Discussion Paper No. 720

AGING, PROBABILITY WEIGHTING,
AND REFERENCE POINT ADOPTION:
AN EXPERIMENTAL STUDY

Ayako Suzuki
andKoichi Kume

September 2008

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Aging, probability weighting, and reference point adoption: an experimental study ∗Ayako Suzuki † and Koichi Kume ‡Abstract
We examine generational differences in risk-taking behavior by means of a laboratory
experiment with monetary incentives. We estimate the parameterized models in the framework of
cumulative prospect theory and examine the risk aversion, probability weightings and reference
point adoption of elderly and young groups. The results of our experiment indicate that the elderlygroup is less sensitive to changes in probability and tends to underestimate large probabilities andoverestimate small probabilities more strongly than does the young group. Furthermore, we find thatthe elderly update their reference point after gains and tend to derive their utility from gains andlosses not from levels of wealth. In sum, we find that the elderly group’s behavior departs more fromthe traditional expected utility theory than does the young group’s behavior.

Keywords: age; cumulative prospect theory; risk aversion; probability weights; reference point.
JEL codes: C91; D81.

∗

This study was conducted while the first author belonged to the Institute of Social and Economic

Research, Osaka University. The study is supported by the Global Center of Excellence (COE)
Program. The authors are grateful to Shinsuke Ikeda, Fumio Ohtake, Yoshiro Tsutsui, and theparticipants at Summer Workshop on Economic Theory. The views expressed here are those of theauthors and do not necessarily reflect those of the Ministry of Economy, Trade, and Industry.
†

School of International Liberal Studies, Waseda University, Email: a-suzuki@aoni.waseda.jp‡

Ministry of Economy, Trade and Industry, Email: kume-koichi@meti.go.jp

1. Introduction
Since the seminal work of Kahneman and Tversky (1979), prospect theory has challenged thedominant expected utility paradigm to the extent that it has been argued that expected utility theory(EUT) should be replaced with prospect theory as the foundation of decision making under risk (e.g.,
Camerer, 1998). There has been a large body of experimental work that supports prospect theory,and economic parameters regarding risk aversion, loss aversion, and probability weighting have been
estimated using such experiments (e.g., Prelec, 1998; Gonzalez and Wu, 1999). Some studies, forexample, Fehr-Duda et al. (2006), have examined gender difference, whereas others, such as
Humphrey and Verschoor (2004), have examined cross-cultural differences in decision making
under risk within the framework of prospect theory. Furthermore, a few studies have tested andsupported prospect theory using real-life datasets (see, for example, Jullien and Salanie (2000),
which used data from the UK betting market for horse races).
Surprisingly, however, little work has been done on generational differences in decision making
under risk, presumably because experimental laboratories normally use students as subjects. 1 How
well older people make economic and risk-taking decisions is an important issue for social policy(see Kovalchik et al. (2004), p. 2) because wealth tends to accumulate over people’s lifetimes, andtherefore, a large portion of wealth is in the hands of older people. The role of this study is to shedlight on the differences in risk-taking behavior between elderly and young individuals.
1

An exception is Kovalchik et al. (2004). They examined the risk-taking behavior of two

populations—old and young—along with other types of decisions with a potential for age effects. In
their studies of risk-taking behavior, subjects were asked to select cards from one of two decks toearn cash. The cards were preorganized so that one deck (B) had the more advantageous composition.
They found that both populations gradually concentrated their choices on deck B. Overall, they
concluded that the older adults’ decision behavior was similar to that of the young adults. Ourcurrent study differs from their study in that we explicitly estimate the parameters of risk-taking
behavior and compare the estimated parameter values between the two populations.

2We obtain elderly and young individuals’ certainty equivalences for risky lotteries using anexperiment based on the procedure of Becker, DeGroot, and Marschak (1964), and we estimate theparameterized value functions and probability weighting functions for the two groups of subjects:
old and young. 2 In addition to risk aversion and probability weighting, we examine how subjects
update their reference points during repeated rounds in the experiment. Although the reference point
plays a prominent role in prospect theory, the manner in which it is updated through time as afunction of the outcomes of past decisions is unknown. The issue of reference point adoption is also
important in real-world situations, such as financial markets: it has been argued that the wellobserved disposition effect—the tendency to sell stocks that have gained value too soon and to keep
stocks that have lost value too long—can be explained by investors’ reference point adoption (see,for example, Kliger and Kudryavtsev (2008)). For each group, we estimate and compare the twoalternative models based on two different assumptions about the reference point adoption.
The results of our experiment indicate that the old group is risk loving, whereas the young one isrisk averse, although this difference disappears after controlling for gender. A more robust difference
was found in probability weighting. We find that the old tend to underestimate large probabilities
and overestimate small probabilities more than do the young. Furthermore, the model selection test
indicates that the old tend to update their reference point after gains, whereas the young do not havethis tendency. In sum, the elderly group’s behavior departs more significantly from the traditional
EUT than does the young group’s behavior.

2. The experiment
The participants were 31 elderly subjects from Silver Jinzai Center (an employment agency for
elderly people) in Osaka and 32 young employed subjects who were recruited through e-mails andfliers. In contrast to the subjects of most existing experimental studies, all our young subjects are2

We assume that individuals report their true willingness to accept at the BDM although it has been

an issue and resulted in a large body of work (see Karni and Safra (1989)).

3

currently employed. We believe that this composition—retired elderly people and young employedpeople—more realistically represents the population whose financial transactions affect society thanwould be the case if students were used as subjects. The subjects were promised a 5000-yen basepayment for participation. In addition, they were told that their gains from winning and selling
lotteries would be added to the 5000-yen participation fee to yield their final payments. Table 1
shows the characteristics of the two subgroups. It can be seen that the old group consists of a greater
proportion of males, less-educated individuals, and lower-income earners.
We used an experiment based on the Becker, DeGroot, and Marschak (1964) procedure. The
Becker–DeGroot–Marschak mechanism is widely used to elicit decision makers’ selling prices forlotteries. The same procedure was used with our participants in the two groups. Each subject takes
part in 20 rounds, following five practice rounds. At the beginning of each round, a subject is given aticket for lottery L , which yields 1000 points with probability p and zero points with probability
1 − p. In each round, the probability p is randomly assigned at the beginning of each round. Afterobserving the probability p , subjects are asked to announce the minimal price for which they arewilling to sell the lottery. After the announcement of the selling price, a random offer price isselected out of a [1, 1000] segment. If it exceeds the subject’s declared selling price for the lottery,he or she has to accept the offer price and receive the monetary payoff of the offer price. If the offer
price is below his or her selling price, the subject plays the lottery. If the lottery is won, the subject
receives 1000 points, whereas if the lottery is lost, the subject receives nothing. After 20 rounds areplayed, earned points are exchanged for yen at the rate of 1000 points for 250 yen.

3. The econometric models
We invoke the concepts of cumulative prospect theory (CPT) (Tversky and Kahneman, 1992)
and estimate the models based on this theory. The model nests the traditional EUT so that we cantest the two alternative theories of risk-taking behavior.

4In our experiment, a two-outcome lottery yields 1000 points with probability p and zero points
with probability 1 − p . Let ai denote the initial wealth of an individual at round i with a1 = 0 . The
variable ai is the accumulation of the past gains by selling or winning lotteries. Then, for our twooutcome lottery, the outcome when the lottery wins is ai + 1000 , whereas when the lottery loses, theoutcome is ai + 0 . One aspect of CPT is that individuals derive utility from gains and losses relative
to a reference point, not from total wealth or consumption. Now let ri denote the reference point and
let wtai be the declared selling price at round i . Then, the certainty equivalence CEi is defined by
the following equation:v(CE i ) = π 1v(ai + 1000 − ri ) + π 2 v(ai + 0 − ri ) ,(1)

where CEi = ai + wtai − ri . Decision weights are denoted by π j ( j = 1,2) , and v is the value function
defined on the points. The decision weights depend on the subject’s domain-specific probability
weighting function w( p ) . In our two-outcome case, the decision weights can be represented asfollows:

π 1 = w( p)π 2 = 1 − w( p).
We have to choose functional forms for v and w to make the model estimable. One of thefunctional forms most frequently used for the probability weighting function w is a one-parameterversion introduced by Quiggin (1982) as well as Tversky and Kahneman (1992). Here, we consider
the latter version of the probability weighting function: 3

3

Other studies such as Lattimore et al. (1992), Prelec (1998), and Gonzalez and Wu (1999) proposed

two-parameter functional forms that are more flexible and probably more appropriate to use.
However, we use a one-parameter form because the number of observations is very small when we5pγw( p) = γ
; γ > 0.
( p − (1 − p ) γ )1 / γ
For the value function, we assume that the valuation of outcomes is represented by the following
power functional (Tversky and Kahneman, 1992):v( x) =xα− λ (− x) β

x≥0
.
x<0

As noted in the following, we assume that an individual’s reference point is either zero for eachround ( ri = 0∀i ) or ai ( ri = ai ) for round i . Therefore, we will not have negative outcomes in ourlottery experiment, and the parameters λ and β are not identified. The parameter α is assumed to
be greater than zero, and the value function is either concave ( α < 1 ), convex ( α > 1 ), or linear( α = 1 ).
Now, we turn to the reference points. The extension of CPT models to dynamics remains acontroversial issue. In particular, the manner in which reference points are updated through time as afunction of the outcomes of past decisions is unknown. 4 A natural hypothesis is that the reference
point adapts in response to past outcomes, shifting upward following a gain and downward
following a loss. To reflect this hypothesis, we first assume that the reference point ri is updated
following gain in each round so that ri = ai . Then, equation (1) is rewritten as follows:v(CE i ) = π 1 v(1000) + π 2 (0),=w( p i )v(1000)

estimate the likelihood function for each individual below and, in such a case, it is preferable that the
number of parameters to be estimated be small.
4

However, recent research such as Arkes et al. (2006) has examined this issue and found theasymmetric adaptation of gains and losses.

6where CEi = ai + wtai − ai = wtai . Note that, under this assumption, the individual derives his utility
from gain from the lottery not from the outcome level. We refer to this model as Model 1. Next, weconsider the case in which a reference point is not adopted at all: ri = a1 = 0 for all i . In this case,equation (1) is rewritten as follows:

π 1v(ai + 1000) + π 2 (ai + 0)v(CEi ) =,= w( pi )v(ai + 1000) + (1 − w( pi ))v(ai )where CEi = ai + wtai − 0 = ai + wtai . We refer to this model as Model 2. Note that under thesefunctional forms of the value function and the probability weighting function, Model 2 nests the
EUT. Under this model, the individual is assumed to value the final asset level, and when γ = 1, there
is no probability weighting, as in EUT. Therefore, EUT is addressed as a special case of Model 2.
We estimate Model 1 and Model 2 and compare which model performs better for the of old andyoung. The two models are polar cases, in the sense that one assumes complete reference point
adoption whereas the other assumes no such adoption occurs. In reality, of course, individuals may
perform somewhere between these two polar cases. Therefore, in order to compare the two models,
we conduct the likelihood ratio test of Vuong (1989), which does not require us to assume that oneof the two models is correctly specified. The Vuong test assumes that both of the competing
nonnested models are incorrect and compares their distance from the correct specification. Therefore,
we can obtain the information that reveals which model is closer to the true model.
Finally, our empirical models for Model 1 and Model 2 are specified as follows:
Model 1: wtai = v −1[ w( pi )v(1000)] + ε i ,Model 2: wtai + ai = v −1[ w( pi )v(ai + 1000) + (1 − w( pi ))v(ai )] + ε i ,where we assume that ε i ~ N (0, σ ) . We estimate parameters α , γ , σ by maximum likelihood
estimation.

7

4. The results
First, we estimate Model 1 and Model 2 using the entire sample, and the two groups: old andyoung. Table 2 shows the results. It can be seen that the estimated values are very similar between
the two models (for the same sample). The results of estimated α suggest that, on average, elderly
individuals are risk loving whereas young individuals are risk averse. The value of γ for the oldgroup is estimated to be significantly lower than that for the young group, implying that elderly
individuals are less sensitive to the change in probability.

Figure 1 presents the probability

weighting functions for the old and young groups using the estimated values of γ . They imply thatthe old tend to overestimate small probabilities and underestimate large probabilities more strongly
than do the young.
Table 3 shows the results from the Vuong tests. The results show that Model 1 is preferred for theestimations using the entire sample and the old sample, at the 10% and 5% levels, respectively. It
seems that old individuals tend to update their reference points in each period. For the young sample,
neither of the alternative models is preferred. However, the likelihood ratio is higher for Model 2,
and hence, other model selection tests such as Akaike information criteria would choose Model 2 forthe young sample. Our selection tests indicate that the old and the young are different in their
methods of reference point adoption. Elderly individuals tend to update their reference points every
time after gains and, therefore, tend not to take their initial wealth a into account in their risk-taking
behavior. Their utility is derived from gains from the lotteries not from the outcome levels. Realizingthat Model 2 is identical to EUT when γ is one, our results, which indicate a lower γ and a better
fit of Model 1 for the old, imply that the behavior of the elderly departs more significantly fromtraditional EUT than does that of the young.
Because the characteristics of our two groups differ significantly, as shown in Table 1, the above
results indicating behavioral differences between the two groups may merely be a reflection of thedifferences in the each group’s characteristics. In order to control for these characteristics, we
conduct the same estimation for each individual and obtain the same parameter values for each8

individual. Then, we regress these parameters on their characteristics, including age, and determine
whether age has significant effects on the parameter values. We conduct a Vuong test for eachindividual and use the parameter values from the selected model for the regression. When the Vuong
test does not prefer either model, we use the one that has a higher likelihood.
Table 4 presents the results from estimations for 23 elderly individuals and 32 young
individuals. 5 We show the average and median values for each parameter. The results continue toindicate that elderly individuals have a higher α and a lower γ . Table 5 presents the results from theregressions on α and γ . Here, we use age, education, income, a male dummy, and a constant forcontrol variables. 6 The results show that once the other characteristics are controlled for, age does
not have a significant effect on α . However, age still has a significant effect on γ , and older
individuals have a lower γ , implying that older individuals tend to have more distorted probability
weighting functions. In addition, the results show that more highly educated individuals are morerisk averse, whereas male individuals are less risk averse at a significant level. Furthermore, we
found that more highly educated individuals have less-distorted probability weighting functions,which is intuitive. In addition, individuals with higher incomes have less-distorted probability
weighing functions.

5. Conclusion
Our experimental study demonstrates that generational differences in risk-taking behavior
crucially depend on probability weights and reference point adoption. We found that elderly
individuals are less sensitive to the probability change. In addition, we found that elderly individuals
5

There are some elderly individuals who obviously do not behave nonstrategically, and whoseparameter values are extremely unrealistic. For example, some individuals always hit the same
number for 20 rounds. We excluded these elderly samples from our further estimation.
6

The education variable in Table 1 is converted to the actual number of years of education. For theincome and age variables in Table 1, the median value in each category is used.

9

tend to update their reference points after gains and do not derive utility from total level of wealth.
That is, elderly individuals seem to depart more significantly from traditional EUT. As mostdeveloped countries face aging populations and the main wealth holders are elderly individuals, such
behavioral differences should be taken into account where social policy is concerned.
We have estimated only one specification of CPT. The next step should be to consider more
flexible functional forms, including weighting functions with two parameters and semiparametricforms. Furthermore, the current study cannot identify the degree of loss aversion. Although we do
not find any difference in risk aversion between the two generations, elderly individuals may exhibit
higher loss aversion, and such behavioral differences will have important policy implications.
Therefore, our next direction is to consider an experimental design that would enable us to determine
the degree of loss aversion.

References
Arkes, H., D. Hirshleifer, D. Jiang and S. Lim, 2006, Reference point adaptation: Tests in thedomain of security trading, mimeo, Ohio State University.
Becker, G.M., M.H. DeGroot and J. Marschak, 1964, Measuring utility by a single-responsesequential method, Behavioral Science 9, 226–232.
Camerer, C.F., 1989, An experimental test of several generalized utility theories, Journal of Risk andUncertainty 2, 61–104.
Fehr-Duda, H., M. De Gennaro and R. Schubert, 2006, Gender, financial risk, and probability
weights, Theory and Decision 60, 283–313.
Gonzales, R. and G. Wu, 1999, On the shape of the probability weighting function, Cognitive
Psychology 38, 129–166.
Jullien, B. and B. Salanie, 2000, Estimating preferences under risk: The case of racetrack bettors,
Journal of Political Economy 108, 503–530.
Humphrey S. H. and A. Verschoor, 2004, The probability weighting function: Experimental
evidence from Uganda, India, and Ethiopia, Economics Letters 84, 419–425.

10Kahneman, D. and A. Tversky, 1979, Prospect theory: An analysis of decision making under risk,
Econometrica 47, 263–292.
Karni, E. and Z. Safra, 1987, “Preference Reversal” and the observability of preferences byexperimental methods, Econometrica 55, 675-68.
Kliger, D. and A. Kudryavtsev, 2008, Reference point formation by market investors, Journal of
Banking and Finance, forthcoming.
Kovalchik, S., C. Camerer, D.M. Grether, C.R. Plott and J.M. Allman, 2005, Aging and decision
making: A comparison between neurologically healthy elderly and young individuals, Journalof Economic Behavior and Organization 58, 79–94.
Lattimore, P. K., J.K. Baker and A.D. Witte, 1992, The influence of probability on risky choice: A
parametric examination, Journal of Economic Behavior and Organization 3, 323–343.
Prelec, D., 1998, The probability weighting function, Econometrica 66, 497–527.
Quiggin, J., 1982, A theory of anticipated utility, Journal of Economic Behavior and Organization 3,
323–343.
Tversky, A. and D. Kahneman, 1992, Advances in prospect theory: Cumulative representation of
uncertainty, Journal of Risk and Uncertainty 5, 297-323.
Vuong, Q.H., 1989, Likelihood ratio tests for model selection and nonnested hypotheses,
Econometrica 57, 307–333.

11

Table 1 Characteristics of the entire sample, old and young subsamples (percentages)

Sex

Male
Female

All
63.49
36.51

Old
74.19
25.81

Young
53.13
46.88

20s
30s
40s
50s
60s
70s
Unknown

7.94
7.94
22.22
11.11
34.92
11.11
4.76

0.00
0.00
0.00
0.00
67.74
22.58
9.68

15.63
15.63
43.75
21.88
3.13
0.00
0.00

Elementary school – Junior high– High school– Polytechnic– Junior college– University (non science)– University (science)– Graduate school

3.17
30.16
1.59
9.52
42.86
9.52
3.17

6.45
58.06
0.00
6.45
19.35
9.68
0.00

0.00
3.13
3.13
12.5
65.63
9.38
6.25

No incomeLess than 1 million yen
1–2 million yen
2–4 million yen
4–6 million yen
6–8 million yen
8–10 million yen
10–12 million yen
12–14 million yen
More than 14 million yen
Unknown

12.7
3.17
11.11
19.05
11.11
1.59
12.7
4.76
3.17
1.59
19.05

19.35
3.23
9.68
25.81
16.13
0.00
0.00
0.00
0.00
0.00
25.81

6.25
3.13
12.5
12.5
6.25
3.13
25
9.38
6.25
3.13
12.5

Age

Education

Income12

Table 2 Estimation results for Models 1 and 2αγ

σModel 1
(full adoption of reference point)AllOld
1.0842***
1.3643***(0.1257)(0.0333)
0.5267***0.3922(0.0569)
239.4308***(45.8356)(0.0681)232.3844***(52.7820)

Young0.9520*(0.5483)
0.6766***

Model 2(no reference point adoption)
AllOld
1.0843***1.3642***(0.1257)(0.0333)
0.5268***0.3925***Young
0.9517*(0.5491)
0.6765***(0.1077)
229.4452***(52.7524)(0.0569)
239.4317***(45.8382)(0.1077)
229.4450***(52.7530)(0.0681)232.3892***(52.7849)

Note: The symbols ***, **, and * denote statistical significance at the 1%, 5%, and 10% levels,
respectively.

13

Table 3
All
Old
Young

Vuong test of Model 1 against Model 2
Vuong statistics Model 1 vs. Model 2
1.81*
2.50**–0.60

Note: The symbols ***, **, and * denote statisticalsignificance at the 1%, 5%, and 10% levels, respectively.

14

Table 4

Individual estimation results

α
γ

All
Mean
1.35730.6614σ135.7924

Median1.00990.6828Old
Mean
1.8646
0.5418112.6086

108.9852

15

Median1.11420.5167Young
Mean
1.00850.7437

Median0.97770.757689.5142154.2223

126.9430

Table 5 Regression results for α and γ

γ

α
Age
Education
IncomeMaleConstant

0.0002(0.0026)–0.0893***(0.0177)0.0001*(0.0001)
0.3737***(0.0703)2.1605***(0.3387)–0.0031***(0.0007)0.0422***(0.0050)0.0001***(0.0000)–0.0348*(0.0201)
0.1985***(0.0968)

Note: The symbols ***, **, and * denote statistical significance at the 1%, 5%, and 10% levels,
respectively.

16

Probability weighting function of young and old

1

0.9

0.8

0.7

w(p)

0.6

← weighting function of young

0.5

0.4

0.3
← weighting function of old
0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5
probability

0.6

0.7

0.8

0.9

Figure 1: Estimated probability weighting functions of the old and the young.

171