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What Lies Beneath?
Pervasive Intransitive Tendencies Revealed by Strength of
Preference Measures
David Butler*
Andrea Isoni†
Graham Loomes‡
Kei Tsutsui§
10th June, 2010
*
University of Western Australia Business School; email: [email protected].
†
Department of Economics, University of Warwick (UK); email: [email protected].
‡
Department of Economics, University of Warwick (UK); email: [email protected].
§
Centre for Behavioural and Experimental Social Science, University of East Anglia (UK); email:
[email protected].
The financial support of the UK Economic and Social Research Council (grant no. RES-051-27-0248) is
gratefully acknowledged. We thank the Centre for Behavioural and Experimental Social Science at the
University of East Anglia for the resources and facilities used to carry out the experiments reported here.
PRELIMINARY VERSION
PLEASE DO NOT CITE OR QUOTE WITHOUT PERMISSION OF THE AUTHORS
What Lies Beneath?
Pervasive Intransitive Tendencies Revealed by Strength of
Preference Measures
Abstract
Seemingly systematic violations of transitivity have been found in experimental data. When
these violations take the form of choice cycles, they are sometimes attributed to regret,
sometimes to similarity effects. However, those cycles that are visible may only be the tip of
the iceberg. In this paper, we use self-reported strength of preference judgments to study the
underlying intransitive tendencies which may not be visible in choice data. We find that the
potential cycles revealed by these judgments are pervasive and systematic, suggesting that
pair-wise choices provide only limited – and potentially, misleading – information about
preferences. [95 words]
Keywords: Strength of preference, preference reversals, repeated choice
JEL classifications: C91, D81
1
Most economic decision theories assume that preferences are transitive, and
transitivity is often regarded as an essential element of rationality. However, experimental
research in psychology and economics has shown that there are many circumstances, not least
those involving risky prospects, in which people appear not to comply with this assumption
(see the survey provided by Rieskamp et al., 2006).
One of the most robust challenges to standard models is the so-called ‘preference
reversal phenomenon’ (e.g. Lichtenstein and Slovic, 1971; Grether and Plott, 1979; Seidl,
2002). This phenomenon occurs when individuals place a higher certainty equivalent value
on prospect X than prospect Y, but prefer Y to X in a straight choice between the two. Some
have attributed this behaviour to intransitive tendencies in people’s preferences of the kind
implied by regret theory (Bell, 1982; Fishburn, 1982; Loomes and Sugden, 1982, 1987) and
cycles of choices consistent with this explanation – ‘regret’ cycles – have been found to
outnumber cycles in the opposite direction in settings typical of preference reversal
experiments (e.g. Loomes et al., 1991; Humphrey, 2001).
However, under different circumstances, it is the opposite cycles which predominate.
Such cycles are consistent with the possibility that, as two prospects become more similar on
the probability dimension, the payoff dimension receives increasing weight (see Tversky,
1969; Lindman and Lyons, 1978; Budescu and Weiss, 1987; Bateman et al., 2006). Hence we
shall refer to cycles in this direction as ‘similarity’ cycles.1
In fact, the same individuals seem liable to violate transitivity in the direction of
regret cycles in some cases and in the direction of similarity cycles in others (Day and
Loomes, 2010). In particular, regret cycles prevail when probabilities are ‘scaled-up’, so that
in any given pairwise choice the prospects have relatively dissimilar probabilities, while
similarity cycles prevail when probabilities are ‘scaled-down’, which makes the two
prospects look more similar on that dimension.
However, many of the studies that have investigated regret cycles have generally
found them to account for only a rather small minority of all observations – typically between
5% and 15% – and to occur substantially less often than ‘standard’ preference reversals
(which, not infrequently, constitute the modal pattern). One possible explanation for this is
that certainty equivalent valuations are subject to influences/biases that are absent from
1
For extensions of the similarity analysis to a broader range of decision phenomena, see Rubinstein (1988),
Mellers and Biagini (1994), Leland (1994, 1998), Loomes (2010).
2
straight choices and that result in systematic overvaluation of some prospects and
undervaluation of others (see Tversky, Slovic and Kahneman, 1990, for an example of such
an explanation). But another possible explanation is that an experimental design which
involves fixing the parameters of the pairwise choices in advance and presenting the same
sets of these choices to samples of heterogeneous individuals may simply miss many
individuals’ critical values and thereby greatly understate the degree of underlying
preferences with the potential to exhibit cycles. To use a metaphor, by looking at pre-set
pairwise choices we may just be observing the tip of the iceberg, while a more substantial
body of intransitive behaviour lies unobserved beneath the surface.
In this study, we probe more deeply to investigate the extent of any underlying
intransitive tendencies. We complement choice data with information about individuals’
strength of preference (SoP) judgments. Elsewhere (Butler et al., 2010), we have provided
extensive evidence that such SoP judgments may usefully supplement the limited information
provided by simple pair-wise choices.
In Section 1, we consider how SoP judgments may allow us to identify and classify
potential intransitivities. Our instrument for eliciting SoP is presented in Section 2. Sections
3 and 4 report two experiments which deploy this instrument and which thereby detect
systematic patterns of response indicative of intransitive propensities that are greatly
understated by pair-wise choices alone. In Section 5, we discuss some implications of our
main findings and offer some concluding remarks.
1. Strength of preference and violations of transitivity
In this paper, we focus on choices between prospects of the form G = (£x, p; £0, 1 – p): i.e.,
G offers payoff x with probability p and nothing otherwise. We denote the strength of
preference for prospect G over prospect H by SoP(G, H), and we suppose that G is strictly
preferred to / indifferent to / strictly less preferred than H according to whether SoP(G, H) is
greater than / equal to / less than 0. More compactly:

>
G ~ H  SoP(G, H) = 0

(1)
<
3
In deterministic models of choice, SoP(. , .) could be given a utility interpretation.
For instance, in transitive models such as expected utility theory or cumulative prospect
theory that assume choice options to be evaluated separately, SoP(. , .) could be interpreted as
the difference between the subjective values, SV, assigned to each of the options, that is,
SoP(G, H) = SV(G) – SV(H). In intransitive models, such as regret theory, SV(. , .) would
not just depend on the option to be evaluated, but also on the alternative.
If a stochastic specification were added to these deterministic ‘cores’, SoP(. , .) would
also be subject to stochastic variability. Because there are many different deterministic core
models in the literature and many possible variants of stochastic specification, we do not here
take any stand on which combination of core model and stochastic specification is the best
interpretation. Rather, we will structure our discussion around a set of possible relations
between the SoP reported for the three pairwise combinations of some triple of prospects {A,
B, C}. Let those three prospects be defined such that SoP(A, B) > 0 and SoP(B, C) > 0. Then
we can identify five possibilities.
Strictly Transitive SoP (STS): SoP(A, C) ≥ SoP(A, B) + SoP(B, C).2
Strongly Transitive SoP (SGS): SoP(A, B) + SoP(B, C) > SoP(A, C) ≥ max[SoP(A, B),
SoP(B, C)].
Moderately Transitive SoP (MS): max[SoP(A, B), SoP(B, C)] > SoP(A, C) ≥ min[SoP(A,
B), SoP(B, C)].
Weakly Transitive SoP (WS): min[SoP(A, B), SoP(B, C)] > SoP(A, C) ≥ 0.
Intransitive SoP (IS): SoP(A, C) < 0. In this case, we observe a choice cycle where A  B,
B  C and C  A.
Notice that we cannot distinguish between the first four classifications simply by
observing actual choices: in all cases, since SoP(A, C) > 0, A  C, so that the set of three
pairwise choices appear to be consistent with the ordering A  B  C. However, what WS,
MS and SGS represent is the potential for a cycle, were the parameters of at least one of the
prospects to be altered.
To see this, consider first an example which satisfies WS: SoP(A, B) > SoP(B, C) >
SoP(A, C) ≥ 0. Now progressively improve C – perhaps by adding some positive amount of
2
Note that most transitive ‘core’ choice models imply that this condition holds as an equality. The weak
inequality sign is more in the spirit of our inherently empirical analysis.
4
money m to one or more of C’s payoffs – while holding A and B constant. Denote this
enhanced prospect by C*. As m progressively increases, both SoP(B, C*) and SoP(A, C*)
fall. But so long as SoP(B, C*) > SoP(A, C*), it will be the case that some value of m will
produce SoP(A, C*) = 0 while SoP(B, C*) is still strictly positive. A further marginal
increase in m has the potential to leave SoP(B, C*) positive but make SoP(A, C*) negative.
Once this occurs, we have a cycle: A  B, B  C*, C*  A. Thus someone who satisfies
WS is someone who does not visibly display intransitivity when presented with prospects A,
B and C, but would (or at least, more probably would3) have displayed a cycle if the
experimenter had pre-set the parameters of the prospects so as to produce A, B and C*.
How are things different if the individual’s SoP satisfies MS rather than WS – for
example, if SoP(A, B) > SoP(A, C) ≥ SoP(B, C) > 0? Progressively improving C, as in the
previous case, will result in the reversal of preference between B and C* while SoP(A, C*)
remains positive, so that the new set of three pairwise choices would now be consistent with a
different ordering, A  C*  B. by A*. Since SoP(A, B) > SoP(A, C), there is the potential
for SoP(A*, C) < 0 while SoP(A*, B) > 0 (with SoP(B, C) remaining unaffected and
positive). Once again, this would produce a cycle: A*  B, B  C, C  A*.
Finally, consider the SGS case where SoP(A, C) ≥ SoP(A, B) > SoP(B, C) > 0, and
where SoP(A, B) + SoP(B, C) > SoP(A, C). Progressively degrading A and also
progressively improving C can, in principle, be done in such a way that both SoP(A*, B) and
SoP(B, C*) remain positive but the joint effect is sufficient to produce SoP(A*, C*) < 0,
resulting in the cycle A*  B, B  C*, C*  A*.
In short, WS, MS and SGS all indicate that the individual’s preferences have the
potential to produce cycles if the parameters of the prospects happen to be set in the ‘critical
range’. Only in cases of strictly transitive SoP, where SoP(A, C) ≥ SoP(A, B) + SoP(B, C), is
it impossible to manipulate the parameters of A and/or C to produce a cycle. Thus, while IS
represents the above-surface part of the iceberg, WS, MS and SGS may together constitute a
substantial body of potentially intransitive behaviour that lies below the surface. This applies
to any experiment involving pairwise choices between prospects with pre-set parameters that
are presented to samples of individuals whose ‘critical ranges’ are heterogeneous. By using
individuals’ self-reported SoP, we seek to take soundings of the main body of the iceberg and
3
In a deterministic model we can say ‘would’; in a stochastic model we have to qualify this by ‘more probably’.
5
get some measure of the possible extent of potential cycles in addition to those that are more
directly visible.
The examples above that we used to illustrate WS, MS and SGS all involved the cycle
A  B, B  C, C  A. But of course cyclical preferences, whether visible or potential, over
pairwise combinations of A, B and C could also take the form C  B, B  A, A  C. As
mentioned in the Introduction, the two directions of cyclical choice have been classified in
the literature as ‘regret’ cycles and ‘similarity’ cycles.
We illustrate what these entail with the aid of Figure 1, which reports the typical
structure of our triples of prospects. In each triple, we have three prospects, a Safe prospect,
S = (£c, p1 + p2 + p3; £0, 1-p1-p2-p3), a Medium prospect, M = (£b, p1 + p2; £0, 1 – p1 – p2),
and a Risky prospect, R = (£a, p1; £0, 1 – p1), with a ≥ b > c > 0.4 In some cases the safe
prospect is a certain amount, which obtains if p1 + p2 + p3 = 1.
[Figure 1 about here]
For each {S, M, R} triple, SoP judgments are reported for three straight choices, from pairs
{S, M}, {M, R} and {S, R} (not necessarily presented in this order), which can give rise to
the eight different combinations presented in Table 1.
[Table 1 about here]
Two of these combinations satisfy IS, and therefore result in visible cycles. The remaining
six appear to be compatible with transitive orderings of S, M and R. Of the two visible cycles,
the first is the regret cycle. In the regret cycle, the safer option is chosen in the choices
involving the two pairs in which the two prospects are ‘adjacent’ to each other (i.e. {S, M}
and {M, R}), while the riskier option is chosen in the choice involving the more ‘distant’
prospects ({S, R}), giving rise to a safe-safe-risky sequence over the three choices (s-s-r in
4
In most choices, a > b. In just two cases a = b.
6
short).5 The visible cycle shown in the last row of Table 1 is the similarity cycle. In the
similarity cycle, the riskier option is chosen in the choices involving the two pairs in which
the two prospects are more similar to each other ({S, M} and {M, R}), while the safer option
is chosen in the choice involving the more dissimilar prospects ({S, R}), giving rise to a
risky-risky-safe sequence over the three choices (r-r-s in short).6
We now turn to potential cycles. These arise when SoP responses satisfy any of WS,
MS or SGS, so that the SoP judgment relating to the pairing of the prospects at each end of
any of the six apparently transitive orderings in Table 1 is not sufficiently big compared with
one or other or both of the SoP judgments for the other two pairings. For example, consider
the second ordering in Table 1, M  R  S. Given SoP(M, R) and SoP(R, S), that is, the
SoP judgments for adjacent prospects in the ordering, there is the potential for a cycle
whenever SoP(M, S) is such that improving S or degrading M (or both) could reverse the
preference between M and S without changing the preference between M and R and between
R and S. This is indicated by the (  M) extension to the ordering in Table 1 where this
notation stands for ‘M is not sufficiently strongly preferred to S’. Note that in this case, the
safer option is chosen in pair {M, R}, while the riskier is chosen in pair {S, R}. Although the
riskier option is chosen between M and S, the SoP judgments signify that the risky prospect is
not sufficiently strongly preferred, as indicated by the r(s) in the fourth column of Table 1.
That is, the SoP cycle deriving from M not being sufficiently strongly preferred to S turns the
observed r-s-r sequence into an s-s-r sequence. This allows us to classify the resulting
5
As we said in the Introduction, we are not primarily interested in trying to discriminate between any particular
‘core’ theories. We use the terms ‘regret cycle’ and ‘similarity’ cycle simply to conform with the previous
literature. Under the usual assumptions of regret theory (see Loomes and Sugden, 1987), the regret associated
with getting £0 instead of £c may outweigh the rejoicing associated with getting £b instead of £c, thus leading to
 M. Similarly, the regret associated with getting £0 instead of £b may outweigh the rejoicing associated
with getting £a instead of £b, thus leading to M  R. However, because of the bigger difference between a and
S
c and the convexity of the net advantage function, the rejoicing associated with getting £a instead of £c may
outweigh the regret associated with getting £0 instead of £c, thus allowing for R
6
 S.
These have come to be known as ‘similarity cycles’ because they typically arise in situations in which the first
two choices contain options that are very similar in one dimension, usually probability. Because of this, in the
first two choices it is hypothesised (see Tversky, 1969) that subjects underweight the probability difference and
choose the prospect offering the higher payoff, which results in M
 S and R  M. However, in the third
choice, in which the probability difference between the two prospects is larger and more salient, they choose the
safer option, giving S
 R.
7
potential cycle as a regret cycle. Applying the same reasoning to each of the six transitive
orders of Table 1, we can identify whether the potential cycle arising when the SoP judgment
for the furthest prospects in the ordering is not sufficiently big is of the regret type or of the
similarity type. This is indicated by the last column of Table 1.
In the experiments described below, we shall report not only the frequencies of visible
cycles satisfying IS, but also the frequencies of potential cycles where SoP judgments satisfy
WS or MS. As defined earlier, IS, WS and MS are mutually exclusive categories, but when it
comes to reporting the data it may be helpful to present cumulative frequencies at three
levels, as follows: 7
F0: SoP(A, B) > 0, SoP(B, C) > 0 but SoP(A, C) < 0. These cases satisfy only IS and
constitute visible cycles.
F1: SoP(A, B) > 0, SoP(B, C) > 0 but SoP(A, C) < min[SoP(A, B), SoP(B, C)]. This category
includes both IS and WS.
F2: SoP(A, B) > 0, SoP(B, C) > 0 but SoP(A, C) < max[SoP(A, B), SoP(B, C)]. This category
includes IS, WS and MS.
2. Eliciting SoP judgments
In the two experiments reported in this paper SoP judgments were elicited using the same
procedures. Each task involved two prospects, indicated by the neutral labels A and B, with
each prospect offering a positive amount with some probability and nothing otherwise. The
prospects were shown as in Figure 2.
[Figure 2 about here]
In Figure 2, prospect A offers a 25% chance of winning £40 and a 75% chance of winning
nothing. The probabilities of winning are reported underneath the money amounts, while the
numbers above the prizes (1–25 and 26–100) refer to the numbers reported on a set of 100
7
In principle, a further category of potential cycles could be defined to include IS, WS, MS and SGS, such that
SoP(A, B) > 0, SoP(B, C) > 0 but SoP(A, C) < SoP(A, B) + SoP(B, C). However, for reasons we explain later,
these cycles are not considered in our empirical analysis and we therefore omit this further category here.
8
discs which were used to play out the prospects. Similarly, prospect B offers a 20% chance
of £60, nothing otherwise. The upper prospect was always labelled A and the lower prospect
was always labelled B, but the assignment of parameters to A and B was counterbalanced
across subjects. So for every occasion when this pair was presented as in Figure 2, there was
another occasion where the 20% chance of £60 was displayed as the upper (A) prospect while
the 25% chance of £40 was displayed as the lower (B) prospect.
In each task, participants were asked to choose either prospect A or prospect B and in
so doing to indicate to what extent they preferred the prospect they chose by moving a slider
on a bar like that presented in Figure 3. They could indicate preference for prospect A by
moving a button along the bar to the left and preference for prospect B by moving it to the
right. The further they moved the button, the stronger their preference for the chosen option.
They were not allowed to report indifference. This SoP option was recorded on a 100 point
scale (i.e. the total length of the bar was 200 points). As soon as the button was moved to the
left, the text ‘You think A is slightly better’ appeared underneath the bar (if 0 < SoP(A, B) ≤
25). As the button was moved further to the left, the text changed into ‘You think A is better’
(25 < SoP(A, B) ≤ 50), ‘You think A is much better’ (50 < SoP(A, B) ≤ 75) and ‘You think A
is very much better’ (75 < SoP(A, B) ≤ 100).8 When the button was moved to the right,
analogous statements involving the preference for B were shown, and the corresponding
SoP(B, A) scores recorded.9
[Figure 3 about here]
Participants were told:10
‘If you feel that both alternatives are almost equally good so that you think the one
you are choosing is only SLIGHTLY better than the other one, just move the button a
8
As soon as the participant clicked on the button, but before this had been moved in any direction, the text
‘Please choose either A or B’ was displayed.
9
The text appearing underneath the bar could turn the 100 point scale into a 4 point scale if subjects used it as
an indication of the extent they have to move the bar to report their judgment. We take this possibility into
account in our data analysis.
10
See the Appendix for the full text of the instructions.
9
little way in the direction of your choice. However, if you think the one you are
choosing is VERY MUCH BETTER than the other one, move the button a long way
along the bar in the direction of your choice, possibly as far as the end if you feel very
strongly indeed.’
The SoP judgments themselves were not linked to incentives, although they were elicited as
part of making the choice which was directly incentive-compatible. That is, participants were
told that at the end of the experiment one of the questions would be selected at random, and
the prospect they chose in that randomly-selected question would be resolved by drawing one
of 100 numbered discs from a bag to determine their earnings. Their understanding of this
random device was tested with several questions during the delivery of the instructions.
3. Experiment 1
3.1 Design
Experiment 1 was designed to explore the extent of violations of transitivity in sets of
prospects which differ in terms of whether the probabilities of winning the positive amount
are scaled-up or scaled-down. This experiment was motivated by the evidence that the same
individuals commit (pure) regret cycles that outnumber similarity cycles in triples of scaledup prospects, and similarity cycles that outnumber regret cycles in triples of scaled-down
prospects (Day and Loomes, 2010).
In Experiment 1, we explore whether this pattern not only occurs for visible cycles,
but also for potential cycles. The sets of prospects we used are reported in Table 2.11
[Table 2 about here]
We used two different sets of prospects (Set I and Set II), each containing four scaled-up
prospects {Wu, Xu, Yu, Zu} and four scaled-down prospects {Wd, Xd, Yd, Zd}. These all
offer a positive amount with some probability and nothing otherwise, and become
increasingly risky moving down from W to Z. Each participant faced only one set of
11
There were other tasks in this experiment, all involving gambles of the same type, which were designed for
other purposes. Some of these are discussed in Loomes (2010).
10
prospects, and made a total of twelve choices, six involving the scaled-up prospects and six
involving the scaled-down prospects.12 In each {W, X, Y, Z} set, there are four triples which
fit the {S, M, R} structure described in Section 1, which will form the basis for our analysis.
3.2 Results
A total of 134 subjects recruited from the general student population of the [deleted for
anonymity] took part in the experiment. Of these, 68 made decisions involving Set I
prospects and 66 with Set II prospects. There was no show-up fee, and the average earning
was £10.23.
Table 3 reports visible (F0) and potential (F1 = IS+WS and F2 = IS+WS+MS) cycles
for scaled-up and scaled-down prospects pooling Set I and Set II.13 Potential cycles are
computed by first assigning preferences to the six transitive orders in Table 1, and then
looking at whether the absolute values of the SoP scores satisfy either WS or MS.14 Our SoP
measures were limited by the length of the SoP bar. This sets an upper bound on SoP(A, C)
which might in some cases constrain it artificially to be less than the sum of SoP(A, B) and
SoP(B, C). Because of this possibility, we restrict our attention to just WS and MS and we do
not consider the further category of potential cycles mentioned in footnote 7.
12
For each {W, X, Y, Z} set there were six choices involving pairs {W, X}, {W, Y}, {W, Z}, {X, Y}, {X, Z},
and{Y, Z}.
13
The patterns we highlight do not depend on this aggregation. The disaggregated data tell exactly the same
story and can be found in the Appendix.
14
Given the definitions of WS and MS, using the SoP score in the 100 point scale in the face of the possibility
that participants use the verbal descriptions as a guide to where to stop the button on the SoP bar is a
conservative method of counting potential cycles. In order to get the intuition for this, consider the following
hypothetical example. Suppose |SoP(A, B) |= 28 (i.e. the participant drags the button just into the second
category because she thinks A is better than B) and |SoP(B, C)|= 52 (i.e. the participant drags the button just into
the third category because she thinks that B is much better than C). Now suppose that |SoP(A, C)| = 27 (i.e. the
participant drags the button just into the second category because she thinks A is better than C). This would be
counted as an F1 under our procedure, because |SoP(A, C)| is less than the minimum of |SoP(A, B)| and |SoP(B,
C)|. If instead |SoP(A, C)|= 29, that is, the participant still thinks A is better than C, which would satisfy WS if
the data were compressed into just four SoP categories, our procedure would not count this as an F1 cycle. In
this respect, our procedure can only underestimate the numbers of potential cycles, but never overestimate them.
11
[Table 3 about here]
Table 3 shows that although visible cycles are not negligible, potential cycles are much more
common. For example, in the triple {Wu, Xu, Yu}, 15 subjects (11% of the whole sample)
exhibit visible F0 cycles; however, F1s and F2s account for 49 and 89 subjects (37% and
66% of the sample) respectively. The proportion of subjects exhibiting visible cycles in any
particular triple ranges from 11% to 28% over the scaled-up and scaled-down prospects,
while F1s account for between 37% and 51% of observations, with the corresponding figures
for F2s being 66% and 78%. These percentages show that actual choices reveal only a
relatively minor fraction of the substantial underlying intransitive tendencies.
Table 3 also shows that the asymmetries between regret and similarity cycles are
systematically related to whether the prospects are scaled up or scaled down. As reported by
Day and Loomes (2010), regret cycles prevail for scaled-up prospects, while similarity cycles
are prevalent for scaled-down prospects. For example, of the 15 visible cycles for the scaledup triple {Wu, Xu, Yu}, 11 are in the regret direction compared with 4 in the similarity
direction; whereas of the 23 visible cycles for the corresponding scaled-down triple {Wd, Xd,
Yd}, 8 are in the regret direction and 15 in the similarity direction. This reversal of the
direction of asymmetry is found in all four matched scaled-up and scaled-down triples.
This pattern is amplified for SoP cycles. For instance, for the {Wu, Xu, Yu} scaledup triple, 25% of the sample exhibit F1 responses in the regret direction, while 11% do so in
the similarity direction, while for F2 the figures are 46% and 21% respectively. In the
corresponding scaled-down prospects, {Wd, Xd, Yd}, regret and similarity F1s account for
15% and 31% of the subjects, while 29% and 49% fall into F2. For all matched triples,
potential cycles are pervasive and show a very consistent pattern: regret cycles outnumber
similarity cycles in the scaled-up prospects, while the opposite is true for scaled-down
prospects.
4. Experiment 2
Experiment 1 shows how SoP judgments may help us to explore the extent of some
underlying tendencies, only a minority of which are visible when we rely on simple choice
data alone. However, we acknowledge that such judgments are still somewhat untried and
untested. So when designing Experiment 2, one of our objectives was to conduct some checks
for reliability and sensitivity: that is, we wanted to check whether SoP judgments are
12
reasonably stable when the same task is repeated, and whether they move in the predicted
direction when the parameters of the prospects are varied systematically. To make the test
particularly demanding, our design involved exposing these measures to between-sample
comparisons of various kinds.15
At the same time, we aimed to use the SoP instrument to explore other aspects of
intransitive tendencies, with particular reference to the standard preference reversal
phenomenon involving certainty equivalents and to the less well-documented and opposite
asymmetry involving probability equivalents (see Butler and Loomes, 2007).
4.1 Design
In Experiment 2, we elicited choices and SoP judgments for the prospects presented in Table
4. In keeping with many previous preference reversal studies, we constructed one prospect
which offered a relatively small probability of a relatively high payoff (and which has
therefore come to be known as a $-bet) and another prospect which offered a more modest
payoff with a substantially larger probability (and which has therefore come to be known as a
P-bet). In our case, $ = (£40, 0.25; £0, 0.75), while we had two P-bets: P1 = (£10, 0.9; £0,
0.1) which was presented to one half of the sample; and P2 = (£10, 0.65; £0, 0.35) which was
presented to the other half.
All subjects made a total of 52 choices involving these prospects16 paired either with
each other or with other prospects listed in Table 4.
[Table 4 about here]
The tasks of Experiment 2 were of three types:
i) Choices between the $-bet and a P-bet, either P1 or P2, which were repeated four
times.17 The use of two different P-bets allows us to check the responsiveness of our
instrument to changes in the parameters of the prospects between samples.
15
Further checks on the reliability and sensitivity of our SoP measures are reported in Butler et al. (2010).
16
There were a total of 100 choices in this experiment. The remaining 48 are reported in Butler et al. (2010).
13
ii) Choices between the $-bet and the P-bet on the one hand and different levels of
certainty (£10–£4) on the other, each repeated three times. These tasks are interesting
for various reasons. First, they allow us to conduct within-subject tests of whether our
SoP measures respond sensibly to unambiguous changes in the choice problems.
Second, when the certainty X is varied in {$, P, X} triples like those used by Tversky,
Slovic and Kahneman (1990), we can check whether there are systematic changes in
the patterns of cyclical choices. Third, since the sequence of tasks {$, £10}, {$, £8},
{$, £6}, {$, £4} and {P, £10}, {P, £8}, {P, £6} {P, £4} can be regarded as a
translation into a series of choices of the valuation task in a preference reversal
experiment,18 we can check whether the spread of SoP judgments is different for the
$-bet and the P-bet as previous findings (McCrimmon and Smith, 1986; Butler and
Loomes, 2007) suggest.
iii) Choices between the $-bet and the P-bet on the one hand and prospects involving
different chances of winning £60 on the other, also repeated three times. These tasks
allow for further within-subject comparative statics tests. For reasons analogous to
those just described in relation to certainties, these tasks, when considered in
sequence, are reminiscent of a probability equivalence task. Our tasks involving the
R0.25–R0.10 prospects allow us to check whether the prevalence of standard reversals
when choices are compared with certainty equivalents and the prevalence of counter
reversals when the same choices are compared with probability equivalents is also
reflected in systematically different patterns of visible and potential cycles.
4.2 Results: Reliability of SoP measures
We recruited a total of 138 subjects, drawn from the general student population of the
[deleted for anonymity]. The sample was split into two equally-sized groups, which differed
only in terms of the P-bet that was used. Half of the subjects made choices involving P1, and
half made choices involving P2. In all other respects, the tasks were the same for the two
17
In both experiments, the order of the two gambles – i.e. whether they were shown as A or B – was varied
when the same task was repeated. We included several repetitions of all tasks in order to check whether the
patterns of SoP judgments changed over time. As we explain in Section 3, there were no systematic trends.
18
Notice, however, that these choices were not presented in sequence, but in random order and separated by
other tasks.
14
groups. The experiment took less than an hour to complete, and the average earning was
£10.25.
Before turning to visible and potential cycles, we consider the reliability of our SoP
data, summaries of which are reported in the Appendix. Figure 4 shows the distribution of
SoP scores in all tasks, excluding those involving dominance, that is, {£10, P1}, {£10, P2},
{$, R0.25}.19 The scores are recorded so that negative numbers reflect preference for the
safer prospect. Because of the tendency of experimental subjects to behave as if risk averse,
the distribution is slightly skewed towards safer choices.20
[Figure 4 about here]
Figure 4 highlights two key aspects of the data. First, subjects do use the SoP bar to a
significant extent. Critics who worry about the non-incentive compatible nature of these
judgments would probably concede that if subjects were only concerned about the reward
they could get from taking part in the experiment they would have no reason to move the bar
more than is necessary to choose their preferred option. The extensive use of the whole
spectrum is a clear indication that subjects tried to give a fair indication of how strongly they
preferred one prospect over the other.
The second key aspect, also evident in Figure 4, is that when making their SoP
judgments subjects were, at least in part, guided by the verbal descriptions that appeared over
the bar. This explains the spikes in the proximity of the score values where the verbal
description changed, i.e. ± 25, ± 50, ± 75 and ± 100. When analyising the data, we will take
this into account when necessary by recoding the SoP scores into eight SoP categories (–4,
..., –1, 1, ..., 4).
19
In tasks involving dominance, some subjects reported strong preference in favour of the dominating gamble,
possibly to signify that they were quite sure about their choice. In order to avoid this confounding
interpretation, these tasks were not included in the histogram. Their inclusion would have resulted in much
more pronounced spikes at the two ends of the distribution.
20
In the experiment, the safe option was sometimes shown as A and sometimes as B. When the same histogram
is drawn using the preference for A and B as reported by the subjects, the picture, which is omitted due to space
constraints, looks remarkably symmetrical.
15
The next possible consistency check on our data involves considering whether or not
the distributions of SoP judgments change systematically when the same tasks are repeated.
Since the actual raw score was not shown on the screen, it would be excessive to expect that
these scores would not change at all with repetition. A similar requirement on the category
data would be less extreme, but expecting no variation at all would also be too strong a
requirement.21 What does seem reasonable to ask is that any change in the distributions is not
systematic, that is, choices do not become consistently more risk seeking or more risk averse
in later repetitions.22 In fact, average SoP scores vary little between repetitions. When the
SoP distributions over the eight categories are compared in within-subject non-parametric
tests, the occasional significant differences do not seem to show systematic patterns.23
Nevertheless, just as a matter of caution, in the remainder of the analysis we focus
exclusively on the median decision of each task.24
We now turn to the patterns we observe in the tasks that match the three prospects ($,
P1 and P2) with various levels of certainty (£10–£4) and various chances of winning £60
(R0.25–R0.10). With respect to these tasks, we expect our SoP data to display some sensible
comparative statics properties. The first of these is that the entire distribution of SoP
judgments should shift towards $, P1 or P2 as the alternative is made progressively worse.
The second is that taking the four certainties, there should be relatively less movement for $
than for the two P-bets, while for R0.25–R0.10 there should be relatively more movement for
the $-bet than for the two P-bets. The reason for this is that certainty equivalents for the $21
Even at the level of actual incentive-linked choice, it is now well-established that when exactly the same
choice is presented in exactly the same format on more than one occasion within the same session, as many as
30% of individuals are liable to choose different alternatives on different occasions – see, for example, Bardsley
et al., (2009), Chapter 7. It would be unreasonable to suppose that SoP judgments are less subject to stochastic
variability than straight choices.
22
All tasks were presented once before any of them was repeated, and all were faced twice before the third
repetition.
23
The limited variation over repetitions is confirmed by the data reported in the Appendix. For any given task,
the median number of SoP categories over which the three or four repetitions are spread is one in all cases,
except for two of the tasks involving dominance, in which case the median is zero.
24
For tasks repeated three times, the median SoP score is easy to determine as the intermediate value. For the
tasks repeated four times, {$, P1} and {$, P2}, this is the average of the two intermediate values. The median
category can be determined by assigning the median SoP score to the corresponding point on the –4, ..., –1, 1,
..., 4 scale.
16
bet can be spread over a wider range (in principle, any value between £0 and £40 does not
violate dominance) than is allowable for P1 and P2 (whose certainty equivalent values must
lie between £0 and £10 if they are to respect dominance) so that the change between £4 and
£10 covers much of the interval for the P-bets but just a fraction of the allowable range for
the $-bet. Similarly, since the range of possible probability equivalents is wider for the Pbets than for the $-bet, the change between 0.1 and 0.25 covers much of the allowable
interval for $ and a rather smaller fraction for the P-bets. The final expectation is that the
SoP distributions should be sensitive to the difference between P1 and P2. Note that since
tasks involving these two prospects are faced by different subjects, this is a more demanding
sensitivity check.
We look at whether the SoP data comply with our expectations with the aid of Figures
5 and 6. Figure 5 depicts the mean SoP score for the three PR prospects when these are
compared with various certainty levels. Figure 6 depicts the corresponding information for
comparisons with R0.25–R0.10. In both figures, the SoP scores are normalised so that
positive values indicate preference for the PR prospect of interest. Problems involving
dominance ({£10, P1}, {£10, P2} and {$, R0.25}) are not included for the purposes of these
figures (see footnote 19 above). The data for the $-bet are pooled across the whole sample
because the distributions of SoP scores did not differ significantly for any of the tasks
between the two subsamples in two-tail Mann-Whitney tests.
[Figures 5 and 6 about here]
It is easy to see that the SoP measures show remarkably consistent comparative statics
properties. All curves are downward sloping, signifying that the PR prospect of interest
becomes progressively less attractive as the certainty is increased (Figure 5) or the probability
of winning £60 is increased (Figure 6). As expected, the curve for $ is flatter for tasks
involving certain amounts and steeper for the R0.25–R0.10 tasks. What is perhaps much
stronger evidence in favour of the meaningfulness of these measures is that, in both figures,
the curve for P1 lies above that for P2, even though these reflect the decisions of different
individuals. The curves in Figure 5, in particular, are strikingly parallel. This is especially
reassuring when one considers that the SoP distributions did not differ between the two
subsamples in any of the tasks involving the $-bet.
17
4.3
Results: Visible and Potential Cycles
The prospects used in Experiment 2 also fit the general framework of Figure 1. In tasks
involving certainties, the {S, M, R} triples are of the form {£x, P, $}, where x is 10, 8, 6 or 4,
and P can either be P1 or P2. In tasks involving different chances of winning £60, the {S, M,
R} triples take the form {P, $, Ry}, where y is 0.25, 0.20, 0.15 or 0.10. As in Experiment 1,
for each triple we can identify visible regret and similarity cycles, as well as potential cycles
in the two directions. These are reported in Table 5.
[Table 5 about here]
Visible cycles are less common than in Experiment 1. A maximum of 5 subjects (7% of the
69 subjects) exhibit visible cycles in the P1 subsample, while at most 7 subjects (10% of the
69 subjects) do so in the P2 subsample. However, potential cycles occur with high
frequencies. Over the 16 triples, the number of F1s ranges between 10 and 30 (i.e. 14% to
43% of the sample), while the number of F2s ranges between 40 and 63 (i.e. between 58%
and 91% of the sample). Even excluding the dominance tasks, in which the pattern of SoP
values is more extreme, F1s are exhibited by between 17% and 43% of participants, F2s by
between 58% and 80%. Once again, substantial intransitive tendencies can be found beneath
the surface.
The most striking feature of these data, however, is the systematic pattern of regret
and similarity cycles they reveal. In tasks involving certainties, potential regret cycles greatly
outnumber similarity cycles for higher certainties, but the pattern tends to be progressively
reversed as the certainty is decreased. In tasks involving chances of winning £60, the
opposite is true. Similarity cycles are prevalent for larger probabilities, while the balance
progressively switches in favour of regret cycles as the probability of winning £60 is reduced.
These patterns are particularly accentuated in the P1 subsample, in which the P-bet more
closely resembles those used in PR experiments. For certainties, regret F1s go down from 23
to 11, while similarity F1s increase from 0 to 5. Regret F2s decrease from 63 to 19, while
similarity F2s increase from 0 to 21. In the R0.25–R0.10 tasks, regret F1s increase from 2 to
18, while similarity F1s go down from 14 to 0. Regret F2s increase from 2 to 38, while
18
similarity F2s decrease from 53 to 6. In the P2 subsample the pattern is similar, except that
there is less action for similarity F1s and F2s in tasks involving certainties.
The relative slopes of the curves for the P-bets and $ in Figures 5 and 6 can give some
intuition for this rather surprising result. Remember that in the tasks involving certainties, the
{S, M, R} triples take the form {£x, P, $} while in the tasks involving R0.25-R0.10 they take
the form {P, $, Ry}. For any triple, the first pair-wise choice is S versus M, the second is M
versus R and the third is S versus R. First consider sub-sample P1 and Figure 5, which
summarises the case involving certainties. The first choice here is £x against P1 and the third
is £x against $. The second choice is given. For low values of £x such as x = 4, P1 is more
strongly preferred to £x than is $. This implies that the first choice is more likely for the
riskier prospect while the third is for the safer prospect. With reference to Table 1, we see
that a similarity cycle requires an r-r-s sequence and a regret cycle requires an s-s-r sequence.
For a given SoP (P1, $), the chances of observing a similarity cycle will clearly outweigh the
chances of observing a regret cycle.
But Figure 5 also reveals that the slopes for the P1 and $-bets cross. For high values
of £x such as x = 8, this implies the certainty is now more strongly preferred to P1 than it is
to $. This means the first choice is more likely to be safe and the third risky, so for a given
SoP (P1, $), the chances of observing the regret cycle sequence s-s-r now exceeds the chance
of observing a similarity cycle. The more minor change in the number of similarity cycles for
sub-sample P2 is due to the relative unattractiveness of this bet. Although the slopes of P1
and P2 are very similar, P2 does not cross $, leaving little room for similarity cycles to occur.
Analogous reasoning can also explain the opposite pattern of cycles for tasks
involving R0.25-R0.10. Figure 6 shows that for low probabilities such as y = 0.1, $ is more
strongly preferred to Ry than is P1. That is, the second choice is for the relatively safe
prospect while the third is for the relatively risky prospect. For a given SoP (P1, $) as the first
choice, the chances of observing the regret cycle sequence s-s-r should exceed the chances of
observing the opposite cycle r-r-s. Figure 6 also shows that the curves for P1 and the $-bet
cross. So for higher probabilities such as y = 0.2, the second choice is now more likely to be
for the riskier prospect and the third choice for the safer prospect. For a given SoP (P1, $) as
the first choice, the similarity cycle sequence r-r-s is now more likely to be observed than the
regret cycle, ceteris paribus. As sub-sample P2 also crosses the $-bet curve in Figure 6, the
same patterns are then observed using P2.
19
5. Discussion and Conclusion
We have reported two experiments in which the information from pair-wise choices is
supplemented by self-reported judgments about how strongly the chosen prospect is preferred
over the alternative. In triples including a relatively safe, a medium, and a relatively risky
prospect, our SoP data highlight very systematic tendencies for preferences to cycle in either
the regret or the similarity direction depending on the parameters of the prospects. Regret
cycles prevail in scaled-up prospects, while similarity cycles prevail when the probabilities
are scaled down. When one of the prospects is made progressively worse, the pattern of
cycles tends to be reversed: the prevalence of regret cycles turns into a prevalence of
similarity cycles and vice versa.
Even though our approach of using only individuals’ median SoP responses is
conservative, the underlying intransitive tendencies revealed by potential cycles are
substantial and systematic. To be more cautious still, we can run the analysis using just the
categories -4 to +4. This makes it yet more difficult to satisfy the inequalities required by WS
and MS. 25 Table 6 reports potential regret and similarity F2 cycles computed on the basis of
these categorical data. In Table 6, individuals are only deemed to exhibit a cycle if the
absolute value of their SoP(A, C) category is strictly less than the maximum of the absolute
values of the SoP(A, B) and SoP(B, C) categories. To distinguish these from the cycles
computed using the median SoP responses, we denote them by F2*.
[Table 6 about here]
It is clear that the same patterns persist. Even if one excludes the cases in which one prospect
is dominated, between 19% and 57% of the subjects are still liable to exhibit a cycle. Our
SoP data clearly show that such a tendency is not limited to a minority of individuals but is
widespread.
25
Notice that when the data are coded in SoP categories there are just four levels of SoP in each direction.
Since SoP(A, C) = max[SoP(A, B), SoP(B, C)] is not counted as a cycle, this method greatly reduces the room
for potential cycles to be observed.
20
These findings pose serious challenges to most economic decision theories, in which
transitivity is regarded as one of the most basic and uncontroversial tenets, and they raise
questions about the adequacy of studies in which transitivity is assessed via straight choices
only. However, developing a fully-fledged theoretical account of our findings goes beyond
the aim of this paper. Accommodating these forms of intransitivity in the context of
deterministic models may be far from straightforward. Even if transitivity’s normative
appeal is over-rated,26 existing intransitive models such as regret theory (e.g. Bell, 1982;
Fishburn, 1982; Loomes and Sugden, 1982, 1987) clearly cannot account for the large
amount of similarity cycles we observe. Other models recently appeared in the psychological
literature (e.g. the perceived relative argument model presented by Loomes, 2010) that allow
choice to be led by the comparison between the alternatives may stand better chances.
However, it is probably in the combination of such comparative core models and some form
of stochastic component that the most promising way ahead can be found. A more thorough
exploration of preferences under the surface may be extremely useful in this respect.
The systematic change in the pattern of potential cycles as one of the options is
progressively altered suggests that the ability to detect violations of transitivity using straight
pair-wise choices may not just be limited, as demonstrated by the large amount of potential
cycles, but also heavily dependent on the parameters of the prospects. Perhaps because of the
heterogeneity in the parameter values for which different individuals would commit a cycle, a
one-size-fits-all selection of pre-set parameters is bound to only pick up a limited number of
underlying intransitivities. Indeed, by appropriately choosing the parameters of the
prospects, it is possible to find cycles prevailing in one direction, while different parameters
could reveal similar proportions of cycles in the opposite direction. There may be
circumstances in which studying preferences via straight choices may not only provide rather
limited information, but also, and more worryingly, information that is misleading.
In this paper, we have made a first attempt to obtain measures of SoP capable of
overcoming such limitations of choice data. Our measurement instrument may be capable of
benefitting from further refinement, but as we have shown, it elicits responses which display
sensitive and appropriate comparative statics properties. We hope that our results
26
There is a tendency to extrapolate the obvious appeal of transitivity over single-attribute objects to multi-
attribute objects. But this extension is valid only when the desirability of each prospect is evaluated in isolation.
It does not hold in general if a choice emerges from a comparative judgment process.
21
demonstrate the potential for SoP measures to provide researchers with much more
informative data about people’s preferences, allowing them to explore beneath the tip of the
iceberg.
22
References
Bardsley, N., Cubitt, R., Loomes, G., Moffatt, P., Starmer, C. and Sugden, R., (2009).
Experimental Economics: Rethinking the Rules, Princeton University Press.
Bateman, I., Day, B., Loomes, G. and Sugden, R., (2006). “Ranking Versus Choice in the
Elicitation of Preferences.” Working Paper, University of East Anglia.
Bell, David E., (1982). “Regret in Decision Making under Uncertainty.” Operations Research
30(5): 961–81.
Budescu, D. and Weiss, W., (1987). “Reflection of Transitive and Intransitive Preferences: A
Test of Prospect Theory.” Organizational Behavior and Human Decision Processes
39: 184–202.
Butler, D. J. and Loomes, G., (2007). “Imprecision as an Account of the Preference Reversal
Phenomenon.” American Economic Review 97(1): 277–297.
Butler, D. J., Isoni, A., Loomes, G. and Tsutsui, K., (2010). “Investigating the Sensitivity
and Validity of Measures of Strength of Preference.” Working paper.
Cubitt, R. P., Munro, A. and Starmer, C., (2004). “Testing Explanations of Preference
Reversal.” Economic Journal 114(497): 709–726.
Day, B. and Loomes, G., (2010). “Conflicting Violations of Transitivity and Where They
May Lead Us.” Theory and Decision 68: 233–242.
Fishburn, P., (1982). “Nontransitive Measurable Utility.” Journal of Mathematical
Psychology 26: 31–67.
Grether, D. and Plott, C., (1979). “"Economic Theory of Choice and the Preference Reversal
Phenomenon." American Economic Review 69: 623–38.
Humphrey, S. J., (2001). “Are Event-Splitting Actually Boundary Effects?” Journal of Risk
and Uncertainty 22: 79–93
Leland, J., (1994). “Generalized Similarity Judgments: An Alternative Explanation for
Choice Anomalies.” Journal of Risk and Uncertainty 9: 151–172.
Leland, J., (1998). “Similarity Judgments in Choice Under Uncertainty: A Reinterpretation of
the Predictions of Regret Theory.” Management Science 44: 659–672.
23
Lichtenstein, S. and Slovic, P., (1971). “Reversals of Preference between Bids and Choices in
Gambling Decisions.” Journal of Experimental Psychology 89: 46–55.
Lindman, H. and Lyons, J., (1978). “Stimulus Complexity and Choice Inconsistency Among
Gambles.” Organizational Behavior and Human Performance 21: 146–159.
Loomes, G., (2010). “Modeling Choice and Valuation in Decision Experiments.”
Psychological Review (in press).
Loomes, G. and Sugden, R., (1982). “Regret Theory: An Alternative Theory of Rational
Choice under Uncertainty.” Economic Journal 92(368): 805–824.
Loomes, G., and Sugden, R., (1987). “Some Implications of More General Form of Regret
Theory.” Journal of Economic Theory 41(2): 270–287.
Loomes, G., Starmer, C. and Sugden, R., (1991). “Observing Violations of Transitivity by
Experimental Methods.” Econometrica 59: 425–439.
MacCrimmon, K. and Smith, M., (1986). “Imprecise Equivalences: Preference Reversals in
Money and Probability.” University of British Columbia Working Paper 1211.
Mellers, B. and Biagini, K., (1994). “Similarity and Choice.” Psychological Review 101:
505–518.
Rieskamp, J., Busemeyer, J. and Mellers, B., (2006). “Extending the Bounds of Rationality:
Evidence and Theories of Preferential Choice.” Journal of Economic Literature 44:
631–661.
Rubinstein, A., (1988). “Similarity and Decision-Making under Risk (Is There a Utility
Theory Resolution to the Allais Paradox?).” Journal of Economic Theory 46(1):
145–153.
Seidl, C., (2002). “Preference Reversal.” Journal of Economic Surveys 16(5): 621–655.
Tversky, A., (1969). “Intransitivity of Preferences.” Psychological Review 76: 31–48.
Tversky, A., Slovic, P. and Kahneman, D., (1990). “The Causes of Preference Reversal.”
American Economic Review 80(1): 204–217.
24
Figure 1 – A generic {S, M, R} triple
p1
p2
S
1-p1-p2-p3
£c
M
R
p3
£0
£b
£a
£0
£0
25
Figure 2 – Prospects display
26
Figure 3 – The SoP bar
27
100
0
50
Frequency
150
Figure 4 – Distribution of SoP Scores
-100
-50
0
Strength of Preference
50
100
28
Figure 5 – Mean SoP Score for PR Prospects and Certainties
29
Figure 6 – Mean SoP Score for PR Prospects and Chances of £60
30
Table 1 – Regret and similarity cycles
S vs. M
M vs. R
S vs. R
safer (s)/riskier (r)
option chosen
Inferred
preference
Type of
cycle
S
M
S
S
M
M
R
M
R
R
R
S
s-s-r
r(s)-s-r
s-r(s)-r
s-s-s(r)
RSMR
M  R  S(  M)
R  S  M(  R)
S  M  R(  S)
Regret
(Regret)
(Regret)
(Regret)
M
M
S
M
R
M
R
R
R
S
S
S
r-r-r(s)
r-s(r)-s
s(r)-r-s
r-r-s
R  M  S (  R)
M  S  R (  M)
S  R  M (  S)
MSRM
(Similarity)
(Similarity)
(Similarity)
Similarity
31
Table 2 – Experiment 1 Prospects
Prospect
Prize (£)
Wu
Xu
Yu
Zu
9
15
25
45
Wd
Xd
Yd
Zd
9
15
25
45
Set I
Prob.
Set II
Prob.
EV
Prize (£)
EV
1
0.8
0.6
0.4
9
12
15
18
8
14
21
35
1
0.8
0.6
0.4
8
11.2
12.6
14
0.25
0.2
0.15
0.1
2.25
3
3.75
4.5
8
14
21
35
0.25
0.2
0.15
0.1
2
2.8
3.15
3.5
32
Table 3 – Regret and Similarity Cycles for Scaled-Up and Scaled-Down Prospects
{S, M, R}
F0 = IS
F1 = IS+WS
F2 = IS+WS+MS
Reg.
Sim.
Reg.
Sim.
Reg.
Sim.
{Wu, Xu, Yu}
{Wu, Xu, Zu}
{Wu, Yu, Zu}
{Xu, Yu, Zu}
11
13
12
10
4
3
3
6
34
42
39
36
15
14
13
20
61
75
72
63
28
25
22
28
{Wd, Xd, Yd}
{Wd, Xd, Zd}
{Wd, Yd, Zd}
{Xd, Yd, Zd}
8
4
4
9
15
16
21
29
20
13
17
23
41
48
45
46
39
24
32
42
66
66
71
59
33
Table 4 – Experiment 2 Prospects
Prospect
Prize (£)
Prob.
EV
$
P1
P2
40
10
10
0.25
0.9
0.65
10
9
6.5
£10
£8
£6
£4
10
8
6
4
1
1
1
1
10
8
6
4
R0.25
R0.20
R0.15
R0.10
60
60
60
60
0.25
0.2
0.15
0.1
15
12
9
6
34
Table 5 – Regret and Similarity Cycles for Preference Reversal Prospects
{S, M, R}
F0=IS
F1=IS+WS
F2=IS+WS+MS
Reg.
Sim.
Reg.
Sim.
Reg.
Sim.
{£10, P1, $}
{£8, P1, $}
{£6, P1, $}
{£4, P1, $}
1
0
2
0
0
0
0
0
23
18
15
11
0
4
6
5
63
35
27
19
0
9
18
21
{P1, $, R0.25}
{P1, $, R0.20}
{P1, $, R0.15}
{P1, $, R0.10}
0
1
3
1
2
1
2
0
2
8
20
18
14
4
9
0
2
22
32
38
53
24
11
6
{£10, P2, $}
{£8, P2, $}
{£6, P2, $}
{£4, P2, $}
0
6
7
6
0
0
0
1
8
27
29
22
2
2
1
2
48
51
50
39
2
3
5
5
{P2, $, R0.25}
{P2, $, R0.20}
{P2, $, R0.15}
{P2, $, R0.10}
0
2
1
1
1
0
3
1
0
6
19
17
18
10
8
3
0
19
39
38
53
24
10
5
35
Table 6 – Regret and Similarity Cycles with Category Data
{S, M, R}
F2*
Reg. Sim.
{S, M, R}
F2*
Reg. Sim.
{S, M, R}
F2*
Reg. Sim.
{Wu, Xu, Yu}
{Wu, Xu, Zu}
{Wu, Yu, Zu}
{Xu, Yu, Zu}
37
48
43
36
16
15
13
17
{£10, P1, $}
{£8, P1, $}
{£6, P1, $}
{£4, P1, $}
49
19
16
4
0
4
7
10
{£10, P2, $}
{£8, P2, $}
{£6, P2, $}
{£4, P2, $}
44
38
31
17
1
1
1
2
{Wd, Xd, Yd}
{Wd, Xd, Zd}
{Wd, Yd, Zd}
{Xd, Yd, Zd}
15
6
9
14
38
38
38
39
{P1, $, R0.25}
{P1, $, R0.20}
{P1, $, R0.15}
{P1, $, R0.10}
2
4
12
16
42
9
2
2
{P2, $, R0.25}
{P2, $, R0.20}
{P2, $, R0.15}
{P2, $, R0.10}
0
3
11
12
50
12
5
2
36