Prospect Theory: Where is the Reference Point?
Serge Blondel a
Abstract. In a new experiment, we compare two cases: a classical choice between two lotteries and
the possibility of substituting one lottery to another one. Although we found the classical preference
reversal phenomenon, there is no endowment effect in the case of selecting one lottery, even if
the subjects start with one of both options. The subjects seem to anticipate and integrate into
their reference point the potential wealth increase. By assuming that the reference point could
is a lottery in the case of the willingness to accept, third-generation prospect theory explains
the disparity between selling and buying prices and the preference reversal phenomenon.
However, indentifying the reference point in the case of a choice requires further investigation
because zero appears too low. Assuming that the reference point is a compounded lottery
(equal chance of having both lotteries) appears as a worthwhile precision of the reference
point in prospect theory, consistent with the preference reversal phenomenon.
Keywords BDM mechanism . Preference reversal . Prospect theory . Reference point
JEL Classification . C91 . D81
a
GRANEM - Université d’Angers - 13 Allée F. Mitterrand F-49036 Angers Cedex - France
e-mail: [email protected]
I am grateful to Alexis Belianin, Gemma Davis, Enrico Diecidue, Nikos Georgantzis, Raphaël Giraud, Louis Lévy-Garboua, Stéphane
Rossignol, Chris Starmer and Peter Wakker for helpful comments. I remain responsible for any errors.
1
1. Introduction
How can preferences be evaluated? At first sight, it is equivalent to collect choices or
certainty equivalents, although the latter give more information on the preferences of the
decision maker. For example, suppose that you are indifferent between $100 and 50% of
chances of winning $250. Hence, you will choose $120 over the lottery, but this choice only
indicates that the certainty equivalent is under $120. However, observed behaviors, whether in
the laboratory or in the field, are not always consistent with this simple reasoning, and there
are some well-known systematic biases.
Over four decades, various methods of measurement of preferences have led to
anomalies. The preference reversal (PR) phenomenon appears when a significant proportion
of people choose P (a large probability of winning a small prize) over $ (a small chance of
winning a large prize), and affects a higher willingness-to-accept (WTA) for $ (Liechtenstein
and Slovic 1971; Lindman 1971). Moreover, there is a significant disparity between
willingness-to-pay (WTP) and WTA (Knetsch and Sinden 1984): WTA is significantly higher
than WTP, and this difference can not be explained by an income effect.
How can these anomalies be explained? The WTA-WTP disparity is seen as a
consequence of an existing reference point (RP) combined with loss aversion. A RP is defined
as the status quo or the initial wealth, and loss aversion assumes that avoiding a loss is much
more important to people than experiencing a gain, or that “losses (…) loom larger than
corresponding gains” (Tversky and Kahneman 1991, page 1039). With the referencedependent model (Tversky and Kahneman, 1991), the WTA-WTP disparity is easy to
understand. When you sell your house, your RP is the following: you will win the selling
price and lose your house. If you are subject to loss aversion, you will ask a higher price for
2
compensating this loss. In contrast, when you buy a house, your RP is the sum of money that
you will lose, whereas the house will be a gain. In this case, loss aversion will lead to a low
buying price. Finally, the endowment effect (Thaler 1980) is explained by the referencedependent model. This model is a revolution in microeconomics because it predicts that the
indifference curves will be secants. The experimental test of Bateman et al. (1997) has
validated the reference-dependent model. More generally, the “Black swan” theory of Taleb
(2007) argues that the RP is highly dependent from an unexpected event, i.e. a surprise.
The WTA-WTP disparity is explained but is the PR the consequence of the same
behavior? Third-generation prospect theory (PT3), proposed by Schmidt et al. (2008), extends
prospect theory (Kahneman and Tversky 1979; Tversky and Kahneman 1992) by allowing
RP’s to be uncertain. This hypothesis is parsimonious since it requires no extra parameter in
relation to the previous version of prospect theory. Hence, PT3 can predict the PR and
therefore the WTA-WTP disparity.
PT3 includes some precise predictions that will be tested in a new experiment. We will
also consider the model of Köszegi and Rabin (2006), where the RP is endogeneized. We
obtain the classical PR, but also a case where the subjects already possessed an option and
were asked to choose between keeping it and substituting it for an alternative option. Under
expected utility (EU), this is exactly the same problem as choosing between two options, but
PT3 will predict a different behavior, because, just as in the case of a WTA, the RP is the
option already in the possession of the agent.
The purpose of this paper is to point out that the PT3 explanation of the PR
phenomenon raises crucial questions regarding the RP, with some new experimental results.
Hence, assuming that the RP is a compounded lottery (equal chance of having both lotteries)
appears as a worthwhile precision of the RP in prospect theory, consistent with PR.
3
In Section 1, we introduce the RP and prospect theory. The experimental design and
the hypotheses are developed in Section 2. Section 3 presents the results. The following
section develops a more realistic hypothesis on the RP. The last section concludes on the
findings.
2. The reference point and prospect theory
As underlined by Wakker (2010, page 292), the notion of RP, combined with loss aversion,
may be the strongest assumption of prospect theory. This central concept has been extensively
investigated over the last decades without its foundations being known. After a presentation
3
of PT (Schmidt et al. 2008), where we examine how this theory can explain PR or WTAWTP disparity, we discuss the hypotheses regarding the RP.
2.1. Third-generation prospect theory (Schmidt et al., 2008)
Schmidt et al. (2008) define RP as the endowment or the status quo, which could be
uncertain, as a lottery in an experiment. The potential options will be evaluated relatively to
this RP, which is neutral, and hence has no effect on the value function: it has a value equal to
0.
Schmidt et al. (2008) assume that the RP is:

0 for a choice between two lotteries L1 and L2 ;

0 for the WTP for a lottery;

the lottery itself, for the WTA.
Under PT3, the RP will not be the same for choosing as for pricing. For choosing, the
RP is the initial endowment, i.e. 0. Consider the classical version of PR (Tversky et al. 1990,
4
pages 208-209), where 83% of subjects have chosen H  4;.0.97  over L  16;0.31 1 while
74% have given a higher WTA to L. The choice of H over L implies:
w.97 v4  w.31v 16 
(1)
Suppose the value (v) and weighting (w) functions are the ones estimated by Tversky
and Kahneman (1992). We retain   2 as value of the parameter of loss aversion, and .6 for
the parameters indicating the curvatures of weighting and value functions:
wr  
r
(2)
r .6
.6

1
.6 .6
 1  r 
 y.6 if y  0
v y   
 2 y .6 if y  0
which leads to w.31  .322 , w.97   .833 , v4  2.297 and v16   5.278 . Hence, (1)
becomes 1.913  1.7 , which reflects a clear preference for H, consistent with 83% of subjects
choosing H: the choice of H is predicted by the strong concavity of v, despite the
underweighting of the probability .97.
Now, how is computed the WTA? Consider a lottery L   y; p  , with y  0 . For the
WTA of L, noted wta L , L is the RP. The certainty equivalence leads to
w p vwta L  y   1  w p v wta L   0  w p v  y  wta L   1  w p vwta L 
(3)
With the parameter values, the WTA for L is:
wta L 
(4)
y
 1  w p  
1  

 2w p  
1 / .6
Hence, for H and L, the WTA values are:
wta $  7.646
wta P  3.915
Concurring with classical PR, we find that wta $ is significantly greater than wta P .
1
In the following,  y; p  denotes a probability p of winning y, and 0 otherwise.
5
(5)
As noted by Schmidt et al. (2008), although the choice is independent from loss
aversion (   1 ), the WTA for $ increases with  . In fact, loss aversion is a sufficient
condition for PR, even if the value and the weighting functions are linear. Note that if the
weighting function is linear, PR is also explained since (1) becomes 2.228  1.636 , reflecting
a preference for P, and the WTA in (4) become wta $  7,572 and wta P  3,939 .
For the maximum buying price wtp L , 0 is the reference and the certainty equivalence
leads to
w p v  y  wtp L   1  w p v  wtp L   0  w p v  y  wtp L    1  w p v wptbL 
(6)
With the parameter values, wtp L is:
wtp L 
(7)
y
 21  w p  
1  

 w p  
1/ .6
Hence, wtp $ and wtp P are respectively 1.61 and 1.332. The disparity between WTA and
WTP is greater for $ than for P. Again, if the weighting function is linear, both WTP in (7)
become wtp $  2.935 and wtp P  3.767 . This reversal of PR is in line with previous findings
of Casey (1991, 1994).
2.2. Where is the RP?
Although this extension of prospect theory is fruitful, it raises a question: Where is the RP?
When you have to choose without paying between L1 and L2 , you are sure to possess either
L1 or L2 after the choice. It could be argued that, just before the choice, the RP was surely 0,
but the possibility of choosing for free must change the RP since the subject is richer than
before. The RP could be, for example, L1 or L2 . It could also be argued that there is a 50%
chance of playing L1 or L2 , since the choice has not yet been made. We summarize the
various cases in Table 1.
6
[Table 1 here]
The more troubling observation is the comparison of Cases 1 and 2 (or 1 and 3). Since
0 is the RP for both decisions, the decision maker must be indifferent between both situations.
However, it is clear that everybody would prefer to choose between L1 or L2 for free (Case
1), rather than to have to pay for a lottery. Amongst the three types of decision framing (Cases
1, 2-3 or 4-5), having to give a WTP (Cases 2-3) is clearly the worst situation for the decision
maker. The decisions of giving the WTA (Cases 4-5) or choosing (Case 1) are close because
in all cases, the lottery is free.
Lastly, an example will underline this point. Suppose that subjects are endowed with a
lottery, say L1 . If you ask the persons’ WTA, L1 will be their RP (Case 5), as assumed in PT3.
Now, you ask them if they want to substitute L2 for L1 (Case 6). Again, the RP will be L1 ,
and they will evaluate L2 from that. However, they will simply keep L1 if they prefer it, or
take L2 otherwise. Case 6 seems almost identical to a simple choice. Although Cases 1 and 6
are very close, the RP varies too much. Contrary to other situations, in Case 1, the RP is far
from the wealth obtained from the decision. There seems to be a large difference between the
RP assumed by PT3 and that of the reasonable RP.
PT3 clearly represents a useful evolution of PT, but it raises the question of the RP in
the case of choosing between lotteries. When prospect theory (versions 1 or 2) applies only to
choices between given lotteries, the RP 0, as proposed by Kahneman and Tversky (1979),
does not raise any questions. Under PT3, the generalization to WTP and WTA decisions must
lead to a more cautious definition of the RP in the case of a choice. The explanation of the PR
is questionable since the various RP’s are not consistent.
In short, defining the RP is not well defined, as underlined by Wakker (2010).
Kahneman and Tversky (1979) have proposed intuition, but there is no formal model of it.
Kahneman (2011) has noted this point when he considers a lottery 1000000;0.9  and observe
7
that “winning nothing is the reference point and its value is zero. Do these statements
correspond to your experience? Of course not”. Recently, Köszegi and Rabin (2006)
developed a model where the RP is determined endogenously by the economic environment.
It could be a way of redefining the RP in the case where an agent has to choose between
options for free, but such a model does not exist yet.
Cases of choosing (1) or buying (2 or 3) are clearly different because the expectations
are not the same: in the first case, there will be a lottery for free, while in the second one, the
RP is clearly nothing or 0. In the case of selling prices mentioned in the introduction, the
expectations are not the same for sellers of their own houses, where the RP is their houses,
and professional sellers trading in the market place. The latter have no loss aversion when
they sell something.
In the model of Köszegi and Rabin (2006), an agent could have various personal states
of equilibrium, where an option F chosen in a set S represents equilibrium if this option F
exists also when F is the RP. With loss aversion, multiple personal states of equilibrium are
possible. The preferred personal equilibrium will be the one maximizing utility.
In our case, the predictions will be the same as PT3 for WTA and WTP but will differ
for the choice between H and L. We will note V L / R  the value of a lottery L when R is the
RP. It is obvious that V H / H   V  L / L   0 and the criterion of choice of H over L will be:
V H / H   V L / H   V L / L   V H / L 
 V  H / L   V L / H 
(8)
It should be noted that this model predicts the same decision for a choice (Case 1, Table1) or a
possibility of substitution (Case 6, Table 1). Since the RP is a lottery, the comparison of both
lotteries requires linking the lotteries with the same states of the world; the problem is then
presented as in Table 2:
[Table 2 here]
8
Hence, (8) becomes:
w 0.31v  12   w0.66v 6  w0.31v 12  w0.66v 6
(9)
(9) reduces to w0.66 v6   w0.31v12 . With the functions defined in (2), we obtain
1.469  1.428 . Hence, here, the choice is predicted to be consistent with the PR.
3. Experimental design
The experiments were run in December 2012 (session 1) and February 2013 (session 2), in a
large amphitheatre. A total of 79 subjects were recruited among students in economics. Each
subject received nothing (the experiment was included in the program of an “Experimental
economics” lecture) for his/her participation and earned €6.31 on average. After a training
session2, each subject answered the 10 decisions on paper: first 4 prices, and then 6 choices.
The total time needed was no more than 45 minutes.
One of the 10 decisions was randomly chosen and played for real. It uses the isolation
effect, namely that the subjects will consider each decision in isolation (Kahneman and
Tversky, 1979). When a lottery was played, the outcome was determined by a card (see
Figures 2 to 4). For a pricing task, the subjects drew one number among 10 numbers from 1 to
10 for determining the selling price, following the BDM procedure (Becker, DeGroot and
Marschak 1964). The random price was always in proportion of the prize of the lottery: for a
prize X, the ten possible random selling prices were 0.1X, 0.2 X, …, X. The numbers and cards
were drawn by each subject, for his/her own decision. Figure 1 (translated from French) was
presented to the subjects in order to summarize how the experiment was constructed.
[Figure 1here]
Three examples of lotteries were presented: 11;0.75 , 17;0.375 and 10;1 . The BDM mechanism was
explained in details, especially why this was in their interest to give their true price, with the same arguments
than in the experimental design of Grether and Plott (pages 637-38), showing that giving a price different from
the true life is a loosing or neutral strategy.
2
9
In each session, there were two groups; composed of 20 subjects (except one, with 19
subjects, in the session 1). The subjects were equally distributed between the left and the right
parts of the amphitheatre, according to the last two digits of their phone number 3, forming the
two groups. In Group 1, the choice was to substitute, or not, a second lottery to the owned
lottery. The latter was seen for 20 seconds before the second lottery that could be substituted
was shown. When all 20 subjects had responded (it was a quick decision, in practice no more
than 30-35 seconds after having seen the owned lottery), the following choice was shown. In
Group 2, the choice was simply between both lotteries appearing simultaneously. When all
subjects of the group had responded, which took no more than 40 seconds, the following
choice was shown. The decisions of the second group were made in a much shorter time span.
Both groups first gave together 4 prices. Just after, Group 1 responded to 6 choices, and then
Group 2 made 6 choices also. When one of the groups had made its choices in silence,
members of the other group turned their back to the screen and looked behind them.
An example of a price decision is given in Figure 2, and examples of decisions made
by Group 1 and 2 appear in Figures 3 and 4 (translated from French).
[Figures 2, 3 and 4 here]
According with PT3, subjects must be more attached to lottery H in Group 1 than in
Group 2. Since the subjects are endowed with lottery H (as “haute” in French, i.e. high4),
preferences must be consistent between prices and decisions in Group 1; hence, the rate of PR
must be lower in Group 1. In short, we test two hypotheses, H1 and H2:
H1. The rate of subjects keeping H (Group 1) is higher than the rate of choice of H (Group
2).
3
Left: 00-49; right: 50-99. Two subjects moved to the other part in order to have an equal number in each
session.
4
B is for “basse” in French, i.e. low.
10
H2. The rate of PR is higher in Group 2 than in Group 1.
4. Results
Table 3 shows the 10 decisions, and gives overall results.
[Table 3 here]
Statistical tests indicate that the prices (Decisions 1 to 4) are significantly greater in
the group 1 for two prices over four (at 5%). However, both groups have close preferences.
Furthermore, statistical tests conclude that the percentages of choices of B are not different
between groups. Hence H1 is not validated.
Result 1. Keeping a lottery or exchanging it for the other one is similar to a classical
choice between both lotteries.
It is worth noting that after the experiment, two reactions of the subjects were
interesting. When we showed them the difference between the framing of both decisions, they
considered that the problems were exactly the same. In other words, 20 seconds were not
enough for being attached to the lottery and considering it as a RP. This is not surprising, but
in this case, why should they have to be attached to the lottery, first seen also during 20
seconds, in the case of a selling price? The second interesting fact was observing their
reactions when they drew their own numbers and cards. In the case of a choice (Group 2), if
the RP was 0, they had to be indifferent in the case of winning nothing; it was clearly not the
case, and they were very disappointed if they won nothing.
The experiment included three tests of PR. The first one was a classical PR with two
lotteries, 6,0.75 and 9,0.5 , and then two prices and one choice (Decisions 3, 4, and 7). The
other two cases of PR were obtained with one lottery by comparing a price and a choice:
8,0.75 with Decisions 1 and 5, and 18,0.5 with Decisions 2 and 6. The observed rates of
11
PR, with a choice of the safer lottery and a higher price for the riskier one, are shown in Table
4.
[Table 4 here]
The average rate of PR, 31.7%, is in line with previous experiments. There is no
higher rate of PR in Group 2, except for one case over three (at 5%). Hence, H2 is not
validated.
Result 2. The rate of PR is no higher in the classical version with a choice than in the
substitution case.
More, the conditional rate of PR, i.e., amongst the subject choosing P, is computed in
the table 5. It is important because, for example, Group 2 has less chosen P in decision 5
(59.5% instead of 75.1%): hence the rate of PR could be due to the first condition of PR,
choosing P. The results of the table 4 are confirmed.
[Table 5 here]
5. PT3 with a new reference point
As we have seen, the RP equal to 0 when you have to choose is not “neutral” as defined by
Kahneman and Tversky (1979, page 274) because after the choice, the decision-maker will
have one of the options. At first sight, as done in prospect theory, it seems natural to use 0
since it separates gains and losses. However, following the principle of indifference (also
called principle of insufficient reason), in a binary choice, we must assign to each of the
possible options a probability of 1/2. Thus, we add to PT that the RP is the neutral position of
the decision-maker before deciding. The principle of indifference states that when the n
possibilities are indistinguishable, and this is the case before the decision, then each option
should be assigned a probability equal to 1/n. It leads to H3 in our case.
12
H3. Prior a binary decision, each lottery will be assigned a probability equal to 1/2. In
other words, the RP for a choice between L1 and L2 is L1 ,0.5; L2 ,0.5 , noted L1L2 .
Consider now the classical PR studied in the sub-section 2.1. Both WTA are the same but the
criterion of choice is modified. From the table 2, the problem is then presented as in Tables 6
and 7.
[Tables 6 and 7 here]
With HL as RP, the choice of H over L implies :
V H / HL   V L / HL 
(11)
w0.155v 12  w.33v 4   w0.155v12  w.33v 4 
(12)
Which leads with to:
 w.33v 4   w0.155v12
With the parameters of (2), (12) leads to 0.7581  0.688 . This is consistent with the PR
phenomenon.
6. Conclusion
This new experiment suggests that there is no endowment effect in a case of choosing
between lotteries, even if the subjects start with one of the options. The subjects seem to
anticipate and integrate into their reference point (RP) the potential wealth increase. By
assuming that the RP could be a lottery, third-generation prospect theory explains the
disparity between selling and buying prices and the preference reversal (PR) phenomenon.
However, indentifying the RP in the case of a choice requires further investigation because
zero appears too low. Assuming that the RP is a compounded lottery (equal chance of having
both lotteries) appears as a worthwhile precision of the RP in prospect theory, consistent with
PR.
13
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14
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15
Table 1. Framing of decision, reference point, wealth
Case Framing of decision
Reference point
PT2
PT3
Wealth
1
Choice between L1 and L2
0
0
L1 or L2
2
WTP of L1
?
0
L1 less the price of L1 , or 0
3
WTP of L2
?
0
L2 less the price of L2 , or 0
4
WTA of L1
?
L1
Price of L1 , or L1
5
WTA of L2
?
L2
Price of L2 or L2
6
Possibility of substituting L2 to L1
?
L1
L1 or L2
Table 2. Choice between H and L
Probability
0.31
0.66
0.03
H
4
4
0
L
16
0
0
16
Table 3. 10 decisions
H
Decision
B
probability Sum (€)
probability €
Group
1 ( N  40 )
2 ( N  39 )
substitution
choice
Average price of Ha
1
0.75
8
6.34 (0.98) ***
5.43 (.99)
2
0.5
18
10.18 (2.39)
10.2 (3.29)
3
0.75
6
4.85 (1.12) *
4.39 (1)
4
0.5
9
5.55 (1.33) **
4.88 (1.05)
% of choice of L
5
0.75
8
1
6
75.1
59.5
6
0.5
18
1
9
72.4
74.6
7
0.75
6
0.5
9
22.5
28.4
8
0.5
6
0.375
8
57.6
48.9
9
0.5
9
0.25
18
30
20.5
10
0.25
9
0.375
6
33.2
23
a
Standard deviation between brackets. The tests are two-tailed for prices and one-tailed for choices, under H1.
Significant at 1%
***
, 5%
**
*
and 10% .
Table 4. Rates of preference reversal
Rate of classical PR (P chosen and higher price of $), in %
Group 1 (N=39) substitution
Group 2 (N=40) choice
Decisions 3,4 and 7
40.7
52.4
Decisions 1 and 5
13.4
29.9 **
Decisions 2 and 6
31.1
39.5
**
The tests are one-tailed, under H2. Significant at 5% .
17
Table 5. Conditional rates of preference reversal
Conditional rate of PR (higher price of $ after a choice of P ), in %
Group 1 (N=39) substitution
Group 2 (N=40) choice
Decisions 3,4 and 7
53.3
69.0
Decisions 1 and 5
17.2
50 ***
Decisions 2 and 6
42.9
53.3
The tests are one-tailed. Significant at 1%
***
.
Table 6. Value of H with HL as RP
0.485
0.155
0.33
0.03
HL
4
16
0
0
H
4
4
4
0
0.155
0.33
0.155
0.36
HL
4
4
16
0
L
16
0
16
0
Table 7. Value of L with HL as RP
18
Figure 1. Organization of the experiment
Random
price €
Random number 1-10
yes
1
Random
price ≥
your price ?
2
?
3
no
0 to 18 €
4
5
6
7
Your decision :
H or B
?
8
9
0 to 18 €
10
Figure 2. An example of minimum selling price
Decision 2
You own the ticket H
Ticket
H
18
18
18
18
0
0
0
0
How much are you willing to accept for selling H ?
A number N will be drawn out of 1 to 10 and determines the price P.
1
2
3
4
5
6
7
8
9
1,8€
3,6€
5,4€
7,2€
9€
10,8€ 12,6€ 14,4€ 16,2€ 18€
 If your price is less than P, you will sell at the price P
 Otherwise, you keep the ticket and play it
19
10
Figure 3. An example of choice, in group 1
Decision 5
You own the ticket H
Ticket
H
8
8
8
8
8
8
0
0
Do you want to exchange H against the ticket B?
Ticket
B
6
6
6
6
6
6
6
6
Figure 4. An example of choice, in group 2
Decision 5
Which ticket will you choose, H or B?
Ticket
H
8
8
8
8
8
8
0
0
Ticket
B
6
6
6
6
6
6
6
6
20