The Silver Lining Effect: Formal Analysis and Experiments

Electronic copy available at: http://ssrn.com/abstract=1957343

MANAGEMENT SCIENCEVol. 55, No. 11, November 2009, pp. 1832–1841issn 0025-1909 �eissn 1526-5501 �09 �5511 �1832

informs ®

doi 10.1287/mnsc.1090.1076©2009 INFORMS

The Silver Lining Effect: Formal Analysisand Experiments

Peter JarnebrantEuropean School of Management and Technology (ESMT), 10178 Berlin, Germany,

[email protected]

Olivier Toubia, Eric JohnsonDepartment of Marketing, Columbia Business School, Columbia University, New York, New York 10027

{[email protected], [email protected]}

The silver lining effect predicts that segregating a small gain from a larger loss results in greater psychologicalvalue than does integrating them into a smaller loss. Using a generic prospect theory value function, we

formalize this effect and derive conditions under which it should occur. We show analytically that if the gainis smaller than a certain threshold, segregation is optimal. This threshold increases with the size of the lossand decreases with the degree of loss aversion of the decision maker. Our formal analysis results in a set ofpredictions suggesting that the silver lining effect is more likely to occur when (i) the gain is smaller (for a givenloss), (ii) the loss is larger (for a given gain), and (iii) the decision maker is less loss averse. We test and confirmthese predictions in two studies of preferences, both in a nonmonetary and a monetary setting, analyzing thedata in a hierarchical Bayesian framework.

Key words : utility-preference; estimation; theory; prospect theory; loss aversionHistory : Received April 7, 2008; accepted August 1, 2009, by George Wu, decision analysis. Published online inArticles in Advance September 28, 2009.

1. IntroductionDecision makers are often faced with mixed out-comes, famously captured by the saying “I have goodnews and I have bad news.” In this paper, we lookat the case where the bad news is larger in magni-tude than the good, and ask: Does the decision makerwant these events combined or presented separately?Thaler, in his seminal paper (Thaler 1985), showedthat a decision maker faced with such a mixed out-come consisting of a loss and a smaller gain shouldgenerally prefer to separate the loss and the gain.That is, evaluating the gain separately from the largerloss is seen more positively than reducing the loss bythe same amount. The small gain becomes a “silverlining” to the dark cloud of the loss, and, pushing theanalogy further, adding a silver lining to the cloudhas a more beneficial impact than making the cloudslightly smaller. The separate evaluation is referred toas segregation and the joint evaluation as integration.The silver lining effect has broad application, because

equivalent information may often be framed to deci-sion makers in either integrated or segregated form.Consider, for example, a retailer who decides todecrease the price of a product: he or she could sim-ply lower the price, reducing the loss to the consumer,and announce the new discounted price. Anotheroption would be to keep charging the same amount,

but then give some of it back to the consumer in theform of a rebate (Thaler 1985). Similarly, a vacationresort could lower its average daily rates for stays ofone week or more, or offer a free night for every sixnights spent in the resort. The two methods, of course,can translate to the exact same dollar saving, and dif-fer only in framing. Consider an investor receiving abrokerage statement containing a net loss: Would heor she want to see only the smaller total loss or to seethe winners separated from the losers, even thoughthe balance would be identical?Despite the relevance of the silver lining effect to

both academics and practitioners, we are not awareof any formal study of when it should occur, beyondThaler’s (1985) intuitive argument that the silverlining effect is more likely when the gain is smallerrelative to the loss. The primary contribution of thecurrent paper is to fill this gap. We assume a genericprospect theory value function and formally showthat if the gain is smaller than a certain threshold,segregation is optimal. Next, we show how the valueof this threshold is affected by both the magnitude ofthe loss, and by the loss aversion parameter of thevalue function. Our formal analysis provides a set ofpredictions suggesting that the silver lining effect ismore likely to occur when (i) the gain is smaller (fora given loss), (ii) the loss is larger (for a given gain),

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Jarnebrant, Toubia, and Johnson: The Silver Lining Effect: Formal Analysis and ExperimentsManagement Science 55(11), pp. 1832–1841, © 2009 INFORMS 1833

and (iii) the decision maker is less loss averse. Wetest and confirm these predictions in two experiments.Finally, we provide a methodological contribution tothe literature on the measurement of loss aversion, byreplacing the deterministic, individual-level approachtraditionally used by behavioral economists witha hierarchical Bayes framework that accounts formeasurement errors and similarities across decisionmakers.This paper is structured as follows. In §2, we briefly

review the silver lining effect and the prospect theoryframework. In §3, we report our theoretical analysisof the silver lining effect. The context and results ofour first empirical study, which is conducted in a non-monetary setting, is reported in §4. In §5, we presentthe second empirical study, which extends the find-ings from the first study to monetary decisions. Sec-tion 6 concludes and provides directions for futureresearch.

2. Prospect Theory and SilverLining Effect

2.1. Silver Lining EffectWhen faced with a decision that involves severalpieces of information, decision makers do not alwaysintegrate them into a whole, but instead may usethem in the form presented by the decision context;Slovic (1972) calls this the concreteness principle. Thus,an individual may treat two amounts of money, forinstance, as separate entities instead of simply sum-ming them. Thaler and Johnson (1990) provide evi-dence that these effects occur for monetary gambles,and Linville and Fischer (1991) demonstrate this phe-nomenon for life effects. Thaler’s (1985) theory ofmental accounting addresses what this phenomenonimplies for subjective value and thus for decisionmaking. The silver lining principle focuses on out-comes that consist of a loss and a smaller gain, knownas mixed losses. When faced with such outcomes,Thaler’s (1985) prescription—which he dubbed the“silver lining principle”—is to keep them separate inmind (i.e., segregate the gain from the loss), so thatthe small gain can provide a silver lining to the largerloss, rather than disappear if used to diminish the loss(i.e., if the gain were integrated with the loss). Animportant implication of concreteness is that decisionscan be materially affected by the presentation of theoutcomes, as either integrated or segregated, becausedecision makers do not spontaneously combine them(Thaler and Johnson 1990).Read et al. (1999) introduce the related concept of

choice bracketing; in broad bracketing, several deci-sions are considered jointly, whereas in narrow brack-eting each individual choice is considered separately.

Figure 1 Illustration of Proposition 1

x

v(x

)

G*G ′ G ′′

l (–L+G ′′)

Gain fromsegregation

Loss reduction from integrationl (–L)

l (–L+G*)l (–L+G ′)

–L –L+G*

g(G*)

Bracketing is distinguished from segregation and inte-gration. Integration and segregation refer to outcomeediting (Kahneman and Tversky 1979, Thaler 1985),the consideration of outcomes within a single choice,but bracketing addresses whether a group of choicesis evaluated jointly or separately.

2.2. Prospect TheoryProspect theory (Kahneman and Tversky 1979,Tversky and Kahneman 1992) proposes a value func-tion v�·� as an alternative to the utility functionsassumed by expected utility theory. The value func-tion is characterized by three key features: referencedependence, meaning that the arguments of the valuefunction are positive and negative deviations fromsome reference level, defined as gains and losses,respectively; loss aversion, reflecting the fact that lossesloom larger than gains of the same magnitude; anddiminishing sensitivity, indicating that the marginalimpact of both gains and losses decrease as theybecome larger. These three characteristics result in avalue function shaped as in Figures 1–3. The origin

Figure 2 Varying the Magnitude of the Loss

x

v(x

)

G0*

–L1–L0

g (G0*)

l (–L0+G0*)– l (–L0)

l (–L1+G0*)– l (–L1)



https://www.researchgate.net/publication/227344939_Gambling_With_the_House_Money_and_Trying_to_Break_Even_The_Effects_of_Prior_Outcomes_on_Risky_Choice?el=1_x_8&enrichId=rgreq-b7223e6f-4673-4e93-b6ee-f8e9ec5b4141&enrichSource=Y292ZXJQYWdlOzIyMDUzNDY4NjtBUzoxMTEyNDM2Mzc1NjMzOTNAMTQwMzUzMzk1OTcxOQ==

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https://www.researchgate.net/publication/4814288_Prospect_Theory_An_Anlysis_of_Decision_Under_Risk?el=1_x_8&enrichId=rgreq-b7223e6f-4673-4e93-b6ee-f8e9ec5b4141&enrichSource=Y292ZXJQYWdlOzIyMDUzNDY4NjtBUzoxMTEyNDM2Mzc1NjMzOTNAMTQwMzUzMzk1OTcxOQ==

https://www.researchgate.net/publication/21159846_Preferences_for_Separating_or_Combining_Events?el=1_x_8&enrichId=rgreq-b7223e6f-4673-4e93-b6ee-f8e9ec5b4141&enrichSource=Y292ZXJQYWdlOzIyMDUzNDY4NjtBUzoxMTEyNDM2Mzc1NjMzOTNAMTQwMzUzMzk1OTcxOQ==


Jarnebrant, Toubia, and Johnson: The Silver Lining Effect: Formal Analysis and Experiments1834 Management Science 55(11), pp. 1832–1841, © 2009 INFORMS

Figure 3 Varying the Degree of Loss Aversion

x

v(x

)

l0(–L+G0*)– l0(–L)

l1(–L+G0*)– l1(–L)

G0*

–L –(L–G0*)

g (G0*)

denotes the reference point relative to which out-comes are categorized as gains and losses. Loss aver-sion results in a “kink” in the function at the origin:for all x > 0, −v�−x� > v�x�. Moreover, diminishingsensitivity in both domains implies that the valuefunction is concave for gains and convex for losses,that is, v′′�x� < 0 for x > 0, and v′′�x� > 0 for x < 0.A large amount of evidence supporting this function,both from field studies and experimental work, canbe found in Kahneman and Tversky (2000).In this paper, we assume the following generic

specification of the value function:

v�x�=g�x� x≥ 0�l�x�=−g�−x� x < 0�

(1)

where > 1 and g is a function from �0�� to �0��that is strictly concave, strictly increasing, twice differ-entiable, and such that g�0�= 0. Note that we assumehere a “reflective” value function (as in Kahnemanand Tversky 1979), i.e., the value function for lossesis the mirror image of the value function for gains.This assumption has received mixed empirical sup-port (see, e.g., Abdellaoui et al. 2007). We leave theextension of our results to nonreflective value func-tions to future research. Such extension would be triv-ial under a power function specification.

3. Theoretical AnalysisIn this section, we derive theoretical predictions forthe silver lining effect. In particular, we character-ize regions where segregation of gains and lossesis preferable to integration, and identify how thetrade-off between integration and segregation is influ-enced by the magnitude of the gain, the magnitude ofthe loss, and the degree of loss aversion of the deci-sion maker.The proofs to all the results are provided in

Appendix A. Our analysis starts with the followingproposition:

Proposition 1. For any fixed loss L > 0, there existsa gain G∗ ∈ �0�L� such that the value derived from seg-regation is greater than that derived from integration forany gain G < G∗, and the reverse is true for any gainG>G∗.• If limx→0+ g′�x�=� or if limx→0+ g′�x� �= � and <

∗ = g′�0�/g′�L�, then G∗ > 0, that is, there exists a regionin which the value derived from segregation is greater thanthat derived from integration.• If limx→0+ g′�x� �= � and > ∗ = g′�0�/g′�L�, then

G∗ = 0, that is, the value derived from integration is greaterthan that derived from segregation for any gain G≤ L.

Proposition 1 shows the existence of a thresholdG∗ such that segregation is optimal for all gainssmaller than this threshold, and integration is opti-mal for all gains larger than this threshold. The intu-ition behind the existence of a gain threshold is bestexplained graphically. Figure 1 gives an example ofa mixed loss �L�G∗� for which the decision maker isindifferent between integration and segregation, i.e.,g�G∗�= l�−L+G∗�− l�−L�. Let us consider a smallergain G′. Both the corresponding gain g�G′� and thecorresponding loss reduction l�−L + G′� − l�−L� aresmaller under G′ than under G∗. Whether integrationor segregation is optimal hence depends on which ofthe two quantities is decreased the least. Because ofthe concavity of the gain function g, the differencebetween the initial gain g�G∗� and the smaller gaing�G′� corresponds to the flattest part of the gain func-tion between 0 and G∗. In other words, the decrease ingain between G∗ and G′ is relatively small comparedto the initial gain g�G∗�. On the other hand, the differ-ence between the initial loss reduction and the smallerloss reduction corresponds to the steepest part of theloss function between �−L+G∗� and −L. Therefore,the decrease in loss reduction l�−L + G′� − l�−L� isrelatively large compared to the initial loss reductionl�−L + G∗� − l�−L�, and hence compared to the ini-tial gain g�G∗� (recall that g�G∗�= l�−L+G∗�− l�−L��.Therefore, the new gain from segregation is largerthan the new gain from integration, and segregationbecomes optimal for G′. The same argument holdsfor G′′ larger than G∗. In this case, both the gain andloss reduction are increased. Because of the concavityof the gain function, the increase in gain is smallerthan the increase in loss reduction, and integrationbecomes optimal for G′′.Proposition 1 also states that for some specifica-

tions of the value function, there may exist situationsin which a decision maker who is very loss aversewill always prefer integration over segregation, i.e.,G∗ = 0.1 Note, however, that such situations do not

1 Because g is strictly concave, limx→0+ g′�x� �= � implies that

limx→0+ g′�x� exists and is finite.

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arise in the common specification in which g is apower function (of the form g�x�= x�). Indeed, in thatcase limx→0+ g′�x�=� and there always exists a rangeof gains for which it is optimal to segregate, no matterhow loss averse the decision maker is.Proposition 1 has the following testable implication:

Hypothesis 1 (Gain Size Hypothesis). For a givenmixed loss �L�G� with L > G, the smaller the gain G, thegreater the value derived from segregation relative to inte-gration, and conversely, the greater the gain, the greater thevalue derived from integration relative to segregation.

Proposition 1 introduced a threshold gain, G∗, suchthat segregation is optimal for gains smaller than G∗,and integration is optimal for gains larger than G∗.G∗ is a function of L and , as well as of any param-eter of the value function. In the region in whichG∗ > 0, G∗ is defined by g�G∗�= l�−L+G∗�− l�−L�=−g�L − G∗� + g�L�. Setting F �G�L�� = g�G� +g�L−G�−g�L�, G∗ is defined by F �G∗�L��= 0.The following proposition uses the envelope theo-

rem to evaluate the influence of L and on G∗.2

Proposition 2. In the region in which G∗ > 0, the fol-lowing comparative statics hold:(a) G∗ is monotonically increasing in the amount of the

loss L�dG∗/dL> 0�.(b) G∗ is monotonically decreasing in the loss aversion

parameter �dG∗/d< 0�.

Like Proposition 1, these results may be illustratedgraphically. Let us first consider Proposition 2(a),illustrated by Figure 2. The gain G∗

0 is such that for aloss L0 a decision maker is indifferent between inte-gration and segregation, i.e., g�G∗

0� = l�−L0 + G∗0� −

l�−L0�. Let us consider a change from L0 to L1 < L0.The gain function being unaffected by L, the gainfrom segregation g�G∗

0� is unaffected as well. How-ever, because of the concavity of the loss function,reducing the loss by a fixed amount G∗

0 leads to agreater increase in value when the loss being reducedis smaller, i.e., l�−L1 + G∗

0� − l�−L1� > l�−L0 + G∗0� −

l�−L0�. As a result, although the decision maker isindifferent between segregation and integration forG=G∗

0 under L0, he or she would prefer integrationfor G=G∗

0 under L1, i.e., g�G∗0� < l�−L1+G∗

0�− l1�−L1�,and therefore G∗�L1�� <G∗�L0��.

2 The implications of Proposition 2, captured by Hypotheses 2and 3, extend to the domain in which G∗ = 0. The more loss aversethe decision maker, the more likely > ∗ = g′�0�/g′�L� is to hold,and therefore, the more likely is G∗ to be 0 and integration tobe always preferred over segregation. Because of the concavityof the function g, ∗ = g′�0�/g′�L� is increasing in L, and for afixed , the larger the loss L, the less likely > ∗ is to hold, andtherefore, the less likely is integration to be always preferred tosegregation.

Second, let us consider Proposition 2(b). Figure 3represents two value functions that differ only onthe value of the loss aversion parameter . The gainfunction is unaffected by the parameter , and thetwo loss functions l0 and l1 on Figure 3 correspond,respectively, to = 0 and = 1 > 0. The gainG∗0 is such that a decision maker with a loss aver-sion parameter 0 is indifferent between integrationand segregation, i.e., g�G∗

0�= l0�−L+G∗0�− l0�−L�. Let

us consider a change from 0 to 1. The gain func-tion being unaffected by , the gain from segregationg�G∗

0� is unaffected as well. However, the loss reduc-tion from integration l�−L+G∗

0�− l�−L� being propor-tional to , it is greater when is greater. As a result,although a decision maker with loss aversion 0 wouldbe indifferent between segregation and integration forG=G∗

0, a decision maker with loss aversion 1 wouldprefer integration, i.e., g�G∗

0� < l1�−L + G∗0� − l1�−L�,

and therefore G∗�L�1� <G∗�L�0�.Like Proposition 1, Proposition 2 has the following

testable implications.

Hypothesis 2 (Loss Size Hypothesis). For a givenmixed loss �L�G� with L > G, the larger the loss L, thegreater the value derived from segregation relative to inte-gration, and conversely, the smaller the loss L, the greaterthe value derived from integration relative to segregation.

Hypothesis 3 (Loss Aversion Hypothesis). For agiven mixed loss �L�G� with L > G, the more loss aversea decision maker, the greater the value derived from inte-gration relative to segregation, and conversely, the less lossaverse a decision maker, the greater the value derived fromsegregation relative to integration.

The remainder of the paper will focus on testingthe above hypotheses experimentally.

4. Experiment 1Our analysis led to a set of hypotheses that togethersuggest that the segregation of a small gain from alarger loss is more appealing when (i) the gain issmaller (for a given loss), (ii) the loss is larger (for agiven gain), and (iii) the decision maker is less lossaverse. Our first study tests these hypotheses in thecontext of choices between different amounts of vaca-tion days.

4.1. MethodThe experiment was conducted using a large onlinepanel of preregistered individuals, who were offeredthe opportunity to participate on their own time andcompensated $2 for their assistance. A total of 53 par-ticipants completed the survey. We employed threemethods to ensure that only participants who paidproper attention were kept for analysis. First, com-pletion times were recorded and screened for par-ticipants who were outside two standard deviations


away from the mean; this led to the exclusion ofone subject. Second, one subject was excluded foranswering “yes” to all 16 of our gamble items (pre-sented below); this response does not provide usabledata, and also suggests inattention. The third screen-ing method was a “trick” item inserted in the mid-dle of our survey. This item consisted of a block oftext apparently containing the instructions for how toanswer the following questions; however, in the mid-dle of the block participants were told that this was anattention-check, and given instructions to ignore thesurrounding text and instead answer the questions ina particular way to show that they were indeed payingattention. The “real” instructions were positioned sothat briefly skimming or ignoring the text altogetherwould lead to an identifiable response to the ques-tion; this method eliminated a further 15 participantsthat we deemed inattentive. The analyzed sample thusconsisted of 36 individuals, with a median age of 37and a median income of $42,500. Our sample goes wellbeyond the student populations often used in exper-imental studies: only 14% of our subjects describedthemselves as students, 11% as unemployed, and 64%as employed outside of the household.We tested our three hypotheses using a rating task

in which each participant read four scenarios. Eachscenario involved a change of jobs necessitated by thecurrent employer going out of business, and an asso-ciated change in the allocation of vacation days. Ineach of the four scenarios, the decision maker facedtwo job offers, both of which involved a net loss ofvacation days, but which differed in their distribution(see Table 1 for details of each scenario).For instance, in scenario 1, offer A would mean a

loss of four summer vacation days, and a gain of threewinter vacation days, whereas offer B would meanonly a loss of one summer vacation day. Both offersthus resulted in a net loss of one vacation day; in onecase, this was presented as a larger loss (−4 days) anda separate (segregated) smaller gain (3 days), and inthe other only as a smaller loss (−4 + 3 = −1 day).After reading the scenario, the participant rated herpreference for one offer over the other on a five-pointscale from −2 (strongly prefer the integrated offer) to2 (strongly prefer the segregated offer).

Table 1 Four Pairs of Job Offers and Their Outcomes

Pair 1 Pair 2 Pair 3 Pair 4

A B C D E F G H

Change in number of summer −4 −1 −4 −3 −7 −4 −7 −6vacation days

Change in number of winter 3 0 1 0 3 0 1 0vacation days

Net change −1 −1 −3 −3 −4 −4 −6 −6

The design of the experiment was a 2 (loss: small(4 days) versus large (7 days))×2 (gain: small (1 day)versus large (3 days)) factorial design, with all manip-ulations within-subject, and the order of presentationof the pairs of job offers varying between-subjects ina Latin square pattern.In addition to the main preference measure, we col-

lected two other individual-level measures: relativepreference for summer vacations over winter vaca-tions, and loss aversion. Because each choice wasbetween a loss in summer vacation days and a gainin winter vacation days (segregated option) versusa smaller loss in summer vacation days (integratedoption), it is important to control for a general prefer-ence for summer vacation days. To do this, we askedparticipants to allocate 20 vacation days betweensummer, winter, and the rest of the year. Our measureof participant i’s relative preference for summer vaca-tion days, summeri, was then defined as the ratio ofdays allocated to summer, divided by the total daysallocated to summer and winter (days allocated to therest of the year were ignored as they do not appearin our scenarios).The other additional measure was loss aversion for

vacation days (the currency used in this experiment).To estimate this parameter, we used two sequencesof gambles, each of which participants were asked toaccept or reject (see Goette et al. 2004 and Tom et al.2007 for similar measures of loss aversion). The gam-bles were introduced by a scenario in which the deci-sion maker had the option of joining a new project atwork, which if successful would result in extra vaca-tion days (5 days in one sequence, 12 in the other), andif failed would result in a loss of vacation days. Theproject’s probability of success was estimated at 0.5,and because it depended on competitors and clients,the decision maker herself would not be able to influ-ence it. The gambles varied on the amount of lost vaca-tion days resulting from the project’s failure; see Table2 for the complete sequences of gambles used.For each subject, each sequence of gambles pro-

vided the number of vacation days lost equivalentto 5 and 12 days gained, respectively. For instance,if a subject accepted gamble 3 but rejected gamble4 in the first sequence, and accepted gamble 7 butrejected gamble 8 in the second sequence, we codedthese responses as if a gain of 5 days was equivalentto a loss of 2 (the average of 1.5 and 2.5) days, settingx5 days = 2, and as if a gain of 12 days was equivalentto a loss of 9 (the average of 8 and 10) days, settingx12 days = 9. This provided us with two measures of theparameter � 5 = 5/x5 days and 12 = 12/x12 days. In theanalysis below, we use both of these measures andtake into account the existence of measurement error.The two measures correlated at Kendall’s � = 0�45


Table 2 Gambles Used to Measure Loss Aversion and TheirOutcomes

Sequence 1 Sequence 2

Gamble Gain Loss Gain Loss

1 5 0�5 12 12 5 1 12 23 5 1�5 12 2�54 5 2�5 12 35 5 3 12 3�56 12 4�57 12 58 12 69 12 810 12 10

Note. All probabilities are 1/2 and all amounts are in numbers of vacationdays.

across subjects. Responses were screened for mono-tonicity; that is, the analysis only included subjectswith at most one switch from acceptance to rejectionas the size of the loss increased. This eliminated oneparticipant. Six participants indicated that they wouldaccept all gambles in the first sequence; in those cases,only the 12-day measure was used in the analysis.Note that the expressions for 5 and 12 assume

that the value function is approximately linear forsmall amounts, and that the probability weighingfunction is similar for gains and losses. These esti-mates are biased upwards if the true value functionexhibits diminishing sensitivity. Indeed, if v�x� ∝ x�,then the correct estimates of would be �5/x5 days��

and �12/x12 days��, which are smaller than 5 and12, respectively, assuming � < 1 (diminishing sensi-tivity) and x5 days < 5 and x12 days < 12 (loss aversion).3

If different probability weighing functions are appliedfor gains and losses, then our estimates are off by afactor of w−�0�5�/w+�0�5�, where w− and w+ are theweighing functions for losses and gains, respectively.These effects would increase the noise and decreasethe fit of our model and would work against ourhypotheses, making our analyses conservative.

4.2. Results and DiscussionThere was an overall preference for integration in oursample (M = −0�49 on our scale from −2 to 2; t =−4�72, p < 0�0001), as well as an overall preference forsummer over winter vacations (M = 0�62 on our scalefrom 0 to 1, with 0.5 being the indifference point; t =−7�95, p < 0�0001). As predicted by our hypotheses,the relative preference for segregation was greater forpairs involving small versus large gains (Msmall_gain =−0�38�Mlarge_gain = −0�60) and large versus small

3 The magnitude of this effect would be modest. For example, giventhe commonly cited value of � = 0�88, an estimated of value of 2would correspond to a true value of 20�88 ≈ 1�84.

losses (Mlarge_ loss =−0�39�Msmall_ loss =−0�58), and wasnegatively correlated with loss aversion as measuredby the average of 5 and 12 (�=−0�19).We now test these hypotheses formally and assess

the level of statistical significance of our results. Thisrequires modeling the impact of loss aversion, losssize, and gain size on the relative preference for segre-gation over integration. We have two noisy measuresof loss aversion for each subject. Our individual-levelestimate of should reflect these two measures andweigh them appropriately based on their respectiveprecisions. Moreover, previous research has shownthat extreme values of are unusual and that thisparameter tends to follow a unimodal distributionacross decision makers (Gaechter et al. 2007). Weuse a Bayesian framework to capture these aspectsof our data (Gelman et al. 1995, Rossi and Allenby2003). This framework allows producing individual-level estimates of that appropriately weigh eachmeasure while capturing similarities across individ-uals by allowing to be shrunk toward a popula-tion mean. Our individual-level estimates of maybe thought of as a weighted average between thetwo measures of loss aversion and the populationmean, where the weights are determined from the dataaccording to the variance of each of component. Theeffect of shrinkage is primarily limited to outliers, cor-recting any unreasonably extreme value of towardthe population mean while effectively “leaving alone”the other subjects (Rossi and Allenby 2003). Additionalbenefits of using a Bayesian framework include flexi-bility in model formulation and the direct quantifica-tion of uncertainty on the parameters. Such a frame-work is particularly well suited for situations such asours in which limited data are available from a moder-ately large sample of subjects. The details of the modeland its estimation can be found in Appendix B. Simi-lar results were obtained with a simpler non-Bayesianlinear model (details are available from the authors).At the subject-level, we modeled the preference

of subject i on gamble pair j , pref ij , as a functionof the size of the gain, the size of the loss (bothcoded orthogonally, with small = −1 and large = 1),their interaction, a subject-specific intercept ai, and anormally distributed error term, �i; we also includedindicator variables for the position (within the Latinsquare design) at which pair j was presented:pref ij = ai + �1gainij + �2lossij + �3gainij × lossij +�4 × 1(positionij = 1� + �5 × 1(positionij = 2� + �6 ×1(positionij = 3�+ �ij with �ij ∼N�0� 2�.Our two measures of individual i’s loss aversion,

5� i and 12� i are modeled as functions of a true under-lying value, i, plus normally distributed errors !iand "i:

5� i = i + !i� 12� i = i + "i

with !i ∼N�0� #25�, "i ∼N�0� #212�.


Table 3 Estimates From the Hierarchical Bayes Model

Parameter Estimate p-value

a1 (loss aversion) −0�402 0�022a2 (summer preference) −4�82 <0�0002

�1 (gain: large= 1, small=−1) −0�113 0�032�2 (loss: large= 1, small=−1) 0�088 0�071�3 (gain× loss interaction) 0�034 0�558�4 (position= 1) 0�339 0�052�5 (position= 2) 0�027 0�876�6 (position= 3) 0�074 0�664

We captured the effect of loss aversion and rela-tive preference for summer vacations on the prefer-ence for segregation versus integration by allowing iand summeri to impact the subject-specific intercept ai.In particular, we used the following Bayesian priorfor ai:

ai ∼N�a0+ia1+ summeri · a2�$2��Finally, we specified a prior on i that captures sim-

ilarities across subjects and allows shrinking towarda population mean:

i ∼N�0� �2��

We estimated this model using Markov chainMonte Carlo, with 100,000 iterations, the first 50,000being used as burn-in. Convergence was assessedinformally from the time-series plots of the param-eters. The mean estimate of i, the individual-leveldegree of loss aversion for vacation days, was 4.34.The results are shown in Table 3. The estimates

shown are the means of the posterior distributionsof each parameter; the reported p-values are poste-rior p-values, based on the draws from the posteriordistribution (one-tailed where our hypotheses makedirectional predictions). The results generally supportour hypotheses; greater gains predict increased pref-erence for integration (Hypothesis 1), greater losses(marginally) predict increased preference for segrega-tion (Hypothesis 2), and greater loss aversion predictsincreased preference for integration (Hypothesis 3).As expected, we also see a significant impact of a gen-eral preference for summer over winter vacations.Experiment 1 provides initial support for our three

hypotheses in a nonmonetary setting, and also intro-duces our modeling framework. We conducted anadditional experiment to look for stronger support forHypothesis 2 (loss size) in a larger sample size and amore canonical context. The second experiment alsoextends the initial findings to a monetary setting.

5. Experiment 2The three hypotheses were tested here in the contextof mixed monetary gambles.

5.1. MethodThe experiment was conducted in the virtual lab of alarge East Coast university, with participants access-ing the experiment over the internet. The presentstimuli and measures were embedded in a longerseries of surveys. Invitations to participate were sentout to a group of preregistered individuals, who hadnot previously participated in surveys from the lab,and were offered $8 for completing the series ofsurveys.The sample consisted of 163 individuals, with a

median age of 35.5; the reported median income rangewas between $50,000 and $100,000. Again the sam-ple was composed of individuals of varied occupa-tions: 13% of our subjects were students, 13% wereunemployed, and 57% were employed outside of thehousehold.We tested our hypotheses using four pairs of gam-

bles, all of which had three possible outcomes, eachwith probability 1/3. Each of the four pairs corre-sponded to one cell in a 2 (loss: small ($30) versuslarge ($60))×2 (gain: small ($5) versus large ($20)) fac-torial design. See Table 4 for a complete description ofthe gambles. Within each pair, one gamble presentedthe gain and the loss separately (as outcomes 1 and 2)while the other combined them in one single outcome(outcome 2). In each pair, a third outcome was addedto both gambles to equate the expected value of allgambles to $20; this ensured that differences acrosspairs were not because of differences in the expectedvalues of the gambles.For each pair of gambles, subjects were asked to

rate their preference for one of the gambles versus theother on a five-point scale from −2 (strongly preferthe integrated gamble) to 2 (strongly prefer the segre-gated gamble), with zero indicating indifference. Theorder of presentation of the gambles was counterbal-anced between subjects using a Latin square design.The loss aversion parameter was measured using

gambles in a similar way to Experiment 1, themain difference being the use of gambles for moneyrather than for vacation days. Subjects were shown2 sequences of 10 gambles each and for each gamblewere asked to indicate whether they would accept toplay it. Again, all gambles were binary, with one loss

Table 4 Four Pairs of Gambles and Their Outcomes

Pair 1 Pair 2 Pair 3 Pair 4

A B C D E F G H

Outcome 1 (gain) 5 0 20 0 5 0 20 0Outcome 2 (loss) −30 −25 −30 −10 −60 −55 −60 −40Outcome 3 (equalizer) 85 85 70 70 115 115 100 100Expected value 20 20 20 20 20 20 20 20

Note. All probabilities are 1/3, and all amounts are in dollars.


Table 5 Two Sequences of Gambles Used to Measure Loss Aversion

Sequence 1 Sequence 2

Gamble Gain Loss Gain Loss

1 6 0�5 20 22 6 1 20 43 6 2 20 64 6 2�5 20 85 6 3 20 106 6 3�5 20 127 6 4 20 148 6 5 20 169 6 6 20 1810 6 7 20 20

Note. All amounts are in dollars.

and one gain, and probabilities were held constant at0.5. Each of the two sequences held the gain amountconstant, at $6 and $20, respectively, while the lossamounts increased as the sequence progressed. SeeTable 5 for the complete set of gambles.For each subject, each sequence of gambles then

provided an amount of loss equivalent to a gain of$6 and $20, respectively. This provided us with two(noisy) measures of the parameter : 6 = 6/x6 and20 = 20/x20. The two measures correlated at Kendall’s� = 0�60 across subjects. Responses were screened formonotonicity; that is, the analysis only includes sub-jects with at most one switch from acceptance to rejec-tion of the gambles as the size of the loss increased.This led to the elimination of seven subjects.

5.2. Results and DiscussionThere was a slight overall preference for segregationover integration (M = 0�09 on our scale from −2 to 2;t = 1�60, p = 0�11). As predicted by our hypotheses,the relative preference for segregration was greater forpairs involving small versus large gains (Msmall_gain =+0�20�Mlarge_gain = −0�03) and large versus smalllosses (Mlarge_ loss =+0�18�Msmall_ loss =−0�01), and wasnegatively correlated with loss aversion as measuredby the average of 5 and 20��=−0�15�.As in Experiment 1, we used a hierarchical linear

model to test our hypotheses formally. The only dif-ference was in the specification of the prior distribu-tion for the subject-specific intercepts ai, where weeliminated the term that captured a preference forsummer over winter vacations, leaving only the inter-cept and the effect of loss aversion:

ai ∼N�a0+ia1�$2��

We again estimated this model using Markov chainMonte Carlo, with 100,000 iterations, the first 50,000being used as burn-in. Convergence was assessedinformally from the time-series plots of the param-eters. Our estimate of the individual-level degree of

Table 6 Estimates From the Full Hierarchical Bayes Model

Parameter Estimate p-value

a1 (loss aversion) −0�400 0�0050

�1 (gain: large= 1, small=−1) −0�115 0�0024�2 (loss: large= 1, small=−1) 0�099 0�0042�3 (gain× loss interaction) 0�105 0�0074�4 (position= 1) 0�257 0�017�5 (position= 2) −0�006 0�96�6 (position= 3) 0�155 0�16

loss aversion for money, i, had a mean of 2.88, whichis consistent with previous findings (see, e.g., Camerer2005), and somewhat smaller than what we observedfor vacation days in Experiment 1.The results are shown in Table 6.4 The estimates

support all three hypotheses, including the loss sizehypothesis. As before, a large gain predicts greaterpreference for integration compared to a small gain(Hypothesis 1), and greater loss aversion predictsgreater preferences for integration (Hypothesis 3).Our loss size hypothesis now also receives significantsupport, as an increase in loss size predicts greaterpreference for integration (Hypothesis 2).5

In conclusion, our second experiment strongly sup-ports all three of our hypotheses. We take this as con-firmation of the accuracy of our analytical results, andultimately of the underlying assumptions of prospecttheory.

6. General Discussion andConclusion

In this paper, we have formalized the silver liningeffect identified by Thaler in 1985, using the basicassumptions of prospect theory. We have identifiedanalytically and tested experimentally a set of condi-tions under which decision makers are more likely tosegregate gains from losses. We have shown that seg-regating a gain from a larger loss is more appealingwhen (i) the gain is smaller (for a given loss), (ii) theloss is larger (for a given gain), and (iii) the decisionmaker is less loss averse.Our first empirical study tested the analytic pre-

dictions in a nonmonetary setting, in the context

4 It is important to note that according to the cumulative prospecttheory (Tversky and Kahneman 1992), the difference between thevalue of the “integrated” and “segregated” gamble is only approx-imately proportional to l�−L+G�− l�−L�−g�G�. We also note thatWu and Markle (2008) have recently documented that the assump-tion of gain-loss separability in mixed gambles is questionable.These caveats do not apply to Experiment 1.5 The interaction between gain size and loss size is significant inthis experiment. This interaction term was included in the modelfor completeness and was not motivated by our theory. Moreover,it was not significant in the first experiment; therefore, we do notelaborate on it.


of vacation days, and found initial support for ourhypotheses. The second study extended and general-ized the results to monetary decisions. Together, thetwo studies suggest that the basic phenomenon of thesilver lining effect, and the moderators we have doc-umented here, are likely to be quite general.Our predictions are highly relevant to researchers,

practitioners and policy makers in a variety of do-mains where different frames of presentation of thesame underlying information may provide differentsubjective values. Our analysis could, for instance,provide guidance to marketers wishing to design pro-motion schemes (where a rebate could provide a sil-ver lining to a base price) or inform economic pol-icy (“stimulus checks” or tax refunds may be silverlinings to overall tax payments). We hope that futureresearch will test our predictions in such decisionenvironments.A final contribution of our current work is the use

of a small set of choices to estimate, at the individ-ual level, the degree to which a decision maker isloss averse. Our results show that loss aversion is animportant individual difference in predicting the reac-tions of decision makers to integration and segrega-tion as predicted by our model. Recent work showsthat individual differences in loss aversion are relatedto demographic variables (Gaechter et al. 2007), reac-tions to changes in wages (Goette et al. 2004) andunderlying neural signals in the striatum (Tom et al.2007). Our estimation method provides a frameworkfor estimating these differences and modeling thissource of heterogeneity.

Appendix A. Proof of PropositionsA.1. Proof of Proposition 1

A.1.1. Preparatory Material.

Lemma 1. For any fixed loss L> 0:• If segregation is optimal for a gain G, it is optimal for any

smaller gain G′ <G.• If integration is optimal for a gain G, it is optimal for any

larger gain G′′ >G.

Proof of Lemma 1. Let us define y0 and y1, respectively,as the loss reduction achieved by integration under G0 andG1 such that G1 <G0:

y0 = l�−L+G0�− l�−L�� y1 = l�−L+G1�− l�−L��and x0 and x1 as the analogous gains from segregation:

x0 = g�G0�� x1 = g�G1��

Monotonicity and concavity of gains and monotonicity andconvexity of losses imply that

y1G1

<y0G0

⇔ y1 <y0G1G0

andx1G1

>x0G0

⇔ x1 > x0G1G0

�

Thus, x0 >y0⇒ x1 > x0�G1/G0� > y0�G1/G0� > y1⇒ x1 >y1.

Recalling that x0 > y0 means that it is optimal to segre-gate, (i.e., the gain from segregation is greater than the lossreduction from integration), we see that if it is optimal tosegregate a gain G0 from a loss L, it is also optimal to seg-regate any smaller gain G1 <G0.Conversely, x1 < y1 ⇒ x0 < x1�G0/G1� < y1�G0/G1� < y0 ⇒

x0 <y0; hence if it is optimal to integrate for G1, it is optimalto integrate for any G0 >G1. �

A.1.2. Proof of the Main Result.Proof of Proposition 1. Lemma 1 insures us that there

are at most two regions: one in which it is optimal to seg-regate (smaller gains) and one in which it is optimal tointegrate (larger gains). Clearly, the latter region is neverempty as it is always optimal to integrate if G= L (as longas > 1). The former region is nonempty if and only if itis optimal to segregate as G goes to 0. The condition underwhich segregation is preferred is g�G� > l�−L+G�− l�−L�.Let us first consider the case in which g′�0� is a finite

number. In that case we can write the following Taylorseries expansions:

g�G�=G · g′�0�+ o�G��

l�−L+G�− l�−L�=G ·L′�−L�+ o�G��

As G goes to 0, segregation is preferred if g′�0� > l′�−L� andintegration is preferred if g′�0� < l′�−L�. Because l′�−L� = · g′�L�, segregation is preferred as G goes to 0 if <g′�0�/g′�L� and integration is preferred if > g′�0�/g′�L�.Let us next consider the case in which limG→0 g′�G�

= +�.In that case, there exists M > 0 and �′ > 0 such that for

all �< �′ and for all G<L:

g′�� > 3 ·M' �L�−L+ ��−L�−L��<M · �G− ��'

L′�−L+ �� <M�

For any � < �′, we can write the following Taylor seriesexpansions:

g�G�= g��+ �G− �� · g′��+ o�G− ��

> �G− �� · 3 ·M + o�G− ��

l�−L+G�− l�−L� = L�−L+ ��−L�−L�+ �G− �� ·L′�−L+ ��+ o�G− ��

< �G− �� · 2 ·M + o�G− ��

Segregation is preferred as G goes to �, i.e., the region inwhich segregation is preferred is nonempty.

A.2. Proof of Proposition 2Proof of Proposition 2. As mentioned in the text, G∗

is defined by F �G∗�L�� = 0, where F �G�L�� = g�G� −g�L�+g�L−G�.Using the envelope theorem, dG∗/dL = ��−(F /(L�/

�(F /(G�� and dG∗/d= ��−(F /(�/�(F /(G��.We have the following:• �(F /(G��G∗� = g′�G∗� − g′�L − G∗� < 0 because

g′�G∗�G∗ < g�G∗� = �g�L� − g�L − G∗�� < g′�L − G∗�G∗,where the inequalities hold because g is monotonicallyincreasing and concave and g�0�= 0, and the equality holdsby definition of G∗.


• (F /(L=−g′�L�+g′�L−G� > 0, because g′ is mono-tonically decreasing.• (F /(=−g�L�+ g�L−G� < 0 because g is monotoni-

cally increasing.Thus, we have dG∗/dL> 0, and dG∗/d< 0.

Appendix B. Details of the HierarchicalBayes Model

B.1. Model

B.1.1. Likelihood.

pref ij = ai +�1gainij +�2lossij +�3gainij × lossij

+�4× 1�positionij = 1�+�5× 1�positionij = 2�+�6× 1�positionij = 3�+ �ij �

6� i = i + !i� 20� i = i + "i�

�ij ∼N�0� 2�� !i ∼N�0� #25�� "i ∼N�0� #212��

B.1.2. First-Stage Prior.

ai ∼N�a0+i · a1+ summeri � a2�$2��

B.1.3. Second-Stage Prior.

i ∼N�0� �2��

B.1.4. Third-Stage Prior. Diffuse on a0, a1, a2, �1, �2, �3,�4, �5, �6, 0:

2∼ IG(r02�

s02

)� #25∼ IG

(r02�s02

)� #212∼ IG

(r02�s02

)�

$2∼ IG(r02�s02

)� �2∼ IG

(r02�s02

)

with r0 = s0 = 1.

B.2. Markov Chain Monte Carlo Estimation• L�ai � rest�∼N�mi�Vi�, where

Vi =(J

2+ 1$2

)−1

and

mi = Vi ·(1 2

( J∑j=1

pref ij−(�1gainij+�2lossij+�3gainij lossij

+�41�positionij = 1�+�51�positionij = 2�

+�61�positionij = 3�))

+ a0+ a1 · i + a2 · summeri$2

)�

• L�$2 � rest� ∼ IG��r0 +∑I

i=1�ai − a0 − i · a1 − summeri ·a2�

2�/2� �s0+ I�/2�.• L� 2 � rest� ∼ IG��r0 + ∑I

i=1∑J

j=1�pref ij − pij ��2/2�

�s0+ I · J �/2�, wherepij = ai+�1gainij+�2lossij+�3gainij lossij+�41�positionij=1�

+�51�positionij=2�+�61�positionij=3��

• L��1'�2'�3'�4'�5'�61 � rest� ∼ N��X ′X�−1 · X ′ · Y ' 2 · �X ′X�−11, where

Xij = �gainij � lossij �gainij · lossij �1�positionij = 1��1�positionij = 2��1�positionij = 3�1�

Yij = pref ij − ai�

• L��a0'a1'a21 � rest� ∼ N��X ′X�−1 · X ′ · Y '$2 · �X ′X�−11,where

Xij = �1�i� summeri1� Yij = ai�

• L�i � rest�∼N�mi�Vi�, where

Vi =(1#25

+ 1#212

+ a21$2

+ 1�2

)−1�

mi = Vi ·(6� i#25

+ 20�i#212

+ �ai − a0− summeri · a2� · a1$2

+ 0�2

)�

• L�0 � rest� ∼ N�m�V �, where V = �2/I2 and m =∑Ii=1 i/I .• L��2 � rest�∼ IG��r0+

∑Ii=1 �i −0�

2�/2� �s0+ I�/2�.• L�#25 � rest�∼ IG��r0+

∑Ii=1 �6� i −i�

2�/2� �s0+ I�/2�.• L�#212 � rest�∼ IG��r0+

∑Ii=1 �20� i −i�

2�/2� �s0+ I�/2�.

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The Silver Lining Effect: Formal Analysis and Experiments