Prospect Theory, Mental Accounting, and Differences in Aggregated and Segregated Evaluation of Lottery Portfolios
Thomas Langer • Martin Weber Universität Mannheim, Lehrstuhl für Bankbetriebslehre, L 5, 2, 68131 Mannheim, Germany [email protected] • [email protected]
f individuals have to evaluate a sequence of lotteries, their judgment is influenced by the Ipresentation mode. Experimental studies have found significantly higher acceptance rates for a sequence of lotteries if the overall distribution was displayed instead of the set of lot- teries itself. Mental accounting and loss aversion provide an easy and intuitive explanation for this phenomenon. In this paper we offer an explanation that incorporates further evalu- ation concepts of Prospect Theory. Our formal analysis of the difference in aggregated and segregated portfolio evaluation demonstrates that the higher attractiveness of the aggregated presentation mode is not a general phenomenon (as suggested in the literature) but depends on specific parameters of the lotteries. The theoretical findings are supported by an experi- mental study. In contrast to the existing evidence and in line with our theoretical results, we find for specific types of lotteries an even lower acceptance rate if the overall distribution is displayed. (Prospect Theory; Mental Accounting; Evaluation Procedures)
1. Introduction decisions? Kahneman and Lovallo (1993) argue that Assume you are asked to participate in two indepen- people show a strong tendency to consider problems dent draws of a simple lottery as unique, neglecting the portfolio context of the over- all choice situation. By this “narrow framing” the two 0 5 $200 draws in the repeated trial offer might be evaluated in 0 5 −$100 isolation, neglecting the overall outcome distribution. In this paper we address the question of whether Would you accept playing such a sequence of gam- such a perception of the portfolio context would bles? What about the alternative offer to play the lot- make the repeated trial offer look more or less attrac- tery tive. Stated in other words: Does the attractiveness 0 25 $400 of a portfolio of lotteries increase (or decrease) by 0 5 $100 explicit provision of aggregated outcome informa- tion? The literature on this point has provided a 0 25 −$200 consistent answer. In a recent paper Benartzi and just once? Needless to say, both outcome distributions Thaler (1999, p. 366) conclude from their experimen- are identical, so the offers should be either rejected or tal data, “When subjects are shown the distributional accepted. facts about repeated plays of a positive expected But do individuals in fact perceive both choice value gamble they like them more rather than less.” situations to be equivalent and thus make identical Benartzi and Thaler (1995) provide a theoretical expla-
Management Science © 2001 INFORMS 0025-1909/01/4705/0716$5.00 Vol. 47, No. 5, May 2001 pp. 716–733 1526-5501 electronic ISSN LANGER AND WEBER Prospect Theory and Aggregated vs. Segregated Evaluations nation for this phenomenon, which is based on men- modes for repeated gambles by Benartzi and Thaler tal accounting and loss aversion. (1995). They gave an explanation for the Equity Pre- In this paper we argue that, despite its intuitive mium Puzzle2 based on the argument that stocks look appeal, the explanation is incomplete and the phe- much less attractive to investors if repeatedly evalu- nomenon is not as general as it might seem. We ated in short intervals than if considered within the present a theoretical analysis of the evaluation prob- overall investment horizon. In a more general formu- lem based on mental accounting and Prospect Theory lation this argument reads as follows: A portfolio or (Tversky and Kahneman 1992). This analysis predicts sequence of gambles (stock price movements) is less the aggregated presentation format to be even less attractive compared to the overall portfolio distribu- attractive if lotteries with specific risk profiles (low tion if each gamble in the set is evaluated separately. probability for high loss) are involved. The predic- Benartzi and Thaler (1995) provided a simple exam- tions are confirmed by an experimental study. ple to demonstrate how an isolated evaluation and The general phenomenon has much practical rele- loss aversion might lead to the proposed difference in vance. New financial products might be considered attractiveness. They consider a decision maker with a highly attractive by individual investors, although value function of the form they are just combinations of existing investment x if x ≥ 0 alternatives, which are less appreciated in isolated vx = 2 5 · x if x<0 evaluation.1 On the other hand, the attractiveness of a product might increase by splitting it and offering its reflecting loss aversion through the 2.5 times stronger components separately. In general, the question to be impact of losses compared to equal-sized gains. This answered is how a portfolio of risky options should decision maker would accept the aggregated gamble be presented to make it look more attractive. 0 25 $400 The remainder of the paper is structured as fol- lows. In §2 we give an overview of the literature. 0 5 $100 In §3 we formalize the evaluation problem, extend 0 25 −$200 the consideration to a more general space of lotteries, and provide a theoretical analysis based on mental because the overall evaluation +25 is positive. How- accounting and Prospect Theory. In §4 we derive ever, he assigns the negative value −25 to each of the hypotheses and report the results of an experimental two lotteries 0 5 $200 study testing these hypotheses. We conclude in §5 with a discussion of the results and suggestions for 0 5 −$100 further research. Thus, for segregated evaluation the gamble is rejected, although it is accepted in aggregated evaluation. Gneezy and Potters (1997) explicitly tested the 2. Related Literature interdependence of evaluation period and risk-taking Most of the early literature on repeated gambling behavior in an experimental study.3 In their set- focused on the different risk-taking behavior in single ting, subjects were repeatedly asked to invest part and multiple plays of a lottery, i.e., risk in the short of their endowment in sequences of mixed gam- and the long run (Samuelson 1963; Lopes 1981, 1996; bles. The willingness to take the risk was found to Tversky and Bar-Hillel 1983; Keren and Wagenaar depend on the frequency of previous outcome feed- 1987; Keren 1991). Recently, however, attention has back. The sequence of risky gambles was considered been drawn to the question of different evaluation 2 The puzzle, as introduced by Mehra and Prescott (1985), states 1 Consider as an example the very popular guarantee funds, which that the historical difference in returns of stocks and bonds is much can be constructed as a simple portfolio of zero-coupon bonds and too high to be explained by any plausible degree of risk aversion. stock options. 3 Thaler et al. (1997) provided a similar test in parallel research.
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more attractive if just the combined return of three 3.1. The Lottery Space 2500 and the consecutive draws was reported, but not each single Value Function outcome. For the analysis we exclusively consider mixed two- A direct experimental test of the influence of pre- outcome lotteries 1 − pg sentation mode on the attractiveness of a lottery port- p folio was reported by Redelmeier and Tversky (1992). They asked subjects to state their willingness to take In the following it will turn out that our qualitative part in five independent draws of a lottery of the form results are not influenced by the absolute size of gains and losses but rather by the relative size of gains com- 0 5$2000 pared to losses.4 Hence we simplify the analysis with- out loss of generality by assuming a fixed difference 0 5 −$500 between the two lottery outcomes. Because we want If the problem was presented as a repeated trial the lottery 0 5$2000 (i.e., segregated), 63% of the subjects accepted the offer. If the aggregated distribution was displayed 0 5 −$500 instead, the acceptance rate increased to 83%. This used in the Redelmeier and Tversky (1992) study to result might also confirm that the higher attractive- be included in our analysis, we choose to be 2,500, ness of an aggregated evaluation is a general phe- i.e., g = + 2500. nomenon. Note, however, that the risk profile used by Given the restriction, each mixed lottery can be Redelmeier and Tversky (1992) is very similar to the described as a pair p of loss probability and loss lottery size. Our general lottery space is defined as = 2500 1/3$2 50 p ∈ −2500 0 p ∈ 0 1. Note that the lotter- ies at the frontier of are not mixed anymore 2/3 −$1 00 2500 but are included for limit considerations. The lotter- used by Gneezy and Potters (1997) and to the one ies in 2500 can be displayed in a p coordinate used in Benartzi and Thaler’s (1995) illustrative theo- system (see Figure 3.1). Each point within the rectan- retical argument. gle corresponds exactly to one lottery in 2500.Atthe left and the right boundary there are lotteries with sure gains and losses, respectively. All the lotteries on the diagonal have an expected value (EV) of 0. The 3. Formal Analysis expected value increases by moving up and to the left. Our formal analysis extends the previous literature in Two specific lotteries are marked in the space. The two ways. First, we incorporate a more general value point R = 0 5 −500 corresponds to the lottery used function, expressing loss aversion and diminishing by Redelmeier and Tversky in their study. The point sensitivity as proposed by Prospect Theory. Second, K = 0 04 −2100, representing the lottery we extend the space of lotteries to cover all kinds of 0 96 $400 gain/loss probabilities and gain/loss sizes. In §3.1 we define the general lottery space and the value func- 0 04 −$2100 tion. In §3.2 we demonstrate how the evaluation dif- is an example for a risk profile with low loss prob- ference for a two-lottery portfolio can be separated, ability and high loss size, which will turn out to be and formally analyze the case of pure loss aversion. particularly interesting in the further analysis. In the In §3.3 the analysis is extended to incorporate both following we will refer to this type of lottery as loan loss aversion and diminishing sensitivity. In §3.4 we discuss the additional impact of probability weight- 4 This is because we assume throughout a specific type of power ing and increasing portfolio size. value function.
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the two concepts (loss aversion and diminishing sen- Figure 3.1 The Lottery Space 2 500 sitivity) influence the overall evaluation difference.
3.2. Separation of the Evaluation Difference and the Case of Pure Loss Aversion We will use the notation X +X to describe a portfolio consisting of two lotteries X (we call this segregated presentation). By 2X we refer to the lottery resulting as the overall distribution of the portfolio X + X (we call this aggregated presentation). For each lottery X = p , the aggregated distribution 2X has the three outcomes 2 + g, and 2g. While the sign of 2 and 2g is obvious, the mixed outcome + g is positive or
negative depending on the size of and g.ByALAG, and AM we notate the values of these outcomes, i.e.,
AL = v2 AG = v2g and AM = v + g lottery because its risk profile is typical for the one Similarly pLpG, and pM are used to refer to the proba- faced by a creditor in a standard loan contract. The bilities for the pure loss, the pure gain, and the mixed lottery K represents a $2100 loan, defaulting with a case. The overall evaluation of the portfolio distribu- 5 probability of 4% and an interest rate of 19%. tion 2X is then obtained as Tversky and Kahneman (1992) proposed the follow- ing functional form for the value function: A = pL · AL + pM · AM + pG · AG x if x ≥ 0 Next we consider the portfolio X + X if both lot- vx = −k−x if x<0 teries are evaluated separately and the values are summed up to derive the overall evaluation of the The parameter k ≥ 1 describes the degree of loss aver- portfolio. This segregated evaluation process results sion and ≤ 1 measure the degree of diminishing in a total value of S = 2 · p · v + 1 − p· vg. sensitivity. = = 1 represents the case of pure loss Defining SL = 2 · v SG = 2 · vg, and SM = aversion discussed in the previous section. Tversky v + vg, we can write and Kahneman (1992) estimated the parameters , 2 2 and k and identified = = 0 88 and k = 2 25 as S = p · 2 · v + 1 − p · 2 · vg median values. We will use these parameters for most + 2 · 1 − p· p · v + vg of our calculations. It should be mentioned that this specific functional = pL · SL + pG · SG + pM · SM form of the value function is not a necessary con- It turns out that the total difference D between the dition for our results. Similar results can be derived two types of evaluation can be separated into three for other value functions reflecting loss aversion and parts (a loss part, a gain part, and a mixed part), diminishing sensitivity. However, the two-parameter form vk (we will assume = throughout and omit D = A − S = pL · AL − SL + pG the index in the following) nicely demonstrates how ·AG − SG + pM · AM − SM
5 This argument assumes that the loan engagement is evaluated = pL · DL + pG · DG + pM · DM ex ante using the status quo as a reference point and offsetting outputs against inputs. Cf. Casey (1994, 1995) for different ways of The relevant expressions for the separation are sum- framing this kind of decision problem. marized in Table 3.1. By definition a positive value of
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Table 3.1 Separation of Evaluation Difference into L- G-, and M-part
L-part G-part M-part
2 2 Probability for the case pL = p pG = 1 − p pM = 2 · 1 − p· p Contribution to value
if evaluated aggregated AL = v2 · AG = v2 · g AM = v + g Contribution to value
if evaluated segregated SL = 2 · v SG = 2 · vg SM = v + vg
Difference of value DL = AL − SL = DG = AG − SG = DM = AM − SM = contributions v2 · − 2 · v v2 · g− 2 · vg v + g− v − vg
D corresponds to a higher attractiveness of an aggre- 2 · vg = 0 and DL = v2 · − 2 · v = 0, s.t. neither gated evaluation. Using this notation we can formally the loss part L nor the gain part G contributes to the state the main question as follows: overall evaluation difference. • For which lotteries X = p in do we get 2500 Further we have DM = v +g−v −vg = k− positive (negative) values of D, i.e., a higher (lower) 1 · ming , thus the mixed part is positive for all attractiveness of an aggregated evaluation?6 lotteries in the interior of 2500. Proposition 1 follows We should further examine the question: immediately: • How does the sign of D, i.e., the preference for one of the presentation modes, depend on the parameters Proposition 1.(The Case of Pure Loss Aversion). and k of the value function v? Assuming a value function To contrast with the later results, we start the anal- x if x ≥ 0 ysis with the case of pure loss aversion = 1. The vx = linearity of the value function in the gain as well as −k−x if x<0 in the loss domain immediately yields DG = v2 · g− with k>1, the difference D between aggregated and seg- 6 At this point it becomes clear that the restriction of the lottery regated evaluation of a portfolio is positive for each mixed space to a fixed gain/loss difference = 2500 does not mean a lottery X in the interior of 2500. qualitative loss of generality. Assuming a power value function vk , a scaling of both outcomes g and by a factor c causes D to change Despite the simplicity of the argument, let us visu- by the factor c. This does not influence the sign of D. alize some of the details to contrast with the further
Figure 3.2 Value Differences DL , DG , DM and Probabilities pLp, pGp, pM p for = 1
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Figure 3.3 Iso D Linesfor the Case = 1 and k = 225 and DG = v2 · g− 2 · vg = v2 · + 5000 − 2 · v + 2500 are decreasing functions in that are positive or negative, respectively (see Figure 3.4). This is a formal description of a general rule first explicitly stated by Thaler (1985): It is more attractive
to consider losses in aggregation DL > 0 but gains in segregation DG < 0. The negativity of DG allows us to state an immedi- ate existence theorem: Proposition 2. (Existence of mixed lotteries with negative evaluation difference D). For any value func- tion vk x with <1, there exist mixed lotteries in with a negative evaluation difference D, i.e., with a higher segregated than aggregated evaluation. The proposition can be proven by a simple continu- ity argument. If we have <1 and pick an arbitrary analysis. Figure 3.2 presents the value and probability ∈ −2500 0, we get a negative value DG . If fol- components of the separation. lows D = DG < 0 for the lottery 0 , as it holds In Figure 3.3 we present Iso-D lines in a p pL = pM = 0 and pG = 1 for p = 0. Because D is a contin- uous function in p on [0, 1], there must exist a small coordinate system representing the space 2500 (for = 1 and k = 2 5). It can be seen that D is positive p>0 such that D is still negative for a lottery p . everywhere and converges to 0 only at the bound- This is an intuitive result. Mixed lotteries that are almost sure gains are more attractive in segregated aries of the space 2500, where we get pure gain (loss) lotteries. evaluation because the negative value difference in the G-part is dominant. With the same arguments it 3.3. The Case of Loss Aversion and can be concluded that we will find positive D values Diminishing Sensitivity if the lotteries have very high loss probabilities. In If we generalize the value function to reflect dimin- fact, it is true that for each loss size ∈ −2500 0 ishing sensitivity as well as loss aversion, we can still there exist lotteries p with positive and negative separate D into an L-, G-, and M-Part. Because of the D values. concavity of v in the gain domain and the convexity in However, these existence theorems are not really helpful for practical considerations because they do the loss domain, the components DL = v2· −2·v
Figure 3.4 DL and DG for k = 25 and = 088
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not provide any information about the size of the Figure 3.5 DM for = 07 and k = 225 effect. In particular, we do not get clear predictions for the lotteries we consider to be most important for practical applications. These are lotteries with moder- ately positive expected values, placed slightly above the diagonal in the p coordinate system. We dis- tinguish three types of such lotteries: (a) lotteries with low probability for a high loss, (b) lotteries with rather moderate probabilities and outcome sizes, and (c) lot- teries with high probability for a small loss. For the first step of the further analysis, let us ignore the mixed M-part of the evaluation difference
D and focus on the term DLG = pL · DL + pG · DG.
We can conclude from the monotonicity of pLpGDL, and DG that DLG grows in as well as in p. Hence, a unique critical loss size , s.t. D is negative on we have low DLG values for very attractive, and high M −2500 and positive on 0 (see Figure 3.5).8 DLG values for very unattractive, lotteries. But what about different risk profiles with similar expected val- Because of loss aversion the positive interval dom- 9 ues as in (a), (b), and (c) above? An examination of inates. Hence we find a negative contribution of the lotteries on the diagonal EV = 0 can provide the M-part to the evaluation difference D if and only if we have lotteries with sufficiently high losses. some basic insight for the situation of simultaneous From the three interesting risk profiles (a), (b), and -increase and p-decrease. It can be shown that there (c) we should thus expect profile (a)—low probability exists a unique p ∈ 0 1 s.t. we have for each lot- 0 for high loss—to be best suited to result in a negative tery p ∈ with EV = 0 D < 0 if and only 2500 LG D value. While these thoughts are informal, an Iso-D if p
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Figure 3.6 space with Iso-D Linesfor = 088 and k = 225 Figure 3.8 Iso-D Linesfor = 088 and k = 35
0.88 to 0.7. As could be expected from our analysis, D as in §3.2. However, because of the increasing num- the area of negative D values, and especially the size ber of parts and the complexity of each value com- of the bulge, grows. ponent, a detailed formal analysis does not provide The consequence of increasing loss aversion is dis- any additional insights. Hence, we just visualize the played in Figure 3.8. For k = 3 5 the bulge almost dis- effect of increasing portfolio size in Iso-D diagrams. Figures 3.9 and 3.10 display the evaluation differ- appears. From the known properties of DLG and DM we can conclude that each mixed lottery in has a ence D for portfolios consisting of 5 and 10 identical positive D value for a sufficiently high degree of loss- lotteries. The larger the portfolio, the less the bulge is aversion k. pronounced in the lower left corner. The proportion of the lottery space with a negative evaluation dif- 3.4. Extensions ference D increases with growing portfolio size. Next we present some results for larger portfolios and Figure 3.11 demonstrates the evaluation difference probability weighting. For an increased portfolio size for even larger portfolios. It presents the relative eval- it is still possible to separate the evaluation difference uation difference D/n for a portfolio of n lotteries R
Figure 3.9 Portfolio of Size 5 for = 088 and k = 225 Figure 3.7 Iso-D Linesfor = 07 and k = 225
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Figure 3.10 Portfolio of Size 10 for = 088 and k = 225 ther monotonic nor required to have a single peak. In fact, for some lotteries the D/n values change their signs several times for increasing n. In the light of the experimental results provided by Benartzi and Thaler (1999), we do not consider our results for the large portfolios to have much practical relevance. For large portfolios (Benartzi and Thaler used sizes between 100 and 150), the effect caused by the value transformation seems to be superimposed by a severe bias in probability judgment. The subjects in the Benartzi and Thaler study strongly overesti- mated the very small probability for an overall loss in the repeated trial format. Prospect Theory does not only predict value trans- formation but also the transformation of probabilities into decision weights. These weights are used in the evaluation procedure instead of the actual probabil- and K. For sufficiently high n, not only the relative ities. Unfortunately, the probability weighting rules difference D/n, but also the absolute difference D, out the simple separation and general analysis as pre- decreases in n. Result 3 in the appendix shows that sented in §3.2. We cannot define “weights” of the any gamble in , which is accepted in single play, G-, L-, and M-part any longer, because the deci- has a higher segregated than aggregated evaluation sion weights depend on the type of portfolio seg- (i.e., D<0) for a sufficiently large portfolio size. Thus, mentation, i.e., they are different in aggregated and even for the lottery R we would eventually (i.e., for segregated evaluation. Informally, one can conclude higher n) find negative D/n values in Figure 3.11. It that the preference for a segregated evaluation of should further be noted that the function D/n is nei- a loan lottery portfolio is weakened if probability
Figure 3.11 D/n asa Function of n for = 088 and k = 225 (the Lines Represent Scale Changes)
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Figure 3.12 Iso-D Linesfor = 088, k = 225, + = 069, and − = Figure 3.13 Iso-D Linesfor = 07, k = 225, + = − = 08 061
few general comments regarding the experimental weighting is incorporated.10 This effect is illustrated in Figure 3.12, where Iso-D lines are presented in design and the choice of risk profiles used in these the space for an evaluation that takes into account experiments. loss aversion, diminishing sensitivity, and probability A comparison of acceptance rates as used by weighting. The computation is based on Cumula- Redelmeier and Tversky (1992) provides a convenient tive Prospect Theory and incorporates probability- and simple test for our theoretical predictions. We can weighting functions easily hypothesize how the sign of the difference in acceptance rates should depend on the risk profiles p% w p = of the involved lotteries. It is a major strength of this % p% + 1 − p% 1/% method that subjects are confronted with very simple with %<1 as introduced by Tversky and Kahneman choice situations and are not unduly influenced by (1992). Using their parameter estimates %+ = 0 69 and coding or other framing effects. A weakness of this %− = 0 61 for gains and losses, the dark bulge in the method is the fact that its use is restricted to a specific lower left corner completely disappeared. Figure 3.13 class of lotteries. For very attractive or very unattrac- presents Iso-D lines for % values closer to 1, e.g., %+ = tive lotteries we must expect to find acceptance rates %− = 0 8 (see Figure 3.7 for % = 1).11 close to 100% or 0% for both presentation modes. As the acceptance decision does not capture the 4. Experimental Study strength of a preference, the method is restricted to In this section the design and the results of two lotteries with moderately positive expected values for experimental studies are reported. We begin with a which we can hope to find switches in acceptance. Although these specific lotteries are surely most
10 This is because the small probability of a loss (with the posi- important for practical applications, an extension of the empirical tests to other regions of the lottery tive DL contribution) is overweighted and the high gain probability
(with the negative DG contribution) is underweighted. However, it space would be interesting to examine the general should be clear that the specific effect is restricted to loan lotteries. correctness of our predictions. There are two possible In the upper right corner of the lottery space a reverse impact of designs for such an extension. First, subjects could be probability weighting should be expected by the same argument (see Figure 3.12). asked to directly compare the two presentations of the 11 Cf. Wu and Gonzalez (1996) for empirical evidence of less curved same lottery portfolio. Second, subjects could provide probability-weighting functions. certainty equivalents for both presentations. A com-
Management Science/Vol. 47, No. 5, May 2001 725 LANGER AND WEBER Prospect Theory and Aggregated vs. Segregated Evaluations parison of these certainty equivalents could serve as but a higher certainty equivalent of ∼388 DM to the a direct test of our theoretical results. aggregated version. Both methods have severe drawbacks. At this point This problem turns out to be particularly severe for we want to point out once again that our research the very attractive lotteries in the upper left corner does not address the question of how the prob- of our lottery space. While these lotteries (with their lem presentation influences an individual’s coding. high negative D values) might seem well suited to Instead, our theoretical predictions are based on spe- generate easy experimental support for our main cific assumptions about the coding (with the central theoretical result (that specific lotteries are preferred assumption that a segregated presentation causes a in segregated presentation), their appropriateness is segregated evaluation). We can only expect to find undermined by the coding problems of the elicitation support for our predictions if the experimental design procedure. It turns out that by the concavity of the induces the participants to code the decision problems gain function the negative D values are transformed in the assumed way. The more lotteries are involved into positive differences of certainty equivalents for all such lotteries.12 in each decision and the more complex the decision We thus restricted our attention to risk profiles with situation is, the more ambiguous the coding proce- moderately positive expected values, enabling us to dure will be. We do not know how individuals code use an acceptance rate comparison as experimental a direct comparison of a three-outcome lottery with procedure. From our analysis in §3, we know that the a sequence of two-outcome lotteries. In our opinion restricted set should still be sufficiently rich (D values it is not very likely, however, that independent eval- with different signs) to allow an empirical test of our uations are derived and compared for the final deci- main theoretical results. sion. We thus do not consider the direct comparison method to represent an appropriate design. Design of Study 1. This introductory study was For the certainty equivalent method and for all set up to directly test the robustness of the other methods in which the lottery (sequence) is com- Redelmeier/Tversky results and to verify the basic pared to a sure amount, there is another severe coding predictions of our theoretical analysis. For that reason problem. If we assume subjects to neglect the over- we chose a 2 × 2 design as presented in Figure 4.1, all decision context given a segregated presentation, varying the lottery type as well as the portfolio size. it is straightforward not only to assume a segre- The entry R/5 is a replication of Redelmeier and gated evaluation of the lotteries but to further assume Tversky (1992): a portfolio consisting of 5 lotteries of type R = 0 5 −500 using DM instead of U.S. dollars. an isolated assignment of certainty equivalents to The entries R/2 and K/2 allow us to test the predic- the lotteries. Hence an elicitated certainty equiva- tions of our theoretical analysis for portfolios of Size 2. lent must be interpreted as a sum of single-play cer- Assuming that a presentation in repeated trial format tainty equivalents. Because of the nonlinearity of the results in a segregated evaluation S, while the aggre- value function (which not only transforms the lot- gated value A is assigned to the overall distribution, tery outcomes but also the certainty equivalents), a we predict comparison of elicitated certainty equivalents thus would not provide an appropriate test for our theory. Hypothesis 1: (Portfolios of Size 2) To illustrate this point, assume an individual assigns a) The acceptance rate for playing the gamble R = a value of +100 to each of the two lotteries in a 0 5 −500 DM) twice will be higher if the overall distri- sequence and a lower total value of +190 to the aggre- bution of the sequence is explicitly mentioned. gated distribution. The resulting negative D value is not reflected in the certainty equivalents. Assuming 12 We did actually run some studies with very attractive lotteries using the certainty equivalent method and recalculated values from a parameter = 0 88 of the value function, the indi- the elicited certainty equivalents. We do not report these results, as vidual assigns a certainty equivalent of ∼374 DM it turned out that their interpretation strongly depends on assump- (= 187 + 187 DM) to the repeated trial presentation, tions about the specific parameters of the value function.
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Figure 4.1 The 2 × 2 Design of Study 1 Table 4.1 Results of Study 1 Accept. Accept. Lottery Rate (%) N Lottery Rate (%) N
Portfolio Size 2 N 5986 K 20 87 R + R 5941 K + K 24 87 ∗∗ ∗∗∗ 2R 76 41 2K 10 87 Portfolio Size 5 R 5986 K 20 87 b) The acceptance rate for playing the gamble K = R + R +···+R 67 45 K + K +···+K 41 41 ∗∗∗ 0 04 −2100 DM) twice will be lower if the overall dis- 5R 88 41 5K 30 46 tribution of the sequence is explicitly mentioned. ∗∗Significant at the 0.05 level. ∗∗∗Significant at the 0.01 level. Finally, the case K/5 was included to show the accep- tance rate increase/decrease to depend on the type of Results of Study 1. The results for the designs R/2 lottery even for a portfolio size as in the Redelmeier and K/2, supporting our hypotheses, are summarized and Tversky study. in Table 4.1. Fifty-one of 86 subjects (59%) were will- Ninety-five subjects (79 advanced business stu- ing to accept a single gamble R. Two independent dents and 16 other students from the University of draws of R were accepted by 24 out of 41 students Mannheim) took part in this study. They filled out (59%) when presented as repeated trial, and by 31 out questionnaires either in combination with another lab of 41 students (76%) when displayed as aggregated experiment or within a classroom lecture. Typical distribution. Hence, we find a higher attractiveness of questions were of the form, “Are you willing to accept aggregated evaluation for the lottery type R as pre- a gamble where you can win 400 DM with a prob- ability of 96% and lose 2,100 DM with a probability dicted by our formal analysis. A t test shows the dif- of 4%?” Additionally, the gambles were displayed in ference of the acceptance rates to be significant at the tree form 5% level. It is further interesting to note that we have 0 96 400 DM the same acceptance rate for a single draw of the lot- 0 04 −2100DM tery R and the segregated presentation of the portfo- lio. This is well in line with our assumption about the We did not provide any monetary incentives. coding of the repeated trial presentation. For each lottery type (K and R), the portfolios of A single play of the lottery K was accepted by 20% different size (2 and 5) had to be evaluated in two of the subjects (17 out of 87). With 24%, the accep- modes (aggregated and segregated). In addition, the tance rate was slightly higher (21 out of 87) if sub- willingness to play the gambles K and R just once jects were offered to play the gamble twice. For the had to be stated. Overall, 10 different questions were relevant within the 2 × 2 design. We did not present aggregated distribution 2K, however, it decreased to all 10 questions to each subject, however. Instead, two 10% (9 out of 87). Here we can provide a within- types of questionnaires were used to cover the full subject analysis. Sixty-five students made consistent set. Within-subject information concerning the differ- decisions (4 accepted both, 61 rejected both offers). ent presentation modes was just collected for the K/2 Five students accepted 2K but rejected K + K, while case, which we considered most interesting from a 17 moved in the other direction. A paired t test shows theoretical point of view. Six subjects were completely the decrease of the acceptance rate to be significant eliminated from the sample because they left open at the 1% level. Again, this result is in line with our answers in all four cells of the 2 × 2 design. Other formal analysis and Hypothesis 1. questionnaires that lacked answers in just some of the The results for Portfolio Size 5 are also presented in tasks were not excluded from the analysis. Table 4.1. Our replication of the study by Redelmeier
Management Science/Vol. 47, No. 5, May 2001 727 LANGER AND WEBER Prospect Theory and Aggregated vs. Segregated Evaluations
Figure 4.2 The Lottery Types K M R, and V in space
and Tversky R/5 found a strong qualitative similar- with our formal analysis and show that the direction ity of the results, although our acceptance rates were a of the effects reported by Redelmeier and Tversky little higher in general. Fifty-one of 86 students (59%) (1992) is not invariant under variations of the lottery were willing to play the gamble R once. Thirty of 45 parameters. (67%) agreed to play it five times in a row, while 36 Design of Study 2. While Study 1 focused exclu- of 41 (88%) accepted the aggregated distribution 5R. sively on differences concerning the lotteries R and The increase of the acceptance rate (67% to 88%) is K, Study 2 was run to analyze the phenomenon significant at the 1% level. Concerning the lottery K, more generally. To fully cover the spectrum of pos- we found the effect to be somewhat weaker for Port- sible risk profiles, we added two new types of lot- folio Size 5 than in the K/2 scenario but still pointing teries to the existing set R K. A lottery type M in the same direction. Seventeen of 41 students (41%) (“Moderate outcomes”) was included with gains and agreed to play the gamble K five times. If the aggre- losses of approximately the same size. Further, we gated outcome distribution was presented, the accep- introduced two lotteries of type V with rather small 13 tance rate decreased to 30% (14 of 46). This decrease probabilities for high gains (as typical for venture is not significant. Nevertheless the results are in line investments). Figure 4.2 displays the location of the
lotteries K M R V1, and V2 within the space . For 13 In the questionnaire the exact distribution of 5K was replaced the V -lotteries we had the following hypothesis: by a simplified lottery. This lottery was constructed in a way, it stochastically dominated the real distribution of 5K. Hence, the real Hypothesis 2: (Lotteries ofType V , Portfolio distribution of 5K should be even less attractive. Sizes 2 and 5). The acceptance rate for playing lotter-
728 Management Science/Vol. 47, No. 5, May 2001 LANGER AND WEBER Prospect Theory and Aggregated vs. Segregated Evaluations ies of type V twice ( five times) will be higher if the overall Table 4.2 Resultsforthe Lotteriesof Type V (Venture Lotteries) distribution of the sequence is explicitly mentioned. Type V1 Type V2 The lottery M was only examined for Portfolio Accept. Accept. Size 2. From our theoretical analysis of the two-lottery Lottery Rate (%) Lottery Rate (%) case we predicted the effect of increasing acceptance 030 2 200 DM 020 2 300 DM V1 = 43 V2 = 30 rate to be especially strong for the risk profile of the 070 −300 DM 080 −200 DM lottery M. V1 + V1 35 V2 + V2 36 0094 400 DM ∗∗∗ 004 4 600 DM Hypothesis 3: (Lottery ofType M, Portfolio 2V1 = 042 1 900 DM 60 2V2 = 032 2 100 DM 35 Size 2). The acceptance rate for playing the lottery 049 −600 DM 064 −400 DM ∗∗∗ ∗ M twice will be higher if the overall distribution of the V1 + V1 + V1 + V1 + V1 63 V2 + V2 + V2 + V2 + V2 59 5V 82∗∗∗ 5V 69∗ sequence is explicitly mentioned. 1 2 ∗∗∗Significant at the 0.10 level. Eighty-three students from the University of Mannheim took part in the computerized experiment. They were recruited through an experimental mailing and V2. Table 4.2 summarizes the results for the V list and majored in different disciplines. For various lotteries. combinations of lottery type and portfolio size they For the lottery V1 we found a strong effect in line had to state their willingness to play the (sequence with Hypothesis 2. While only 29 of 83 subjects were of) gambles. The questions were formulated as in willing to play the lottery V twice, the acceptance Study 1. Each portfolio in the set was presented in seg- 1 rate increased to 50 of 83 for an aggregated presenta- regated as well as in aggregated form to all subjects. tion of the portfolio distribution. This increase (35% Thus, within-subject information is available for all to 60%) is significant at the 1% level. For the port- type/size combinations. We presented the questions in folio consisting of five lotteries V , the effect was of four different orders to control for order effects. Identi- 1 similar strength. The number of subjects accepting the cal (but differently framed) questions never appeared portfolio increased from 52 to 68 (63% to 82%). This in close temporal vicinity. As in Study 1, there were no difference is significant at the 1% level. Hypothesis 2 monetary incentives given. is thus confirmed for V1. However, a similarly strong
Results of Study 2. Two types of lotteries V effect could not be found for the lottery V2. In case of with rather high loss-probabilities, but comparably the Portfolio Size 5, the increase from 59% to 69% is small losses, were used. The lottery V1 was defined just weakly significant at the 10% level. For the Port- as (0 7 −300 DM), the lottery V2 was given as (0 8; folio Size 2 the acceptance rate did not even move in −200 DM). The risk profiles of these two lotteries the hypothesized direction. look similar. Nevertheless, we included both types Comparing the acceptance rates for the distribu- in our study because we expected the difference in tions 2V1 and 2V2, we come to the conclusion that the loss probability to be important in the case of Port- observed difference between V1 and V2 is driven by folio Size 2. Note that for the aggregated distribution the fact that subjects pay special attention to the over-
2V1 the loss probability is lower than 50%, while it all probability of losses. For the lottery M, defined is above 50% for 2V2. Although this kind of effect as (0.30; −1200 DM), Hypothesis 3 is confirmed (see cannot be explained by Prospect Theory, it is known Table 4.3). Of 83 subjects, 38 accepted a single play of from the literature that people pay special attention to M, 46 were willing to play two independent draws, the overall probabilities of gains and losses and less and 56 stated their willingness to participate if the to the outcome sizes (cf. Lopes 1996). Because 50% portfolio distribution 2M was displayed. The increase is especially salient, we expected the attractiveness of the acceptance rate (55% to 67%) from segregated of the distributions 2V1 and 2V2 to be strongly influ- to aggregated presentation is significant at the 2% enced by the small difference in loss probability of V1 level.
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Table 4.3 Results for Lottery M the portfolio. It turned out that the simple lottery (0.5, Lottery Accept. Rate (%) $2,000; 0.5, −$500) used by Redelmeier and Tversky in their experimental design (and frequently used in 070 1 300 DM M = 46 similar form in the literature for related questions) is 030 −1 200 DM M + M 55 especially suited to result in a higher evaluation of 0492 600 DM ∗∗ the aggregated distribution. 2M = 042 100 DM 67 However, there also exist portfolios of mixed lotter- 009 −2 400 DM ies with a higher segregated evaluation. A portfolio of ∗∗Significant at the 0.05 level. lotteries with moderate to low probabilities for rather high losses should appear less attractive to decision 5. Conclusion makers if an aggregated evaluation is performed. This If individuals have to evaluate a portfolio or sequence type of lottery can be found in a bank loan setting. of lotteries, their judgment is influenced by the port- In case of a typical loan the investor has to bear a folio presentation mode. In an experimental study, high loss (compared to the possible interest gain) in Redelmeier and Tversky (1992) found a significantly the rather unlikely event of default. higher acceptance rate for a sequence of lotteries Because we could derive analytical results for only if the overall distribution was displayed instead of small portfolios of simple lotteries, we ensured the the set of lotteries itself. There seems to be an easy robustness of our findings by computer-supported explanation for this response pattern. A tendency for simulations. The full range of lottery types (low to a narrow framing (Kahneman and Lovallo 1993) of high loss probability and low to high loss size) in the decision problem causes individuals to neglect a given lottery space was examined and graphi- the portfolio context if the overall outcome distri- cally displayed for portfolios with up to 10 identical bution is not explicitly mentioned. Because of loss lotteries and for different parameter settings. The aversion, this type of mental accounting leads to a general findings of our formal analysis translate to lower overall evaluation of a portfolio of mixed lot- bigger portfolio sizes. While for the loan-type lot- teries as exemplarily demonstrated by Benartzi and teries segregated evaluation remains favored (even Thaler (1995). In the “segregated evaluation” of the for bigger portfolios), for most other lotteries with mental accounts (i.e., the isolated evaluation of each moderately positive expected values the aggregated lottery), the asymmetric impact of losses and gains evaluation is much higher. This holds in particular is fully taken into account, while some of the gains for the Redelmeier/Tversky type of lotteries (median and losses would cancel in an “aggregated evalu- ation” of the overall distribution. The difference in probability for moderate loss) and a probability/size aggregated and segregated evaluation of portfolios combination we called “venture lotteries.” These lot- has obvious implications for economic problems, par- teries offer a high probability of loss but extremly ticularly in the financial sector. The basic question is high gain chances in case of success (typical profile of whether a bank or an investment fund (e.g., venture venture capital investments). capital fund) can make its loan or investment portfo- The different theoretical predictions for loan, ven- lio look more attractive to customers by (not) provid- ture, and other lotteries were finally investigated in ing isolated details about the single risks within the two experimental studies. The experimental results portfolio. were in line with our theoretical predictions. It should In the theoretical part of our paper we extended be mentioned, however, that the strength of the the loss-aversion explanation and assumed the value effect seems to be very sensitive to small changes function to reflect diminishing sensitivity too. We of the lottery parameters. For two similar risk pro- formally analyzed the sign and the size of the evalu- files within the venture category we found the dif- ation difference dependent on the gain/loss probabil- ference in acceptance rates for the two presentation ities and the relative gain/loss sizes of the lotteries in modes to be surprisingly distinct. These patterns can
730 Management Science/Vol. 47, No. 5, May 2001 LANGER AND WEBER Prospect Theory and Aggregated vs. Segregated Evaluations
hardly be explained within a Prospect Theory frame- Further it holds for a value function vk : work. Obviously, simple heuristics (e.g., a maximum loss-probability rule) might be used by some subjects DL = v2 − 2v = k2 − 2 − and to decide about the acceptance of outcome distribu- D g = v2g− 2vg =−2 − 2g tions and lottery portfolios. These rules of thumb will G surely gain even more importance if the evaluation It follows: tasks get more complex. It is not clear how well a DL − 1 − p Prospect Theory–based explanation will predict indi- = k · = k · −DGg g p vidual decisions if more complicated (and not iden- tical) lotteries within larger portfolios are considered. and we can conclude: First experimental evidence for large portfolios was DL pGp provided by Benartzi and Thaler (1999). In their study, DLGp < 0 ⇔ pLpDL + pGgDGg < 0 ⇔ < −DGg pLp the effect of nonlinear value transformation is super- 1 − p 1 − p 2 ⇔ k · < ⇔ p · k1/2− < 1 − p imposed by a strong bias in probability judgment. p p Hence, they do not observe the dependence of the −1 1/2− evaluation difference on the risk profiles of the lotter- ⇔ p< 1 + k = p0 ies, which we identify for smaller portfolios. Further The monotonicity of p in k and in is obvious. experimental research is needed to be able to make 0 reliable predictions of individual behavior in more Result 2. For k>1 there exists a unique loss size ∈ complex real-world decision situations. −2500 −1250 s.t. We see another challenging area for further research in the experimental examination of evaluation dif- DM < 0 ⇔ ∈ −2500 decreases in k and in ferences for lotteries that do not allow a simple acceptance rate comparison. For these extremely Proof. We notate the gain part of the value function v by v⊕ and the loss part by v . The existence and uniqueness of can be (un)attractive lotteries the complex coding problems have to be analyzed, then an appropriate experimen- proven for all continuous value functions with the properties: (i) Loss aversion: v⊕x < −v−x for all x>0. tal method must be developed to provide a more gen- (ii) Concavity [convexity] of the continuously differentiable eral test for the empirical correctness of our theoretical function v⊕v. predictions. (iii) A sufficiently high v⊕x for x 0 (explicitly: v−2500< limx→0 v⊕x). Acknowledgments Note that these properties are given in particular for value func- tions v. From (iii) and the continuity of v in the gain and in This research was financially supported by the SFB 504 k “Rationalitätskonzepte, Entscheidungsverhalten und ökonomische the loss domain we derive the existence of some small g with: v −2500 + 2g
Appendix Defining = g − 2500, this proves the existence of some ∈ −2500 0 with DM < 0. From (i) and (ii), we further derive for Result 1. There exists a p0 ∈ 0 1, s.t. for all mixed lotteries p with EVp = 0: all ∈ −1250 0
DLGp < 0 ⇔ p Proof. For a mixed lottery p ∈ 2500 with EVp = 0we ≥−v⊕− − v > −v⊕− + v⊕− = 0 have p + 1 − pg = 0, thus: − 1 − p By the continuity of DM , this suffices to conclude the existence = g p of some ∈ −2500 −1250 with DM = 0. Management Science/Vol. 47, No. 5, May 2001 731 LANGER AND WEBER Prospect Theory and Aggregated vs. Segregated Evaluations To prove the uniqueness of , we show for all ∈ Proof. Let p ∈ 2500 with s = Sp > 0. Choose n0 > + 1/1− −2500 −1250 with DM = 0 2500 /s , and it follows for all n>n0 DM < 0on−2500 n · s>n · + 2500 For each with D = 0 and ∈ −2500 , we define M Now note that the total outcome of n draws of a lottery p g = + 2500 and g = + 2500. Because we have g ∈ 0 1, the g cannot exceed n · + 2500. Thus, the sure amount n · + 2500 convexity of v yields cannot have a lower evaluation than the aggregated distribution of g g a portfolio of n lotteries p , i.e., · v + g + 1 − · v g g A p ≤ vn· + 2500 = n · + 2500 g g n >v · + g + 1 − · = v + g g g With Snp = n · Sp = n · s the proposition is immediate. Subtracting v on both sides gives g · v + g − v >v + g− v g References Benartzi, S., R. H. Thaler. 1995. Myopic loss aversion and the equity Using the property DM = 0, i.e., v⊕g = v +g −v we finally conclude premium puzzle. Quart. J. Econom. 110 73–92. , . 1999. Risk aversion or myopia? Choices in repeated g g v g > · v g = · v + g − v gambles and retirement investments. Management Sci. 45 ⊕ g ⊕ g 364–381. >v + g− v >v + g− v Casey, J. 1994. Buyers’ pricing behavior for risky alternatives: Encoding processes and preference reversals. Management Sci. This proves D < 0. M 40 731–749. The proof of monotonicity of in k and in relies on the specific . 1995. 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Advances in prospect theory: Cumulative myopia and loss aversion on risk taking: An experimental test. representations of uncertainty. J. Risk and Uncertainty 5 Quart. J. Econom. 112 647–661. 297–323. Tversky, A., M. Bar Hillel. 1983. Risk: The long and the short. Wu, G., R. Gonzalez. 1996. Curvature of the probability weighting J. Experiment. Psych.: Human Learning, Memory and Cogn. 9 function. Management Sci. 42 1676–1690. 713–717. Accepted by Mako Bockholt; received July 13, 1999. This paper was with the authors 7 months for 2 revisions. Management Science/Vol. 47, No. 5, May 2001 733