PROJECTION BIAS IN PREDICTING FUTURE UTILITY

PROJECTION BIAS IN PREDICTING FUTURE UTILITY

GEORGE LOEWENSTEIN

TED OrsquoDONOGHUE

MATTHEW RABIN

People exaggerate the degree to which their future tastes will resemble theircurrent tastes We present evidence from a variety of domains which demon-strates the prevalence of such projection bias develop a formal model of it and usethis model to demonstrate its importance in economic environments We showthat when people exhibit habit formation projection bias leads people to consumetoo much early in life and to decide as time passes to consume moremdashand savelessmdashthan originally planned Projection bias can also lead to misguided pur-chases of durable goods We discuss a number of additional applications andimplications

The great source of both the misery and disorders of humanlife seems to arise from over-rating the difference betweenone permanent situation and another Avarice over-rates thedifference between poverty and riches ambition that be-tween a private and a public station vain-glory that betweenobscurity and extensive reputationmdashAdam Smith The The-ory of Moral Sentiments [2002 p 173 IIIiii31]

I INTRODUCTION

Optimal decision-making often requires a prediction of fu-ture tastes and future tastes may differ from current tastes dueto such factors as habit formation day-to-day mood uctuationssocial inuences maturation and changes in the environment

For helpful comments we are grateful to Erik Eyster Christopher Harrisand members of the Russell Sage Foundation Behavioral Economics Roundtableseminar participants at Cornell University Yale University Harvard UniversityUniversity of Michigan University of Texas Syracuse University London Schoolof Economics University of Zurich the Toulouse Conference on Psychology andEconomics and the Jerome Levy Institute and Lawrence Katz Edward Glaeserand an anonymous referee We especially thank Colin Camerer and Drazen Prelecfor very helpful discussions at the formative stages of this project For researchassistance we thank Kitt Carpenter Erik Eyster Jeffrey Holman David Huff-man Christopher Meissner and Mandar Oak For nancial support Loewensteinthanks the Center for the Study of Human Dimensions of Global Change atCarnegie Mellon University (NSF Grant SBR-9521914) OrsquoDonoghue and Rabinthank the National Science Foundation (Awards SBR-9709485 SES-0078796and SES-0079266) and Rabin thanks the Russell Sage MacArthur and SloanFoundations This research was started while Loewenstein and Rabin were Fel-lows at the Center for Advanced Study in the Behavioral Sciences supported byNSF Grant SBR-960123 and they are very grateful for the Centerrsquos hospitalityand the National Science Foundationrsquos support

copy 2003 by the President and Fellows of Harvard College and the Massachusetts Institute ofTechnologyThe Quarterly Journal of Economics November 2003

1209

When making summer vacation plans during the cold of winterpeople must predict what vacations will be most enjoyable duringthe heat of summer When ordering food at the beginning of ameal people must predict how hungry they will be at the end ofthe meal When contemplating smoking cigarettes or indulging inother habit-forming substances people must predict how thisconsumption will affect their future desire for and enjoyment ofthese substances

In this paper we provide evidence for formalize and explorethe implications of a general bias in the prediction of futuretastes people tend to understand qualitatively the directions inwhich their tastes will change but systematically underestimatethe magnitudes of these changes Hence they tend to exaggeratethe degree to which their future tastes will resemble their currenttastes Such projection bias may cause people making summervacation plans in the winter to choose overly warm destinationsdiners to order too much food at the beginning of meals andpeople unaddicted to cigarettes to underestimate the power ofand drawbacks of addiction

In Section II we review evidence from a variety of domainssupporting the existence of projection bias People underappreci-ate the effects of long-term changes in tastes such as those thatresult from adaptation to a shifting standard of living People alsounderappreciate the effects of frequently uctuating tastes suchas uctuating hunger Indeed virtually all evidence we are fa-miliar with on misprediction of future tastes is consistent withprojection bias

In Section III we develop a formal model of projection bias Tox ideas suppose that a personrsquos instantaneous utility can bewritten as u(cs) where c is her consumption and s is a ldquostaterdquothat parameterizes her tastes Suppose further that the personwith current state s9 must predict her tastes at a time in thefuture when her state will be s Consistent with evidence thatpeople tend to understand the qualitative nature of changes intastes but underestimate the degree of change we assume thatthe personrsquos prediction of her own future preferences u(csus9)lies somewhere ldquoin betweenrdquo her true future tastes u(cs) and hercurrent tastes u(cs9) Our formal analysis in this paper assumesthat u(cs us9) is a simple linear combination of u(cs) and u(cs9)which we refer to as simple projection bias

Because projection bias leads to discrepancies between pre-dicted and subsequently realized utilities it implies that a per-sonrsquos behavior need not correspond to correct intertemporal util-

1210 QUARTERLY JOURNAL OF ECONOMICS

ity maximization For instance if current consumption has dele-terious effects on future well-being and projection bias leads theperson to underappreciate these effects she may overconsumerelative to what would maximize her true intertemporal utilityMoreover as tastes change over time in ways she does not pre-dict a person makes plans that she may end up not carrying outthat is projection bias can lead to dynamic inconsistency Astressed undergraduate who underappreciates the addictivenessof cigarettes for instance might start smoking with the plan ofquitting upon graduation only to continue smoking after gradu-ation once she becomes addicted

To demonstrate the potential economic importance of projec-tion bias in Sections IV and V we formally analyze two economicenvironments Section IV explores the implications of projectionbias in a life-cycle consumption model with habit formationWhen consumption is habit-forming a person should rationallypursue an increasing consumption prole so that she is alwaysconsuming more than she is accustomed to Projection bias leadsa person to underappreciate the impact of current consumptionon future utility and hence to consume too much early in life andtoo little late in life relative to what would be optimal Moreinteresting as time passes and the person habituates to higherconsumption levels she may decide to consume more than shehad earlier planned hence projection bias can cause saving to fallshort of intentions Finally as the person gets accustomed tohigher consumption levels she also values income more highlyand hence might decide to work more (or retire later) than shehad earlier planned

In Section V we show how projection bias can cause mis-guided purchases of durable goods The satisfaction that a personderives from a durable good often uctuates from day to day andprojection bias leads a person to underappreciate how much herfuture valuations may differ from her current valuation As aresult people will overvalue the good on high-value days andundervalue it on low-value days A person making a one-timebuying decision is therefore equally likely to buy when she shouldnot or not to buy when she should However if the person hasmultiple opportunities to buy and (as is typically the case) un-buying is more difcult than buying projection bias will lead onaverage to overpurchasing of durable goods

We believe that projection bias is important for many eco-nomic applications and that it can provide an intuitive andparsimonious account for many phenomena that are otherwise

1211PROJECTION BIAS

difcult to explain In Section VI we extrapolate from our formalanalysis in Sections IV and V and discuss some of these addi-tional implications We conclude in Section VII

II EVIDENCE OF PROJECTION BIAS

In this section we review evidence from a variety of domainssupporting the existence of projection bias1 A common type oftaste change is adaptation people have a remarkable ability toadapt to major changes in their life circumstances such as ac-quiring serious medical conditions moving to different climatesand changing occupations (see Helson [1964] and Frederick andLoewenstein [1999] for a recent review)2 Moreover there is agreat deal of evidence that people underappreciate the extent ofsuch adaptation Specically by comparing a ldquocontrolrdquo grouprsquospredictions for how some major change would affect their lives tothe self-reports of people who have actually experienced thatchange a number of studies suggest that people overestimate theimpact of major changes on their long-run level of happiness

In the medical domain cross-sectional studies have consis-tently found that nonpatientsrsquo predictions of the quality of lifeassociated with serious medical conditions are lower than actualpatientsrsquo self-reported quality of life For instance Sackett andTorrance [1978] nd that nonpatients predict that chronic dialy-sis would yield a quality of life of 039 whereas dialysis patientsreport a quality of life of 056 (on a 0 to 1 scale on which 0 meansas bad as death and 1 means perfect health) Boyd et al [1990]nd analogous cross-sectional results with regard to colostomiesThe same pattern also shows up in longitudinal studies JepsonLoewenstein and Ubel [2001] asked people waiting for a kidneytransplant to predict what their quality of life would be one yearlater if they did or did not receive a transplant and then asked

1 See Loewenstein and Schkade [1999] for a summary of much of the evi-dence presented in this section as well as for a discussion of the psychologicalmechanismsthat underlie projection bias Also see Loewenstein OrsquoDonoghue andRabin [2002] for a more extensive discussion of this evidence

2 There are some exceptions to this rule First there are a variety of factorsthat impede adaptation such as uncertainty about whether a situation is perma-nent and repeated reminders of the original situation Second some studies havefound that people do not seem to adapt to noise indeed if anything they seem tobecome increasingly irritated by it (for an overview see Weinstein [1982]) More-over noise is the one example we know of that might contradict our assertion thatpeople understand the direction in which tastes change because people seem topredict that they will adapt when in fact they tend to become more irritated


those same people one year later to report their quality of lifePatients who received transplants predicted a higher quality oflife than they ended up reporting and those who did not predicteda lower quality of life than they ended up reporting Sieff Dawesand Loewenstein [1999] nd similar longitudinal results for peo-ple testing for HIV

Outside the medical domain Gilbert et al [1998] compared(among other things) assistant professorsrsquo predictions of the im-pact of getting or being denied tenure to the self-reports of formerassistant professors and Loewenstein and Frederick [1997] com-pared the predictions by survey respondents of how variousevents (eg a decline in sport shing and an increase in thenumber of coffee shops) would affect their well-being over thenext decade to the self-reports of other respondents about howactual events in the past decade had affected their well-being Aclear pattern emerged in both studies those making prospectivepredictions expected future changes to affect their well-beingmore than those making retrospective evaluations reported thatmatched changes in the past had affected their well-being

While there are alternative explanations for the resultsabove other research suggests that they are driven in large partby underappreciation of adaptation First in the medical domainrecent research by Ubel Loewenstein and Jepson [2003] showsthat it is sometimes possible to ldquodebiasrdquo peoplemdashto bring nonpa-tientsrsquo predictions closer to patientsrsquo self-reportsmdashby inducingthem to think more carefully about adaptation which suggeststhat underappreciation of adaptation plays a signicant role inthe discrepancy Second a number of ongoing studies are rulingout other explanations For instance a commonly mentionedalternative is ldquoresponse normingrdquomdashchronic dialysis patients forinstance might interpret a 08 on a 0-to-1 scale differently fromnonpatientsmdashbut Baron et al [forthcoming] found that makingthe scales more precise only increases the discrepancy3 Finallyandperhaps more importantly analogousresults are found in experi-ments on shorter term changes in tastes for which these alter-native explanations do not hold we turn to such evidence next

3 The other main explanation that has been offered is a ldquofocusing illusionrdquomdashthat people exaggerate the impact of anything their attention is focused onincluding disabilities [Schkade and Kahneman 1998 Wilson et al 2000] How-ever Ubel Loewenstein and Jepson [2003] also found that a wide range ofldquodefocusingrdquo interventions actually decreased rather than increased nonpatientsrsquoestimates of patientsrsquo quality of life

1213PROJECTION BIAS

A prevalent experimental nding is the endowment effectpeople tend to value an object (such as a coffee mug) more highlyif they possess it than if they do not4 The usual explanation isthat people adapt to owning or not owning the object and thatthere is more pain upon parting with the object than there is joyupon obtaining the object An underappreciation of this adapta-tion implies that unendowed subjects should underestimate byhow much becoming endowed will increase their valuation andthat endowed subjects should underestimate by how much be-coming unendowed will decrease their valuation Van BovenDunning and Loewenstein [2000] nd cross-sectional evidence ofboth predictions In one experiment the usual endowment effectwas replicated by eliciting selling prices from subjects endowedwith coffee mugs and buying prices from subjects not endowed(average selling price 5 $637 average buying price 5 $185)Sellers were then asked to estimate how much buyers would payand buyers were asked to estimate how much sellers wouldcharge with subjects rewarded for accurate predictions Consis-tent with projection bias the average estimate of sellers ($393)was less than their own average selling price but more than theaverage buying price and the average estimate of buyers ($439)was more than their own average buying price but less than theaverage selling price Loewenstein and Adler [1995] provide lon-gitudinal evidence of the former prediction In one study subjectswere shown a coffee mug told to imagine that they had beengiven one but had the opportunity to exchange it for cash andthen lled out a form that elicited their predicted reservationvalues After a delay they were actually given the mug and thenasked to complete an identical form that elicited their actualreservation values Again consistent with projection bias thepredicted selling prices were signicantly lower than the actualselling prices

There is also considerable evidence on underappreciation ofthe effects of hunger This evidence is particularly valuable be-cause it demonstrates that the same basic pattern of mispredic-tionmdash understanding the direction of taste changes but underap-preciating the magnitude of the changesmdashshows up for othertypes of taste changes besides adaptation and it can show up

4 The endowment effect was rst discussed by Thaler [1980] see KahnemanKnetsch and Thaler [1991] for a review


even for taste changes with which people have ample experienceand hence ought to understand well

Several studies lend support to the folk wisdom that shop-ping on an empty stomach leads people to buy too much [Nisbettand Kanouse 1968 Gilbert Gill and Wilson 2002] This phe-nomenon can be interpreted as a manifestation of projection biaspeople who are hungry act as if their future taste for food willreect such hunger Read and van Leeuwen [1998] provide evensharper evidence of projection bias with respect to hunger Ofceworkers were asked to choose between healthy snacks and un-healthy snacks that they would receive in one week either at atime when they should expect to be hungry (late in the afternoon)or satiated (immediately after lunch)5 Subjects were approachedto make the choice either when they were hungry (late in theafternoon) or satiated (immediately after lunch) As depicted inTable I people who expected to be hungry the next week weremore likely to opt for unhealthy snacks presumably reecting anincreased taste for unhealthy snacks in the hungry state but inaddition people who were hungry when they made the choice weremore likely to opt for unhealthy snacks suggesting that peoplewere projecting their current tastes onto their future tastes

Indeed if we interpret the main diagonalmdashthe hungry-hun-gry and satiated-satiated conditionsmdashas reecting true prefer-ences the data t exactly the pattern of projection bias For thosesubjects who are currently hungry but expect to be satiated theyunderstand the direction in which their tastes will change as theybecome satiatedmdashfewer choose the unhealthy snack than in thehungry-hungry conditionmdash but they underestimate the magni-

5 The healthy snacks were apples and bananas the unhealthy snacks werecrisps borrelnoten Mars Bars and Snickers Bars We adopt the terminologyhealthy and unhealthy from the experimenters but none of the snacks werethusly labeled to the subjects

TABLE IPERCENTAGE OF SUBJECTS CHOOSING UNHEALTHY SNACK

(FROM READ AND VAN LEEUWEN [1998])

Future hunger

Hungry Satiated

Current Hungry 78 42Hunger Satiated 56 26

1215PROJECTION BIAS

tude of this changemdashmore choose the unhealthy snack than in thesatiated-satiated condition An analogous conclusion holds forsubjects who are currently satiated and expect to be hungry

While we have limited our detailed discussion to a fewrealms there is considerable further evidence that projection biasoperates across a broad array of domains Indeed virtually allevidence that we are aware of is consistent with projection bias(except possibly noise as discussed in footnote 2)6 Our goal in theremainder of this paper is to demonstrate its potential impor-tance for economics

III THE MODEL

In this section we build a formal model of projection bias Todescribe changes in tastes we use the apparatus of state-depen-dent utility Suppose that a personrsquos instantaneous utility inperiod t which captures her tastes is given by u(ctst) where ct

is her period t consumption The variable st her ldquostaterdquo param-eterizes her tastes The state might reect past behavior as whenpast consumption of a good determines current addiction to thatgood or exogenous factors as when uctuations in serotoninlevels affect mood or when peer pressure affects the benets andcosts of current behavior7

Next consider a person currently with state s9 who is at-tempting to predict her future instantaneous utility from consum-ing c in state s that is she is trying to predict u(cs) Let u(cs us9)denote her prediction If she were accurate her predicted utilitywould equal true utility or u(cs us9) 5 u(cs) But the evidence inSection II suggests that while people understand the qualitative

6 Other domains for which there is evidence consistent with projection biasinclude sexual arousal [Loewenstein Nagin and Paternoster 1997] pain [Readand Loewenstein 1999] thirst [Van Boven and Loewenstein 2003] fear [VanBoven et al 2003] and heroin craving [Giordano et al 2001] See also Loewen-steinrsquos [1996 1999] discussion of hotcold empathy gaps wherein individuals whoare in cold visceral states underappreciate the impact of hot visceral states ontheir own behavior

7 By ldquoconsumptionrdquo we mean any current physical experience that is rele-vant for current well-beingmdashin addition to literal consumption of goods thismight include experiencing a health outcome being exposed to noise or owning anobject Just as the utility from consuming goods might change over time theutility from these other types of experiences might change over time and wecapture such effects with the ldquostaterdquo variable For instance the utility (quality oflife) from being a chronic-dialysis patient might depend on how accustomed theperson is to being a chronic-dialysis patient in this case consumption is being achronic-dialysis patient and the state reects how accustomed the person is tobeing a chronic-dialysis patient


nature of changes in their tastes they underestimate the magni-tude of these changes Roughly speaking this projection biasmeans that a personrsquos predicted utility u(cs us9) lies ldquoin betweenrdquoher true future utility u(cs) and her utility given her currentstate u(cs9)8 In this paper we consider a particularly simpleform of projection bias

DEFINITION 1 Predicted utility exhibits simple projection bias ifthere exists a [ [01] such that for all c s and s9 u(cs us9) 5(1 2 a)u(cs) 1 au(cs9)

With this formulation if a 5 0 the person has no projectionbias she predicts her future instantaneous utility correctly Ifa 0 the person has projection bias the bigger is a the strongeris the bias When a 5 1 the person perceives that her futuretastes will be identical to her current tastes9

Our model says nothing about how tastes change rather itmakes predictions as a function of how tastes change Hence itmight be that a personrsquos happiness tends to mean-revert overtime due to adaptation in which case projection bias would leadher to expect some but not enough mean reversion It could bethat a person develops a taste for certain types of consumptionmdasheg her enjoyment of coffee might grow over timemdashin which caseprojection bias would lead her to underappreciate how much herenjoyment will grow Or it could be that a personrsquos tastes uctu-ate from day to day in which case projection bias would lead herto underappreciate the magnitudes of these uctuations Ourformulation permits us to analyze the implications of projectionbiasmdash of understanding the direction of taste changes but under-

8 Our formal assumption is that people correctly anticipate changes instates but underappreciate how these changes map into changes in utility Butsince states are merely a means of parameterizing utility functions it would makelittle difference if we assumed instead that people fully appreciate how changes instates map into changes in utility but underappreciate the degree to which thestates will change

9 While simple projection bias is sufcient for our analysis in this paper itis too restrictive for use as a general denition One problem is that when thereare multiple states it requires that the magnitude of the bias be identical fordifferent types of states eg that a person who is currently not thirsty andcurrently unaddicted to cocaine be just as bad at predicting her preferences whenshe is thirsty as she is at predicting her preferences when addicted to cocaine Asecond problem is that the magnitude of the bias cannot depend on the currentstate eg it does not permit that a satiated person can predict well her prefer-ences when hungry whereas a hungry person cannot predict well her preferenceswhen satiated See Appendix A in Loewenstein OrsquoDonoghue and Rabin [2002] fora more general formulation of projection bias

1217PROJECTION BIAS

estimating magnitudesmdashfor these and other possible types oftaste changes10

Most economic decisions involve more than merely predictingfuture tastes they involve making choices with intertemporalconsequences We next embed our framework above within anintertemporal-choice environment Suppose that a person mustchoose a path of consumption (ct cT ) when her (true) inter-temporal preferences are given by

Ut~c t cT 5 Ot5t

T

dtu~ctst

where d 1 is her discount factor Standard economic models ofstate-dependent preferences typically assume that people are ldquora-tionalrdquo in the sense that they correctly anticipate how their behaviorinuences the evolution of states Formally for any period t andinitial state st a rational person chooses a path of consumption(ct cT) correctly anticipating the associated path of states(st sT) to maximize true intertemporal utility Ut

A person with projection bias attempts to maximize her in-tertemporal utility but may fail to do so because she mispredictsher future instantaneous utilities More precisely if a personexhibits projection bias and her state in period t is st then sheperceives her period t intertemporal preferences to be

Ut~c t cT ust 5 Ot5t

T

dtu~ctstust

We assume that for any period t and initial state st a person withprojection bias chooses a path of consumption (ct cT ) cor-rectly anticipating the associated path of states (st sT ) tomaximize her perceived intertemporal utility Ut That is shebehaves exactly as a rational person would except that (possibly)Ut THORN Ut

To incorporate uncertainty over future consumption or futurestates we make the standard assumption that a person maxi-mizes her expected discounted utility For instance suppose thatin period t the person expects her period t consumption-statecombination to be (c9s9) with probability p and (c0s0) with prob-

10 For a discussion of many different types of taste changes see Loewen-stein and Angner [2003]


ability 1 2 p Just as true period t expected utility is Et[u(ctst)] 5pu(c9s9) 1 (1 2 p)u(c0s0) a person with projection bias predictsperiod t expected utility to be Et[u(ctstust)] 5 pu(c9s9ust) 1(1 2 p)u(c0s0ust) Similarly true expected intertemporal utility isEt[U

t(ct cT )] 5 Et[St5 tT dtu(ctst)] and a person with pro-

jection bias perceives her expected intertemporal utility to beEt[U

t(ct cT ust)] 5 Et[St5 tT dtu(ctst ust)]

11

While the personrsquos true intertemporal preferences Ut aretime-consistent because she incorrectly predicts how her tasteschange over time her perceived intertemporal preferences Ut canbe time-inconsistent Because this time inconsistency derivessolely from misprediction of future utilities it would make littlesense to assume that the person is fully aware of it12 We assumethroughout the paper that the person is completely unaware ofthe time inconsistencymdashthat at all times the person perceives herpreferences to be time-consistent and therefore at all times sheplans to follow the consumption path that maximizes her currentperceived intertemporal preferences As a result projection biascan lead to dynamic inconsistency a person may plan to behave acertain way in the future but later in the absence of new infor-mation revise this plan13

Given any particular set of state-dependent preferences andparticular economic environment our model of projection biasmakes specic predictions about how actual behavior differs fromrational behavior To demonstrate this point and to highlight the

11 Research has of course documented a number of inadequacies of ex-pected-utility theory (for an overview see Starmer [2000]) To the extent that onefeels the need to modify expected-utility theory for rational types one could usethe same modications for people with projection bias

12 Another psychological phenomenon that has received increasing atten-tion in research on intertemporal choice is hyperbolic discounting (see in par-ticular Laibson [1994 1997] and OrsquoDonoghue and Rabin [1999a]) Under hyper-bolic discounting true preferences are time-inconsistent and hence a personcould be fully aware of this fact as much of the literature has assumed

13 Given the logic of our model it is inherent that a person is unaware of hercurrent misprediction But one could imagine a variant of the model where theperson is aware of her future propensity to mispredict She could for instance beaware of her general propensity to overshop when hungry while still committingthe error on a case-by-case basis The coexistence of day-to-day mispredictionswith a ldquometa-awarenessrdquo of these mispredictions is similar to the discussion inOrsquoDonoghue and Rabin [1999b] of how people can simultaneously be aware oftheir general tendency to procrastinate and yet still procrastinate on a case-by-case basis A model of ldquosophisticated projection biasrdquo could plausibly better de-scribe behavior in some circumstances such as when sophisticated shoppers knowthat they should not shop on an empty stomach but we choose our currentformulation as a simple and realistic starting point

1219PROJECTION BIAS

potential importance of projection bias for economics Sections IVand V formally analyze two economic environments

IV PROJECTION BIAS AND HABIT FORMATION

For half a century though most intensively recently econo-mists have explored life-cycle consumption models with habitformation Habit formationmdashwherein increases in current con-sumption increase future marginal utilitymdashwas discussed byDuesenberry [1949] and later formalized by Pollak [1970] andRyder and Heal [1973] In recent years habit-formation modelshave been used in specic applications see Becker and Murphy[1988] Constantinides [1990] Abel [1990] Campbell and Coch-rane [1999] Jermann [1998] Boldrin Christiano and Fisher[2001] Carroll Overland and Weil [2000] and Fuhrer [2000] Allof these recent researchers have examined habit formation withinthe rational-choice framework14

In this section we formally analyze the implications of pro-jection bias over habit formation in a simple ldquoeat-the-cakerdquo modelSuppose that a person has income Y to allocate over consumptionin periods 1 T which we denote by c1 cT For sim-plicity we assume that there is no discounting and that theperson can borrow and save at 0 percent interest neither of theseassumptions is important for our qualitative conclusions Thepersonrsquos true instantaneous utility in period t is u(ctst) wherethe state st can be thought of as her ldquohabit stockrdquo The personrsquosinitial habit stock s1 is exogenous and her habit stock evolvesaccording to st 5 (1 2 g)st2 1 1 gct2 1 for some g [ (01] Hencethe more the person consumes in a given period the higher is hersubsequent habit stock The parameter g represents how quicklythe person develops (and eliminates) her habit

We assume that instantaneous utility takes a particularlysimple functional form

14 The early literature on habit formation distinguishes between two polarcases ldquorational habitsrdquo wherein consumers fully account for how current con-sumption affects future well-being and ldquomyopic habitsrdquo wherein consumers do notaccount at all for how current consumption affects future well-being Of the paperscited in the text all assume rational habits except for Pollak [1970] which(implicitly) assumes myopic habits Our model is equivalent to rational habitswhen a 5 0 and to myopic habits when a 5 1 Muellbauer [1988] provides anexcellent overview of the two extremes and concludes that the empirical evidenceseems to favor myopic habits We return to this and other empirical evidence inSection VI


u~c ts t 5 v~ct 2 st where v9 0 and v0 0

This formulation is potentially restrictive but it captures the keyfeature of habit formation and is common in the literature15

There are actually two key features that play a role in our resultsbelow First the marginal utility from consumption is increasingin the habit stock (][]u]c]]s 0) which implies habit forma-tionmdashan increase in current consumption increases the futurehabit stock and therefore increases the marginal utility fromfuture consumption Second the level of utility is declining in thehabit stock (]u]s 0) which implies that an increase in currentconsumption reduces the utility from future consumption Al-though this negative ldquointernalityrdquo [Herrnstein et al 1993] is notan inherent part of habit formation it is present in most formalanalyses and real-world instances of habit formation

In period 1 the person faces the following choice problemwhere s1 is exogenous

max~c1 cT U1~c1 cTus1 5 Ot51

T

~1 2 av~ct 2 st 1 av~ct 2 s1

such that

s t 5 ~1 2 gst21 1 gc t21 for t [ $2 T and Ot51

T

ct Y

For ease of presentation let (c1 cT) denote rational behaviorwhich solves this maximization when a 5 0 and let (c1

A cTA)

denote planned behavior from the period 1 perspective for a personwith a 0 with the value of a suppressed in the notation Ouranalysis throughout assumes interior solutions for both rationaland actual behavior

A pattern typically emphasized in models of habit formationis that people choose an increasing consumption prolemdashthat isc1 cT mdashso that they are always consuming more than

15 This formulation is equivalent to that used by Pollak [1970] Constantin-ides [1990] Jermann [1998] Campbell and Cochrane [1999] and BoldrinChristiano and Fisher [2001] indeed all these papers except Pollak furtherassume that v takes a CRRA specication Another formulation proposed by Abel[1990] and used by Fuhrer [2000] and Carroll Overland and Weil [2000]is u(cts t) 5 (c tst

g)12 s (1 2 s) Yet a third formulation suggested by Kahnemanand Tverskyrsquos [1979] prospect theory is to assume that v0( x) 0 for x 0 butv0( x) 0 for x 0 Bowman Minehart and Rabin [1999] use a variant of thisapproach

1221PROJECTION BIAS

they are accustomed to This conclusion holds however only ifthe personrsquos initial habit stock s1 is not too large Otherwise itmight be optimal to break the initial habit and the optimal wayto do so might involve a declining consumption path that lowersthe habit stock gradually over time16 But since breaking a habitis both least painful and most benecial when done early in lifebefore the habit has been further developed and when the bene-ts will be spread over a large number of years a rational personwill break a habit only at the beginning of life Lemma 1 formallyestablishes this conclusion by demonstrating that once a personstarts further developing her habitmdashby consuming more than herhabit stockmdashshe will follow an increasing consumption prolefrom that period onward

LEMMA 1 If ct $ st for some t T then ct ct1 1 cT

We focus on the implications of projection bias for situationsin which rational behavior does not involve early-life habit-break-ing episodes our results below only apply to parameter valuessuch that a rational person would choose an increasing consump-tion prole Lemma 1 implies that a sufcient condition for arational person to choose an increasing consumption prole iss1 5 0 more generally this outcome will occur as long as theinitial habit stock s1 is small enough

Projection bias creates two types of distortions in this envi-ronment because the person underappreciates both the negativeinternality and the habit formation The implication of projectionbias over the negative internality is straightforward Because itimplies that early consumption decreases utility in all later peri-ods the negative internality motivates a person to delay con-sumption Hence an underappreciation of the negative internal-ity makes the person prone to consume too much early in life andtoo little late in life relative to rational behavior The implicationof projection bias over habit formation is in principle more com-plicated because the basic effect of habit formation is complicatedBut for the case in which rational behavior does not involve ahabit-breaking episode and therefore involves an increasing con-sumption prole the personrsquos habit stock will be increasing overtime and therefore habit formation makes her marginal utility

16 Indeed for s1 YT the person must have a habit-breaking episode andthis episode might last her entire life that is she might have c1 c2 cT


increase over time As a result habit formation also motivates theperson to delay consumption Hence an underappreciation ofhabit formation like an underappreciation of the negative inter-nality makes the person prone to consume too much early in lifeand too little late in life relative to rational behavior Proposition1 reects this intuition establishing that whenever rational be-havior does not involve a habit-breaking episode projection biasleads a person to (plan to) consume too much early in life and toolittle late in life relative to what would be optimal

PROPOSITION 1 If c1 $ s1 then for any a 0 St5 1t ct

A St5 1t ct

for all t T

Hence projection bias causes a person to plan a consump-tion prole that consumes her income too quickly Perhaps thecleanest illustration is in the extreme case where a 5 1 wherethe person will plan to consume the same amount in all periodsrather than increase consumption over time as would beoptimal17

More interesting is what happens as time passes and thepersonrsquos tastes change in ways she did not predict To study sucheffects we examine how a personrsquos plans change in period 2 Inperiod 2 the person reoptimizes given her new perceived prefer-ences that is she faces the following choice problem where s1and c1

A are exogenous

max~c2 cT U 2~c2 cTus2 5 Ot52

T

~1 2 av~ct 2 st 1 av~ct 2 s2

such that

s2 5 ~1 2 gs1 1 gc1A

s t 5 ~1 2 gst21 1 gc t21 for t [ $3 T

and Ot52

T

ct Y 2 c1A

17 While the assumption that rational behavior does not involve a habit-breaking episode is sufcient for overconsumption it is not necessary Proposition1 might fail because during a habit-breaking episode habit formation and adeclining habit stock mean the personrsquos marginal utility declines over time whichin turn means that habit formation motivates the person to accelerate consump-tion and so projection bias over habit formation leads the person to consume herincome too slowly But Proposition 1 need not fail because projection bias over thenegative internality still motivates the person to consume her income too quickly

1223PROJECTION BIAS

Rational behavior of course does not change over time andhence the solution to this problem for a 5 0 is (c2 cT ) Fora person with projection bias the solution for this problem whichwe denote by (c2

A A cTAA ) may differ from her period 1

plans (c2A cT

A) Proposition 2 characterizes this revision ofplans in the case where she is developing a habit and T 5 3

PROPOSITION 2 Suppose that T 5 3 and c1A s1 Then v- 0

implies that c2AA c2

A v- 0 implies that c2A A c2

A andv- 5 0 implies that c2

A A 5 c2A

As the personrsquos habit stock changes over time her (perceived)marginal utilities from consumption in each period also changeWhen the person is developing a habit these marginal utilities allincrease18 Hence the relative magnitudes of these changes inmarginal utility determine the revision of plans If v- 5 0 theincrease in marginal utility is the same for all periods whichimplies that the personrsquos marginal trade-offs have not changedand hence she does not revise her consumption plan If v- 0the increase in marginal utility is larger for period 2 than period3 and as a result she revises her period 2 consumption upwardIf v- 0 the increase in marginal utility is smaller for period 2and she revises her period 2 consumption downward19

Any utility function that satises nonincreasing absoluterisk aversion which includes the CARA and CRRA families musthave v- 0 Because this seems a plausible restriction on theinstantaneous utility function Proposition 2 suggests that pro-jection bias leads people to repeatedly readjust their immediateconsumption upwards relative to their most recent plans Henceif people experience habit formation in consumption projectionbias represents a possible source for actual saving being smallerthan planned saving Laibson Repetto and Tobacman [1998]

18 Formally from a period t perspective the (perceived) marginal utilityfrom period 2 consumption is (1 2 a)v9(c2 2 s2 ) 1 av9(c2 2 st) 1 (1 2a)gv9(c3 2 s3) and since s2 s1 implies that v9(c2 2 s2) v9(c2 2 s1)this marginal utility is larger from a period 2 perspective Similarly from aperiod t perspective the (perceived) marginal utility from period 3 consumptionis (1 2 a)v9(c3 2 s3 ) 1 av9(c3 2 st) and since s2 s1 implies that v9(c3 2 s2) v9(c3 2 s1) this marginal utility is also larger from a period 2 perspective

19 We conjecture but have not proved that this conclusion holds for T 3The result that v- 5 0 yields dynamic consistency is quite general For the casev- 0 it is straightforward to show that marginal utility increases most forperiod 2 and least for period T and so perhaps subject to additional regularityconditions after reoptimization we should expect period 2 consumption to in-crease and period T consumption to decrease Analogous conclusions hold for thev- 0 case


review considerable evidence that the actual saving of manyhouseholds falls short of their plans The authors posit self-control problems and naivete about those self-control problems asprimary sources of this shortfall Our analysis suggests thatprojection bias in the form of underappreciation of how increas-ing consumption in the present will raise onersquos consumption stan-dard in the future might also contribute to such mispredictions

While our analysis assumes that a personrsquos lifetime income isexogenous our model suggests implications for how projectionbias might inuence decisions about how hard to work to increaseincome Specically let lA be the marginal utility of lifetimeincome as perceived from period 1 and let lA A be the marginalutility of lifetime income as perceived from period 2 Again lim-iting ourselves to the case when a person is developing a habitand the horizon is T 5 3 Proposition 3 establishes that themarginal utility of lifetime income increases over time

PROPOSITION 3 Suppose that T 5 3 and c1A s1 Then lAA lA

Proposition 3 reects a simple intuition as time passes andthe personrsquos real and perceived marginal utilities from consump-tion increase income becomes more valuable Extrapolating be-yond our formal framework this result suggests that projectionbias over habit formation might lead people to pursue higherincome than planned as time passes Projection bias might forinstance create a force toward choosing a later and later plannedretirement date as time passes using the proceeds to increaseconsumption20 Similarly with endogenous per-period labor-lei-sure decisions projection bias might create a tendency to repeat-edly increase labor and decrease leisure relative to earlier plansWe are wary of pushing this intuition too far without furthertheoretical and empirical analysis however because the logic ofthe argument assumes that there is no reference dependence inleisure But we do note that this intuition parallels the argu-ments of many previous researchers such as Scitovsky [1976]and Frank [1999] who have argued that people spend too muchtime and energy generating wealth and too little time on leisureactivities and that people enjoy increases in their material con-sumption less than they think they will

20 There is some evidence however that people are somewhat accurate atpredicting their retirement dates (see Bernheim [1989]) although this may in partbe due to the existence of focal retirement ages

1225PROJECTION BIAS

V PROJECTION BIAS AND DURABLE GOODS

For most durable goodsmdashsuch as a tent a golf-swing traineror a Johnny Depp videomdashpeople experience day-to-day uctua-tions in their valuations For rational consumers such uctua-tions are virtually irrelevant because they will purchase durablegoods based almost exclusively on their expected daily valuationsfor the goods and virtually ignore their valuations on the daythey happen to be in the store But for people with projection biasbuying decisions are oversensitive to the momentary feelings theyexperience when they happen to be in the store and thus thenature of day-to-day uctuations becomes important In this sec-tion we present a stylized model that identies some implicationsof such effects

Suppose that a personrsquos valuation of a durable good in periodt is given by a random variable mt where mt is distributed iden-tically and independently across periods and has nite mean m The person learns the realization of mt at the start of period t Forsimplicity we further assume that the durable good lasts forexactly D days and that the person cannot consume the good onthe day she purchases it21

Consider rst a consumer who has just one opportunity onday 1 to purchase the item if she does not purchase it on day 1she cannot purchase it at all We normalize the personrsquos inter-temporal utility to be zero when she does not buy the product Ifshe buys the product at price P she will enjoy the benets ofownership but must forgo the consumption of other goods thatshe could have nanced with wealth P22 We assume that thepersonrsquos utility from the durable good is additively separable fromher utility for other goods and that the price P represents thetotal utility value of the other goods forgone by purchasing thedurable good The personrsquos state in period t is her current valua-tion or st 5 mt Finally we assume that there is no discountingor d 5 1 none of our conclusions depend on this assumption

If the person buys the durable good in period 1 then given

21 While it is often unrealistic to assume that the person cannot consumethe good on the day she purchases it none of our qualitative conclusions dependon this assumption and it vastly simplies our analysis

22 We take the price P to be exogenous In Loewenstein OrsquoDonoghue andRabin [2000] we formulate a more complicated model that derives a monopolistrsquospricing and valuation-changing sales-hype policies in the face of projection bias byconsumers


the information available her true expected intertemporal utilityis

E1U1 5 E1F Ok51

D

m11k 2 PG 5 Dm 2 P

A person exhibiting simple projection bias perceives her expectedintertemporal utility to be

E1U1 5 E1F Ok51

D

~1 2 am11k 1 am1 2 PG5 Dm 1 aD~m1 2 m 2 P

m1 m implies that E1[U1] E1[U1] and m1 m implies thatE1[U1] E1[U1] Hence an underappreciation of day-to-dayuctuations can lead variously to underbuying or overbuying Ifher day 1 valuation is larger than average and she projects thisabove-average valuation onto the future the person is prone toovervalue the durable good If in contrast her day 1 valuation issmaller than average and she projects this below-average valua-tion onto the future she is prone to undervalue the durable goodIn other words a person with projection bias is too sensitive toher valuation at purchase time23

While projection bias has ambiguous effects in one-shot buy-ing decisions things change dramatically in the more realisticcase where the person has multiple opportunities to buy a dura-ble good To make this point in a particularly stark way wesuppose that the consumer will purchase the good at most onceand can buy the good in any period t [ 1 2 In thissituation a rational person either will buy the durable good im-mediately in period 1 or never buy the durable good and she buysthe durable good if and only if Dm 2 P $ 0 Intuitively given ourassumption that the person cannot consume the good on the dayshe purchases it the net expected value of the durable good is

23 If we allowed immediate consumption a rational type would also besensitive to her day 1 valuation But a projector would still be oversensitive to herday 1 valuation indeed the conclusion generalizes that an underappreciation ofday-to-day uctuations leads a person to overvalue the good when m1 m andundervalue it when m1 m

1227PROJECTION BIAS

independent of the valuation on the date purchased Hence thegood is either worth purchasing immediately or not at all24

A person with projection bias like a rational person alwaysperceives that the good is either worth purchasing immediately ornot at all But her perception of whether it is worth purchasingimmediately is inuenced by her current valuation As a resultshe ends up purchasing the good in the rst period that Dm 1aD(mt 2 m ) 2 P $ 0 If we let mH denote the largest value thatmt might possibly take on then there will eventually be someperiod in which the person perceives the good to be worth pur-chasing if and only if Dm 1 aD(mH 2 m ) 2 P 0 Because mH m a person with projection bias is unambiguously more prone tobuy the durable good than is a rational person she will alwaysbuy when she should buy and sometimes when she should not

The intuition behind this conclusion is an inherent asymme-try in purchases of durable goods A decision not to buy is revers-ible so if the person does not buy today when she should she canstill buy in the future But a decision to buy is irreversible so ifshe buys today when she should not she cannot unbuy in thefuture With multiple buying opportunities a person is prone notto buy when she should only in the unlikely event that she has aparticularly low valuation on every buying opportunity whereasshe is prone to buy when she should not in the quite likely eventthat she has a particularly high valuation on at least one buyingopportunity Hence projection bias represents a source of ldquoim-pulse purchasesrdquo wherein people overbuy durable goods in re-sponse to transitory desire for that good Many prior theoreticaltreatments of impulse purchases have attributed the phenomenato hyperbolic discounting But for durable goods projection bias ismore relevant than hyperbolic discounting Hyperbolic discount-ing provides a compelling explanation for overconsumption oncumulative small-scale consumption decisions such as purchasesof potato chips where the net effects of repeated purchases can bevast overconsumption of potato chips The purchase of a durablegood however is by its very nature a long-term-consumptiondecision As such self-control problems are less likely to be im-plicated in the purchase of durable than nondurable goodswhereas projection bias is more likely to be implicated

24 Formally we assume that when indifferent between buying now versusbuying in the future people choose to buy now (which would be optimal if wereplace d 5 1 with d 1 but very close to 1)


Our analysis suggests that certain types of sales tacticsmight be understood as attempts by businesses to exploit projec-tion bias If consumers overestimate the longevity of their currentfeelings sellers will have an incentive to induce high valuationswhen people are making buying decisions via sales hype enticingdisplays or mood-inducing music Sellers will also have an incen-tive to pressure people to make purchase decisions when hot andto facilitate rapid purchases by consumers who are in a hot statethat is unlikely to last such as one-click shopping on the internetFinally projection bias might motivate rms to turn nondurablegoods into durable goods via ldquointertemporal bundlingrdquo eg sell-ing memberships in health clubs golf clubs vacation time sharesor season ski passes Consider for instance a person who be-comes enthusiastic about exercise and makes a visit to a healthclub Rather than making a prot solely on that one visit thehealth club may exploit the consumerrsquos tendency to project hercurrent enthusiasm into the future by offering a more expensiveldquoclub membershiprdquo that entitles the person to additional free (orlow-cost) visits in the future Indeed Della Vigna and Mal-mendier [2002] empirically document that people overpay forhealth club memberships Using a panel data set that tracksmembers of three New England health clubs they nd thatmembers who chose a contract with a at monthly fee paid a priceper visit of $17 and members who chose a contract with a atyearly fee paid a price per visit of $15 even though a $10-per-visitcontract was also available Della Vigna and Malmendier at-tribute these ndings to partially naive self-control problemspeople sign up in an attempt to ldquocommitrdquo themselves to futureexercise but then do not have enough self-control to carry outthese plans Our model suggests an additional possible explana-tion people plan to attend frequently because they project theircurrent enthusiasm into the future but then decide not to attendin the future when their enthusiasm has waned25

In addition to helping to explain sales tactics our analysismay also shed light on laws designed to counteract them Cooling-off laws enacted at both the state and federal level allow consum-ers to rescind certain types of purchases within a few days of the

25 We suspect that another contributory factor is that people dislike payingon the margin for consumption [Prelec and Loewenstein 1998] Neither this norprojection bias is likely to explain Della Vigna and Malmendierrsquos evidence ofprocrastination in canceling memberships which is more consistent with naiveself-control problems

1229PROJECTION BIAS

transaction26 Such laws can be viewed as devices for combattingthe effects of projection bias Cooling-off periods that force con-sumers to reect on their decisions for several days can decreasethe likelihood that they end up owning products that they shouldnot Cooling-off laws may also have the benet of reducing sales-personsrsquo incentives to hype If consumers can return productsonce they cool down and if such returns are costly for the sellersellers will have an incentive to put buyers in a long-run averagemood rather than an overenthusiastic state

Although our analysis focuses solely on random uctuationsin tastes more generally durable goods might involve other typesof taste changes Projection bias over such changes could yieldfurther interesting conclusions For some durable goods a per-sonrsquos valuation systematically declines over time as the ldquonoveltyrdquowears off Projection bias over such taste changes would create atendency to overbuy and hence rms might engage in attempts tocreate increased feelings of novelty Alternatively for other du-rable goods a personrsquos valuation increases over time as the per-son develops a taste for the good (or becomes attached to thegood) Projection bias over these taste changes would create atendency to underbuy In such cases rms might in fact engagein behaviors designed to overcome projection bias such as offer-ing a free-trial period

VI OTHER APPLICATIONS

Sections IV and V derive the implications of projection bias intwo specic economic environments These implications highlighttwo types of errors to which projection bias can give rise Firstthe failure to predict future taste changes can lead to misguidedchoices for current consumption eg overconsumption due tounderappreciation of habit formation and oversensitivity to cur-rent valuations as a result of exaggerating the longevity of day-to-day uctuations in tastes Second as perceived tastes changeover time in ways that people do not predict people make plansthat they may end up not carrying out eg people may consumemore (and save less) than earlier planned We believe that pro-jection bias is important for many economic applications andthat it can provide an intuitive and parsimonious account formany phenomena that are otherwise difcult to explain In this

26 For a detailed discussion of such laws see Camerer et al [2003]


section we extrapolate from our formal analysis in Sections IVand V to discuss additional implications of projection bias

There are many implications of projection bias for recentmodels of habit formation beyond the formal analysis in SectionIV In recent years economists have often invoked habit formationin consumption as an explanation for empirical phenomena thatare hard to understand within a stationary-utility framework Aswe discuss in Section IV for instance habit formation is some-times invoked as an explanation for why people choose consump-tion proles that increase over time In addition Constantinides[1990] argues that habit formation can provide an explanation forthe equity-premium puzzle because it leads people to expect tomaintain high levels of risk aversion even with rising levels ofconsumption and wealth (see also Abel [1990] and Campbell andCochrane [1999]) Fuhrer [2000] shows that habit formationmight explain the ldquoexcess smoothnessrdquo of consumption (docu-mented by Campbell and Deaton [1989]) and Carroll Overlandand Weil [2000] demonstrate that habit formation can explainrecent empirical evidence that periods of high aggregate incomegrowth seem to cause periods of high aggregate saving Theseexplanations derive from the fact that because people expect toadapt to their changing consumption levels they adjust slowly toshocks to their permanent income

Habit formation is a compelling explanation for these phe-nomena because it accords well with introspection and commonwisdom and is consistent with psychological evidence on adap-tation Even so it has been hard to nd direct evidence of habitformation in time-series consumption data Dynan [2000] reviewsthe mixed results from tests using aggregate consumption dataand describes how such tests might be prone to overstate thedegree of habit formation Dynan then tests for habit formationusing household data on food consumption and nds no evidenceof habit formation Our model suggests an explanation even ifpeople are characterized by habit formation projection bias maylead people to not react to that habit formation as strongly as therational model suggests Indeed in our simple eat-the-cakemodel we saw exactly this point whereas introducing habitformation would lead rational consumers to switch from smoothconsumption to an increasing consumption prole projection biasundermines this effect In fact for a person with complete pro-jection bias or a 5 1 the introduction of habit formation does notchange her behavior at all Muellbauer [1988] makes a similar

1231PROJECTION BIAS

point when he compares rational versus myopic habitsmdashwhich isequivalent to a 5 0 and a 5 1 in our model Specically he pointsout that habit formation would show up in cross-sectional evi-dence under either assumption but it would show up in time-series evidence only under the assumption of rational habitsGiven that it is hard to nd evidence of habit formation in time-series evidence Muellbauer concludes that the evidence supportsmyopic habits

Hence perhaps a better hypothesis is that people are char-acterized by a combination of habit formation and projection biasIf so our model suggests where to look for additional evidenceSpecically it predicts specic patterns of dynamic inconsistencyWe already described one such prediction in Section IV peoplewith projection bias over habit formation will plan to save more inthe future than they actually end up saving (under the plausibleassumption of nonincreasing absolute risk aversion) Extrapolat-ing from our model an additional area on which there might bedynamic inconsistency is charitable giving Charitable giving de-pends on a trade-off between the benets of giving and the cost offorgone personal consumption While the marginal utility of con-sumption may decline with wealth habit formation reduces themagnitude of this change As a result people with projection biasmay plan to increase their charitable giving as their wealthincreases by more than they actually end up doing27

An obvious application of projection bias is addiction Ratio-nal-choice models of addiction provide plausible explanations formany different patterns associated with addiction but often havedifculty accounting for the most central problem why do peoplebecome addicted in the rst place Because habit formation is anatural way to formalize ldquoaddictivenessrdquomdashindeed most models ofaddiction use this formulationmdashour analysis in Section IV sug-gests two reasons why people with projection bias might be over-prone to develop harmful addictions First projection bias maylead people to underappreciate the degree to which current con-sumption has negative consequences for their future health em-ployment and personal lives that is they may underappreciate

27 A difculty with identifying projection bias via dynamic inconsistency isthat there are other sources of dynamic inconsistency such as naive hyperbolicdiscounting Here saving less than planned and giving less than planned alsoseem consistent with naive hyperbolic discounting but in fact one can distinguishthe two sources because of the asymmetry in the predictions Projection biaspredicts saving less than planned and giving less than planned when wealth isincreasing but the opposite effects when wealth is declining


the negative internality associated with addictive products Sec-ond and perhaps more important projection bias may lead peo-ple to underappreciate the degree to which current consumptionchanges their future desire for addictive products that is theymay underappreciate the habit formation associated with addic-tive products This second error is particularly pernicious becauseit can lead people to consume addictive products in the short runplanning not to continue in the long run but then end up becom-ing addicted28

Our analysis in Section V of day-to-day uctuations in tastesis also relevant for addiction In particular it suggests that peo-ple might overreact to transitory changes in the craving for ad-dictive products When a personrsquos craving is particularly highprojection bias will lead her to overestimate her future craving forthe drug and therefore may discourage her from any efforts toquit Analogously when her craving is particularly low projec-tion bias will lead her to underestimate her future craving for thedrug and therefore may cause her to make repeated painfulefforts to quit only to fail in these endeavors when her cravingreturns to average or high levels There is in fact empiricalsupport for addicts believing that their future craving will besimilar to their current craving Giordano et al [2001] studiedheroin addicts who came to a clinic for a maintenance dose ofBuprenorphine (similar to methadone) These addicts were askedto choose between extra BUP or extra cash on a visit scheduled forve days later where half were given the choice right beforereceiving the BUP and half were given the choice right afterThose making the choice before receiving the BUP valued thefuture BUP dose by almost twice as much as those who made thechoice after receiving the BUP

28 There is in fact evidence that unaddicted cigarette smokers signicantlyunderappreciate their own risk of becoming addicted For instance only 15 per-cent of high school students who were occasional smokers (less than one cigaretteper day) predicted that they might be smoking in ve years when in fact 42percent were still smoking ve years later and 28 percent were daily smokersBut there is also evidence that addicted cigarette smokers underappreciate theirrisk of staying addicted For instance 68 percent of high school students who wereheavy smokers (more than one pack per day) predicted that they would still besmoking in ve years while 80 percent were still smoking at least half a pack perday ve years later [U S DHHS 1994] The mispredictions of occasional smokersare arguably larger than those for heavy smokers and even the mispredictions ofheavy smokers could be due to projection bias if they made predictions while in anicotine-sated state Even so the average mispredictions of heavy smokers sug-gests that the mispredictions of even the occasional smokers might in part be dueto overoptimism about self-control or other factors rather than projection bias

1233PROJECTION BIAS

Projection bias over the endowment effect has further impli-cations (beyond those in Section II) The usual explanation for theendowment effect is that people adapt to owning or not owningobjects and that there is more pain upon parting with objectsthan there is joy upon obtaining objects Projection bias over thisadaptation has several interesting implications First becauseprojection bias leads to exaggerated feelings of loss aversion itmagnies the size of the endowment effect In other words whilethe endowment effect may be caused by valid expectations thatlosses will hurt more than gains will help the endowment effectthat we observe in experiments may be an exaggerated responseto these real preferences29 Perhaps more important is that peo-ple may fail to predict the endowment effect At a purely individ-ual level this failure can lead individuals to make purchases witha false sense of their reversibility and undoubtedly lowers thecost to retailers of offering money-back guarantees and free re-turns In bilateral economic transactions it can cause distortionsor break down in bargaining because buyers will tend to under-estimate ownersrsquo reservation prices and owners will tend tooverestimate buyersrsquo reservation prices Indeed Van Boven Dun-ning and Loewenstein [2000] experimentally demonstrate bar-gaining inefciency due to ldquobuyersrsquo agentsrdquo underestimating sell-ersrsquo reservation values30

As highlighted by our analysis in Section V projection biaspredicts suboptimal patterns of behavior when people make de-cisions with long-term consequences but experience highly vari-able day-to-day feelings While our formal analysis concerneddurable goods more important life decisions such as marriagedivorce and especially fertility all display such a pattern Forsuch long-term decisions momentary uctuations in feelingsshould be virtually irrelevant but projection bias will cause peo-ple to exaggerate their longevity and therefore to give them too

29 Kahneman [1991 p 143] and Tversky and Kahneman [1991] argue thatthe endowment effect is a ldquobiasrdquo because peoplersquos actual pain when losing an objectis not commensurate with their unwillingness to part with that object Evidencefrom Strahilevitz and Loewenstein [1998] supports this interpretation Loewen-stein OrsquoDonoghue and Rabin [2002] present a detailed analysis of the role ofprojection bias in the endowment effect

30 Also see Genesove and Mayer [2001] who nd evidence of nancial lossaversion in housing marketsmdashof people experiencing ldquopainrdquo when they realize anominal loss on their home In particular they nd that sellers subject to nominallosses set higher asking prices and exhibit a lower hazard rate of sale Projectionbias suggests that the magnitudes of these effects may be larger than justied byany true feelings of pain that people experience if and when they do sell theirresidence at a loss


much weight in their decisions Hence projection bias might leadpeople to get marriedmdash or make proposals of marriage that arecostly to rescindmdashin the thralls of love say things they later wishthey had not in a t of rage and fail to use birth control or tofollow safe sex practices in the heat of passion There are in factpolicies that seek to circumvent such tendencies for example inmany states there are mandatory time delays between ling formarriage or divorce and actual changes in status And thereseems to be a demand for methods of birth control that allow oneto control fertility without making decisions in the heat of pas-sion such as Norplant and the ldquomorning afterrdquo pill

Whereas marriage divorce and fertility decisions are dif-cult to reverse suicide is totally irreversible Yet much suicidalbehavior seems to occur on impulse or after only a relativelyshort period of misery Projection bias may well contribute to thisphenomenon The literature on depression documents a tendencyfor people who are depressed to project their depressed feelingsnot only on the future but also on the past As Solomon [1998 p49] expresses it ldquoWhen you are depressed the past and thefuture are absorbed entirely by the present You can neitherremember feeling better nor imagine that you will feel betterrdquoOther research documents that the will to live varies dramati-cally over time In a study of 168 cancer patients admitted to ahospital for end-of-life care the patientsrsquo will to live as measuredon a 100-point scale uctuated an average of 30 percent over a12-hour period and more than 60 percent over a 30-day period[Chochinov et al 1999]

The combination of such uctuations and projection bias alsohas important ramications for end-of-life care in particular forthe use of mechanisms such as ldquoadvanced directivesrdquo and ldquolivingwillsrdquo that permit people to make decisions that will apply whenthey are in a health state that renders them unable to makedecisions for themselves The premise of such tools is that healthypeople can make decisions that will reect their own preferenceswhen sick but the presence of projection bias would challenge thevalidity of this assumption (see Coppola et al [1999] and Druleyet al [1993]) Indeed in one study [Slevin et al 1988] respon-dents were asked whether they would accept a grueling course ofchemotherapy if it would extend their lives by three monthsWhile only 10 percent of healthy people said that they wouldaccept the chemotherapy 42 percent of current cancer patientssay they would A natural interpretation is that the value of a day

1235PROJECTION BIAS

of life is larger when the prospect of death is close at hand butprojection bias leads people to underappreciate this change

Projection bias also has further implications in the mundaneworld of consumer theory Although economists usually capturesatiation with a static model that assumes diminishing marginalutility we usually have in mind a dynamic notion that the utilityof current consumption depends on recent consumption The sat-isfaction from eating a pint of ice cream is smaller if one has justconsumed another pint of ice cream The satisfaction from eatingsalmon is smaller if one has already consumed salmon for severalevenings in a row Projection bias over such effects has diverseconsequences Most straightforwardly it leads to overconsump-tion or at least overordering of appetite-dependent goods Moregenerally people may be prone to overpurchase activities thatthey currently do not engage in People may plan overly longvacations believing the ninth day lying on the beach will benearly as enjoyable as the rst and professionals who have littletime for reading or traveling may falsely anticipate the blissful-ness of spending their retirement years with nonstop reading andtraveling Firms may of course take advantage of such mispre-dictions by selling large quantities in advance restaurants maytake advantage of projection bias by offering all-you-can-eatmeals to hungry diners who underestimate how quickly they willbecome satiated

We conclude with perhaps the broadest implication of projec-tion bias much as in the opening quotation from Adam Smithprojection bias will lead to a general ldquoover-rating the differencebetween one permanent situation and anotherrdquo Projection biasover habit formation for instance can lead people to overrate thedifferences between ldquopoverty and richesrdquo In simple terms poorpeople will overestimate how good it would be to become rich andrich people will overestimate how bad it would be to become poorPerhaps more important because projection bias makes peoplemispredict how they themselves would behave in other situa-tions it can lead to misunderstandings between these two groupsof people For instance projection bias can lead low-wealth indi-viduals to nd the behavior of wealthy individuals reprehensiblebecause they expect the rich to engage in more charitable givingthan they actually do

Although on a smaller scale there is experimental evidenceof such misunderstandings between groups As mentioned abovewith regard to projection bias over the endowment effect Van


Boven Dunning and Loewenstein [2000] experimentally demon-strate bargaining inefciency due to ldquobuyersrsquo agentsrdquo underesti-mating sellersrsquo reservation values They further show that whenthe buyersrsquo agents were asked to explain the high prices of thesellers they rejected explanations that resembled the endowmenteffect in favor of explanations that hinged on greed on the part ofthe sellers Much in the same way that the rich might seemgreedy to the poor projection bias over the endowment effect canlead to negative judgments of other peoplersquos characters To theextent that such negative judgments might make people willingto incur losses to hurt others [Gibbons and Van Boven 2001Loewenstein Thompson and Bazerman 1989] projection biasmight have both direct and indirect consequences in everydayeconomic behavior

Smith also suggests that people are likely to exaggerate theimportance between ldquoprivate and public stationrdquo ie social sta-tus In recent years social-comparison theory which studies theways a person cares about her status relative to comparisongroups has received increasing attention from economists Whenpeople make decisions that cause their comparison groups tochangemdashsuch as switching jobs or buying a house in a newneighborhoodmdashprojection bias predicts that people will underap-preciate the effects of a change in comparison groups and henceconsistent with Smithrsquos assertion overestimate the long-termsatisfaction that would accompany such a change As a resultpeople may be too prone to make reference-group-changing deci-sions that give them a sensation of status relative to their currentreference group If a person buys a small house in a wealthyneighborhood in part because it has a certain status value in herapartment building she may not fully appreciate that her frameof reference may quickly become the larger houses and biggercars that her new neighbors have

VII DISCUSSION AND CONCLUSION

Our goal in this paper has been to introduce a formal modelthat can improve the realism of the economic analysis of inter-temporal decision-making by incorporating a common form ofmisprediction of future preferences The psychological evidencepresented in Section II provides support for the existence ofprojection bias and our analysis and discussion in Sections IV Vand VI demonstrate the potential importance of projection bias

1237PROJECTION BIAS

for economics We conclude by putting projection bias in broadereconomic context and discussing some shortcomings and poten-tial extensions of our model

How might one empirically identify projection bias in eco-nomic data According to our model while people may be wrongin their predicted utility they still obey the axioms of ldquorationalrdquochoice in one-shot decisions they have well-dened predictedpreferences and make decisions to maximize those preferences31

Hence if all we observe is a single decision by each personprojection bias may be difcult to identify except insofar as wecan nd eld-data analogues of Read and van Leeuwenrsquos [1998]experimentsmdashinstances in which peoplersquos current state plays ldquotoolargerdquo a role in their decisions However if we observe multipleobservations for each decision-maker researchers can identifyprojection bias through dynamic inconsistency We might com-pare directly peoplersquos plans and their later behavior as in theLoewenstein and Adler [1995] experiments Or we might indi-rectly infer dynamic inconsistency from intertemporal behavioras in the health-club evidence from Della Vigna and Malmendier[2002]32

Our review of evidence and our analysis in this paper leave anumber of open questions One is the extent to which projectionbias diminishes with experience That projection bias operates onstates such as hunger with which people should have ampleexperience suggests that it does not disappear with experienceMoreover an explicit test of the effect of repeated experiencefailed to produce any appreciable learning [Van Boven Loewen-stein and Dunning 2003] In this study ldquobuyersrsquo agentsrdquomdashwithincentives to facilitate exchange but at the lowest possible pricemdashmade take-it-or-leave-it offers for an object to sellers There wereve rounds of possible trade with feedback after each roundabout whether the bids were too high or too low Bids which wereinitially too low due to buyersrsquo agentsrsquo underappreciation of sell-ersrsquo attachments to objects increased over the ve rounds andconverged toward the prot-maximizing level However when a

31 Kahneman [1994] distinguishes between ldquoexperienced utilityrdquo which re-ects onersquos welfare and ldquodecision utilityrdquo which reects the attractiveness ofoptions as inferred from onersquos decisions projection bias represents a reason whydecision utility may deviate from experienced utility

32 When we observe dynamic inconsistency the question arises whether thesource is projection bias or some other error such as naive hyperbolic discountingBut as we discuss in footnote 27 it is often possible to nd situations wheredifferent behavioral errors predict different types of dynamic inconsistency


new object of similar value was substituted for the original objectthe same pattern occurred Subjects learned to adjust their bidsupward but they did not learn to anticipate the endowmenteffect

A related second open question is how aware are people of thebias The existence of advice such as ldquocount to ten before yourespondrdquo or ldquonever shop on an empty stomachrdquo suggests thatpeople are aware of projection bias on a meta-level In additionwe suspect that many rules people develop are designed to dealwith moment-by-moment projection bias For instance in thecontext of our durable-good model people might develop rulessuch as never buy a car on a rst visit to a dealer The need forsuch rules provides further evidence that people suffer from pro-jection bias but also implies that its damaging effects may bemitigated in many circumstances

A third open question concerns our treatment of projectionbias as a pure error We believe that perhaps the most importantreason to incorporate projection bias into economics is to improvewelfare analysis (rather than solely to improve behavioral pre-dictions)mdashto study for instance whether addicts are making anoptimal lifetime decision to become addicts As such we haveemphasized the ways in which people behave suboptimally Theremay however be reasons to be more cautious about treating allchanges in behavior as suboptimal33

As models that reect the reality of both short-term uctua-tions and long-term changes in preferences become more wide-spread in economics economists must seriously address the ques-tion of whether people accurately predict how their preferenceswill change We hope our analysis and examples illustrate thepotential benets for both behavioral and welfare economics ofincorporating mispredictions of utilities in general and projectionbias in particular into formal economic analysis

33 Projection bias might for instance serve the interests of the human raceto the detriment of the individual eg the failure to appreciate adaptation toparaplegia or blindness may help to limit the number of disabled people in thepopulation A full normative analysis should as always take into account suchexternalities and it is possible that projection bias mitigates them Similarly atthe individual level it is possible that projection bias serves to mitigate othererrors eg to work against factors such as self-control problems or underappre-ciation of risks that might cause people to exert too little effort at avoidingparaplegia But we see no reason to expect projection bias to more often mitigateexternalities and other errors as opposed to exacerbate them and we believe thatin any event full articulation of all errors and externalities including projectionbias is the appropriate way to conduct welfare analysis

1239PROJECTION BIAS

APPENDIX PROOFS

Proof of Lemma 1 To ease our notation we use vt [ v9(ct 2st) for all t Also for any function g(i) we say Si5 a

b g(i) 5 0 whena b

Given a 5 0 the rst-order conditions are vt 2 Xt 5 l forall t where l is the multiplier on the income constraint andXt [ g St5 t1 1

T (1 2 g)t2 (t1 1 )vt Hence for all t vt2 1 2 Xt2 1 5vt 2 Xt or vt2 1 2 vt 5 Xt2 1 2 Xt Because Xt2 1 2 Xt 5 g(vt 2Xt) and because vt 2 Xt 5 vT it follows that for all t vt2 1 2vt 5 gvT 0 which in turn implies vt 5 (1 1 (T 2 t)g)vT Hence v1 vT which given v0 0 implies c1 2 s1 cT 2 sT

For any t ct $ st implies that ct $ st11 $ st combining theseconditions with ct11 2 st11 ct 2 st implies that ct11 ct $

st11 Hence if ct $ st for some t T then ct11 ct $ st11 whichin turn implies that ct12 ct11 $ s

t12 and so forth The resultfollows

Proof of Proposition 1 We use vt as in the proof of Lemma 1and note that c1 $ s1 implies c1 cT and also ct 2 st 0 for all t 1 We also use vt

A [ v9(ctA 2 st

A ) and vt [ v9(ctA 2

s1) and note that vt vs if and only if ctA cs

A The rst-orderconditions are vt

A 2 XtA 1 a(1 2 a)vt 5 lA (1 2 a) where lA is

the multiplier on the income constraint and X tA [ g St5 t1 1

T (1 2g)t2 (t1 1 )vt

A Hence for all t vt2 1A 2 vt

A 5 Xt2 1A 2 X t

A 1 a(1 2a)[vt 2 vt2 1] Because X t2 1

A 2 XtA 5 g(vt

A 2 XtA ) and because

vtA 2 X t

A 5 vTA 1 a(1 2 a)[vT 2 vt] it follows that for all t vt21

A 2vt

A 5 gvTA 1 a(1 2 a)[g vT 2 (vt2 1 2 (1 2 g)vt)] By starting

with the condition for t 5 T and iterating backwards we canderive that for all t

v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i

It is useful to rewrite this condition as vtA (1 1 (T 2 t)g) 1

a(1 2 a) Rt (1 1 (T 2 t)g) 5 vTA 1 a(1 2 a)vT where Rt 5

(vt 1 g Si5 t11T vi) Also note that for all t and s


(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg

We next establish two claimsClaim 1 There exist t and s such that vt

A vt and vsA vs

Proof Suppose otherwise First consider the case in whichvt

A 5 vt for all t which implies that ctA 5 ct for all t Applying

equation (1) vtA 5 vt for all t implies that Rt(1 1 (T 2 t)g) 5

Rs(1 1 (T 2 s)g) for all t and s but this requires that vt 5 vs

and therefore ctA 5 cs

A for all t and s which contradictsc1 cT Next consider the case in which vt

A vt andtherefore ct

A 2 stA $ ct 2 st for all t where the inequalities are

strict for some t For any t if stA $ st then ct

A 2 stA $ ct 2 st

implies that ctA $ ct and therefore st1 1

A $ st1 1 where either stA

st or ctA 2 st

A ct 2 st implies that ctA ct and st1 1

A st1 1 Inaddition c1

A 2 s1A $ c1 2 s1 implies that c1

A $ c1 and therefores2

A $ s2 where c1A 2 s1

A c1 2 s1 implies that c1A c1 and s2

A s2 It follows that ct

A $ ct for all t and ctA ct for some t which

contradicts that St5 1T ct

A 5 St5 1T ct 5 Y Finally an analogous

logic rules out the case in which vtA $ vt for all t and vt

A vt forsome t

Claim 2 There exists t [ 1 T 2 1 such that vtA vt

for t [ 1 t and vtA vt for t [ t 1 1 T

Proof Suppose otherwise Let x [ max tuvtA vt which

exists given Claim 1 and let z [ maxt xuvtA vt which must

exist if the Claim 2 is not true Applying equation (1) vzA vz and

vz1 1A vz1 1 together imply that Rz(1 1 (T 2 z)g) Rz1 1 (1 1

(T 2 z 2 1)g) which means [vz 1 gvz1 1 1 g Si5 z1 2T vi](1 1

(T 2 z)g) [vz1 1 1 g Si5 z1 2T vi](1 1 (T 2 z 2 1)g) or

(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i

We prove that inequality (2) cannot hold from which Claim 2follows

We rst establish that ctA cz

A and therefore vt vz for all

1241PROJECTION BIAS

t [ z 1 1 x Because vt vt1 1 for all t it follows thatvz

A vz vt $ vtA for all t [ z 1 1 x Since vt

A vtimplies that ct

A 2 stA $ ct 2 st ct 2 st 0 implies that ct

A stA

and therefore st1 1A st

A and so stA sz1 1

A for all t [ z 1 2 x If cz

A szA then sz1 1

A czA and therefore ct

A stA $ sz1 1

A cz

A If instead czA $ sz

A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz

A implies that ctA cz

A If x 5 T then vt vz for all t [ z 1 2 T and

therefore g2 Si5 z1 2T vi g2(T 2 z 2 1)vz But then vz vz1 1

implies that g2(T 2 z 2 1)vz [1 1 (T 2 z 2 1)g]vz 2 [1 1(T 2 z 2 1)g(1 2 g)]vz1 1 which contradicts inequality (2)

Consider instead x T Given (vtA 2 vt) 1 a(1 2 a) Rt 5

(1 2 (T 2 t)g)[vTA 2 vT 1 a(1 2 a)vT ] it follows that for all t

and s 1(s 2 t)[(vtA 2 vt) 2 (vs

A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT

A 2 vT 1 a(1 2 a)vT] Hence (vzA 2 vz) 2 (vz1 1

A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2

vx1 n) 1 a(1 2 a)(Rx 2 Rx1 n )] Given vzA vz vz1 1

A vz1 1 vx

A vx and vx1 nA vx1 n it follows that (Rx 2 Rx1 n ) 2 n(Rz 2

Rz1 1) 0 Because Rx 2 Rx1 n 5 vx 1 g Si5 1n 2 1 vx1 i 2 (1 2

g)vx 1 n and Rz 2 Rz1 1 5 vz 2 (1 2 g)vz1 1 this conditionbecomes (Rx 2 Rx1 n ) 2 n(Rz 2 Rz1 1) 5 (1 2 g)(vz1 1 2 vx1 n ) 1[vx 1 (n 2 1)vz1 1 2 nvz] 1 g Si5 1

n 2 1 (vx1 i 2 vz1 1) 0 Since vx vz and vz1 1 vz applying this condition for n 5 1 yields vz1 1 vx1 1 and then applying it for n 5 2 yields vz1 1 vx1 2 and soforth It follows that vx1 n vz1 1 vz for all n [ 1 T 2x and therefore vt vz for all t [ z 1 1 T But then g2

Si5 z1 2T vi g2(T 2 z 2 1)vz [1 1 (T 2 z 2 1)g]vz 2 [1 1

(T 2 z 2 1)g(1 2 g)]vz1 1 which contradicts inequality (2)Claim 2 follows

Finally we prove the main result Posit otherwise and denew [ mintuSi5 1

t c iA Si5 1

t ci Claims 1 and 2 together implythat v1

A v1 and therefore c1A c1 Hence w 1 and cw

A cw Note that if w t (where t dened as in Claim 2) then v1

A v1 and vt

A vt for all t [ 2 w 2 1 which implies that swA

sw (using logic identical to that in proof of Claim 1) But thencw

A cw implies that vwA vw which contradicts that w t It

follows that w t and therefore vwA vw

Dene y [ min t wuctA $ ct such a y must exist We can

write the state st as


s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1

Then Si5 1w ci $ Si5 1

w c iA and Si5 1

t ci Si5 1t c i

A for all t wtogether imply that

sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0

Moreover when y w 1 1 sw 1 1 s w 1 1A combined with ct c t

A

for all t [ w 1 1 y 2 1 implies that sy s yA Since by

the denition of y c yA $ cy it follows that c y

A 2 s yA cy 2 sy and

therefore vyA vy But given vw

A vw this contradicts Claim 2The result follows

Proof of Proposition 2 As a preliminary step we prove c 1A

c 3A and c 2

A c 3A Posit otherwise and suppose that z [ arg

maxt[1 2 c tA Hence c z

A $ c z1 1A and c z

A $ c TA which implies that

vz vz1 1 and vz vT Recall that vzA 2 vz2 1

A 5 gvTA 1 a(1 2

a)[gvT 2 (vz 2 (1 2 g)vz1 1)] Because gvT 2 (vz 2 (1 2g)vz 1 1) 5 (1 2 g)(vz1 1 2 vz) 1 g(vT 2 vz) $ 0 vz

A 2 vz1 1A 0

Given v0 0 this implies that c zA 2 s z

A c z1 1A 2 s z1 1

A and givenc z

A $ c z1 1A this holds only if s z

A s z1 1A which in turn holds only

if c zA s z

A But if z 5 1 this contradicts c 1A s1 and if z 5 2

this contradicts c 2A c 1

A s 2A (c 1

A s1 implies that c 1A s 2

A)In period 1 true utility is U1(c1 c2 c3) 5 St5 1

3 v(ct 2 st) andperceived utility is

U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS

Period 1 behavior (c 1A c 2

A c 3A ) must satisfy ]U1(c 1

A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3

Aus1)]c2 5 ]U1(c1Ac2

Ac3Aus1)]c3 Because ]U1(c1c2c3us1)]ct 5

(1 2 a)]U1(c1c2c3)]ct 1 av9(ct 2 s1) for t [ 123 ]U1(c1Ac2

Ac3Aus1)

]c2 5 ]U1(c1Ac2

Ac3Aus1)]c3 implies that (1 2 a)[]U1(c1

Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2

A 2 s1)]After choosing c 1

A s1 in period 2 the state is s 2A 5 (1 2

g)s1 1 gc 1A true utility is U2(c2 c3 us 2

A ) 5 St5 23 v(ct 2 st) and

perceived utility is U2(c2 c3 us 2A ) 5 St5 2

3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2

3 v(ct 2 s 2A ) Period

2 behavior (c 2AA c 3

A A) must satisfy ]U2(c 2AA c 3

A A us 2A)]c2 5

]U2(c 2AA c 3

A A us 2A)]c3 Note that for t [ 23 ]U1(c1 c2 c3)]ct 5

]U2(c2 c3 us 2A )]ct for all c2 and c3 Hence because ]U2(c2 c3 us 2

A)]ct 5 (1 2 a)]U1(c 1

A c2 c3)]ct 1 av9(ct 2 s 2A) for t [ 23

]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]

v- 0 s 2A s1 (which follows from c 1

A s1) and c 2A c 3

A

together imply v9(c2A 2 s2

A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3

A 2 s1)which in turn implies that ]U2(c 2

A c 3A us2)]c2 ]U2(c 2

A c 3A us2)

]c3 Given the concavity of U2 we must have c2AA c2

A and c3AA c3

AAn analogous argument holds for v- 0v- 5 0 implies that v9(c 2

A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3

A 2 s1) 5 k(s2 2 s1) for some constant k (ie v- 5 0implies that v9 is linear and decreasing so 2k is the slope of v9)and so ]U2(c 2

A c 3A us2)]c2 5 ]U2(c 2

A c 3A us2)]c3 It follows that

(c 2A A c 3

AA ) 5 (c 2A c 3

A ) (The conclusion that v- 5 0 yields dynamicconsistency would hold for any T and for any c 1

A )

Proof of Proposition 3 Using the notation from the proof ofProposition 2

lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3


The concavity of U2 implies that lA A $ min ]U2(c 2A c 3

A us 2A)]c2

]U2(c 2A c 3

A us 2A )]c3 For t [ 23 we have ]U2(c 2

A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3

A us1)]ct 1 a[v9(c tA 2 s 2

A) 2 v9(c tA 2 s1)] Then s 2

A s1 (which follows from c 1

A s1) combined with v0 0 impliesthat v9(c t

A 2 s 2A ) v9(c t

A 2 s1) Hence for t [ 23 ]U2(c 2A

c 3A us 2

2)]ct ]U1(c 1A c 2

A c 3A us1)]ct 5 lA The result follows

CARNEGIE MELLON UNIVERSITY

CORNELL UNIVERSITY

UNIVERSITY OF CALIFORNIA BERKELEY

REFERENCES

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Baron Jonathan David A Asch Angela Fagerlin Christopher Jepson GeorgeLoewenstein Jason Riis Margaret G Stineman and Peter A Ubel ldquoEffect ofAssessment Method on the Discrepancy between Judgments of Health Dis-orders People Have and Do Not Have A Web Studyrdquo Medical DecisionMaking forthcoming

Becker Gary and Kevin Murphy ldquoA Theory of Rational Addictionrdquo Journal ofPolitical Economy XCVI (1988) 675ndash700

Bernheim B Douglas ldquoThe Timing of Retirement A Comparison of Expectationsand Realizationsrdquo in David A Wise ed The Economics of Aging (Chicago ILUniversity of Chicago Press 1989)

Boldrin Michele Lawrence Christiano and Jonas Fisher ldquoHabit PersistenceAsset Returns and the Business Cyclerdquo American Economic Review XCI(2001) 149ndash166

Bowman David Deborah Minehart and Matthew Rabin ldquoLoss Aversion in aConsumption-Savings Modelrdquo Journal of Economic Behavior and Organiza-tion XXXVIII (1999) 155ndash178

Boyd Norman Heather J Sutherland Karen Z Heasman David L Tritcher andBernard Cummings ldquoWhose Utilities for Decision Analysisrdquo Medical Deci-sion Making X (1990) 58ndash 67

Camerer Colin Samuel Issacharoff George Loewenstein Ted OrsquoDonoghue andMatthew Rabin ldquoRegulation for Conservatives Behavioral Economics andthe Case for lsquoAsymmetric Paternalismrsquo rdquo University of Pennsylvania LawReview CLI (2003) 1211ndash1254

Campbell John and John Cochrane ldquoBy Force of Habit A Consumption-BasedExplanation of Aggregate Stock Market Behaviorrdquo Journal of Political Econ-omy CVII (1999) 205ndash251

Campbell John and Angus Deaton ldquoWhy Is Consumption So Smoothrdquo Review ofEconomic Studies LVI (1989) 357ndash374

Carroll Christopher Jody Overland and David Weil ldquoSaving and Growth withHabit Formationrdquo American Economic Review XC (2000) 341ndash355

Chochinov Harvey M Douglas Tataryn Jennifer J Clinch and Deborah Dud-geon ldquoWill to Live in the Terminally Illrdquo The Lancet CCCLIV (1999 Issue9181 September 4) 816ndash 819

Constantinides George ldquoHabit Formation A Resolution of the Equity PremiumPuzzlerdquo Journal of Political Economy XCVIII (1990) 519ndash543

Coppola Kristen M Jamila Bookwala Peter H Ditto Lisa K Lockhart JosephH Danks and William D Smucker ldquoElderly Adultsrsquo Preferences for Life-Sustaining Treatments The Role of Impairment Prognosis and Painrdquo DeathStudies XXIII (1999) 617ndash 634

Della Vigna Stefano and Ulrike Malmendier ldquoOverestimating Self-Control Evi-dence from the Health Club Industryrdquo Working Paper University of Califor-nia at Berkeley and Stanford University 2002

1245PROJECTION BIAS

Druley Jennifer A Peter H Ditto Kathleen A Moore Joseph H Danks ATownsend and William D Smucker ldquoPhysiciansrsquo Predictions of Elderly Out-patientsrsquo Preferences for Life-Sustaining Treatmentrdquo Journal of FamilyPractice XXXVII (1993) 469ndash475

Duesenberry James Income Saving and the Theory of Consumer Behavior(Cambridge MA Harvard University Press 1949)

Dynan Karen ldquoHabit Formation in Consumer Preferences Evidence from PanelDatardquo American Economic Review XC (2000) 391ndash406

Frank Robert Luxury Fever (New York NY Free Press 1999)Frederick Shane and George Loewenstein ldquoHedonic Adaptationrdquo in Daniel

Kahneman Edward Diener and Norbert Schwarz eds Well-Being TheFoundations of Hedonic Psychology (New York NY Russell Sage FoundationPress 1999)

Fuhrer Jeffrey ldquoHabit Formation in Consumption and Its Implications for Mone-tary-Policy Modelsrdquo American Economic Review XC (2000) 367ndash390

Genesove David and Christopher Mayer ldquoLoss Aversion and Seller BehaviorEvidence from the Housing Marketrdquo Quarterly Journal of Economics CXVI(2001) 1233ndash1260

Gibbons Robert and Leaf Van Boven ldquoContingent Social Utility in the PrisonersrsquoDilemmardquo Journal of Economic Behavior and Organization XLV (2001)1ndash17

Gilbert Daniel T Michael J Gill and Timothy D Wilson ldquoThe Future Is NowTemporal Correction in Affective Forecastingrdquo Organizational Behavior andHuman Decision Processes LXXXVIII (2002) 430ndash444

Gilbert Daniel T Elizabeth C Pinel Timothy D Wilson Stephen J Blumbergand Thalia P Wheatley ldquoImmune Neglect A Source of Durability Bias inAffective Forecastingrdquo Journal of Personality and Social Psychology LXXV(1998) 617ndash638

Giordano Louis A Warren K Bickel George Loewenstein Eric A Jacobs GaryJ Badger and Lisa A Marsch ldquoMild Opioid Deprivation and Delay toConsequences Affects How Opioid-Dependent Outpatients Value an ExtraMaintenance Dose of Buprenorphinerdquo Working Paper Substance AbuseTreatment Center Psychiatry Department University of Vermont Burling-ton 2001

Helson Harry Adaptation-Level Theory An Experimental and Systematic Ap-proach to Behavior (New York NY Harper and Row 1964)

Herrnstein Richard George Loewenstein Drazen Prelec and William VaughanJr ldquoUtility Maximization and Melioration Internalities in IndividualChoicerdquo Journal of Behavioral Decision Making VI (1993) 149ndash185

Jepson Christopher George Loewenstein and Peter Ubel ldquoActual versus Esti-mated Differences in Quality of Life before and after Renal TransplantrdquoWorking Paper Department of Social and Decision Sciences Carnegie MellonUniversity 2001

Jermann Urban ldquoAsset Pricing in Production Economiesrdquo Journal of MonetaryEconomics XLI (1998) 257ndash275

Kahneman Daniel ldquoJudgement and Decision-Making A Personal Viewrdquo Psycho-logical Science II (1991) 142ndash145

mdashmdash ldquoNew Challenges to the Rationality Assumptionrdquo Journal of Institutional andTheoretical Economics CL (1994) 18ndash36

Kahneman Daniel and Amos Tversky ldquoProspect Theory An Analysis of Decisionunder Riskrdquo Econometrica XLVII (1979) 263ndash291

Kahneman Daniel Jack L Knetsch and Richard H Thaler ldquoAnomalies TheEndowment Effect Loss Aversion and Status Quo Biasrdquo Journal of Eco-nomic Perspectives V (1991 Winter) 193ndash206

Laibson David ldquoEssays on Hyperbolic Discountingrdquo PhD thesis Department ofEconomics Massachusetts Institute of Technology 1994

mdashmdash ldquoGolden Eggs and Hyperbolic Discountingrdquo Quarterly Journal of EconomicsCXII (1997) 443ndash 477

Laibson David Andrea Repetto and Jeremy Tobacman ldquoSelf-Control and Savingfor Retirementrdquo Brookings Papers on Economic Activity 1 (1998) 91ndash196

Loewenstein George ldquoOut of Control Visceral Inuences on Behaviorrdquo Organi-zational Behavior and Human Decision Processes LXV (1996) 272ndash292


mdashmdash ldquoA Visceral Account of Addictionrdquo in Jon Elster and Ole-Joslashrgen Skog edsGetting Hooked Rationality and Addiction (Cambridge UK Cambridge Uni-versity Press 1999)

Loewenstein George and Daniel Adler ldquoA Bias in the Prediction of TastesrdquoEconomic Journal CV (1995) 929ndash937

Loewenstein George and Erik Angner ldquoPredicting and Indulging ChangingPreferencesrdquo in George Loewenstein Daniel Read and Roy Baumeister edsTime and Decision Economic and Psychological Perspectives on Intertempo-ral Choice (New York NY Russell Sage Foundation Press 2003)

Loewenstein George and Shane Frederick ldquoPredicting Reactions to Environmen-tal Changerdquo in Max Bazerman David Messinck Ann Tenbrunsel and Kim-berly Wade-Benzoni eds Environment Ethics and Behavior (San FranciscoCA New Lexington Press 1997)

Loewenstein George and David Schkade ldquoWouldnrsquot It Be Nice Predicting Fu-ture Feelingsrdquo in Daniel Kahneman Edward Diener and Norbert Schwarzeds Well-Being The Foundations of Hedonic Psychology (New York NYRussell Sage Foundation Press 1999)

Loewenstein George Daniel Nagin and Raymond Paternoster ldquoThe Effect ofSexual Arousal on Predictions of Sexual Forcefulnessrdquo Journal of Crime andDelinquency XXXIV (1997) 443ndash473

Loewenstein George Ted OrsquoDonoghue and Matthew Rabin ldquoProjection Bias inPredicting Future Utilityrdquo Department of Economics Working Paper E00-284 University of California at Berkeley 2000

Loewenstein George Ted OrsquoDonoghue and Matthew Rabin ldquoProjection Bias inPredicting Future Utilityrdquo Center for Analytic Economics Working Paper02-11 Cornell University 2002

Loewenstein George Leigh Thompson and Max Bazerman ldquoSocial Utility andDecision Making in Interpersonal Contextsrdquo Journal of Personality andSocial Psychology LVII (1989) 426ndash 441

Muellbauer John ldquoHabits Rationality and Myopia in the Life Cycle ConsumptionFunctionrdquo Annales drsquoEconomie et de Statistique IX (1988) 47ndash70

Nisbett Richard E and David E Kanouse ldquoObesity Hunger and SupermarketShopping Behaviorrdquo Proceedings of the Annual Convention of the AmericanPsychological Association III (1968) 683ndash684

OrsquoDonoghue Ted and Matthew Rabin ldquoDoing It Now or Laterrdquo American Eco-nomic Review LXXXIX (1999a) 103ndash124

OrsquoDonoghue Ted and Matthew Rabin ldquoIncentives for Procrastinatorsrdquo QuarterlyJournal of Economics CXIV (1999b) 769ndash 816

Pollak Robert A ldquoHabit Formation and Dynamic Demand Functionsrdquo Journal ofPolitical Economy LXXVIII (1970) 745ndash763

Prelec Drazen and George Loewenstein ldquoThe Red and the Black Mental Ac-counting of Savings and Debtrdquo Marketing Science XVII (1998) 4 ndash28

Read Daniel and George Loewenstein ldquoEnduring Pain for Money DecisionsBased on the Perception of Memory of Painrdquo Journal of Behavioral DecisionMaking XII (1999) 1ndash17

Read Daniel and Barbara van Leeuwen ldquoPredicting Hunger The Effects ofAppetite and Delay on Choicerdquo Organizational Behavior and Human Deci-sion Processes LXXVI (1998) 189ndash205

Ryder Harl and Geoffrey Heal ldquoOptimal Growth with Intertemporally Depen-dent Preferencesrdquo Review of Economic Studies XL (1973) 1ndash33

Sackett David L and George W Torrance ldquoThe Utility of Different Health Statesas Perceived by the General Publicrdquo Journal of Chronic Diseases XXXI(1978) 697ndash704

Schkade David and Daniel Kahneman ldquoDoes Living in California Make PeopleHappy A Focusing Illusion in Judgments of Life Satisfactionrdquo PsychologicalScience IX (1998) 340ndash346

Scitovsky Tibor The Joyless Economy An Inquiry into Human Satisfaction andConsumer Dissatisfaction (Oxford UK Oxford University Press 1976)

Sieff Elaine Robyn Dawes and George Loewenstein ldquoAnticipated versus ActualResponses to HIV Test Resultsrdquo American Journal of Psychology CXII(1999) 297ndash311

Slevin Maurice L H Plant D Lynch J Drinkwater and W M Gregory ldquoWho

1247PROJECTION BIAS

Should Measure Quality of Life the Doctor or the Patientrdquo British Journalof Cancer LVII (1988) 109ndash112

Smith Adam The Theory of Moral Sentiments Knud Haakonssen ed (Cam-bridge Cambridge University Press 2002)

Solomon Andrew ldquoA Personal History Anatomy of Melancholyrdquo New YorkerLXXIII (January 12 1998) 46ndash 61

Starmer Chris ldquoDevelopments in Non-Expected Utility Theory The Hunt for aDescriptive Theory of Choice under Riskrdquo Journal of Economic LiteratureXXXVIII (2000) 332ndash382

Strahilevitz Michal and George Loewenstein ldquoThe Effect of Ownership Historyon the Valuation of Objectsrdquo Journal of Consumer Research XXV (1998)276ndash289

Thaler Richard ldquoToward a Positive Theory of Consumer Choicerdquo Journal ofEconomic Behavior and Organization I (1980) 39ndash 60

Tversky Amos and Daniel Kahneman ldquoLoss Aversion in Riskless Choice AReference-Dependent Modelrdquo Quarterly Journal of Economics CVI (1991)1039ndash1061

Ubel Peter George Loewenstein and Christopher Jepson ldquoDisability and Sun-shine Can Predictions Be Improved by Drawing Attention to Focusing Illu-sions or Emotional Adaptationrdquo Working Paper University of MichiganMedical Center 2003

U S Department of Health and Human Services (U S DHHS) PreventingTobacco Use Among Young People A Report of the Surgeon General (Wash-ington DC U S Government Printing Ofce 1994)

Van Boven Leaf and George Loewenstein ldquoProjection of Transient Drive StatesrdquoPersonality and Social Psychology Bulletin XXIX (2003) 1159ndash1168

Van Boven Leaf David Dunning and George Loewenstein ldquoEgocentric EmpathyGaps between Owners and Buyers Misperceptions of the Endowment EffectrdquoJournal of Personality and Social Psychology LXXIX (2000) 66ndash76

Van Boven Leaf George Loewenstein and David Dunning ldquoMispredicting theEndowment Effect Underestimation of Ownersrsquo Selling Prices by BuyerrsquosAgentsrdquo Journal of Economic Behavior and Organization LI (2003) 351ndash365

Van Boven Leaf George Loewenstein Ned Welch and David Dunning ldquoTheIllusion of Courage Underestimating Social-Risk Aversion in Self and Oth-ersrdquo Working Paper University of Colorado Boulder 2003

Weinstein Neil ldquoCommunity Noise Problems Evidence against AdaptationrdquoJournal of Environmental Psychology II (1982) 87ndash97

Wilson Timothy D Thalia P Wheatley Jonathan M Meyers Daniel T Gilbertand Danny Axsom ldquoFocalism A Source of Durability Bias in Affective Fore-castingrdquo Journal of Personality and Social Psychology LXXVIII (2000)821ndash836


When making summer vacation plans during the cold of winterpeople must predict what vacations will be most enjoyable duringthe heat of summer When ordering food at the beginning of ameal people must predict how hungry they will be at the end ofthe meal When contemplating smoking cigarettes or indulging inother habit-forming substances people must predict how thisconsumption will affect their future desire for and enjoyment ofthese substances

In this paper we provide evidence for formalize and explorethe implications of a general bias in the prediction of futuretastes people tend to understand qualitatively the directions inwhich their tastes will change but systematically underestimatethe magnitudes of these changes Hence they tend to exaggeratethe degree to which their future tastes will resemble their currenttastes Such projection bias may cause people making summervacation plans in the winter to choose overly warm destinationsdiners to order too much food at the beginning of meals andpeople unaddicted to cigarettes to underestimate the power ofand drawbacks of addiction

In Section II we review evidence from a variety of domainssupporting the existence of projection bias People underappreci-ate the effects of long-term changes in tastes such as those thatresult from adaptation to a shifting standard of living People alsounderappreciate the effects of frequently uctuating tastes suchas uctuating hunger Indeed virtually all evidence we are fa-miliar with on misprediction of future tastes is consistent withprojection bias

In Section III we develop a formal model of projection bias Tox ideas suppose that a personrsquos instantaneous utility can bewritten as u(cs) where c is her consumption and s is a ldquostaterdquothat parameterizes her tastes Suppose further that the personwith current state s9 must predict her tastes at a time in thefuture when her state will be s Consistent with evidence thatpeople tend to understand the qualitative nature of changes intastes but underestimate the degree of change we assume thatthe personrsquos prediction of her own future preferences u(csus9)lies somewhere ldquoin betweenrdquo her true future tastes u(cs) and hercurrent tastes u(cs9) Our formal analysis in this paper assumesthat u(cs us9) is a simple linear combination of u(cs) and u(cs9)which we refer to as simple projection bias

Because projection bias leads to discrepancies between pre-dicted and subsequently realized utilities it implies that a per-sonrsquos behavior need not correspond to correct intertemporal util-






1211PROJECTION BIAS












1213PROJECTION BIAS











Future hunger

Hungry Satiated


1215PROJECTION BIAS



III THE MODEL













1217PROJECTION BIAS



Ut~c t cT 5 Ot5t

T

dtu~ctst




T

dtu~ctstust









11






1219PROJECTION BIAS













T

~1 2 av~ct 2 st 1 av~ct 2 s1

such that


T

ct Y


A cTA)





1221PROJECTION BIAS









A St5 1t ct

for all t T



A are exogenous


T

~1 2 av~ct 2 st 1 av~ct 2 s2

such that

s2 5 ~1 2 gs1 1 gc1A

s t 5 ~1 2 gst21 1 gc t21 for t [ $3 T

and Ot52

T

ct Y 2 c1A


1223PROJECTION BIAS



plans (c2A cT






A A 5 c2A











1225PROJECTION BIAS










E1U1 5 E1F Ok51

D

m11k 2 PG 5 Dm 2 P


E1U1 5 E1F Ok51

D

~1 2 am11k 1 am1 2 PG5 Dm 1 aD~m1 2 m 2 P




1227PROJECTION BIAS









1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS





















1211PROJECTION BIAS












1213PROJECTION BIAS











Future hunger

Hungry Satiated


1215PROJECTION BIAS



III THE MODEL













1217PROJECTION BIAS



Ut~c t cT 5 Ot5t

T

dtu~ctst




T

dtu~ctstust









11






1219PROJECTION BIAS













T

~1 2 av~ct 2 st 1 av~ct 2 s1

such that


T

ct Y


A cTA)





1221PROJECTION BIAS









A St5 1t ct

for all t T



A are exogenous


T

~1 2 av~ct 2 st 1 av~ct 2 s2

such that

s2 5 ~1 2 gs1 1 gc1A

s t 5 ~1 2 gst21 1 gc t21 for t [ $3 T

and Ot52

T

ct Y 2 c1A


1223PROJECTION BIAS



plans (c2A cT






A A 5 c2A











1225PROJECTION BIAS










E1U1 5 E1F Ok51

D

m11k 2 PG 5 Dm 2 P


E1U1 5 E1F Ok51

D

~1 2 am11k 1 am1 2 PG5 Dm 1 aD~m1 2 m 2 P




1227PROJECTION BIAS









1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS




























1213PROJECTION BIAS











Future hunger

Hungry Satiated


1215PROJECTION BIAS



III THE MODEL













1217PROJECTION BIAS



Ut~c t cT 5 Ot5t

T

dtu~ctst




T

dtu~ctstust









11






1219PROJECTION BIAS













T

~1 2 av~ct 2 st 1 av~ct 2 s1

such that


T

ct Y


A cTA)





1221PROJECTION BIAS









A St5 1t ct

for all t T



A are exogenous


T

~1 2 av~ct 2 st 1 av~ct 2 s2

such that

s2 5 ~1 2 gs1 1 gc1A

s t 5 ~1 2 gst21 1 gc t21 for t [ $3 T

and Ot52

T

ct Y 2 c1A


1223PROJECTION BIAS



plans (c2A cT






A A 5 c2A











1225PROJECTION BIAS










E1U1 5 E1F Ok51

D

m11k 2 PG 5 Dm 2 P


E1U1 5 E1F Ok51

D

~1 2 am11k 1 am1 2 PG5 Dm 1 aD~m1 2 m 2 P




1227PROJECTION BIAS









1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS





















1213PROJECTION BIAS











Future hunger

Hungry Satiated


1215PROJECTION BIAS



III THE MODEL













1217PROJECTION BIAS



Ut~c t cT 5 Ot5t

T

dtu~ctst




T

dtu~ctstust









11






1219PROJECTION BIAS













T

~1 2 av~ct 2 st 1 av~ct 2 s1

such that


T

ct Y


A cTA)





1221PROJECTION BIAS









A St5 1t ct

for all t T



A are exogenous


T

~1 2 av~ct 2 st 1 av~ct 2 s2

such that

s2 5 ~1 2 gs1 1 gc1A

s t 5 ~1 2 gst21 1 gc t21 for t [ $3 T

and Ot52

T

ct Y 2 c1A


1223PROJECTION BIAS



plans (c2A cT






A A 5 c2A











1225PROJECTION BIAS










E1U1 5 E1F Ok51

D

m11k 2 PG 5 Dm 2 P


E1U1 5 E1F Ok51

D

~1 2 am11k 1 am1 2 PG5 Dm 1 aD~m1 2 m 2 P




1227PROJECTION BIAS









1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS



























Future hunger

Hungry Satiated


1215PROJECTION BIAS



III THE MODEL













1217PROJECTION BIAS



Ut~c t cT 5 Ot5t

T

dtu~ctst




T

dtu~ctstust









11






1219PROJECTION BIAS













T

~1 2 av~ct 2 st 1 av~ct 2 s1

such that


T

ct Y


A cTA)





1221PROJECTION BIAS









A St5 1t ct

for all t T



A are exogenous


T

~1 2 av~ct 2 st 1 av~ct 2 s2

such that

s2 5 ~1 2 gs1 1 gc1A

s t 5 ~1 2 gst21 1 gc t21 for t [ $3 T

and Ot52

T

ct Y 2 c1A


1223PROJECTION BIAS



plans (c2A cT






A A 5 c2A











1225PROJECTION BIAS










E1U1 5 E1F Ok51

D

m11k 2 PG 5 Dm 2 P


E1U1 5 E1F Ok51

D

~1 2 am11k 1 am1 2 PG5 Dm 1 aD~m1 2 m 2 P




1227PROJECTION BIAS









1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS























Future hunger

Hungry Satiated


1215PROJECTION BIAS



III THE MODEL













1217PROJECTION BIAS



Ut~c t cT 5 Ot5t

T

dtu~ctst




T

dtu~ctstust









11






1219PROJECTION BIAS













T

~1 2 av~ct 2 st 1 av~ct 2 s1

such that


T

ct Y


A cTA)





1221PROJECTION BIAS









A St5 1t ct

for all t T



A are exogenous


T

~1 2 av~ct 2 st 1 av~ct 2 s2

such that

s2 5 ~1 2 gs1 1 gc1A

s t 5 ~1 2 gst21 1 gc t21 for t [ $3 T

and Ot52

T

ct Y 2 c1A


1223PROJECTION BIAS



plans (c2A cT






A A 5 c2A











1225PROJECTION BIAS










E1U1 5 E1F Ok51

D

m11k 2 PG 5 Dm 2 P


E1U1 5 E1F Ok51

D

~1 2 am11k 1 am1 2 PG5 Dm 1 aD~m1 2 m 2 P




1227PROJECTION BIAS









1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS



















III THE MODEL













1217PROJECTION BIAS



Ut~c t cT 5 Ot5t

T

dtu~ctst




T

dtu~ctstust









11






1219PROJECTION BIAS













T

~1 2 av~ct 2 st 1 av~ct 2 s1

such that


T

ct Y


A cTA)





1221PROJECTION BIAS









A St5 1t ct

for all t T



A are exogenous


T

~1 2 av~ct 2 st 1 av~ct 2 s2

such that

s2 5 ~1 2 gs1 1 gc1A

s t 5 ~1 2 gst21 1 gc t21 for t [ $3 T

and Ot52

T

ct Y 2 c1A


1223PROJECTION BIAS



plans (c2A cT






A A 5 c2A











1225PROJECTION BIAS










E1U1 5 E1F Ok51

D

m11k 2 PG 5 Dm 2 P


E1U1 5 E1F Ok51

D

~1 2 am11k 1 am1 2 PG5 Dm 1 aD~m1 2 m 2 P




1227PROJECTION BIAS









1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS























1217PROJECTION BIAS



Ut~c t cT 5 Ot5t

T

dtu~ctst




T

dtu~ctstust









11






1219PROJECTION BIAS













T

~1 2 av~ct 2 st 1 av~ct 2 s1

such that


T

ct Y


A cTA)





1221PROJECTION BIAS









A St5 1t ct

for all t T



A are exogenous


T

~1 2 av~ct 2 st 1 av~ct 2 s2

such that

s2 5 ~1 2 gs1 1 gc1A

s t 5 ~1 2 gst21 1 gc t21 for t [ $3 T

and Ot52

T

ct Y 2 c1A


1223PROJECTION BIAS



plans (c2A cT






A A 5 c2A











1225PROJECTION BIAS










E1U1 5 E1F Ok51

D

m11k 2 PG 5 Dm 2 P


E1U1 5 E1F Ok51

D

~1 2 am11k 1 am1 2 PG5 Dm 1 aD~m1 2 m 2 P




1227PROJECTION BIAS









1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS



















Ut~c t cT 5 Ot5t

T

dtu~ctst




T

dtu~ctstust









11






1219PROJECTION BIAS













T

~1 2 av~ct 2 st 1 av~ct 2 s1

such that


T

ct Y


A cTA)





1221PROJECTION BIAS









A St5 1t ct

for all t T



A are exogenous


T

~1 2 av~ct 2 st 1 av~ct 2 s2

such that

s2 5 ~1 2 gs1 1 gc1A

s t 5 ~1 2 gst21 1 gc t21 for t [ $3 T

and Ot52

T

ct Y 2 c1A


1223PROJECTION BIAS



plans (c2A cT






A A 5 c2A











1225PROJECTION BIAS










E1U1 5 E1F Ok51

D

m11k 2 PG 5 Dm 2 P


E1U1 5 E1F Ok51

D

~1 2 am11k 1 am1 2 PG5 Dm 1 aD~m1 2 m 2 P




1227PROJECTION BIAS









1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS





















11






1219PROJECTION BIAS













T

~1 2 av~ct 2 st 1 av~ct 2 s1

such that


T

ct Y


A cTA)





1221PROJECTION BIAS









A St5 1t ct

for all t T



A are exogenous


T

~1 2 av~ct 2 st 1 av~ct 2 s2

such that

s2 5 ~1 2 gs1 1 gc1A

s t 5 ~1 2 gst21 1 gc t21 for t [ $3 T

and Ot52

T

ct Y 2 c1A


1223PROJECTION BIAS



plans (c2A cT






A A 5 c2A











1225PROJECTION BIAS










E1U1 5 E1F Ok51

D

m11k 2 PG 5 Dm 2 P


E1U1 5 E1F Ok51

D

~1 2 am11k 1 am1 2 PG5 Dm 1 aD~m1 2 m 2 P




1227PROJECTION BIAS









1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS





























T

~1 2 av~ct 2 st 1 av~ct 2 s1

such that


T

ct Y


A cTA)





1221PROJECTION BIAS









A St5 1t ct

for all t T



A are exogenous


T

~1 2 av~ct 2 st 1 av~ct 2 s2

such that

s2 5 ~1 2 gs1 1 gc1A

s t 5 ~1 2 gst21 1 gc t21 for t [ $3 T

and Ot52

T

ct Y 2 c1A


1223PROJECTION BIAS



plans (c2A cT






A A 5 c2A











1225PROJECTION BIAS










E1U1 5 E1F Ok51

D

m11k 2 PG 5 Dm 2 P


E1U1 5 E1F Ok51

D

~1 2 am11k 1 am1 2 PG5 Dm 1 aD~m1 2 m 2 P




1227PROJECTION BIAS









1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS






















T

~1 2 av~ct 2 st 1 av~ct 2 s1

such that


T

ct Y


A cTA)





1221PROJECTION BIAS









A St5 1t ct

for all t T



A are exogenous


T

~1 2 av~ct 2 st 1 av~ct 2 s2

such that

s2 5 ~1 2 gs1 1 gc1A

s t 5 ~1 2 gst21 1 gc t21 for t [ $3 T

and Ot52

T

ct Y 2 c1A


1223PROJECTION BIAS



plans (c2A cT






A A 5 c2A











1225PROJECTION BIAS










E1U1 5 E1F Ok51

D

m11k 2 PG 5 Dm 2 P


E1U1 5 E1F Ok51

D

~1 2 am11k 1 am1 2 PG5 Dm 1 aD~m1 2 m 2 P




1227PROJECTION BIAS









1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS

























A St5 1t ct

for all t T



A are exogenous


T

~1 2 av~ct 2 st 1 av~ct 2 s2

such that

s2 5 ~1 2 gs1 1 gc1A

s t 5 ~1 2 gst21 1 gc t21 for t [ $3 T

and Ot52

T

ct Y 2 c1A


1223PROJECTION BIAS



plans (c2A cT






A A 5 c2A











1225PROJECTION BIAS










E1U1 5 E1F Ok51

D

m11k 2 PG 5 Dm 2 P


E1U1 5 E1F Ok51

D

~1 2 am11k 1 am1 2 PG5 Dm 1 aD~m1 2 m 2 P




1227PROJECTION BIAS









1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS



















A St5 1t ct

for all t T



A are exogenous


T

~1 2 av~ct 2 st 1 av~ct 2 s2

such that

s2 5 ~1 2 gs1 1 gc1A

s t 5 ~1 2 gst21 1 gc t21 for t [ $3 T

and Ot52

T

ct Y 2 c1A


1223PROJECTION BIAS



plans (c2A cT






A A 5 c2A











1225PROJECTION BIAS










E1U1 5 E1F Ok51

D

m11k 2 PG 5 Dm 2 P


E1U1 5 E1F Ok51

D

~1 2 am11k 1 am1 2 PG5 Dm 1 aD~m1 2 m 2 P




1227PROJECTION BIAS









1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS



















plans (c2A cT






A A 5 c2A











1225PROJECTION BIAS










E1U1 5 E1F Ok51

D

m11k 2 PG 5 Dm 2 P


E1U1 5 E1F Ok51

D

~1 2 am11k 1 am1 2 PG5 Dm 1 aD~m1 2 m 2 P




1227PROJECTION BIAS









1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS






















1225PROJECTION BIAS










E1U1 5 E1F Ok51

D

m11k 2 PG 5 Dm 2 P


E1U1 5 E1F Ok51

D

~1 2 am11k 1 am1 2 PG5 Dm 1 aD~m1 2 m 2 P




1227PROJECTION BIAS









1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS


























E1U1 5 E1F Ok51

D

m11k 2 PG 5 Dm 2 P


E1U1 5 E1F Ok51

D

~1 2 am11k 1 am1 2 PG5 Dm 1 aD~m1 2 m 2 P




1227PROJECTION BIAS









1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS


















E1U1 5 E1F Ok51

D

m11k 2 PG 5 Dm 2 P


E1U1 5 E1F Ok51

D

~1 2 am11k 1 am1 2 PG5 Dm 1 aD~m1 2 m 2 P




1227PROJECTION BIAS









1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS

























1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS




















1229PROJECTION BIAS










1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS


























1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS




















1231PROJECTION BIAS









1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS

























1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS




















1233PROJECTION BIAS









1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS

























1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS




















1235PROJECTION BIAS










1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS


























1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS





















1237PROJECTION BIAS













1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS





























1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS






















1239PROJECTION BIAS

APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS

















APPENDIX PROOFS


b g(i) 5 0 whena b







A [ v9(ctA 2 st






T (1 2g)t2 (t1 1 )vt


A 5 Xt2 1A 2 X t


A 2 XtA 5 g(vt


vtA 2 X t


A 2vt



v tA 5 ~1 1 ~T 2 tgvT

A

1 S a

1 2 aD ~1 1 ~T 2 tgvT 2 ~vt 1 g Oi5t11

T

v i





(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS

















(1)v t

A 2 vt~1 1 ~T 2 tg

1a

1 2 a

Rt

~1 1 ~T 2 tg

5vs

A 2 vs~1 1 ~T 2 sg

1a

1 2 a

Rs

~1 1 ~T 2 sg


A vt and vsA vs










A 2 stA $ ct 2 st



st or ctA 2 st












A vt forsome t









(2) 1 1 ~T 2 z 2 1gvz 2 1 1 ~T 2 z 2 1g~1 2 gvz11

g2 Oi5z12

T

v i




1241PROJECTION BIAS



A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS



















A vtimplies that ct


A stA


A and so stA sz1 1


A szA then sz1 1


A stA $ sz1 1

A cz


A then sz1 1A $ sz

A and since vzA vt

A impliesthat cz

A 2 szA ct

A 2 stA st

A $ sz1 1A $ sz








A 2 vs) 1 a(1 2 a)(Rt 2Rs)] 5 2g[vT


A 2vz1 1) 1 a(1 2 a)(Rz 2 Rz1 1) 5 1n[(vx

A 2 vx) 2 (vx 1 nA 2


A vz1 1 vx








t c iA Si5 1




A v1 and vt








s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS

















s t 5 gc t21 1 ~1 2 ggc t22 1 ~1 2 g2gc t23 1

1 ~1 2 gt22gc1 1 ~1 2 gt21s1

5 g Oi51

t21

ci 2 g2 Oj51

t22 F ~1 2 gj21 Oi51

t212j

ciG 1 ~1 2 gt21s1


w c iA and Si5 1

t ci Si5 1t c i


sw11 2 sw11A 5 gS O

i51

w

ci 2 Oi51

w

ciAD

2 g2 Oj51

w21 F ~1 2 gj21S Oi51

w2j

ci 2 Oi51

w2j

c iAD G 0


A







c 3A and c 2



A $ c z1 1A and c z



A 5 gvTA 1 a(1 2


A 2 vz1 1A 0


A c z1 1A 2 s z1 1

A and givenc z



if c zA s z



A s 2A (c 1




U1~c1c2c3us1 5 Ot51

3

~1 2 av~ct 2 st 1 av~ct 2 s1

5 ~1 2 aU1~c1c2c3 1 a Ot51

3

v~ct 2 s1

1243PROJECTION BIAS



A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS



















A c 2A c 3

A us1)]c1 5]U1(c1

Ac2Ac3




Ac3Aus1)

]c2 5 ]U1(c1Ac2


Ac2Ac3

A)]c2 2]U1(c1

Ac2Ac3

A)]c3] 5 a[v9(c3A 2 s1) 2 v9(c2






3 [(1 2 a)v(ct 2 st) 1av(ct 2 s 2

A )] 5 (1 2 a)U2(c2 c3 us 2A ) 1 a St5 2




A A us 2A)]c2 5

]U2(c 2AA c 3



A)]ct 5 (1 2 a)]U1(c 1


]U2(c2A c3

Aus2A)]c2 2 ]U2(c2

A c3Aus2

A)]c3 5 a[v9(c3A 2 s1) 2 v9(c2

A 2s1)] 1 a[v9(c 2

A 2 s 2A) 2 v9(c 3

A 2 s 2A)]


A s1) and c 2A c 3

A


A) 2 v9(c2A 2 s1) v9(c3

A 2 s2A) 2 v9(c3


A c 3A us2)]c2 ]U2(c 2

A c 3A us2)


A and c3AA c3


A 2 s 2A ) 2 v9(c 2

A 2 s1) 5 v9(c 3A 2

s 2A ) 2 v9(c 3


A c 3A us2)]c2 5 ]U2(c 2


(c 2A A c 3

AA ) 5 (c 2A c 3


A )


lA 5]U1~c1

Ac2Ac3

Aus1

]c1

5]U1~c1

Ac2Ac3

Aus1

]c2

5]U1~c1

Ac2Ac3

Aus1

]c3

and lAA 5]U2~c2

AA c3AAus2

A

]c2

5]U2~c2

AA c3AAus2

A

]c3



A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS


















A us 2A)]c2

]U2(c 2A c 3


A c 3A us 2

A )]ct 5]U1(c 1

A c 2A c 3


A) 2 v9(c tA 2 s1)] Then s 2



A 2 s 2A ) v9(c t


c 3A us 2

2)]ct ]U1(c 1A c 2



CORNELL UNIVERSITY


REFERENCES
















1245PROJECTION BIAS
















































1247PROJECTION BIAS
































































1247PROJECTION BIAS








































1247PROJECTION BIAS

































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PROJECTION BIAS IN PREDICTING FUTURE UTILITY