Peak–End rule versus average utility: How utility aggregation affects evaluations of experiences

Journal of Mathematical Psychology 52 (2008) 326–335

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Peak–End rule versus average utility: How utility aggregation affects evaluations of experiences Irina Cojuharenco a,∗,1 , Dmitry Ryvkin b,1 a

FCEE, Universidade Católica Portuguesa, Portugal

b

Department of Economics, Florida State University, United States

article

info

Article history: Received 18 April 2007 Received in revised form 5 May 2008 Available online 15 July 2008 Keywords: Peak–End rule Average utility Total utility Utility dynamics

a b s t r a c t The ‘‘Peak–End rule’’ which averages only the most extreme (Peak) and the final (End) impressions, is often a better predictor of overall evaluations of experiences than average impressions. We investigate the similarity between the evaluations of experiences based on Peak–End and average impressions. We show that the use of the Peak–End rule in cross-experience comparisons can be compatible with preferences for experiences that are better on average. Two conditions are shown to make rankings of experiences similar regardless of the aggregation rule: (i) the individual heterogeneity in the perception of stimuli, and (ii) the persistence in impressions. We describe their effects theoretically, and obtain empirical estimates using data from previous research. Higher estimates are shown to increase correlational measures of association between the Peak–End and average impressions. The high association per se is shown to be not only a theoretical possibility, but an empirical fact. © 2008 Elsevier Inc. All rights reserved.

A day at work, a medical procedure, a meal — all are examples of experiences lived across time. Each is characterized by a series of impressions. A unique overall evaluation of these impressions determines which experiences are preferred (Kahneman, Fredrickson, Schreiber, & Redelmeier, 1993; Wirtz, Kruger, Napa Scollon, & Diener, 2003). How the overall evaluation is made is important because it will affect the rankings of experiences and, possibly, the choice. It is common to think that overall evaluations represent average impressions throughout experiences. And yet, a sizable body of research shows that some moments within experiences are systematically overweighted in overall evaluations (Kahneman, 2000a). How significantly do the ways of evaluating overall impressions differ? This article addresses the question for two most researched ways of evaluating overall impressions: averaging and the Peak–End rule. In his Mathematical Psychics, Edgeworth (1881) has first suggested how to measure and compare experiences. He advanced the idea of a ‘‘hedonimeter’’, a tool that could register continually

∗ Corresponding address: FCEE, Universidade Catolica Portuguesa, Palma de Cima, 1649-023 Lisboa, Portugal. E-mail address: [email protected] (I. Cojuharenco). 1 We are grateful to Robin Hogarth, Ricardo Nunes, David Patient, Anastasia Semykina, Marc Le Menestrel, Albert Satorra, and participants of IAREP/SABE Congress in Behavioral Economics and Economic Psychology 2006. Special thanks go to Hans Baumgartner and Dan Ariely for making their data available for reanalysis. 0022-2496/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jmp.2008.05.004

‘‘the height of pleasure experienced by an individual’’ (p. 101), thus providing information on moment utilities. Integrating pleasure intensity across the duration of an experience would give a measure of its overall evaluation, or, total utility. Modern research in the ‘‘hedonimeter’’ paradigm has used affective self-reports and physiological measures of affect to elicit moment utilities (Larsen & Fredrickson, 1999). At the same time, researchers have begun to examine the role of memory for self-reports and measures of total utility with an underlying assumption that accurate reports would reflect average moment utility (Kahneman, 1999, 2000b; Stone, Shiffman, & DeVries, 1999). It has been found that, in fact, people take a more selective approach to the impressions they consider in reaching overall evaluations. There is considerable support for people’s use of the ‘‘Peak–End rule’’, which averages only the most extreme (Peak) and final (End) moment utilities. In various experimental settings Peaks and Ends have been manipulated to favor experiences without changing their average utility or utility sum, and participants exhibited preferences consistent with the Peak–End rule (Ariely, 1998; Ariely & Loewenstein, 2000; Baumgartner, Sujan, & Padgett, 1997; Carmon & Kahneman, 1996; Diener, Wirtz, & Oishi, 2001; Fredrickson, 2000; Fredrickson & Kahneman, 1993; Langer, Sarin, & Weber, 2005; Redelmeier & Kahneman, 1996; Rozin, Rozin, & Goldberg, 2004; Schreiber & Kahneman, 2000; Varey & Kahneman, 1992). In a hospital field study with bone marrow transplant patients, select features of daily pain experiences, such as their End and the trend, have predicted significantly reports of total pain, while average pain added nothing to the prediction (Ariely & Carmon, 2000).

I. Cojuharenco, D. Ryvkin / Journal of Mathematical Psychology 52 (2008) 326–335

Yet, in spite of the strong findings in support of the Peak–End rule, it is surprising that very little research has been focused on whether Peak–End and average utility reveal similar preferences over experiences. In this article we first adopt a stimulus–response framework to shed light on the properties of moment utilities in experiences lived across time. Second, we identify two specific properties that will affect the relationship between the evaluations of experiences based on the Peak–End rule and average utility. A puzzling observation, and one that makes the relationship between average and Peak–End utility of practical significance, is the fact that people do not seem to intuit the Peak–End rule and believe that averaging should be used instead. Ariely (1998) noted that experimental participants tended to average moment utilities when these were documented in writing and readily available. In a study of lay intuitions about total utility, participants who had to guess the overall evaluation of an experience lived by another person inquired for piecewise evaluations to average across these (Cojuharenco, 2007). Baumgartner et al. (1997) conducted tests of the Peak–End rule because of a widespread assumption among advertisement researchers that the overall ad liking reflected the average momentary reaction to the ad. Recently, Wirtz et al. (2003) assumed that averaged reports of affect across the duration of a spring break represented the objective experience of college students undertaking vacations. In the comparison of experiences of equal duration, assumptions of this sort are consistent with the ideas of Edgeworth. In fact, comparison of experiences of equal duration in terms of average utility is demonstrably the only ‘‘rational’’ comparison of total utilities as it is equivalent to the comparison of utility sums (Kahneman, Wakker, & Sarin, 1997). As Kahneman (1999) notes, ‘‘the principle is consistent with the intuition that it is imprudent to seek short and intense pleasures that are paid for by prolonged mild distress’’, p. 6. In this article, we address the following question: are rankings of experiences (e.g., advertisements) very different if total utility is based on the Peak–End rule, rather than the average of all moment utilities? By better understanding differences in implied preferences over experiences, we will be able to establish whether the unconscious use of the Peak–End rule represents a bias outside the realm of the experimental laboratory where the two rankings are easily ‘‘set up’’ to be different. This is important because the rule saves on cognitive costs of recalling experiences in detail. If in most situations, it is aligned with preferences for experiences of higher average utility, its use can be justified. The Peak–End rule can then be viewed as a ‘‘smart’’ simple heuristic for crossexperience comparisons (Gigerenzer & Todd, 1999). As a first step, we identify two conditions that characterize situations when this could be the case. One has to do with a stable individual heterogeneity in the perception of stimuli that make the experience. We discuss situation-specific variables that moderate individual heterogeneity in experiencing. The other is the persistence in moment utilities, as when subsequent moment utilities carry over portions of previous moment utilities. The psychological processes of adaptation and anchoring-andadjustment (Frederick & Loewenstein, 1999; Hogarth & Einhorn, 1992) lead naturally to persistence of various magnitudes. We adopt a stimulus–response framework that allows us to analyze quantitatively the psychological processes of experiencing. Analytically and by means of simulations, we explore the implications of individual heterogeneity and persistence for the association between evaluations of experiences based on different aggregation rules. Analysis of panel data on experiences documented in previous research provides empirical evidence for the magnitudes of individual heterogeneity and persistence, as well as the resulting levels of association. Interested primarily in the similarity of preferences that different aggregation rules produce,

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we use the Spearman rank-order correlation coefficient as an ordinal measure of association that characterizes the concordance between total utilities based on the Peak–End and Average aggregation rules. Thus, we focus on the similarity of rankings that the two alternatives produce and not the similarity of absolute levels. Rankings characterize preferences over experiences. We further complement the analysis of association between Peak–End and Average utility by examining the Pearson correlation coefficient and the root mean squared deviation between total utilities based on the Peak–End and Average utility. It is of interest to consider these measures given a cardinal notion of utility. The remainder of the article is organized in three sections. In section ‘‘Theoretical analysis’’ we introduce the stimulus–response framework and explore how psychological processes of experiencing influence the relationship between the Peak–End and Average utility. In section ‘‘Empirical analysis’’ we estimate the parameters of the psychological processes in the data on experiences, and relate the estimates to the high association between the Peak–End rule and Average utility on correlational measures. Our findings are summarized and discussed in section ‘‘Discussion’’. 1. Theoretical analysis 1.1. The stimulus–response framework We represent experiences as series of instantaneous impressions of satisfaction, or moment utilities, ut , with t = 1, . . . , T , where T is the duration of the experience. Moment utilities are responses to the changing exogenous states of nature, or stimuli, st . Moment utilities (u1 , . . . , uT ) are aggregated into total utility, U, the overall evaluation of the experience. As discussed in the introduction, different aggregation rules can lead to different values of total utility for identical experiences. We will focus on two aggregation rules, the ‘‘Peak–End rule’’ and the ‘‘Average rule’’. Following Kahneman (2000a), we define the Peak–End utility of experience (u1 , . . . , uT ) as 1

(1) (uP + uT ) , 2 where the Peak moment utility is defined as uP = max1≤t ≤T ut for positive experiences, and uP = min1≤t ≤T ut for negative experiences (Ariely & Carmon, 2003).2 Further, we define the Average utility as UPE =

T 1X

ut . (2) T t =1 Suppose individuals in the population are exposed to idiosyncratic random series of stimuli (s1 , . . . , sT ), with each stimulus drawn independently from a distribution with probability density function (pdf) f (s). These series of stimuli are transformed by each individual into the series of moment utilities (u1 , . . . , uT ) through a psychological process of perception. Next, moment utilities are aggregated into total utilities UPE and UA defined by Eqs. (1) and (2). We examine three measures of association between the two values of total utility: (i) Spearman rank-order correlation coefficient, ρS ; (ii) Pearson correlation coefficient, ρ ; and (iii) root mean squared deviation, d. Spearman correlation coefficient, ρS , measures the degree of association between rankings of experiences produced by the two aggregation rules. This measure is invariant with respect to monotonically increasing transformations of utility. Pearson correlation coefficient, ρ , is an important benchmark. It measures the strength of linear relationship between the two total utilities. UA =

2 We only consider experiences with moment utilities of the same sign. The Peak–End rule has been tested only for experiences of this kind.

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Finally, the root mean squared deviation measures the expected absolute distance between the two total utilities. Depending on the psychological process involved, moment utility due to a given exogenous stimulus may (i) differ across individuals and (ii) depend on previous moment utilities, and thus, the complete history of stimuli, (s1 , . . . , st −1 , st ). We refer to property (i) as individual heterogeneity and to property (ii) as persistence. Individual heterogeneity depends both on the presence of differences in the characteristics of decision makers, such as their personalities, moods or attitudes to the particular type of stimuli, and the extent to which a given experience ‘‘admits’’ plural responses. Personality psychologists distinguish between ‘‘strong’’ and ‘‘weak’’ situations (Weiss & Adler, 1984). ‘‘Strong’’ situations constrain the heterogeneity of responses, and personality variables fail to predict those responses. The nature of the experience may similarly constrain individual heterogeneity in moment utilities that the stimuli produce. For example, Hsee (2006) reported evidence for a ‘‘potentially fundamental hypothesis about human judgment’’ (p. 17) that people agree more on what they dislike than on what they like. Persistence, too, depends both on the characteristics of the decision maker, such as the decision maker’s willingness to attend to changes in stimuli, and situational constraints (Hogarth & Einhorn, 1992). For example, adaptation to pleasant stimuli is generally found to be faster than adaptation to unpleasant stimuli (Frederick & Loewenstein, 1999). Slow adaptation means that previous moment utilities continue to play a role in the perception of a current stimulus (Manis, 1971).3 In the extreme case, an initial moment utility does not change no matter what stimuli realizations occur. Persistence may also result from a process of ‘‘anchoring’’, when the anchor is history-dependent (Ariely, Loewenstein, & Prelec, 2003; Hogarth & Einhorn, 1992), whereas static anchors that are exogenous to the experience contribute to individual heterogeneity rather than persistence. To summarize, both the decision maker and the nature of the experience (e.g., pleasant versus unpleasant) shape the psychological process underlying moment utilities. Individual heterogeneity and persistence in moment utilities will affect the relationship between Peak–End and Average utility. We organize our discussion of this relationship by the psychological process involved. There are four distinct cases to consider given the presence/absence of individual heterogeneity/persistence. For each case, measures of association describing the relationship between Peak–End and Average utility are affected by a distinct set of parameters that characterize the underlying psychological process. 1.2. Case A: Non-persistent moment utilities and no individual heterogeneity We start with the simplest possible case, in which moment utility ut is completely determined by current stimulus: ut = g (st ). Here g (·) is a strictly increasing and bounded function. The perception of stimuli is at their ‘‘face value’’ (no persistence). There are no stable differences in personalities, moods, attitudes or exogenous ‘‘anchors’’ between individuals. Although somewhat unrealistic, this case is important theoretically. We can show that utility profiles with independent and identically distributed (i.i.d.) stimuli cannot generate high correlation

3 Fredrickson and Kahneman (1993) analyzed data from an experiment in which participants viewed pleasant and aversive film clips. They noted that changes in real-time affect reported by the participants ‘‘were not caused by changes in film content’’, and that ‘‘people can discriminate endogenous processes of affective escalation and satiation from the exogenous effects of stimulus changes’’ (p. 50).

between UPE and UA , unless experiences are only a few moments long. By the virtue of the psychological process involved, for i.i.d. stimuli, the moment utilities will be i.i.d. The length of the experience, T , is the sole determinant of the similarity between rankings of experiences based on the Peak–End rule and Average utility. For simplicity, suppose g (s) = s. Any other function g (·) would effectively lead to a modification of the distribution of stimuli f (s), which does not affect our results. Assume also that the distribution of stimuli has a bounded support. This assumption is obviously realistic in applications. Thus, moment utilities are also bounded from above by some value umax . Further, let µu = E(ut ) and σu2 = Var(ut ). Our predictions are formulated in the following propositions (all proofs are relegated to Appendix A). Proposition 1. For i.i.d. moment utilities (u1 , . . . , uT ), Spearman correlation ρS between UPE and UA goes to zero as T → ∞. Proposition 2. For i.i.d. moment utilities (u1 , . . . , uT ), Pearson correlation ρ between UPE and UA goes to zero as T → ∞. Proposition 3. For i.i.d. moment utilities (u1 , . . . , uT ), root mean squared deviation d between UPE and UA has the limit limT →∞ d =



(umax − µu )2 + σu2

1/2

/2.

As an illustration, consider independent moment utilities distributed uniformly. For such experiences, Pearson correlation ρ can be found in the closed form (see Appendix B): 1 (T + 1)(T + 2) + 6T ρ= √ p . T (T + 2)[(T + 1)2 (T + 2) + 12(2T + 1)]

(3)

As seen from Eq. (3), the correlation between UPE and UA decays as ρ ∝ T −1/2 for T → ∞, and practically vanishes in long experiences. It follows from Proposition 1 that if moment utilities are i.i.d., rankings of long experiences based on Peak–End utility have nothing in common with rankings based on Average utility. This can be explained intuitively by the fact that, due to the intertemporal independence, no transfer of information is possible across moment utilities. As a result, the information contained in the Peak–End utility becomes increasingly disconnected from the information contained in the Average utility as the length of the experience grows. So, in the case of i.i.d. moment utilities, to choose experiences of greater average utility, decision makers have to avoid focusing only on Peaks and Ends (e.g., use memory aids). For independent uniformly distributed moment utilities, one can also calculate the root mean squared deviation (Appendix B): d=

1 2



1 3

−

1 T +2

1/2

.

(4)

Eq. (4) agrees with Proposition 3 in the limit of large T . As seen from Eq. (4) and, more generally, from Proposition 3, root mean squared deviation between UPE and UA does not vanish in long experiences. Its asymptotic value is determined by the distance between the maximal attainable and mean levels of moment utility as well as by the variance in moment utilities. It is also clear intuitively that one should not expect small differences in absolute levels of UPE and UA because the latter total utility is a simple average while the former assigns a high weight to the Peak moment utility. In line with the conclusions based on correlational measures ρS and ρ , this completes the picture of dissociation between Peak–End and Average utility in experiences with large T under the assumptions of case A.

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1.3. Case B: Non-persistent moment utilities and individual heterogeneity In case B, moment utilities are formed according to the following process: ut = g ( s t ) + c ,

t = 1, . . . , T .

(5)

The additive individual-specific term c corresponds to the time invariant (average or background) component of the experience that can be related to individual characteristics/‘‘anchors’’ of the decision maker. Individual heterogeneity is characterized by σc2 = Var(c ), the dispersion of individual-specific terms in the population. It is also convenient to introduce σs2 = Var(g (st )), the dispersion of transformed stimuli. The following propositions summarize the implications of individual heterogeneity for the association between Peak–End and Average utility. Proposition 4. For utility profiles with moment utilities of the form (5), Spearman correlation coefficient ρS between UPE and UA approaches unity for σc2 /σs2 → ∞. Proposition 5. For utility profiles with moment utilities of the form (5), Pearson correlation coefficient ρ between UPE and UA is bounded from below by ρ 0 , the value of ρ without individual heterogeneity (σc = 0), and increases with σc . For σc2 /σs2 → ∞, ρ approaches unity. Proposition 6. For utility profiles with moment utilities of the form (5), root mean squared deviation d is unaffected by individual heterogeneity. Note that case B may apply to intra-personal comparisons of experiences too. Within-individual heterogeneity can be observed if an individual has distinct experience-specific attitudes/‘‘anchors’’ that are time-invariant. For example, Nunes and Novemsky (2007) found that the evaluation of experiences, such as chocolate tasting and the viewing of film clips, tended to drift towards generic category attitudes when such attitudes existed and were made salient to people. 1.4. Case C: Persistent moment utilities and no individual heterogeneity In this section we analyze how persistence changes the relationship between UPE and UA observed in case A. Two prominent psychological processes of subjective judgment, a form of anchoring-and-adjustment (Hogarth & Einhorn, 1992) and adaptation, (Frederick & Loewenstein, 1999), lead naturally to persistence. The anchoring-and-adjustment model takes the form (Hogarth & Einhorn, 1992) ut = ut −1 + (1 − α)(g (st ) − ut −1 ),

t = 2, . . . , T .

(6)

Here (1 − α) is the adjustment weight, 0 ≤ α ≤ 1. In the adaptation model reviewed by Frederick and Loewenstein (1999), moment utility in period t is ut = g ( s t ) − at ,

t = 1, . . . , T

(7)

where (a1 , . . . , aT ) is a sequence of ‘‘adaptation levels’’. Adaptation levels themselves follow an anchoring-and-adjustment process, at = at −1 + β(g (st ) − at −1 ),

t = 2, . . . , T .

(8)

Parameter β determines the speed of adaptation, 0 ≤ β ≤ 1. The initial adaptation level for Eq. (8) is a1 = 0. As seen from Eqs. (6) to (8), a dynamic model that nests both processes is a linear lagged dependent variable model of the form ut = η ut − 1 + x t ,

t = 2, . . . , T .

(9)

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Here η is the persistence parameter, 0 ≤ η ≤ 1, and xt describes exogenous stimuli. Specifically, η = α and xt = (1 − α)g (st ) for anchoring-and-adjustment, and η = 1 −β and xt = g (st )− g (st −1 ) for adaptation. To examine the relationship between Peak–End and Average utility when moment utilities evolve according to processes described by Eq. (9), we revert to simulations. In simulations, we use g (s) = s, and stimuli st , distributed uniformly on the interval [0, 1]. The results are shown in Figs. 1 and 2. Case A is now a special case that corresponds to α = 0 for anchoring-and-adjustment and β = 0 for adaptation (for this case ρ and d can be found using Eqs. (3) and (4), respectively, and their dependence on T is shown in Figs. 1 and 2 with solid squares). For α, β > 0 moment utilities are no longer i.i.d., and Eqs. (3) and (4) are not applicable. To evaluate ρS , ρ , and d, we considered the values of T = 5, 10, . . . , 100, and α and β shown in Figs. 1 and 2. For each combination (T , α) and (T , β) we simulated 106 experiences using Eqs. (6)–(8), summarized them in terms of UPE and UA and computed sample estimates of ρS , ρ , and d.4 The results of the simulations for the correlation coefficients are shown in Fig. 1 with crosses for ρS and solid lines for ρ . The simulation results for the root mean squared deviation are shown with solid lines in Fig. 2. We expressed ρS , ρ , and d as a function of experience length, T , in separate curves built for each value of α and β considered. As seen from Fig. 1, Spearman and Pearson correlation coefficients are very close over the entire range of parameters. For anchoring-and-adjustment, both ρS and ρ decrease with the length of the experience T and increase with α . Larger values of α correspond to more persistence in moment utilities, and, as a consequence, more significant information transfer across moments. The Peak and End utilities capture information about all previous moments, which leads to higher values of ρS and ρ . For adaptation, ρS and ρ tend to a positive constant for large T and increase with β . Coefficient β describes the speed of adaptation to stimuli. When β = 0, the adaptation is perfect, and stimuli are perceived at their face value; when β = 1, the adaptation is slow, and unless a stimulus is better than the stimulus at a previous moment, it cannot cause a positive moment utility. As β increases, ut approaches a moving-average process, ut = g (st )− g (st −1 ), which leads to higher persistence and higher values of ρS and ρ . Thus, even simple forms of persistence may lead to a significant correlation between the Peak–End and Average utility. Preference orders based on the Peak–End rule will resemble the rankings of experiences in terms of their Average utility. The behavior of root mean squared deviation d given the two types of persistence is shown in Fig. 2. For anchoring-andadjustment, d decreases with α and becomes nonmonotonic as a function of T for α close to 1. For adaptation, d increases with β . Thus, for moment utilities evolving according to Eq. (9) our simulations predict that d is a decreasing function of coefficient η regardless of the process governing exogenous terms xt . And yet, the lowest value of the root mean squared deviation (in anchoringand-adjustment) is of the same order as the standard deviation of a uniformly distributed random number, which shows that there is practically no similarity between the levels of Peak–End and Average utility.

4 We used u = 0 as the initial value of utility. The choice of u is without 1 1 loss of generality because the impact of u1 becomes exponentially small in long experiences.

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Fig. 1. Spearman correlation ρS (crosses) and Pearson correlation ρ (solid lines) between Peak–End and Average utility for anchoring-and-adjustment and adaptation with g (s) = s and uniform distribution of stimuli. The solid lines and crosses show the results of simulations (connected by splines for ρ ); the squares are obtained for i.i.d. utility profiles (α = 0 and β = 0) using Eq. (3).

Fig. 2. Root mean squared deviation d between Peak–End and Average utility for anchoring-and-adjustment and adaptation with g (s) = s and uniform distribution of stimuli. The solid lines show the results of simulations (connected by splines); the squares are obtained for i.i.d. utility profiles (α = 0 and β = 0) using Eq. (4).

1.5. Case D: Persistent moment utilities and individual heterogeneity Case D is the most general case, in which persistence and individual heterogeneity are present simultaneously: ut = η ut − 1 + x t + c ,

t = 2, . . . , T .

(10)

√

It is instructive to look at the explicit solution of Eq. (10): ut = u1 η t − 1 +

t −2 X k=0

η k x t −k + c

1 − η t −1 1−η

the term with c in Eq. (11), the variance of which is of the order of D2c = σc2 /(1 − η)2 for large enough T . The dispersion within a utility profile is determined by the variation of ut over time after averaging across individuals. This variation only comes from the term with transformed stimuli xt . The variance is of the order of D2s = k2 σs2 /(1 −η2 ), where k = 1 −α

.

(11)

We can evaluate the impact of different factors on association measures ρS , ρ , and d. The term with the initial condition decays very fast with increasing t, at least for η not too close to 1. It will not contribute much to total utility, regardless of the aggregation rule. The correlation between Peak–End and Average utility depends on how much dispersion there is in moment utilities between as compared to within individual utility profiles. The dispersion between individuals is determined by the variation of ut across individuals after averaging over time. Such variation comes from

for anchoring-and-adjustment and k = 2 for adaptation. Higher dispersion between utility profiles due to individual heterogeneity obviously leads to higher correlation between UPE and UA , or, for that matter, between total utilities produced by any two aggregation rules (as long as these rules are based on moment utilities). For Ds  Dc , when the within-profiles dispersion dominates, the correlation is mainly driven by the persistence in moment utilities as in Case C. When Dc becomes of the same order as Ds or exceeds it, the correlation increases and can become arbitrarily close to 1. Under such circumstances, it matters little whether the decision maker’s preferences for experiences are based on the Peak–End rule or the Average utility. This is in spite of the fact that the effect of dispersion in individual characteristics on root mean squared deviation d in case

I. Cojuharenco, D. Ryvkin / Journal of Mathematical Psychology 52 (2008) 326–335

D is ambiguous. Individual-specific terms are no longer filtered out by differencing as in case C and due to the complex nature of UPE it is hard to come up with a clear-cut prediction. As we show in Section ‘‘Empirical analysis’’ below, in the data the effect of Dc on d is not statistically significant when other factors are accounted for.

331

Our theoretical analysis predicts that correlational measures of association between Peak–End and Average utility will be higher when moment utilities exhibit higher persistence over time and higher individual heterogeneity relative to the dispersion of stimuli within the experience. The root mean squared deviation between Peak–End and Average utility should decrease as a function of persistence parameter η.

between the Peak–End and average impressions. Nevertheless, the high correlation is present in all data sets that came into our possession. This finding is surprising, and suggests that the rankings of experiences based on the two aggregation rules are very similar, even for experiences with large T . As discussed in the previous section, this is an indication of persistence and/or the presence of individual heterogeneity. ˜ the rescaled root mean squared deviation Table 1 also reports d, between UPE and UA . The scales of reported moment utilities are different across studies, therefore to make comparisons across data sets meaningful we rescaled d by the middle value of the scale in each study. ˆ c /Dˆ s in Table 1 report our estimates of Columns with ηˆ and D the magnitude of persistence and individual heterogeneity in each data set. The estimation procedure is described in the next section.

2. Empirical analysis

2.2. Estimation

To test the theoretical predictions and bring evidence of the psychological processes in existence, we revert to the analysis of data on experiences documented in previous research.

For each data set, we estimated the parameters of the underlying psychological process of experiencing. The estimation was based on the most general formulation (case D), which allows for the presence of both individual heterogeneity and persistence. Following Eq. (10), we used the following population model:

1.6. Summary

2.1. Data We collected 35 data sets on reported moment utilities from other authors. In our data collection effort, we targeted published experimental and field studies of the Peak–End rule in which researchers measured and documented impressions throughout experiences, and either the Peak–End rule or average impressions have been shown to predict overall evaluations (Ariely & Carmon, 2003; Baumgartner et al., 1997).5 Additionally, 19 experimental and field data sets came from our own unpublished research on the Peak–End rule, giving us a total of 54 data sets. The data sets are described in Table 1. T and N denote, respectively, the number of time periods and the number of cross-sectional units (individuals). Data sets 1 through 34 are due to Baumgartner et al. (1997). In that study, subjects reported their per-second impressions from viewing advertisements of various lengths. Data sets 1–30 refer to the same group of people (with minor exceptions) viewing the same sequence of advertisements varying in length. We treat them as separate data sets for the estimation of persistence and individual heterogeneity but take the possible interdependence into account in cross-data sets estimation. Data sets 31–34 refer to different people viewing different advertisements. Data set 35 is due to Ariely and Carmon (2003) and represents hourly reports of pain intensity from a day-long hospital field study. The remaining data sets are our own. Data sets 36–39 and 54 are classroom evaluations (evaluations of explanations or class discussions every 5–10 min); data sets 40, 41, 52, and 53 are evaluations of pleasant and aversive images in image-viewing experiments; data sets 42–51 are evaluations of life aspects in a month-long life satisfaction study (reports every three days). Data sets 42–51 refer to the same group of people and are, therefore, treated similarly to data sets 1–30. Table 1 reports ρS and ρ , the empirical Spearman and Pearson correlation coefficients between UPE and UA . Both correlations are high and significant in all data sets (average Spearman correlation across data sets is 0.84; average Pearson correlation across data sets is 0.85). We emphasize that we did not intentionally target or restrict our attention to the data sets that exhibit high correlation

5 Other studies that can be of interest from this perspective include Fredrickson and Kahneman (1993) and Redelmeier and Kahneman (1996).

yit = ηyi,t −1 + xt + ci + it ,

t = 2, . . . , T ; i = 1, . . . , N .

(12)

Here, yit is the reported moment utility of subject i after being exposed to an unobserved stimulus sit . We controlled for the effect of the unobserved stimuli using variables xt , that are timedependent components common to all individuals (to be captured by time dummies). ci was the unobserved individual-specific effect (to be captured by dummies for individuals). For error terms it , we made the following standard assumption (Wooldridge, 2002): E(it |yi,t −1 , yi,t −2 , . . . , yi1 , x1 , . . . , xT , ci ) = 0.

(13)

According to assumption (13), stimuli are strictly exogenous, while yit is completely determined by its lagged value yi,t −1 given the stimuli and the unobserved effect. We estimated model (12) by implementing the ArellanoBond panel data estimator (Arellano, 2003).6 It involves firstdifferencing Eq. (12) to remove the unobserved effect, and then using the GMM estimation. We used yi,t −2 and yi,t −3 as instruments. We also included time dummies to control for the unobserved stimuli in the most general way. The resulting estimates ηˆ of persistence parameter η are reported in Table 1. Most of them are positive and highly significant in spite of the extremely conservative estimation approach we use. Given the parameter estimate ηˆ , we then recovered the underlying time-specific stimuli, xˆ t , and the individual fixed effects, cˆi , in a regression of a new variable zit = yit − ηˆ yi,t −1 on time dummies and individual dummies (see Eq. (12)). The dispersion in the coefficients on individual dummies obtained in the latter regression is the estimate of the dispersion in the individual-specific effect, Dc . Similarly, the dispersion in the coefficients on time dummies is the estimate of the dispersion in stimuli, Ds .7 We estimated both dispersions and report their ratio, ˆ c /Dˆ s , in Table 1. D

6 In three data sets, 52–54, the number of observations was not sufficient to serve for estimation. 7 Although we are not able to identify any dynamic relationship that may exist between the stimuli (e.g., if the stimuli were chosen to be an increasing sequence), our estimates of coefficients on time dummies allow us to evaluate the dispersion, Ds . At the same time, our estimates of persistence, ηˆ are consistent regardless of the presence/absence of a dynamic relationship in the stimuli.

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Table 1

ˆ c /Dˆ s , and Description of data sets in terms of experience length, T , number of individuals, N, estimated persistence, ηˆ , relative dispersion of the individual-specific effect, D three measures of association between UPE and UA – Spearman rank-order correlation coefficient, ρS , Pearson correlation coefficient, ρ , and the root mean squared deviation normalized by the mid-scale response, d˜ Data set 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

T 60 60 60 30 60 50 45 30 30 30 60 90 30 60 60 45 30 30 75 90 90 60 60 90 30 30 45

N 26 26 27 26 27 27 27 27 27 27 27 26 27 27 27 27 27 27 27 27 27 27 27 26 27 27 26

ηˆ

ˆ c /Dˆ s D a

0.51 0.71a 0.71a 0.56a 0.89a 0.72a 0.72a 0.87a 0.33c 0.61a 0.87a 0.76a 0.39a 0.72a 0.82a 0.72a 0.88a 0.62a 0.65a 0.75a 0.91a 0.76a 0.74a 0.77a 0.76a 0.58a 0.81a

2.50 2.83 2.03 1.57 3.20 2.47 2.05 4.81 8.48 2.36 7.68 1.38 2.46 2.03 2.86 1.72 2.86 2.53 1.40 1.55 4.38 1.81 3.01 1.47 2.45 2.67 4.36

ρS 0.97 0.97 0.92 0.89 0.93 0.95 0.93 0.88 0.98 0.81 0.97 0.89 0.94 0.89 0.80 0.93 0.93 0.93 0.62 0.89 0.87 0.94 0.88 0.98 0.83 0.59 0.96

ρ 0.96 0.97 0.92 0.87 0.86 0.94 0.91 0.93 0.93 0.84 0.96 0.94 0.91 0.89 0.94 0.90 0.95 0.91 0.66 0.87 0.88 0.95 0.93 0.93 0.92 0.61 0.95

d˜ 0.06 0.09 0.11 0.13 0.14 0.11 0.12 0.11 0.07 0.07 0.08 0.09 0.08 0.13 0.13 0.10 0.08 0.09 0.19 0.12 0.11 0.09 0.11 0.08 0.10 0.21 0.13

Data set 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

T 30 50 60 42 57 52 73 11 7 7 18 6 30 30 10 10 10 10 10 10 10 10 10 10 3 3 4

N 27 27 27 23 22 24 25 36 46 42 28 37 23 23 35 35 35 35 35 35 35 35 35 35 20 20 42

ηˆ a

0.81 0.92a 0.44a 0.81a 0.76a 0.90a 0.62a −0.05 0.03 0.20c 0.33a −0.11 0.23a 0.21a 0.06 0.17c 0.07 −0.19 0.08 0.00 0.12 0.15b 0.16 0.25a – – –

ˆ c /Dˆ s D

ρS

ρ

d˜

3.05 7.86 1.95 2.39 1.00 2.10 1.25 4.17 3.63 5.10 2.35 1.74 1.54 1.48 5.12 6.17 8.49 6.60 9.29 7.43 5.46 4.72 0.95 6.19 – – –

0.97 0.99 0.89 0.93 0.96 0.93 0.87 0.85 0.79 0.85 0.75 0.65 0.58 0.40 0.87 0.74 0.86 0.86 0.88 0.60 0.72 0.72 0.56 0.81 0.84 0.75 0.75

0.97 0.95 0.87 0.93 0.98 0.92 0.88 0.84 0.80 0.76 0.80 0.67 0.73 0.51 0.90 0.83 0.90 0.89 0.91 0.62 0.72 0.69 0.69 0.79 0.74 0.75 0.81

0.13 0.08 0.10 0.11 0.13 0.08 0.09 0.37 0.10 0.12 0.13 0.19 0.35 0.56 0.11 0.14 0.08 0.06 0.13 0.13 0.11 0.10 0.12 0.12 0.43 0.50 0.22

Data sets 1–34 are reactions to advertisements (Baumgartner et al., 1997). Data set 35 is hospital pain reports (Ariely & Carmon, 2003). Data sets 36–39, 54 are classroom evaluations; data sets 40, 41, 52, 53 are evaluations of images. Data sets 42–51 are evaluation of life aspects in a longitudinal study of life satisfaction. In data sets 52–54 the number of time periods is insufficient to allow for estimation. a Significance levels: −1%. b −5%. c −10%.

We are now in the position to assess the impact of persistence and individual heterogeneity on the association between UPE and UA . We do this by regressing the empirical correlation coefficients ˜ on ρS and ρ , and the rescaled root mean squared deviation d, the estimated persistence ηˆ and the relative dispersion of the ˆ c /Dˆ s . We control for the length of individual specific effect D experiences, T , and a possible dependence among data sets 1–30 and 42–51 by using two dummy variables, d1−30 and d42−51 . The results of the three regressions are given in Table 2. ˆ c /Dˆ s on ρS As seen from Table 2, the effects of both ηˆ and D and ρ are positive and statistically significant. A 0.1 increase in the persistence parameter leads to a 0.017 increase in the Spearman correlation coefficient ρS , and a 0.018 increase in the Pearson correlation coefficient ρ . A unit increase in the relative dispersion of individual heterogeneity leads to a 0.024 increase in ρS and a 0.016 increase in ρ . The coefficient estimates on both dummy variables d1−30 and d42−51 in the regressions for ρS and ρ are not significant, which suggests that the possible dependence among data sets 1–30 and 42–51 has no effect. The coefficient estimate on the length of experiences, T , is also not significant, suggesting that after controlling for persistence and individual heterogeneity, the length of the experience has no significant partial effect. ˜ the effect of ηˆ is negative and statistically In the regression for d, significant. A 0.1 increase in the persistence parameter leads to a ˜ The effects of T and d1−30 are not significant. 0.011 decrease in d. Interestingly, there is a negative and significant effect of dummy ˜ which suggests that the root mean squared variable d42−51 on d, deviation between Peak–End and Average utility was lower in daily life satisfaction reports.

Table 2 Estimation results for OLS regressions of Pearson correlation coefficient, ρ , Spear˜ on man correlation coefficient, ρS , and normalized mean root squared deviation, d, the estimates of persistence coefficient, ηˆ , and the relative dispersion of individual ˆ c /Dˆ s ) fixed effects, (D Explanatory variables

Regressions

ρS

ρ

d˜

ηˆ

0.17 (2.00)

0.18 (2.53)

−0.11c (−1.84)

(Dˆ c /Dˆ s )

0.024a (2.86)

0.016b (2.22)

−0.006 (−0.92)

T

0.00 (0.53)

0.00 (0.37)

0.00 (0.01)

d1−30

0.032 (0.67)

0.018 (0.47)

−0.052 (−1.58)

d42−51

−0.047 (−0.81)

−0.009 (−0.19)

−0.10b (−2.51)

Intercept

0.64a (12.5)

0.69a (16.5)

0.25a (7.15)

b

b

The length of experiences, T , and dummy variables taking into account possible clustering of data sets are included as controls; t-statistics in parentheses. a Significance levels: −1%. b −5%. c −10%.

Our results in this section demonstrated how properties of psychological processes observed empirically affect the correlation and the root mean squared deviation between Peak–End and Average utility.

I. Cojuharenco, D. Ryvkin / Journal of Mathematical Psychology 52 (2008) 326–335

3. Discussion The contribution of this work to the research on the evaluation of experiences is twofold. First, we have analyzed different psychological processes that may underlie experiences lived across time within the stimulus– response framework. The stimulus–response framework has been instrumental to partition the universe of psychological processes into four categories derived from crossing the persistence in moment utilities with individual heterogeneity. Psychological research to date helps relate particular experiences to these four categories. For example, pleasant compared to unpleasant stimuli can be expected to lead to experiences with lower persistence and higher individual heterogeneity. Second, we have shown how magnitudes of persistence and individual heterogeneity mitigate differences in rankings of experiences that are based on distinct conceptions of total utility. Such differences can be huge potentially for long experiences with many moment utilities. And yet, in experiences with high individual heterogeneity or mild individual heterogeneity coupled with moderate levels of persistence decision makers are ‘‘safe’’ in relying on the Peak–End rule even when aiming at experiences of higher average utility. One methodological implication of our results is that one needs to account for the nature of experiences in comparing different aggregation rules. For example, researchers may define the ability to recall an affective experience accurately as the ability to report the experience’s average moment utility. However, even if all the participants remembered their experiences selectively, in terms of Peaks and Ends, the correlational method can suggest falsely that some experiences are remembered better than other experiences. Peaks and Ends may happen to carry more information about the other moments of the experience, depending on the nature of the experience and independent of the accuracy of one’s memory. Our work is related to a broader literature on unit weighting schemes for decision making. Einhorn and Hogarth (1975) examine the general problem of predicting composite variables from select components as opposed to the best linear combination of all components. The success of the prediction based on a select component is found to increase with intercorrelation of the components. In this sense, our work is informative about simple one-parameter dynamic processes inducing a particular structure of such intercorrelation (moment utilities in our analysis are the ‘‘components’’ of total utility). It is shown to impact the relationship between select components (Peaks and Ends) and the average of all components. When the Peak–End rule was first found to describe retrospective accounts of total utility, researchers have looked for alternatives to complement evaluations of experiences that are memory-based. One alternative has been the measure of ‘‘objective happiness’’ (Kahneman, 1999, 2000b). ‘‘Objective happiness’’ would take into account all moment utilities through summation or averaging. Experience sampling methodologies could be used to obtain the necessary information about moment utilities. This was necessary given obvious differences in the absolute levels of overall evaluations of experiences. We have shown that for all types of psychological processes analyzed, these differences are considerable indeed. And yet, our work indicates that for a range of experiences, differences in relative overall evaluations are small. In fact, rankings of experiences derived from ‘‘objective happiness’’ and the Peak–End rule may be rather similar. It has been argued that each moment utility deserves consideration due to its ‘‘inclusive’’ nature, i.e., because it represents ‘‘all the aspects of an experience that are relevant to its evaluation, including the residual effects of prior events (e.g., satiation, adaptations, fatigue) and the affect associated with anticipation of future events (fear, hope)’’ (Kahneman, 2000b). We suggest that some forms of inclusiveness and

333

individual heterogeneity, in fact, allow one to neglect portions of experiences in cross-experience comparisons without running the danger of preferring experiences of lower average utility. Appendix A. Proofs of Propositions 1–6 Proof of Proposition 1. The result follows from the asymptotic independence between order statistics (Hürlimann, 2004) and the fact that ρS = 0 for independent variables (Kruskal, 1958).  Proof of Proposition 2. The result follows immediately from the asymptotic independence between order statistics (Hürlimann, 2004). Here, we provide an explicit estimate of the convergence rate and analytical expressions that can be used to calculate ρ . Consider i.i.d. random variables X1 , . . . , XT with pdf f (x) and cdf F (x). Assume, for simplicity, that f (x) has a bounded support and finite variance σX2 = Var(Xt ), which is always the case for stimuli and/or reported moment utilities. Further, let Y = max{X1 , . . . , XT } denote the Peak value. Define PT UPE = (Y + XT )/2 and UA = T −1 t =1 Xt . We need to show that ρ = Corr(UPE , UA ) → 0 for T → ∞. Clearly, Cov(Y , Xt ) is the same for all t, therefore Cov(UPE , UA ) =

=

" T X

1 2T 1 2

# Cov(Y , Xt ) + σ

2 X

t =1

Cov(Y , Xt ) +

1 2T

σX2 .

To evaluate Cov(Y , Xt ), we need the joint distribution of Y and Xt : Pr(Y ≤ y, Xt ≤ x) = Pr(X1 ≤ y, . . . , XT ≤ y, Xt ≤ x)

= F (min{x, y})F (y)T −1 . The joint density of Y and Xt is given by the cross partial derivative of this expression whenever it is twice continuously differentiable (i.e. for x 6= y). The resulting density g (x, y) is zero for y < x and is equal to g (x, y) = f (x)F (x)T −1 δ(x − y) + (T − 1)f (x)f (y)F (y)T −2 ,

y ≥ x.

Here δ(·) is the Dirac delta-function that arises because of the nondifferentiability of the joint cumulative density for x = y. Then, Cov(Y , Xt ) =

∞

Z

Z

−∞

Z

∞

=

Z

dxg (x, y)xy − E(Y )E(Xt )

−∞

y

dxf (x)F (x)T −1 δ(x − y)xy

dy −∞

y

dy

−∞

Z

∞

Z

y

+ (T − 1) dy dxf (x)f (y)F (y)T −2 xy − E(Y )E(Xt ) −∞ −∞ Z ∞ ≤ dyf (y)F (y)T −1 y2 −∞   Z ∞ 1 + (T − 1) dyf (y)F (y)T −2 y − E(Y ) E(Xt ) ≤ E(Y 2 ). T

−∞

(14)

Ry

The first inequality follows from the fact that −∞ dxf (x)x ≤ E(Xt ) for all y. The second inequality follows from the fact that E(Y ) is nondecreasing in T . Thus, we have shown that Cov(UPE , UA ) ≤

 1  E(Y 2 ) + σX2 ,

2T

therefore 1 E(Y 2 ) + σX2 0≤ρ≤ √ q . T σ 2 Var(Y + X ) t X

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It is easy to see that the right hand side of this expression goes to zero for T → ∞ as const · T −1/2 if E(Y 2 ) is bounded.  Proof of Proposition 3. Consider i.i.d. random variables X1 , . . . , XT with bounded support and finite mean µ = E(Xt ) and variance σX2 = Var(Xt ). Let xmax denote the upper bound of the support of Xt , that is, xmax = inf{y : f (x) = 0 for all x ≥ y}. Let Y = max{X1 , . . . , XT } denote the Peak value, and d denote the root mean squared deviation between UPE = (Y + XT )/2 and PT UA = X¯ = T −1 t =1 Xt . Then d2 = E (UPE − UA )2



 =E

Y2 4

+

XT2 4



+ X¯ 2 +

 − Y X¯ − XT X¯ .

YXT 2

(15)

In the limit T → ∞, we have E(Y 2 ) → x2max , E(X¯ 2 ) → µ2 , E(XT X¯ ) → µ2 , E(YXT ) → xmax µ and E(Y X¯ ) → xmax µ (the latter two limits follow from asymptotic independence of order statistics (Hürlimann, 2004)). Thus, from Eq. (15), d2 → x2max /4 + (µ2 + σX2 )/4 − xmax µ/2, which gives Proposition 3.  Proof of Proposition 4. The result follows immediately from Proposition 5.  Proof of Proposition 5. Consider moment utilities ut = Xt + c, where Xt are i.i.d. across individuals and time, and c are i.i.d. across individuals and time-constant. The Peak–End utility in this case 0 is UPE = UPE + c and the average total utility is UA = UA0 + c, 0 where UPE and UA0 are the Peak–End and average utilities in the absence of individual heterogeneity (c = 0). Let σc2 = Var(c ) and 0 ρ 0 = Corr(UPE , UA0 ). 0 0 Then Cov(UPE , UA ) = Cov(UPE , UA0 )+σc2 , Var(UPE ) = Var(UPE )+ 0 2 σ and Var(UA ) = Var(UA ) + σc . As a result, the correlation coefficient between UPE and UA becomes 2 c ,

√ 1 + ρ 0 vPE vA ρ= √ , (1 + vPE )(1 + vA ) Var(UA0 ) vA = . σc2

vPE =

0 Var(UPE )

σc2

,

Thus, as σc2 increases relative to the variance of stimuli within experiences (i.e. vPE and vA decrease), the correlation coefficient approaches unity. In the opposite limit of small σc2 , ρ approaches ρ0.  Proof of Proposition 6. The result follows immediately from the definition of d and Eq. (5). 

Appendix B. Derivations for the unform distribution of stimuli For the uniform distribution of moment utilities, f (x) = I (0 ≤ x ≤ 1) (where I (·) is the indicator function), covariance Cov(Y , Xt ) can be found in the closed form using Eq. (14): Cov(Y , Xt ) =

1 2(T + 1)(T + 2)

.

This gives, after elementary transformations, expression (3). For the root mean squared deviation, Eq. (15) then gives expression (4).

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