheDMallocatesresourcesbetweenapotentiallyaddictive goodx ∈

{ }

0 1, thepriceofwhichisq,andastandardgood

(

e_s ≥0

)

.Byassumption theDMcannotborroworsave.Eachperiodbeginsincoldmodeandthechoiceof lifestyle,togetherwiththestartingaddictivestatesgivestheprobabilityp_saofcues

triggeringthehotmode.Withsometransitionprobabilityp_T,consumptionofthe

addictivesubstanceinstatesattimetmovestheindividualtoahigheraddictive

states+ 1attimet+ 1,andabstentionmoveshimtoaloweraddictivestates−1at timet+ 1.ThereareS+ 1addictivestateslabeleds=0 1, , ,…S.Thesystemdynamics

isdescribedbytheevolutionofstates_taccordingtothefollowingequation: s min p s p s S if x a E A Max p s t T t T t t t T t + + + − ∈ 1 = { ( 1) (1 ) , } = 1, { , } {1, ( −− + − ∈          1) (1 p s_T) }_t if x_t = 0,a_t { , , }E A R  (1)

Equation(1)impliesthatconsumptioninstatesleadstostatemin S s

{

, _t+1

}

inthe nextperiodwithprobabilityp_T.NouseleadstostateMax

{

1,s_t−1

}

withprobability p_Tfromstates> 1_{andtostate}s= 0_{fromstate}s = 0 .Thevolumeofsubstance-relatedenvironmentalcuesc a,ω

( )

dependsonthelifestyleandonanexogenousstate

ofnatureωdrawnrandomlyfromastatespaceΩaccordingtosomeprobability measureµ.Weassumethefunctionc a,ω

( )

tobedrivenbyanormallydistributed

randomprocesswithvarianceandmeandependingonthelifestylea.Impulsesc a,

( )

ω 

place the DM in hot mode when their intensityM a s a

(

, , ,ω

)

, denoting the DM’s

sensitivitytothecues,exceedssomeexogenouslygiventhresholdMT.Sincepeople

becomesensitizedtocuesthroughrepeateduseM a s a

(

, ', ,ω

)

<M a s a

(

, ", ,ω

)

_fors">s'_, andM a

(

_{, , ,0}a _ω

)

_<MT_{. Moreover,M a s R}

(

_{, , ,ω}

)

_{≤M a s A}

(

_{, , ,ω}

)

_{≤M a s E}

(

_{, , ,ω}

)

_{, i.e. the}

lifestyleaffectstheDMsensitizationtoenvironmentalcues.WhenM a

(

_{, , ,0}a _ω

)

_>MT_ theDMentersthehotmode.WeassumethepowerfunctionM tobelogistic,strictly increasingandtwicecontinuouslydifferentiableins: M c a s a c a M e M e a a a s s ,ω , , ,ω ,ω λ λ

(

)

(

)

=

(

)

+ +

(

−

)

0 0 1 1  (2)

wherea∈

{

R E A, ,

}

andM₀ =M s

(

=0

)

andλisthegrowthrateoftheHFMgenerated

impulses.

AnexampleofthepowerfunctionM isinFigure3depictingfourdifferentphases

drugaddiction,post-addictionstage.InFigure3M ismeasuredontheverticalaxis andsismeasuredonthehorizontalaxis.Inthefirststage,theDMstartsconsuming adrugdrivenbycuriosity,peerpressure,socialfactors(lifestylesaandenvironmental cuesc.) SensitizationtriggersfurtherexperimentationandincreasestheHFM(M weakly increasingins).Inthefirststagesubjectsreacttothedrugstimuliinacontrolled way.Repeateddrugconsumptionleadstothedrugabusestageinwhichsensitization isverypowerfulandcravingisgeneratedbydrugstimuli(M stronglyincreasingin s).Thedrugaddictionstageischaracterizedbytoleranceandphysicaldependence.

Finally, post-addiction is characterized by decreasing sensitization and periods of abstinenceeventhoughtheHFM-generatedimpulsesarestillactive.

WeconsiderT s a

( )

_{, =}

{

_ω∈Ω:M a s a

(

_{, , ,ω}

)

_>MT

}

_{.TheDMentersthehotmodeif}

andonlyifω ∈T s a

( )

, .Lettingp_sa _=µ

(

T s a

( )

_,

)

_{denotetheprobabilityofenteringthe}

hotmodeattimetinaddictivestatesandlifestylea,anincreaseintheaddictive

statesraisestheprobabilityofenteringthehotmodeatanymoment,becausethe

sensitivity to random environmental cues has increased. Sop_sa p s a +1 ≥ ,p a 0 =0 and p_sE p p s A s R ≥ ≥ _{ineachtimeperiod.} InstatestheDMreceivesanimmediatehedonicpayoffw e x a_s

(

_s, ,

)

=u e

( )

_s +v x a_s

( )

,  whereutilityderivedfromnonaddictivegoodsu e

( )

_s isassumedtobeseparablefrom utilityderivedfromaddictiveconsumptionv_s.w_sisincreasing,unbounded,strictly concaveandtwicedifferentiablewithboundedsecondderivativeinthevariablee_s. Moreoverv x a_s u_sa b s a ,

( )

≡ + ,whereu_sarepresentsthebaselinepayoffassociatedwith

successful abstention in states and activity a , andb_sa represents the marginal

instantaneousbenefitfromusetheindividualreceivesinstatesaftertakingactivity a.Bythesameassumption,atanyinstantu_sE u u s A s R ≥ ≥ andu_sE u u u s E s A s A + > + .Future hedonicpayoffsarediscountedusinganexponentialdiscountfactorδ.Choicesin coldmodecorrespondtothesolutionofadynamicstochasticprogrammingproblem withavaluefunctionV_s

( )

θ _{andBellmanequationequalto:} V_h u b V V a x C h a h a x h a h a x h h a x h θ σ δ σ θ σ θ

( )

= + +

(

−

)

( )

+

(

( )max, ∈ − + , ₁ , , 1 1

))

   (3) s t.. 0≤ ≤h S  h− =1 Max

{

1,s−1

}

 h+ =1 min S s

{

, +1

}



C  is the set of decision states

{

( ) ( ) ( ) ( )

E, ,1 E, , , , ,0 A0 R 0

}

;σ_sa x, _{ represents the}

probabilityofconsumingthesubstanceinstatexwithcontingentplan

( )

a x, andθ

isavectorspecifyingthemodelparameters.ThestationarityofEquation(3)follows fromtheassumptionthattheDMtakesherdecisionatthebeginningofeachperiod4_.

Weareinterestedinthechoiceset

( )

E, 0 .Inthiscaseimpulsestousearenot

forcedlycontrolledthroughrehabilitation,butabstinenceoccursforhighenoughMT_,

thethresholdleveloftheimpulses’intensityrequiredtodefeatcognitivecontrol.

4.1. Expected Losses and Past Addiction Histories

DrawingfromSuranovicetal.(1999)weassumetheDMtobeY yearsoldandT Y

( )



isanonaddict’slifeexpectancyatageY .T Y

( )

islinearinY withT Y'

( )

< 0.An addict’slifeexpectancyatageY canberepresentedasT Y

( )

− βH withβbeinga

parameterweightingthereductioninlifeexpectancycausedbythemaximumaddictive stateH . The present value of an addict’s expected future utility streamV  from

V_{Y H} s e b dt t Y T Y Y H s r t Y s ,

( )

= = ( )+ + +        −(− )

∫

β 2  (4)

wherer isthediscountrate;e−r t Y(− ) = δisthediscountfactorattimetandb_sisthe

individual’sexpectedutilityofconsumingtheaddictivegoodattimet.βH +s        2 is theaveragelostlifecausedbythemaximumaddictivestatereachedinthepastand bythecurrentaddictivestates.ForaDMagedY andmaximumaddictivestateH

thepresentvalueoftheexpectedfuturelossesattimetisgivenby5_: L_{Y H} s V_{Y H} s V_{Y H} s T Y Y H s T Y ,

( )

= ,

( )

− ,

(

+

)

= ( )+ + + +        ( )+ 1 1 2 β Y Y H s r t Y s e b dt + +        −(− )

∫

β 2  (5) DifferentiationofEquation(5)withrespecttosleadsto: L_{Y H} s e r T Y b H s s T Y Y ' _, ,

( )

= − − ( )− +                   ( )+ − β β 2 2 ββ β β H s r T Y H s e +         − _{( )}− +                  + 2 2 2 bbs T Y, ( )+ −Y H s+ +        β 1 2  (6) whereL’isthetimederivativeofL.(6)isweaklypositivebecause: e r T Y H s _er T Y H s − ( )− +                 < − ( )− + +    β β 2 1 2                 and: b b s T Y, ( )+ −Y H s+ s T Y, Y H s        ( )+ − ++         ≤ β β 2 2 1  Asaconsequenceof(5)and(6),futurelossesincreasewiththeaddictivestateas higheraddictivestatescutofftheexpectedbenefitsofthefinalmomentsoflife.As theDMgetsolderthelossfunctionL_{Y H,}

( )

s rises: ∂ ∂ =

(

( )

+

)

−

(

( )

+

)

( )+ − − ( )− ( )+ − L Y T Y b e T Y b Y s T Y Y s rT Y s s T Y Y ' ' , , 1 1 β β β ss rT Y s T Y Y s r t Y s e re b dt + ( ) − ( )−( )+ ( )+ + −(− ) +

_∫

≥ 1 1 β β 00  (7)

Inaddition,futurelossesalsoincreasewithage,asthediscountfactorincreases asonegestsclosertotheendoflife.Ontheotherhand,end-of-lifeutilitieswillbe weightedlessatyoungeragesbecausetheyarefartoodistant. Futureexpectedlossesaffectconsumptionbehaviorinthefollowingways:(i)by increasingMT_{andbydecreasingtheprobabilityofenteringthehotmode;(ii)by} affectingtheBellmanEquation(3)throughthedecreasedprobabilityofuseσ_;(iii) byerodingthemarginalbenefitinV_Sasb_sa L Y H − _, .Theeffectofpastexperiencesis accountedforintroducingthevariableH =Max s

{ }

_i ,i=0 1, , ,… −t 1indicatingthe DM’smaximumaddictivestatereacheduptothecurrentperiodt.

4.2. Increasing Cognitive Appraisal

WeconsideravariationoftheM functionandaninitialconditiondenotingthea

priorilevelofcognitivecontrol.ThepopulationofDMsissplitbetweennon addicts

I₀ ≥M₀andpotential addictsI₀ <M₀(OrphanidesandZervos1995).Non addicts

mayneverbecomeaddictediftheircompetingcognitiveincentivesarehighenough

Figure 4. M and I functions corresponding to different assumptions on I₀. Dashed line: I₀ <M₀ (for

γL_{Y H,} <M₀−I₀); solid line: I₀ =M₀ (for γL_{Y H,} =M₀−I₀); dashdot line: I₀ >M₀ (for

toavoidthehotmode.WhenI₀ <M₀theDMisnotyetacquaintedwiththesubstance

anditspotentialconsequences.InthiscasetheI functionforpotential addictsis

relatedtothelossfunctionL_{Y H,}

( )

s inthefollowingway: I s Y I e I e s s ,

( )

= +

(

−

)

0 0 1 1 λ λ  (8) whereλisthesameasinEquation(2).Theinitialconditionisdefinedas: I =I +γg L

( )

 (9)

whereI0andgaccountforDMsheterogeneity,andtheincreasinginLY H, functiong

isdefinedastheadditionalcognitivecontrolarisingfromthepresentvalueoffuture expectedlossesL_{Y H,} .Isatisfiesthefollowingproperties:I s Y

(

',

)

<I s Y

(

",

)

_fors'<s"_;

I s Y

(

, '

)

<I s Y

(

, "

)

_forY'<Y"_{.Moreover,itisstrictlyincreasingin}L_{Y H,}

( )

s andtwice

continuouslydifferentiableins.γindicatesthepresenceoflearningprocessesrelated

tothepasthistoryofconsumption,ageandawarenessoffutureexpectedlosses.We assume0≤ ≤γ 1withγ=1implyingperfectlearningandγ = 0implyingabsence

Figure 5. Frequency of decisions for each addictive state s

of learning. GivenI₀, the presence of learning may drive cognitive incentives to

overridetheHFMimpulsestouseforsufficientlyhighY andH.InFigure4weplot

theIfunctionagainsttheaddictivestatescorrespondingtodifferentvaluesofthe

initialconditionI₀.

Astimetandaddictivestatesincrease,theI functionmovesupforanyγ_so

that different values ofI  are associated with the sames. When theI  function

overridestheHFM,theprobabilityofenteringthehotmodeisdriventozero.Similar

Table 3. Summary statistics on initial level of cognitive control l0

dynamicsoccurwhenI₀increases.Naturalrecoverycanthusbethelong-termoutcome

ofacompetitionbetweenHFM-generatedimpulsesandcognitiveincentivesI. AppendixAdetailstheconditionsunderwhichI s Y

( )

, >M s a

(

, ,ω_a

)

_{andshowsthat}

the equilibrium solutionseq _{= 1 is globally asymptotically stable for the dynamic}

systemdescribedin(1).Inparticular,weprovethefollowingpropositionsinAppendix A.

Proposition 1:

(1)HighervaluesofI₀decreasetheprobabilityp_sa andthustheprobabilities_σ s E,0

andσ_sA,0_;

(2)Highervaluesofγdecreasetheprobabilityp_saandthustheprobabilities_σ s E,0

andσ_sA,0_.

Proposition 2:

Theequilibriumsolutions₌seqisgloballyasymptoticallystable.

4.3. Simulations and discussion

Thissectionshowsnumericalsimulationsoftheextendedmodelanddiscussesthe theoreticalresults.Thenumericalsolutionsareobtainedsolvingthestochasticdynamic programmingproblemimpliedbythemodelanddescribedinAppendixB.Table1 containsadescriptionofthemainvariablesandthemodel’sparameters. Thenumericalassumptionsonthecharacteristicsofthesubstanceandtheuser aretakenfromBernheimandRangel(2005).WeconsiderS = 50;y_s=800$;t=1 week;simulationlength=1000periods(20years);costofaddictivesubstance= 200$;costofrehabilitation=250$;decisionsset

( ) ( ) ( )

E, ,1 E, , ,0 A0 ,

( )

R, 0 .Figure5 showstheprobabilityofeachchoiceasafunctionoftheaddictivestates. Tables2and3showsomesummarystatisticsofthesimulationsperformedby varyingincomeyandinitialcognitivecontrolI₀withrespecttotheirbaselinevalue y I*_, * 0

(

)

. Means (1st column), Standard Deviations (STD, 2nd column), Absolute Maxima(3rdcolumn)andtimeperiodsatwhichnaturalrecoveryoccurs(5thcolumn) areshown.STDofMax(4thcolumn)isthestandarddeviationoftheabsolutemaximum correspondingtoeachrun.Allstatisticsareevaluatedon10runsoftheevolutionof theaddictivestates.ChoicesovertimeareshowninFigure6.Figure7showsthe evolutionoftheaddictivestatesleadingtonaturalrecovery. Additionalresultsconcerntherelationshipbetweenthemodelsolutionsandthe parameters.Inparticular,weprovethefollowingpropositionsinAppendixA. Proposition 3: AssumefixedalltheparametersinϕexceptforI₀: (1)OnaverageanincreaseinI₀lengthensthetimeintervalbetweentheinitial useandthemaximumaddictivestateHandshortenstheintervalbetweenH andnaturalrecovery. (2)OnaverageanincreaseinI₀lowersthemaximumaddictivestateH . Proposition 4:

Assumefixedalltheparametersinϕ_{exceptfor}γ_{.Anincreasein}γ_{shorthensthe}

intervalbetweentheinitialuseandthemaximumaddictivestateHandanticipates

naturalrecovery.

Both the numerical simulations and the theoretical results show that natural recoveryisapossibleoutcomeofanextendedmodelaccountingforlearning,cognitive controlandself-appraisal.Identifyingfactorsleadingtonaturalrecoveryanditslong-rundynamicsmayhelpdesigningpolicymeasuresaimedatreducingconsumption ofaddictivegoods.Ifconsumersaresometimesrationalandsometimesdrivenby cue-triggeredmistakes,measuressuchastaxationoflegaladdictivesubstancesor strictregulationmayonlyraisethecostofconsumption.However,ifspontaneous remissionoccursthroughincreasedawarenessoffutureexpectedcostsandlearning frompastexperiences,standardpublicpolicyapproachescanstillplayarole.The

implication of our model is that more attention should be paid to education and information policy measures relative to health and pharmacological ones. Policy strategiesdifferentiatedbytheageprofileofthepatientscouldbeuseful.Inyoung consumers cognitive therapies, education and information campaigns can have a positiveimpactnotonlytodiscourageinitialexperimentation,butalsoonI0andγ, andcanhelpindividualsactivatecognitivecontrolmechanisms.Trosclairetal.(2002) stressthatmoreeducatedindividualsarefarmorelikelytosuccessfullyquitsmoking, forexample,aseducationhelpsthemactivatingthecompetingcognitiveincentives necessarytooverridetheHFM.Oldestconsumers,ontheotherhand,maybemore responsivetoregulationbecausetheircognitivesystemismoredevelopedandmore likelytoprevailoverimpulsestouse.

5. CoNCLUSIoN

Weproposeadecisionmakingmodelexplaininghowevenlongtermaddictsmay findtheirwayoutofsubstanceabusewithouttheutilizationofprofessionalhelp. Eventhoughnaturalrecoverycharacterizesasubstantialfractionofindividualswith ahistoryofpathologicaladdiction,researchisstillscarce.Spontaneousremission becomes a possibility when additional decision making factors, neglected by the previousliterature,aretakenintoaccount.Drawingfromclinicalandexperimental research,weintroduceacognitiveappraisalfunctiondependingonpastaddiction historiesaswellasonfutureexpectedconsequencesofaddictiveconsumption.This affectsthedecisionmakerintwoways:iterodesthepayofffromuseasthedecision makergrowsolderanditincreasesthecognitivecontrolcompetingwiththehedonic impulsestousethusreducingtheprobabilityofenteringthehotmode. Futureresearchcouldfocusonempiricaltestsandcalibrationofthemodel,if appropriate longitudinal data are available. The estimated parameters incorporate informationonindividualtraitsthatmaybecrucialfornaturalrecovery.Moreover, parameters estimation would allow classification of population groups based on theiraddictivebehavior.Thisinformationcouldthenbeusedtodesignappropriate addictioncontrolmeasures.

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ENdNoTES

1_{ BernheimandRangel(2004)analysisisrelatedtopreviousworkbyLoewenstein} (1996and1999)onthe“cold-to-hotempathygap”. 2_{ FromBernheimandRangel2004.} 3_{ Bernheim&Rangel(2004)actuallymentionthepossibilityoftheirmodelto} generatenaturalrecovery.

4_{  Bernheim and Rangel (2004) show that this model generates a number of}

addiction patterns. Unsuccessful attempts to quit occur when there is an unanticipated or anticipated and sufficiently slow shift in parameters

θ_s =

(

p u b_s, ,_{s s}

)

from

θ '

to

θ "

.Cue-triggered recidivismisassociatedwithhigh exposuretorelativelyintensecues,e.g.highrealizationsofc(a,ω).Self-described

mistakesinwhichtheDMchooses(E, 0)or(A, 0)incoldmode,butthenhe

entersthehotmode.Self-control through pre-commitmentgivenbythechoice (R,0)implyingacostlypre-commitment.Self-control through behavioral and

cognitive therapythroughchoice(A,0)implyingcostlycueavoidance.

5_{ Inwritingequation(5)wedonotaccountfortransitionprobabilitiesaffecting}

the evolution of addictive state

s

, because the DM evaluates future losses independentlyofthespeedoftransitionbetweenaddictivestates.

6_{ Stateddifferently,thereexistsasubsetoftherelevantparameterssatisfyingthe}

conditionsleadingtonaturalrecovery.

7_{ Tosimplifythenotationweomit,henceforth,thetimeindexfromvariablesand}

APPENdIX A: STABILITy oF THE EQUILIBRIUM SoLUTIoN

Proposition 1:

(1)HighervaluesofI₀decreasetheprobabilityp_sa _{andthustheprobabilitiesσ} s E,0_

andσ_sA,0_;

(2)Highervaluesofγ_{decreasetheprobability}p_sa_{andthustheprobabilitiesσ} s E,0_ andσ_sA,0. Proof: (1)LetI₀'_andI 0 "_{betwodistinctinitialconditionsoftheI function,suchthat}

I₀' <I₀" . From Equation (A8) _{I s Y I( , , ) < ( , , )I s Y I}

0 0

' "  _{∀s= 0,1, ,…S}  and

T s a I( , , )₀" _⊂T s a I( , , )₀' _{.Itfollowsthatµ( ( , , ')) > ( ( , ,µ ))}

0 0

T s a I T s a I " .

(2)Analogously,I s Y( , , ') < ( , ,γ I s Y γ ")forγ' < "γ andµ( ( , , ')) > ( ( , ,T s a γ µT s a γ")). Wenextshowthattheequilibriumsolutionseq _{= 1}isgloballyasymptoticallystable forthedynamicsystemdescribedby(1).(1)isahybriddynamicsystemsasitevolves accordingtodifferentdynamicsdependingonthespecificpointinthestate-input spaceunderconsideration.Ingeneral,PiecewiseAffineSystemsallowtoconsider fundamentalhybridfeaturessuchaslinear-thresholdeventsandmodeswitching.In ourcasetheregimeshiftsdependontheDM’schoicesateachtimeperiodandthe resultingdynamicsystemsis: s s if x s if x s if x s S x s t t t t t t t t t t + + − ∧ ∨ ∧  1 = 1, = 1, 1, = 0, , ( = 1 = ) ( = 0 = 1)        Inthefirsttworegimes,noequilibriumsolutionsexistandthedynamicsisalways increasingordecreasing;inthethirddynamicstherearetwoequilibria:seq1 _{= 1}and seq2 ₌S.Asymptoticstabilityofoneofthemcorrespondstoeitheraddictivebehavior

leadingtochronicaddiction

(

s =S

)

ortonaturalrecovery

(

s = 1

)

.Anyoscillating

dynamics is due to shifts or to transient behavior. In order to study the stability propertiesofthedynamicsystemwefocusonlyonthethirdregimeandonthetwo single-pointsetsM = {seq1}and_{N = {s}eq2_}.

Proposition 2:

Theequilibriumsolutions=seq1 isgloballyasymptoticallystable. Proof:

LetL s( ) =_t V_max−V s_t( )_{t beafunctiondefinedintheopenset}G= {0,1,2, ,… S−1}_of thevaluesreachedbythestatevariables.LisacontinuousonGLiapunov

LonthesetGofthestatevariables.Differentcolorscorrespondtosimple

moving averages(SMA) ofL:L (green);SMA L( , 3) (blue);SMA L( , 5) (red);

SMA L( , 7)(black).

M isthelargestinvariantsetinGandGisaboundedopenpositivelyinvariantset.

Then,thetheoremonasymptoticstabilityofthesetM (LaSalle,1997)showsthat

theequilibriums=seq1 isasymptoticallystableon_G.Thiscompletestheproof.

Since the loss function decreases the instantaneous marginal benefit from use we expecttheDMtochoose(E,0)whenincoldmodeandforanumberoftimeperiods sufficienttogeneratenaturalrecovery6_.

Nowletϕbetheparameters’vector,ϕ= ( , , , , ,δr q y I M_{s 0 0}, )γ suchthatnaturalrecovery

mayoccur.

Proposition 3:

AssumefixedalltheparametersinϕexceptforI₀:

(1)OnaverageanincreaseinI₀lengthensthetimeintervalbetweentheinitial

useandthemaximumaddictivestateHandshortenstheintervalbetweenH

andnaturalrecovery.

(2)OnaverageanincreaseinI₀lowersthemaximumaddictivestateH .

Proof:

(1)GivenI₀ =I₀+ γL_{Y H,} ,anincreaseinI₀isdeterminedbyachangeinthea

priori level of cognitive controlI₀. For a given stochastic processω and

lifestylea, this causes p_sa_{ to decrease (see Proposition 1) at eacht thus}

reducing consumption in hot mode and reducing the velocity at whichs

increases. (2)LetI₀'and_I

0

"betwodistinctinitialconditionsofthe_I function,suchthat

I₀' <I₀"_{. The maximum levels ofsH I'( )}'

0  andH I" "

( )₀  are reached at two differenttimeinstantst 'andt ".From(i)itfollowsthatt'≤t".Sinceby definitionL H Y( , )isincreasingintime,H I"_{( )}" H I"_{( )}'

0 ≤ 0 .

Proposition 4:

Assumefixedalltheparametersinϕ_{exceptfor}γ_{.Anincreasein}γ_{shorthensthe}

intervalbetweentheinitialuseandthemaximumaddictivestateHandanticipates

naturalrecovery.

Proof:

AdecreaseinγshiftstheI functiondownwards.FromProposition1thisimplies

anincreaseinp_sa _{whichcausesadelayintheeffectsofthelossfunction.}

Propositions3and4implythattheprocessleadingtoadvancedaddictionstagescan besloweddownbyincreasingI₀orγ_.

APPENdIX B: THE SToCHASTIC dyNAMIC

PRoGRAMMING PRoBLEM

Numericalsimulationsareobtainedbyassigningvaluestothemodelparametersand bymaximizingthevalueFunction(3).

TheparametersoftheM andI functionsare:λ = 0.1,M₀ = 0.09,I₀ = 0.07and

γ = 1.c a( , )ω isspecifiedbyc a( ,ω_a) =k₁+k₂ω_a,whereω_aisanormallydistributed

random process with varianceσ2 _{= 1} and mean depending on lifestyle_a. The

parametersk₁andk₂dependonlifestylea.

Takingaquadraticapproximationinalltheargumentsexcepte_stheinstantaneous payofffunctionw_sis7_: w e x a_{s s} b_sa w s u e b u s s a s a , ,

(

)

= +

( )

+

( )

= +  (B1) with:

b_sa x x xs x a xx a xs a =α +α +α 2 2 2 _ w s _sas sss xs a xs a

( )

=α +α +α 2 2 2 _ u e

( )

_s =α_elog

( )

e_s +α_eelog

( )

e_s +α_xexe+α_sese u_sa w s u e s =

( )

+

( )



w x( )andu e( )_s areincreasingandconcaveinx(potentiallyaddictivegood)and e(nonaddictivegood);w s( )isdecreasinginsandtheinteractiontermsα_xeandα_se

arezerobytheseparabilityassumption.Monotonicityandconcavityofw x( )andu e( )_s  followfromstandardarguments,whereasthepropertiesofw s( )incorporatetheeffect ofpastuseoncurrentwellbeing,i.e.tolerance,deteriorationofhealth,illness(Figure 9). Thepayofffunctionw_{s t,} isspecifiedbyEquation(B1),whereα_x = 10,α_xx = 0.5− _, α_s = 1.0− _,α_xs = 0.9,α_ss = 0.1− _,α_e = 30,α_ee = 1− _,e_s =y_s.