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The Anna Karenina Effect in Risky Choice 1
RUNNING HEAD: The Anna Karenina Effect in Risky Choice
The Anna Karenina Effect in Risky Choice:
Anomalies to Markowitz’s Hypothesis and a Prospect-Theoretical Interpretation
Marc Scholten
ISPA University Institute,
Rua Jardim do Tabaco 34,
1149-041 Lisboa, Portugal.
Tel: 00 351 21 8811700
Fax: 00 351 21 8860954
E-Mail: [email protected].
Daniel Read
Warwick Business School
Scarman Road
The University of Warwick
Coventry, CV4 7AL, United Kingdom
Tel: 00 44 24 76524306
Fax: 00 44 24 7652 3719
Email: [email protected]
The Anna Karenina Effect in Risky Choice 2
Abstract
Harry Markowitz hypothesized a fourfold pattern of risk preferences, with risk aversion for large
gains and small losses, but risk seeking for small gains and large losses. To test this hypothesis,
we conduct three experiments in which 1,013 participants choose between sure things and
gambles with the same expected monetary value. The major finding is what we call an Anna
Karenina effect: Uniformity in gains and diversity in losses. In gains, most people change from
risk seeking to risk aversion as the stakes increase, consistent with Markowitz‟s hypothesis. In
losses, we observe two kinds of diversity. First, preference heterogeneity: As the stakes increase,
some people change from risk aversion to risk seeking (the „Markowitzians‟), others change in
the opposite direction (the „non-Markowitzians‟), still others are risk averse (the „Cautious‟), and
some are risk seeking (the „Audacious‟). Second, preference uncertainty: Many people exhibit
preferences that do not confirm to a definite pattern, and these people take longest to choose in
losses. We offer a prospect-theoretical interpretation of the Anna Karenina effect. We show what
assumptions must be invoked about the value function and the probability-weighing function to
accommodate preference heterogeneity. We test and confirm this interpretation. We further
suggest how prospect theory can be incorporated into decision field theory to accommodate
preference uncertainty.
Keywords: Risk aversion, risk seeking, choice time, choice inconsistency, Markowitz, prospect
theory.
The Anna Karenina Effect in Risky Choice 3
The Anna Karenina Effect in Risky Choice:
Anomalies to Markowitz’s Hypothesis and a Prospect-Theoretical Interpretation
The question of how to describe individual decision making under risk has been subject to
a great deal of investigation, with published articles now numbering in their thousands. Of the
numerous models that have been proposed, most are representative-agent models, in which the
typical decision maker is assumed to exhibit the modal preference patterns that are observed. This
is true of the earliest models (Bernoulli, 1738/1954) and continues through to the present (e.g.,
Brandstätter, Gigerenzer, & Hertwig, 2006; Hogarth & Einhorn, 1990; Kahneman & Tversky,
1979; Loomes, 2010; Markowitz, 1952). However, it is not clear whether representative-agent
models actually describe the decisions people make, rather than accommodate a preference
pattern emerging from an aggregate of people who may not individually exhibit the modal
preference pattern.
In this paper, we examine individual choices under risk, and ask how well two
representative-agent models can account for these choices. We focus on series of choices that
have rarely been investigated at the individual level. These are choices between a sure thing and a
gamble with the same expected value, in a series where the sure thing varies from very small to
very large. The models are the one proposed by Markowitz (1952), who originally designed the
choice series, and designed his model to accommodate the preference patterns that he observed
“informally,” and prospect theory (Kahneman & Tversky, 1979), which produces a broader set of
predictions than Markowitz‟s model does. We find that the predictions of both models are far off
the mark, both at the aggregate level (modal preference patterns) and at the disaggregate level
(individual preference patterns).
The major finding of our investigation is what we call the Anna Karenina effect, because
it can be aptly summarized by a suitable rewriting of that novel‟s famous first sentence: “Happy
The Anna Karenina Effect in Risky Choice 4
[choices] are all alike; every unhappy [choice] is unhappy in its own way.”1 We find a remarkable
uniformity in people‟s preferences among certain and probable gains. Their “happy choices” are
very much alike, and very much akin to the predictions made by Markowitz. However, we find a
much greater diversity in people‟s preferences among certain and probable losses. The diversity
in these “unhappy choices” has both systematic and unsystematic components. The systematic
component is preference heterogeneity: Different people exhibit different preference patterns.
The unsystematic component is preference uncertainty: Many people exhibit preferences that do
not conform to a definite pattern.
Confronted with the Anna Karenina effect, we discuss how prospect theory must be
modified to accommodate preference heterogeneity. Our prospect-theoretical interpretation has
testable implications for the way in which different people react differently to changes in the
probability of gaining or losing, and we confirm these implications. We also discuss how
prospect theory, a deterministic, static, and psychophysical approach to risky choice, can be
incorporated into decision field theory (Busemeyer & Townsend, 1993), a probabilistic, dynamic,
and motivational approach to risky choice, to accommodate greater preference uncertainty for
losses than for gains.
The plan of this paper is as follows. We first discuss the preference pattern hypothesized
by Markowitz and the explanation offered by him. We then show how prospect theory explains
the hypothesized pattern, and how it produces a broader set of preference patterns. Following
that, we discuss the existing evidence on Markowitz‟s hypothesis, and show how the Anna
Karenina effect can account for it. This is followed by two experiments, in which we confirm the
Anna Karenina effect. We then modify prospect theory in order to account for our evidence. The
implications of our analysis are tested and confirmed in a third experiment. Finally, we discuss
The Anna Karenina Effect in Risky Choice 5
the implications of our analysis for real-world behaviors, and focus on the paradigmatic cases of
purchasing insurance and gambles.
Markowitz’s Theory:
A Fourfold Pattern of Risk Preferences
Markowitz (1952) questioned the common assumption that people are globally risk
averse, meaning they will prefer a sure thing to a gamble with the same expected monetary value.
He reported that respondents in an informal survey were risk averse for large gains and small
losses, but risk seeking for small gains or large losses. As shown in Table 1, they preferred a 1/10
chance of gaining $1 to gaining 10 cents for sure (risk seeking), but preferred gaining $1 million
for sure to a 1/10 chance of gaining $10 million (risk aversion); conversely, they preferred losing
10 cents for sure to a 1/10 chance of losing $1 (risk aversion), but preferred a 1/10 chance of
losing $10 million to losing $1 million for sure (risk seeking). To explain this fourfold pattern of
risk preferences, which we call the „M4 pattern,‟ Markowitz proposed a triply inflected,
reference-dependent value function, as shown in Figure 1. Like the prospect theory value
function, Markowitz‟s is defined over positive and negative deviations from current wealth (gains
and losses). Unlike the prospect theory value function, which is concave in gains and convex in
losses, Markowitz‟s is first convex and only then concave in gains, and first concave and only
then convex in losses. Markowitz‟s model is an expected value model, meaning that outcome
values are weighted by outcome probabilities, and it is for this reason that it needs the triply
inflected value function to produce the M4 pattern. As we discuss next, prospect theory can
produce the M4 pattern by combining a non-linear probability-weighing function with a special
value function.
<Insert Table 1 about here>
<Insert Figure 1 about here>
The Anna Karenina Effect in Risky Choice 6
Prospect Theory:
The Fourfold Pattern of Risk Preferences
Prospect theory is a non-expected value model. This means that outcome values are
weighted by a non-linear transformation of probabilities, so that the value of a gamble is not its
expectation, pv(x), as in Markowitz‟s theory, but the product of a probability weight and the
outcome value, w(p)v(x). In prospect theory, the probability-weighing function overweighs low
probabilities and underweighs moderate to high ones. If it overweighs Markowitz‟s 1/10, as is
generally assumed, it will contribute to risk seeking in gains and risk aversion in losses. The
value function, on the other hand, is concave over gains and convex over losses (diminishing
sensitivity), contributing to risk aversion in gains and risk seeking in losses. For Markowitzian
choices, therefore, the probability-weighing function and the value function work in opposite
directions.
The M4 pattern requires that the probability-weighing function outweighs the value
function for small outcomes, i.e., w(1/10) > v(x) / v(10x), but is outweighed by it for large ones,
i.e., v(mx) / v(10mx) > w(1/10) for some m > 1. If we combine the two inequalities, we get
)10(
)(
)10(
)(
xv
xv
mxv
mxv for m > 1.
This property of the value function is called decreasing elasticity (al-Nowaihi, & Dhami, 2009;
Scholten & Read, 2010; see also Loewenstein & Prelec, 1992): As the magnitude of x increases
by constant proportions, the magnitude of v(x) increases by decreasing proportions. For instance,
doubling $100 yields a smaller proportional increase in value than doubling $1.
Decreasing elasticity has not previously been associated with prospect theory. A power
value function, as proposed by Tversky and Kahneman (1992), and “by far the most popular form
for estimating money value” (Prelec, 2000, p. 84), has constant elasticity (within the domains of
The Anna Karenina Effect in Risky Choice 7
gains and losses). Examples of reference-dependent value functions that exhibit decreasing
elasticity are the normalized exponential function discussed by Köbberling and Wakker (2005;
see also Luce, 1996, and Moffatt, 2005) and the normalized logarithmic function proposed by
Scholten and Read (2010). For gains, the normalized logarithmic function satisfies the two
conditions for utility functions discussed by Abdellaoui, Barrios, and Wakker (2007), and Holt
and Laury (2002): Decreasing absolute risk aversion and increasing relative risk aversion. Figure
2 compares the power value function and the logarithmic value function. In log-log coordinates,
the power value function is a straight line (constant elasticity), whereas the logarithmic value
function is concave (decreasing elasticity). Appendix A discusses the concept of elasticity more
extensively.
<Insert Figure 2 about here>
Prospect theory can therefore produce the M4 pattern if it combines an overweighing of
low probabilities with a decreasingly elastic value function. However, it can also produce other
preference patterns, depending on whether the probability-weighing function outweighs the value
function for small outcomes, and on how the elasticity in gains compares with the elasticity in
losses. We next discuss these predictions.
Prospect Theory:
A Twofold and Two Threefold Patterns of Risk Preferences
In this section, we discuss how prospect theory can produce four preference patterns in
Markowitzian choices between x and (10x, .1), as summarized in Table 2. We assume that, as in
original prospect theory (Kahneman & Tversky, 1979), the probability weighing function is the
same for gains and losses. This assumption has been relaxed by cumulative prospect theory
(Kahneman & Tversky, 1992; see Wakker, 2010), but doing so would not change the predictions
we derive from prospect theory; it would only sometimes open different possibilities to arrive at
The Anna Karenina Effect in Risky Choice 8
the same predictions. In any event, prospect theory needs decreasing elasticity to produce the
preference patterns in Table 2, and it does not need domain-specific probability weighing to
accommodate the evidence we obtain in our experiments. In the remainder of this paper, and
unless otherwise indicated, „prospect theory‟ will refer to original prospect theory augmented
with increasing elasticity.
<Insert Table 2 about here>
A twofold pattern of risk preferences
As just discussed, prospect theory combines diminishing sensitivity to outcomes with the
overweighing of low probabilities. Diminishing sensitivity contributes to risk aversion in gains
and risk seeking in losses; the overweighing of low probabilities contributes to risk seeking in
gains and risk aversion in losses. The predictions of prospect theory depend on the relative
strength of these opposing forces. Risk preferences will reverse only if the probability-weighing
function outweighs the value function for small outcomes but is outweighed by it for large ones.2
That is, when
)10(
)()10/1(
)10(
)(
xv
xvw
mxv
mxv for m > 1.
Risk preferences will never reverse if the probability-weighing function never outweighs the
value function. This would yield a twofold pattern of global risk aversion in gains and global risk
seeking in losses:
)10/1()10(
)(
)10(
)(w
xv
xv
mxv
mxv for m > 1.
3
The M4 pattern and the twofold pattern can occur regardless of how elasticity in gains
compares with elasticity in losses. There are three logical possibilities: Either elasticity is equal
for losses and gains, or it is greater for losses, or it is greater for gains. If elasticity is equal for
The Anna Karenina Effect in Risky Choice 9
losses and gains (symmetric elasticity), the value ratio (between the values of the sure thing and
the risky outcome) is unchanged by a sign reversal:
10
1
)10(
)(
)10(
)(
xv
xv
xv
xv for x > 0. (1)
Alternatively, if elasticity is greater for losses (loss amplification; see Prelec & Loewenstein,
1991), the value ratio is smaller for losses:
10
1
)10(
)(
)10(
)(
xv
xv
xv
xv for x > 0. (2)
Finally, if elasticity is greater for gains (gain amplification), the value ratio is smaller for gains:
10
1
)10(
)(
)10(
)(
xv
xv
xv
xv for x > 0. (3)
Symmetric elasticity, loss amplification, and gain amplification relate to how close to
linearity the value function is in gains and in losses. We can illustrate this with the power value
function proposed by Tversky and Kahneman (1992):
,0if)(
0if)(
xx
xxxv
where 0 < , < 1 are diminishing sensitivity (the value function is concave in gains and convex
in losses, meaning that, as discussed in Appendix A, elasticity lies between 0 and 1) and > 1 is
constant loss aversion (see also Tversky & Kahneman, 1991). Symmetric elasticity means = .
Loss amplification means > , and gain amplification means > . The evidence reviewed by
Wakker, Köbberling, and Schwieren (2007, Note 1) includes cases of both symmetric and
asymmetric elasticity, with loss amplification being more prevalent than gain amplification.
Regardless of whether the elasticity of the value function is symmetric or asymmetric, it is
possible that, in both gains and losses, the probability-weighing function outweighs the value
The Anna Karenina Effect in Risky Choice 10
function for small outcomes but is outweighed by it for large ones, yielding the M4 pattern, or
never outweighs the value function, yielding the twofold pattern. Under symmetric elasticity, and
in the absence of domain-specific probability weighing, these are the only two possibilities: The
value ratio v(x) / v(10x) is the same for gains and losses (Expression 1), so that, if w(1/10)
exceeds the value ratio for small gains, it will exceed the value ratio for small losses as well, or,
alternatively, if w(1/10) never exceeds the value ratio for gains, it will never exceed the value
ratio for losses either. Under asymmetric elasticity, however, two other preference patterns can
emerge.
Two threefold patterns of risk preference
Under loss amplification, the value ratio is smaller for losses than for gains (Inequality 2),
so w(1/10) can exceed the value ratio for small losses but never exceed the value ratio for gains.
This would yield a threefold pattern of global risk aversion in gains, alongside a reversal from
risk aversion to risk seeking in losses. Under gain amplification, the value ratio is smaller for
gains than for losses (Inequality 3), so w(1/10) can exceed the value ratio for small gains but
never exceed the value ratio for losses. This would yield a threefold pattern of global risk seeking
in losses, alongside a reversal from risk seeking to risk aversion in gains.
Prospect theory can therefore produce the M4 pattern, but can also produce a twofold
pattern and two threefold patterns. We next discuss existing evidence on Markowitz‟s hypothesis.
Existing Evidence
The earliest evidence comes from Hershey and Schoemaker (1980, Table 3, rows 13-17).
Letting x be the sure outcome and (x/p, p) be a gamble yielding x/p with probability p, they set p
at .01, and x at $1, $100, $1,000, and $10,000 (in this section, x stands for the magnitude of x, or
its absolute value). The results supported Markowitz‟s hypothesis: As x increased, aggregate
choices changed from risk seeking to risk aversion for gains, but from risk aversion to risk
The Anna Karenina Effect in Risky Choice 11
seeking for losses. However, choice probabilities were closer to chance level (i.e., closer to a
50/50 distribution) for losses than for gains.4
Hogarth and Einhorn (1990) conducted three studies, in which p was set at .1, .5, and .8
(Study 1, Table 2) or at .1, .5, and .9 (Studies 2 and 3, Table 4), and x was set at $2, $200, and
$20,000 (Study 1, hypothetical payoffs), at $1 and $10,000 (Study 2, hypothetical payoffs), or at
$0.10 and $10 (Study 3, real payoffs). The results of these studies show strong signs of an Anna
Karenina effect.
In gains, Markowitz‟s hypothesis was supported at all probability levels: As x increased,
aggregate choices moved away from risk seeking and toward risk aversion. For p = .1, there was
either global but decreasing risk seeking or a reversal from risk seeking to risk aversion; for p =
.5 and .8, there was a reversal from risk seeking to risk aversion; for p = .9, there was global and
increasing risk aversion. The reversals for moderate to high probabilities (i.e., p = .5 and .8) are
inconsistent with prospect theory, because the probability-weighing function underweighs those
probabilities, and thus contributes, together with the concave value function, to risk aversion.
In losses, the results showed a great diversity. In Study 1, the direction of the results
depended on the probability level. For p = .1, there was global but decreasing risk aversion,
consistent with Markowitz‟s hypothesis and prospect theory. For p = .5, there was a reversal from
risk seeking to risk aversion, when Markowitz would predict a reversal in the opposite direction,
and prospect theory would predict global and increasing risk seeking, because (1) the probability-
weighing function underweighs moderate probabilities, and thus contributes, together with the
convex value function, to risk seeking, and (2) the value function is decreasingly elastic. For p =
.8, there was global but decreasing risk seeking, when Markowitz and prospect theory would
predict increasing risk seeking. In Study 2, the results did not depend on the probability level:
There was an almost constant level of risk seeking, when Markowitz and prospect theory would
The Anna Karenina Effect in Risky Choice 12
predict increasing risk seeking. Finally, in Study 3, the results again depended on the probability
level, but were always inconsistent with Markowitz‟s hypothesis and prospect theory. For p = .1,
there was a reversal from risk seeking to risk aversion; for p = .5, there was global but decreasing
risk seeking; for p = .9, there was an almost constant level of risk seeking. Across all studies, and
as in Hershey and Schoemaker‟s (1980) study, choice probabilities were closer to chance level for
losses than for gains.
The results of the studies reviewed above confirm Markowitz‟s hypothesis for gains: As
outcome magnitude increased, aggregate choices moved away from risk seeking and toward risk
aversion. This trend has also been observed in matching (Fehr-Duda, Bruhin, Epper, & Schubert,
2010) and choice-based matching (Du, Green, & Myerson, 2002; Green, Myerson, Ostaszewski,
1999; Holt, Green, & Myerson, 2003; Myerson, Green, Hanson, Holt, & Estle, 2003). However,
the results showed a great diversity for losses. Aggregate choices either moved away from risk
aversion and toward risk seeking, as Markowitz hypothesized, or they moved in the opposite
direction, or they did not move at all. Moreover, choice probabilities were generally closer to
chance level for losses than for gains (see also Kühberger, Schulte-Mecklenbeck, & Perner,
1999). The combination of a clear trend in gains and an unclear trend in losses has also been
observed in matching (Fehr-Duda et al., 2010). We next offer a possible interpretation of these
results.
An Interpretation of Existing Evidence:
The Anna Karenina Effect in Risky Choice
The Anna Karenina effect is that uniformity in choices between certain and probable gains
coincides with diversity in choices between certain and probable losses, where the „diversity‟
involves both preference heterogeneity and preference uncertainty. So let us make an ideal
representation of the existing evidence, in which all people exhibit the Markowitzian reversal
The Anna Karenina Effect in Risky Choice 13
from risk seeking to risk aversion in gains, but exhibit different preference patterns in losses. We
thus have the following five groups: (1) The Markowitzians, who exhibit a reversal from risk
aversion to risk seeking, as hypothesized; (2) The non-Markowitzians, who exhibit a reversal
from risk seeking to risk aversion; (3) The Audacious, who are globally risk seeking; (4) the
Cautious, who are globally risk averse, and (5) the Wavering, who do not exhibit a definite
preference pattern. In our ideal interpretation of the existing evidence, we will assume that the
Wavering have an uncontrollably trembling hand in losses, so that their choices in that domain
are completely at random.
One systematic result is that choice probabilities are closer to chance level for losses than
for gains. One explanation is preference uncertainty: The Wavering bring the aggregate choices
closer to a 50/50 distribution. The other explanation is preference heterogeneity: The aggregate
choices are brought closer to a 50/50 distribution by the Markowitzians and non-Markowitzians,
who reverse their preferences in opposite directions, and by the Audacious and the Cautious, who
have globally opposite preferences. Both preference uncertainty and preference heterogeneity
contribute to „risk neutrality‟ at the aggregate level where such neutrality may not exist at the
disaggregate level. Risk neutrality would mean that the value of the sure thing and the value of
the gamble are close together, so that people would feel more or less „indifferent‟ between the
two options (Schneider, 1992). Neither Markowitz‟s theory nor prospect theory, whether in its
original form or in a form that accommodates the Anna Karenina effect, implies such indifference
between certain and probable losses.
Preference heterogeneity can also explain why, in Hogarth and Einhorn‟s (1990) studies,
modal preference patterns change across studies and, within studies, across probability levels. In
our prospect-theoretical interpretation of the Anna Karenina effect, the four preference groups
that exhibit a definite preference pattern in losses (1 to 4) are a behavioral reflection of four
The Anna Karenina Effect in Risky Choice 14
evaluation groups, each with its unique combination of a value function and a probability-
weighing function. Assuming the existence of these evaluation groups, whose prominence is
likely to vary across studies, we can explain modal preference patterns, and how the composition
of preference groups, and, therefore, modal preference patterns, changes across probability levels.
The existence of the five preference groups poses a severe challenge to both Markowitz‟s theory,
which only covers the Markowitzians, and prospect theory, which only covers the Markowitzians
and the Audacious.
We will report three studies in which we directly address the Anna Karenina effect by
examining preference patterns both at the aggregate and at the disaggregate level. In Experiment
1, participants make Markowitzian choices between x and (10x, .1), or „10x choices,‟ for which
both Markowitz and prospect theory predict the M4 pattern. In Experiment 2, participants make
choices between x and (2x, .5), or „2x choices,‟ for which Markowitz continues to predict the M4
pattern, but prospect theory predicts a twofold pattern of global risk aversion in gains and global
risk seeking in losses. In both studies, the majority of the participants fall into the five preference
groups identified above, with the Wavering taking longest to decide. This is a confirmation of the
Anna Karenina effect. Apart from preference heterogeneity and preference uncertainty, there is
variability in the composition of the preference groups across 10x and 2x choices. Our prospect-
theoretical interpretation of the results covers all this. Our interpretation has testable implications
for the way in which different evaluation groups react differently to changes in probability level
(.1 and .5). However, a comparison between Experiment 1 (10x choices) and Experiment 2 (2x
choices) does not allow us to directly address this. We therefore conduct Experiment 3, in which
the same participants make both 10x and 2x choices. This study confirms our prospect-theoretical
interpretation of the results.
The Anna Karenina Effect in Risky Choice 15
Experiment 1
Method
Participants. The participants were 189 members of the Yale School of Management
virtual laboratory (eLab), including 73 men and 116 women, averaging 35 years of age. The great
majority had some college or a Bachelor‟s degree. They were entered in a lottery offering a 1 in
50 chance to win a $50 Amazon.com gift certificate.
Stimuli. Participants made six 10x choices for gains and for losses. The magnitude of the
sure thing ranged from $0.25 to $25,000 in multiples of 10, and in that order. We fixed the order
so as to recreate Markowitz‟s thought experiment as closely as possible. In Experiment 3, we
evaluate whether fixing or randomizing the order has any effect on the results, which it has not.
The order of gains and losses was counterbalanced.
Measures. We observed both choice and choice time. The latter is a conventional measure
of preference uncertainty, or choice conflict (e.g., Diederich, 2003; Fischer, Luce, & Jia, 2000;
Scholten & Sherman, 2006).
Results
Figure 3 shows the aggregate results. Aggregate choices for gains were monotonically
related to their magnitude, and reversed from risk seeking to risk aversion. Aggregate choices for
losses were globally risk averse, but were not monotonically related to outcome magnitude. The
results for losses contradict both Markowitz‟s theory, which would predict a reversal from risk
aversion to risk seeking, and prospect theory, which would predict either the Markowitzian
reversal or global risk seeking (see Table 2).
<Insert Figure 3 about here>
The Anna Karenina Effect in Risky Choice 16
Preference heterogeneity. Table 3 shows the preference groups that emerged at the
disaggregate level. A preference pattern with more than one preference shift is classified as
„inconsistent.‟
<Insert Table 3 about here>
The great majority (75%) changed from risk seeking to risk aversion in gains, but differed
in losses. This majority was composed of the Cautious (26%), who were globally risk averse, the
Wavering (23%), who were inconsistent, the Markowitzians (12%), who changed from risk
aversion to risk seeking, the non-Markowitzians (12%), who changed in the opposite direction,
and the Audacious (3%), who were globally risk seeking. Apart from the Markowitzians and the
Audacious, there were only 5 cases (3%) who exhibited the remaining two prospect theory (PT)
compatible preference patterns.
We explored how the Wavering would most likely have been classified had they not been
inconsistent in losses. We did so by counting, for each participant, the number of matches with
(1) each of the five Markowitzian preference patterns (e.g., two choices of the sure thing followed
by four choices of the gamble), (2) each of the five non-Markowitzians patterns (e.g., two choices
of the gamble followed by four choices of the sure thing), (3) the Cautious pattern (six choices of
the sure thing), and (4) the Audacious pattern (six choices of the gamble), and assigning the
participant to the group with the highest number of matches. The results are included in Table 3.
Fourteen out of 43 participants could not be reclassified because they matched the patterns of two
or more groups equally well. The Markowitzians benefited most from the reclassification of the
Wavering, becoming the second largest group (20%), after the Cautious (28%).
Preference uncertainty. The Wavering were the second largest preference group, which
may indicate greater preference uncertainty for losses than for gains. This was confirmed by an
analysis of choice times. For each participant, we took geometric averages of choice times over
The Anna Karenina Effect in Risky Choice 17
the six gain trials and six loss trials, and then computed a loss/gain ratio. Across all participants,
the loss/gain ratio was 1.13, t(187) = 4.38, p = .00, indicating greater preference uncertainty for
losses than for gains, and the loss/gain ratio was greater for the Wavering (1.31) than for the other
participants (1.08), t(187) = 3.00, p = .00 (regression analysis performed on the logarithms of the
loss/gain ratios).5 The Wavering took longer than the other participants in losses, t(187) = 3.00, p
= .00, but took no longer in gains, t(187) = 0.30, p = .77 (regression analyses performed on the
logarithms of choice times).
Discussion
The great majority exhibited the Markowitzian reversal from risk seeking to risk aversion
in gains, but dispersed into five preference groups in losses. The Cautious were the largest group.
The Wavering, prior to their reclassification, were the second largest group. In addition to their
inconsistency in losses, they took longest to decide in that domain. Upon reclassification of the
Wavering, the Markowitzians became the second largest group. The other three groups exhibiting
PT compatible preference patterns, including the Audacious, were very small. In sum, we confirm
the Anna Karenina effect, with Markowitz‟s theory and prospect theory accounting for a minority
of the participants.
The evidence from Hogarth and Einhorn‟s (1990) studies suggests that the composition of
the preference groups may change across probability levels. Therefore, in the next experiment,
we change from choices between x and (10x, .1), or 10x choices, to choices between x and (2x,
.5), or 2x choices.
Experiment 2
Introduction
In this experiment, we examine not only 2x choices, in which outcome magnitude varies,
but also „p choices,‟ in which outcome probability, rather than outcome magnitude, varies. For 2x
The Anna Karenina Effect in Risky Choice 18
choices, Markowitz predicts a fourfold pattern of risk preferences while prospect theory does not.
Conversely, for p choices, prospect theory predicts a fourfold pattern of risk preferences while
Markowitz does not. We included p choices as a check on our method: The fourfold pattern in
outcome probability has proven to be very robust in past studies, and so our method should be
able to replicate it.
A twofold pattern in outcome magnitude
In Markowitz‟s theory, the value function changes from convex to concave in gains and
from concave to convex in losses, so that, for 2x choices as well as 10x choices, the prediction is
the M4 pattern. In prospect theory, however, the value function and the probability-weighing
function both contribute to risk aversion in gains and risk seeking in losses: The value function
because of diminishing sensitivity, and the probability-weighing function because it underweighs
the moderate probability of .5. Prospect theory therefore predicts a twofold pattern of global risk
aversion in gains and global risk seeking in losses.
A fourfold pattern in outcome probability
The M4 pattern, which concerns how risk preferences change as a function of outcome
magnitude, differs from the fourfold pattern of risk preferences hypothesized by Kahneman and
Tversky (1979; Tversky and Kahneman, 1992), the „KT4 pattern, ‟ which concerns how risk
preferences change as a function of outcome probability. In Tversky‟s illustration of this pattern,
(see Tversky & Fox, 1995; Tversky & Wakker, 1995), people will prefer a 1/20 chance of gaining
$100 to gaining $5 for sure (risk seeking), but will prefer gaining $95 for sure to a 19/20 chance
of gaining $100 (risk aversion). Conversely, they will prefer losing $5 for sure to a 1/20 chance of
losing $100 (risk aversion), but will prefer a 19/20 chance of losing $100 to losing $95 for sure
(risk seeking). As mentioned above, the KT4 pattern has proven to be very robust in past studies
(e.g., Cohen, Jaffray, & Said, 1987, Table 2; Fehr-Duda et al., 2010, Figure 2; Hershey &
The Anna Karenina Effect in Risky Choice 19
Schoemaker, 1980, Table 3; Kahneman & Tversky, 1979, Table 1; Tversky & Kahneman, 1992,
Table 4).
Prospect theory explains the KT4 pattern as follows. The probability-weighing function
overweighs the low probability of 1/20, contributing to risk seeking in gains and risk aversion in
losses, but underweighs the high probability of 19/20, contributing to risk aversion in gains and
risk seeking in losses. The value function is concave in gains and convex in losses, contributing
to risk aversion in gains and risk seeking in losses. The KT4 pattern will occur when, for the low
probability of 1/20, the probability-weighing function outweighs the value function, i.e., w(p) >
v(px) / v(x).
Alternative patterns in outcome probability
Both prospect theory and Markowitz‟s theory produce alternative patterns in outcome
probability, which are summarized in Table 4. Comparison of Tables 2 and 4 reveals that the
predictions from prospect theory for p choices strictly parallel its predictions for 10x choices,
with the KT4 pattern paralleling the M4 pattern.
<Insert Table 4 about here>
Prospect theory predicts the KT4 pattern when the probability-weighing function
outweighs the value function for low probabilities, but this does not necessarily happen. When
the probability-weighing function is outweighed by the value function for both gains and losses,
prospect theory predicts a twofold pattern of global risk aversion in gains and global risk seeking
in losses. Under symmetric elasticity, and in the absence of domain-specific probability weighing,
this is the only possible alternative to the KT4 pattern, because, if the probability-weighing
function does not outweigh the value function in gains, it does not outweigh the value function in
losses either, i.e., v(px) / v(x) = v(-px) / v(-x) > w(p) for x > 0.
The Anna Karenina Effect in Risky Choice 20
Under loss amplification, the probability-weighing function may outweigh the value
function for losses but not for gains, i.e., v(px) / v(x) > w(p) > v(-px) / v(-x), producing a threefold
pattern of global risk aversion in gains and a reversal from risk aversion to risk seeking in losses.
Conversely, under gain amplification, the probability-weighing function may outweigh the value
function for gains but not for losses, i.e., v(-px) / v(-x) > w(p) > v(px) / v(x), producing a threefold
pattern of global risk seeking in losses and a reversal from risk seeking to risk aversion in gains.
In sum, there are four PT compatible preference patterns, only one of which is the KT4 pattern.
Markowitz does not predict the KT4 pattern. For him, the probability-weighing function is
an identity function, i.e., w(p) = p, so that risk preferences are entirely determined by the location
of the inflection points in gain and losses. There are four logical possibilities. When px and x are
evaluated in the convex part for gains and the concave part for losses, there will be global risk
seeking in gains and global risk aversion in losses. Conversely, when px and x are evaluated in
the concave part for gains and the convex part for losses, there will be global risk aversion in
gains and global risk seeking in losses, a preference pattern shared with prospect theory. If the
inflection points are equally far removed from the origin in gains and losses, these are the only
possibilities.
When the inflection point is further removed from the origin in losses than in gains, it is
possible that both gains and losses are evaluated in the concave part, yielding global risk aversion
in both domains. Conversely, when the inflection point is further removed from the origin in
gains than in losses, it is possible that both gains and losses are evaluated in the convex part,
yielding global risk seeking in both domains. In sum, there are four Markowitz (M) compatible
preference patterns, one of which is PT compatible as well.
The Anna Karenina Effect in Risky Choice 21
Summary and conclusion
Markowitz‟s theory and prospect theory make different predictions about 2x choices, in
which the magnitude and sign of x change, and p choices, in which the magnitude of p and the
sign of x change.
Concerning 2x choices, Markowitz predicts the M4 pattern, whereas prospect theory
predicts a twofold pattern of global risk aversion in gains and global risk seeking in losses.
Drawing on Hogarth and Einhorn‟s (1990) results, we expect to confirm the Anna Karenina
effect: Uniformity in gains (a reversal from risk seeking to risk aversion) and diversity in losses
(dispersion of those who are uniform in gains into distinct preference groups).
Concerning p choices, prospect theory and Markowitz each predict four preference
patterns, with both predicting a twofold pattern of global risk aversion in gains and global risk
seeking in losses. At the aggregate level, we expect to confirm the KT4 pattern, thus replicating
past results. At the disaggregate level, we expect a majority to exhibit PT compatible preference
patterns, with both the KT4 pattern and the two threefold patterns contributing to the KT4 pattern
at the aggregate level.
Method
Participants. Population and recruitment were the same as in Experiment 1. The
participants were 255 members of eLab, including 92 men and 163 women, averaging 38 years of
age. The great majority had some college or a Bachelor‟s degree. They were entered in a lottery
offering a 1 in 50 chance to win a $50 Amazon.com gift certificate.
Stimuli. Participants first made two p choices and then seven 2x choices for gains and for
losses. In the series of two choices, the magnitude of the risky outcome was $100 and its
probability was .05 and .95, in that order. In the series of seven choices, the magnitude of the sure
The Anna Karenina Effect in Risky Choice 22
outcome ranged from $0.25 to $250,000 in multiples of 10, and in that order. In each series, the
order of gains and losses was counterbalanced.
Results
Figure 4.1 shows the aggregate results for the p choices. There is strong evidence for the
KT4 pattern (a reversal from risk seeking to risk aversion in gains and from risk aversion to risk
seeking in losses). Figure 4.2 shows the aggregate results for the 2x choices. There is no evidence
for the M4 pattern (a reversal from risk seeking to risk aversion in gains and from risk aversion to
risk seeking in losses) or the twofold pattern predicted by prospect theory (global risk aversion in
gains and global risk seeking in losses). Aggregate choices for gains were monotonically related
to their magnitude, and reversed from risk seeking to risk aversion. Aggregate choices for losses
also reversed from risk seeking to risk aversion, but were not monotonically related to outcome
magnitude.
<Insert Figure 4 about here>
Preference heterogeneity. Table 5 shows the preference groups that emerged at the
disaggregate level for the p choices. A majority (55%) exhibited the four PT compatible
preference patterns. Three of these groups contributed to the KT4 pattern at the aggregate level:
Those exhibiting the KT4 pattern, i.e., a reversal from risk seeking to risk aversion in gains and
from risk aversion to risk seeking in losses (28%), those exhibiting a reversal in losses but global
risk aversion in gains (16%), and those exhibiting a reversal in gains but global risk seeking in
losses (6%). A minority (21%) exhibited the four M compatible preference patterns. Because 5%
of these cases exhibited the PT / M compatible preference pattern of global risk aversion in gains
and global risk seeking in losses, Markowitz uniquely accounted for only 16% of the cases.
<Insert Table 5 about here>
The Anna Karenina Effect in Risky Choice 23
Table 6 shows the preference groups that emerged at the disaggregate level for the 2x
choices. A preference pattern with more than one preference shift is classified as „inconsistent.‟
<Insert Table 6 about here>
As in Experiment 1, a majority (60%) changed from risk seeking to risk aversion in gains,
but differed in losses. The largest groups were the Wavering (25%) and the non-Markowitzians
(22%). The other groups comprising the majority emerged in small numbers: The Cautious (6%),
the Audacious (4%), and the Markowitzians (4%). Only two participants (1%) exhibited the
preference pattern predicted by prospect theory (global risk aversion in gains and global risk
seeking in losses).
A majority (61%) of those who, in 2x choices, changed from risk seeking to risk aversion
in gains exhibited, in p choices, a PT compatible preference pattern, 2(1) = 7.51, p = .01, and the
incidence rate of PT compatible preference patterns was significantly lower (47%) among those
who, in 2x choices, did not change from risk seeking to risk aversion in gains, 2(1) = 5.19, p =
.02. Therefore, confirmatory cases in 2x choices tended to be confirmatory cases in p choices.
We also explored how the Wavering would most likely have been classified had they not
been inconsistent in losses. The results are included in Table 6. The non-Markowitzians benefited
most from the reclassification of the Wavering, becoming by far the largest group (31%). This is
reflected in the aggregate results: Risk preferences reversed from risk seeking to risk aversion in
both gains and losses.
Preference uncertainty. We analyzed choice times separately for the p choices and the 2x
choices. In the p choices, choices took longer in losses than in gains, t(254) = 6.52, p = .00 (t-test
for dependent samples performed on the logarithms of the choice times), indicating greater
preference uncertainty for losses than for gains.
The Anna Karenina Effect in Risky Choice 24
In the 2x choices, the Wavering were the largest preference group, which, as previously,
may indicate greater preference uncertainty for losses than for gains. This was confirmed by an
analysis of choice times. Across all participants, the ratio between choice times in losses and
choice times in gains was 1.17, t(253) = 6.24, p = .00, indicating greater preference uncertainty
for losses than for gains. This is also reflected in the aggregate results: Choice probabilities were
closer to chance level for losses than for gains. The loss/gain ratio was significantly greater for
the Wavering (1.39) than for the other participants (1.10), t(253) = 3.94, p = .00.
Discussion
Prospect theory both succeeded and failed. On the one hand, it succeeded in predicting p
choices: Most participants exhibited the four PT compatible preference patterns, including the
KT4 pattern. Markowitz did substantially worse. On the other hand, prospect theory failed in
predicting 2x choices: Almost no-one exhibited the twofold pattern of global risk aversion in
gains and global risk seeking in losses. Markowitz did slightly better. In 2x choices, a majority
exhibited uniformity in gains, i.e., a reversal from risk seeking to risk aversion, but diversity in
losses, i.e., preference heterogeneity and preference uncertainty. In sum, we confirm the Anna
Karenina effect in 2x choices and prospect-theoretical predictions in p choices. An additional
finding was that confirmatory cases in 2x choices tended to be confirmatory cases in p choices.
The composition of the preference groups differed between Experiment 1 (10x choices)
and Experiment 2 (2x choices). For instance, the Cautious were the largest group in the former,
whereas the non-Markowitzians were by far the largest group in the latter. This leaves open the
question of whether the same theoretical account can cover both 10x and 2x (as well p) choices.
We will propose such a theoretical account, and test it in Experiment 3.
The evidence on p choices shows that an expected value model like Markowitz‟s cannot
be correct, so we focus our attention on prospect theory, a non-expected value model. We ask
The Anna Karenina Effect in Risky Choice 25
what the minimum modifications of original prospect theory must be in order to explain our
findings. The modifications involve non-standard assumptions about the shape of the value
function and the probability-weighing function, and about how their shape differs between
decision makers. Our prospect-theoretical interpretation has testable implications for the way in
which different decision makers react differently to changes in probability level (.1 and .5). We
test these implications in Experiment 3.
The Anna Karenina Effect in Risky Choice:
A Prospect-Theoretical Interpretation
Table 7 shows the distribution of the five preference groups that reversed from risk
seeking to risk aversion in gains, constituting 75% of the 10x choice sample (Experiment 1) and
60% of the 2x choice sample (Experiment 2). Four of these groups combine the reversal in gains
with a definite pattern in losses: The Markowitzians reverse from risk aversion to risk seeking,
the Audacious exhibit global risk seeking, the non-Markowitzians reverse from risk seeking to
risk aversion, and the Cautious exhibit global risk aversion. When going from 10x to 2x choices,
however, the shares of the Markowitzians and the Cautious fell while the shares of the Audacious
and the non-Markowitzians rose. We will discuss how prospect theory can accommodate the four
groups, and why the shares of the respective groups rose or fell.
<Insert Table 7 about here>
We view the four preference groups in Table 7 as a behavioral reflection of four
evaluation groups, which we name M (after the Markowitzians), O (Audacious), N (Non-
Markowitzians), and D (Cautious). The names refer to the preference pattern exhibited by the
evaluation group in 2x choices. The groups differ in their value function and probability-weighing
function, and have the following Characteristics:
a. All groups have a value function that is decreasingly elastic in gains.
The Anna Karenina Effect in Risky Choice 26
b. Groups M and A have a value function that is decreasingly elastic in losses.
c. Groups M and A have a value function that is more elastic in gains than in losses (gain
amplification).
d. Groups N and C have a value function that is increasingly elastic in losses.
e. Group C has the most elevated probability-weighing function, and group A has the
least elevated probability-weighing function (see Figure 5).6
f. All four groups have a more elevated probability-weighing function than in
conventional prospect theory, overweighing low to moderate probabilities and
underweighing high ones (see Figure 5).7
While the names of the evaluation groups are the initials of the preference pattern exhibited in 2x
choices, we will show it is possible for group A to exhibit the Markowitzian pattern, and for
group C to exhibit the non-Markowitzian pattern in 10x choices. This explains the differences in
preference-group composition between Experiments 1 and 2.
<Insert Figure 5 about here>
We begin with the case of gains, in which all evaluation groups exhibit the Markowitzian
reversal from risk seeking to risk aversion. This, as discussed earlier, requires that the probability-
weighing function outweighs the value function for small outcomes, but is outweighed by it for
large ones. Defining a value-ratio function as R(x, p) = v(x) / v(x / p), we have
R(mx, p) > w(p) > R(x, p) for m > 1 and x > 0,
The case of gains is shown in Figure 6.1. In what we call an „Rw-mapping plot,‟ probability-
weighing functions are superimposed on value-ratio functions. Two value-ratio functions, one for
small x and another for large x (i.e., mx), are plotted as a function of p. Because the value function
is concave over gains, both R(x, p) and R(mx, p) are greater than p. Moreover, because the value
function is decreasingly elastic in gains (Characteristic a), R(mx, p) is greater than R(x, p). The
The Anna Karenina Effect in Risky Choice 27
three probability-weighing functions from Figure 5, which overweigh low to moderate
probabilities (Characteristic f), are superimposed on the value-ratio functions. For low to
moderate probabilities, w(p) is greater than R(x, p) but smaller than R(mx, p), yielding the
Markowitzian reversal from risk seeking to risk aversion.
<Insert Figure 6 about here>
We now turn to losses, where the four evaluation groups have distinct preference patterns.
The Rw-mapping plots are shown in Figure 6.2 (groups M and A) and Figure 6.3 (groups N and
C).
We first consider groups M and A. Their value function is decreasingly elastic in losses
(Characteristic b), so that, as in the case of gains, R(mx, p) is greater than R(x, p). Furthermore,
their value function is less elastic in losses than in gains (gain amplification; Characteristic c), so
that the value-ratio functions are more elevated in losses than in gains (compare Figures 6.1 and
6.2). This has no consequences in group M, whose probability-weighing function is more
elevated than that of group A (Characteristic e): For low to moderate probabilities, w(p) is greater
than R(x, p) but smaller than R(mx, p), yielding the Markowitzian reversal from risk aversion to
risk seeking. In group A, however, gain amplification does have consequences. For moderate
probabilities, w(p) is smaller than R(x, p) as well as R(mx, p), yielding the Audacious pattern of
global risk seeking. For low probabilities, which are overweighed more than moderate ones, w(p)
is greater than R(x, p) but smaller than R(mx, p), yielding the Markowitzian pattern. Thus, for low
probabilities, group A exhibits the Markowitzian pattern, but, for moderate probabilities, group A
exhibits the Audacious pattern of global risk seeking. Because group A goes from Markowitzian
to Audacious when going from 10x to 2x choices, but group M remains Markowitzian, we should
see a fall in the share of the Markowitzians and a rise in the share of the Audacious (as we see in
Table 7). This leads us to the first prediction of our prospect-theoretical analysis: When going
The Anna Karenina Effect in Risky Choice 28
from 10x to 2x choices, there should be many cases changing from Markowitzian to Audacious
but few cases changing in the opposite direction.
For groups N and C, the situation is similar to that of groups M and A, except that they
have an increasingly elastic value function over losses (Characteristic d), so that R(x, p) is greater
than R(mx, p).8 The probability-weighing function of group C is more elevated than that of group
N (Characteristic e). For low to moderate probabilities, w(p) is greater than R(x, p) as well as
R(mx, p), yielding the Cautious pattern of global risk aversion.9 The probability-weighing
function of group N is less elevated than that of group C. For moderate probabilities, w(p) is
smaller than R(x, p) but greater than R(mx, p), yielding the non-Markowitzian reversal from risk
seeking to risk aversion. For low probabilities, which are overweighed more than moderate ones,
w(p) is greater than R(x, p) as well as R(mx, p), yielding the Cautious pattern. Thus, for low
probabilities, group N exhibits the Cautious pattern, but, for moderate probabilities, group N
exhibits the non-Markowitzian pattern. Because group N goes from Cautious to non-
Markowitzian when going from 10x to 2x choices, but group C remains Cautious, we should see a
fall in the share of the Cautious and a rise in the share of the non-Markowitzians (as we see in
Table 7). This leads us to the second prediction of our prospect-theoretical analysis: When going
from 10x to 2x choices, there should be many cases changing from Cautious to non-Markowitzian
but few cases changing in the opposite direction.
In Experiment 3, we test the above predictions by asking the same participants to make
both 10x and 2x choices.
Experiment 3
Method
Participants. Population and recruitment were the same as in Experiments 1 and 2. The
participants were 569 members of eLab, including 193 men and 365 women (11 participants did
The Anna Karenina Effect in Risky Choice 29
not fill in their gender), averaging 38 years of age. The great majority had some college or a
Bachelor‟s degree. They were entered in a lottery offering a 1 in 50 chance to win a $50
Amazon.com gift certificate.
Stimuli. There were five conditions. Four conditions counterbalanced the order of choice
tasks. Participants first made two p choices for gains and for losses. The magnitude of the risky
outcome was $100 and its probability was .05 and .95, in that order. The order of gains and losses
was counterbalanced between participants. Participants then made two series of seven 10x and 2x
choices for gains and for losses. The magnitude of the sure outcome ranged from $0.25 to
$250,000 in multiples of 10, and in that order. The order of 10x and 2x choices, and, within them,
the order of gains and losses, was counterbalanced between participants. The fifth condition
randomized all choices. This condition was included as a check on the robustness of our results,
and potential order effects (see Harrison, Johnson, McInnes, & Rutström, 2005). Randomization
did not affect the results (see Discussion), and our analyses will therefore collapse over all five
conditions.
Results
Figure 7.1 shows the aggregate results for the p choices, which replicate the KT4 pattern
from Experiment 2. Figure 7.2 shows the aggregate results for the x choices. The results for the
10x choices replicate those from Experiment 1, and the results for the 2x choices replicate those
from Experiment 2.
<Insert Figure 7 about here>
Preference heterogeneity. The results at the disaggregate level were also generally
consistent with those from the first three experiments, and can be summarized as follows.
In the p choices, a majority (51%) exhibited the four PT compatible preference patterns.
Three of these groups contributed to the KT4 pattern at the aggregate level: Those exhibiting the
The Anna Karenina Effect in Risky Choice 30
KT4 pattern, i.e., a reversal from risk seeking to risk aversion in gains and from risk aversion to
risk seeking in losses (26%), those exhibiting a reversal in losses but global risk aversion in gains
(14%), and those exhibiting a reversal in gains but global risk seeking in losses (6%). A minority
(24%) exhibited the four M compatible preference patterns. Because 6% of these cases exhibited
the PT / M compatible preference pattern of global risk aversion in gains and global risk seeking
in losses, Markowitz uniquely accounted for only 18% of the cases. In sum, the results are almost
identical to those from Experiment 2.
In the 10x choices, the five preference groups exhibiting a reversal from risk seeking to
risk aversion in gains included 62% of the sample, compared to 75% in Experiment 1. In the 2x
choices, the five preference groups included 51% of the sample, compared to 60% in Experiment
2. The complete breakdown by choice type and preference group, before and after reclassification
of the Wavering, is provided in Table 8. We can see that, when going from 10x to 2x choices, the
shares of the Markowitzians and the Cautious fell while the shares of the Audacious and the non-
Markowitzian patterns rose, as we saw in the comparison between Experiments 1 and 2. While
our non-standard modification of prospect theory accounts for a majority of the cases (62% in the
10x choices and 51% in the 2x choices), standard prospect theory, augmented with generically
decreasing elasticity, accounts for only 31% of the cases in the 10x choices (21% Markowitzians,
6% Audacious, and 4% others), and for only 2% of the cases in the 2x choices (those exhibiting
global risk aversion in gains and global risk seeking in losses).
<Insert Table 8 about here>
As in Experiment 2, confirmatory cases in x choices tended to be confirmatory cases in p
choices. On the one hand, a majority (60%) of those who, in 10x choices, changed from risk
seeking to risk aversion in gains exhibited, in p choices, a PT compatible preference pattern, 2(1)
= 14.81, p = .00, and the incidence rate of PT compatible preference patterns was significantly
The Anna Karenina Effect in Risky Choice 31
lower (37%) among those who, in 10x choices, did not change from risk seeking to risk aversion
in gains, 2(1) = 28.14, p = .00. On the other hand, a majority (58%) of those who, in 2x choices,
changed from risk seeking to risk aversion in gains exhibited, in p choices, a PT compatible
preference pattern, 2(1) = 7.64, p = .01, and the incidence rate of PT compatible preference
patterns was significantly lower (45%) among those who, in 2x choices, did not change from risk
seeking to risk aversion in gains, 2(1) = 5.19, p = .02. Overall, the results are very similar for the
two choice types.
Table 9 reports observed minus expected number of transitions between preference groups
across 10x and 2x choices, upon reclassification of the Wavering. Our predictions are confirmed.
First, those who were Markowitzians in 10x choices tended to either remain Markowitzians or
become Audacious in 2x choices (they also tended to become Wavering), whereas the Audacious
tended to remain Audacious. Among those who made a transition between these two preference
groups across 10x and 2x choices, a large majority (86%) were Markowitzians in 10x choices but
Audacious in 2x choices, 2(1) = 7.14, p = .00. Second, those who were Cautious in 10x choices
tended to either remain Cautious or become non-Markowitzians in 2x choices, whereas non-
Markowitzians tended to remain non-Markowitzians. Among those who made a transition
between these two preference groups across 10x and 2x choices, a large majority (86%) were
Cautious in 10x choices but non-Markowitzians in 2x choices, 2(1) = 15.21, p = .00. This is
strong support for our prospect-theoretical interpretation of the results.
<Insert Table 9 about here>
Preference uncertainty. In the p choices, choices took longer in losses than in gains,
t(568) = 19.81, p = .00, indicating greater preference uncertainty for losses than for gains.
The Anna Karenina Effect in Risky Choice 32
In the 10x and 2x choices, the Wavering were the largest preference group, which, as
previously, may indicate greater preference uncertainty for losses than for gains. This was
confirmed by an analysis of choice times. Across all participants, the ratio between choice times
in losses and choice times in gains was 2.19 for 10x choices, t(567) = 8.07, p = .00, and 2.87 for
2x choices, t(257) = 9.82, p = .00, indicating greater preference uncertainty for losses than for
gains. The loss/gain ratio was greater for the Wavering (3.16) than for the other participants
(1.94) in 10x choices, t(567) = 2.19, p = .03, and greater for the Wavering (5.04) than for the
other participants (2.51) in 2x choices, t(567) = 2.53, p = .01.
Discussion
The results for preference heterogeneity in p choices and preference uncertainty in both
choice types (10x and 2x) confirm the results from Experiments 1 and 3. Moreover, the results for
preference heterogeneity in 10x and 2x choices confirm our prospect-theoretical interpretation,
meaning we arrive at a successful extension of prospect theory to individual differences in risky
choice. This individual-difference analysis accounts for the Anna Karenina effect (uniformity in
gains and diversity in losses) and the variability in the specific patterns of diversity across choice
types.
Our analysis covers a significantly greater percentage of participants in 10x choices than
in 2x choices, both in the between-participants analysis (75% in Experiment 1 vs. 60% in
Experiment 2, see Table 7) and in the within-participant analysis (62% vs. 51% in Experiment 3,
see Table 8). This may indicate that the overweighing of a .1 probability is more common than
the overweighing of a .5 probability, as standard prospect theory would imply. However, only a
very small percentage of participants (2%) exhibited the preference pattern predicted by prospect
theory in 2x choices (global risk aversion in gains and global risk seeking in losses). It thus seems
that, when going from 10x to 2x choices, about 10% of the participants slip through the net, and
The Anna Karenina Effect in Risky Choice 33
start doing things that neither standard prospect theory nor our non-standard modification of
prospect theory accommodates.
As mentioned earlier, we included a randomized order condition. Randomization did not
affect our conclusions about the differential composition of preference groups in 10x and 2x
choices. To substantiate this, we computed, separately for the randomized order condition and the
four non-randomized order conditions, the difference in incidence rates of the Markowitzians, the
non-Markowitzians, the Cautious, and the Audacious between 10x and 2x choices. Figure 8
shows the result: Our conclusions are the same, regardless of whether choices were randomized
or not.
<Insert Figure 8 about here>
General Discussion
Harry Markowitz hypothesized a fourfold pattern of risk preferences, the „M4 pattern,‟
with risk aversion for large gains and small losses, but risk seeking for small gains and large
losses. Daniel Kahneman and Amos Tversky later hypothesized another fourfold pattern of risk
preference, the „KT4 pattern,‟ with risk aversion for likely gains and unlikely losses, but risk
seeking for unlikely gains and likely losses. To explain the M4 pattern, Markowitz preserved the
expectation principle, i.e., w(p) = p, but replaced a concave utility function over final states of
wealth by a triply inflected value function over changes of wealth, which, as depicted in Figure 1,
is concave over large gains and small losses, but convex over small gains and large losses. To
explain the KT4 pattern, Tversky and Kahneman replaced the expectation principle by probability
weighing, with an overweighing of low probabilities, i.e., w(p) > p, and an underweighing of
moderate to high ones, i.e., w(p) < p. Because Tversky and Kahneman also replaced the triply
inflected value function by a singly inflected value function, which is concave over gains and
convex over losses, prospect theory can explain the M4 pattern only by combining the
The Anna Karenina Effect in Risky Choice 34
overweighing of low probabilities with a value function that is decreasingly elastic over gains
and losses of increasing magnitude.
We confirmed the KT4 pattern, which was already well-established, but not the M4
pattern. While Markowitz‟s hypothesis is correct for gains, it is generally wrong for losses.
Different preference groups emerge, which are depicted in Figure 9 from a Markowitzian
perspective: Preserving the expectation principle, each preference group has its appropriately
inflected value function. This proposal, however, does not explain why the composition of
preference groups changes across probability levels. Nor does it explain the KT4 pattern. To
explain this, we need probability weighing, and, therefore prospect theory.
<Insert Figure 9 about here>
We have shown that prospect theory can accommodate our evidence if we make non-
standard assumptions about both the value function and the probability-weighing function. These
assumptions however, do not change the central tenets of prospect theory. For instance, there is
nothing in prospect theory that prohibits an overweighing of moderate probabilities, or a non-
constant elasticity of the value function. Moreover, different experimental setups and individual
differences may affect the functional forms of prospect theory, which have, indeed, shown wild
variability across studies, and across individuals within any particular study (e.g., Gonzalez &
Wu, 1999). In our experimental setup, individuals dispersed into several groups, each with its own
set of functional forms. However, our analysis, like most analyses of risky choice, applies to pallid,
monetary outcomes. It has been shown that both the probability-weighing function (Rottenstreich &
Hsee, 2001) and the value function (Hsee & Rottenstreich, 2004) can change depending on whether
the outcomes are pallid or vivid. More generally, the stability of functional forms in decision making
has been called into question (e.g., Ungemach, Stewart, & Reimers, 2011).
The Anna Karenina Effect in Risky Choice 35
Markowitz framed his theory as an explanation of why people simultaneously purchase
insurance and gambles. Upon identifying implausible implications of the analysis offered by
Friedman and Savage (1948), Markowitz proposed his triply inflected value function over
changes of wealth, in which a small loss (the price of an insurance or a gamble) is barely different
from the status quo, whereas the large potential calamity against which one insures and the large
potential prize for which one gambles are not. Our analysis of the Anna Karenina effect in risky
choice shows that Markowitz‟s proposal is not viable either. So we must sketch the implications
of our analysis for the behavior that he set out to explain.
In standard prospect theory, the purchase of insurance and gambles is promoted by the
overweighing of low probabilities (Kahneman & Tversky, 1979). The willingness to purchase
insurance is mitigated by the convexity of the value function over losses, and the willingness to
purchase gambles is mitigated both by the concavity of the value function over gains and by loss
aversion when the loss (the price of the gamble) is segregated from the potential gain (the prize;
see Kahneman & Tversky, 1979). Apart from these general principles, however, there are, in our
non-standard modification of prospect theory, particularities of different groups in the evaluation
of risky prospects.
We identified four groups, each with their unique combination of a value function and a
probability-weighing function. Let us name them by the preferences they expressed in the 50/50
gambles involving losses. Using the same functional forms as in Figure 6, the reservation prices
for insuring against a .01 probability of losing $1,000 (fair price is $10) are:
$16 (Markowitzians),
$11 (Audacious),
$193 (non-Markowitzians), and
$255 (Cautious).
The Anna Karenina Effect in Risky Choice 36
Three relations can be established. First, the Markowitzians and the Audacious react in the same
way to losses, but the Markowitzians put a greater weight on the .01 probability of losing than the
Audacious, so that their reservation price is higher. Second, the Markowitzians and the non-
Markowitzians put the same weight on the .01 probability of losing, but the non-Markowitzians,
with their increasingly elastic value function over losses, react more strongly to the $1,000 loss
than the Markowitzians, with their decreasingly elastic value function over losses, so that their
reservation price is higher. Third, the non-Markowitzians and the Cautious react in the same way
to losses, but the Cautious put a greater weight on the .01 probability of losing than the non-
Markowitzians, so that their reservation price is higher. Altogether, we have a differentiated
market for insurance, with the willingness to participate depending on probability weighing and
outcome valuation. In realistic situations, with companies selling insurance well above the fair
price, the Markowitzians and the Audacious would, in this simulation, probably not participate,
whereas the non-Markowitzians and the Cautious probably would. This is good news for the
insurance companies, because the non-Markowitzians and the Cautious were relatively large
groups in our experiments.
In making the transition from insurance to gambles, note that those who are „Audacious‟
in losses are, because they put less weight on risky outcomes, more cautious than Markowitzians
in gains, and that those who are „Cautious‟ in losses are, because they put more weight on risky
outcomes, more audacious than non-Markowitzians in gains. Using the same functional forms as
before, and assuming a loss-aversion coefficient equal to 2, the reservation prices for a gamble
offering a .01 probability of winning $1,000 are:
$38 (Markowitzians),
$23 (Audacious, now more cautious),
$15 (non-Markowitzians), and
The Anna Karenina Effect in Risky Choice 37
$21 (Cautious, now more audacious).
Three observations can be made. First, the non-Markowitzians put a higher price on the insurance
than the Markowitzians, but put a lower price on the gamble, and for the same reason. These two
groups react in the same way to gains, and put the same weight on the .01 probability of winning,
but the non-Markowitzians, with their increasingly elastic value function over losses, react more
strongly to the price of the gamble than the Markowitzians, with their decreasingly elastic value
function over losses, so that their reservation price is lower. Second, the Markowitzians and the
„Audacious‟ put a higher price on the gamble than on the insurance, which means that, in this
simulation, gain amplification (a stronger reaction to the potential prize than to the potential
calamity) outweighs loss aversion (a disproportionately stronger reaction to the price of the
gamble than to the potential prize). Third, the non-Markowitzians and the „Cautious‟ put a
significantly higher price on the insurance than on the gamble. This has two reasons. One is that
their value function is increasingly elastic in losses but decreasingly elastic in gains (a stronger
reaction to the potential calamity than to the potential prize). The other reason is loss aversion (a
disproportionately stronger reaction to the price of the gamble than to the potential prize). The
combined influence of these two factors yields, in these groups, the pronounced discrepancy
between willingness to insure and willingness to gamble. Altogether, our analysis implies not
only inter-individual differences, but also pronounced intra-individual differences in the reaction
to insurance and gambles.
Kahneman and Tversky (1979) recognized that the purchase of insurance and gambles can
be determined by many factors, and that the psychophysics of risk and value is only one of them.
The same caveat applies to our analysis. Moreover, the above analysis of insurance and gambles
does not cover all realistic situations of decision under risk. If insuring is a form of „playing safe,‟
so is „safe sex,‟ in which a relatively small inconvenience (a condom) averts a large potential cost
The Anna Karenina Effect in Risky Choice 38
(premature death). Similarly, if gambling is a form of „taking a chance,‟ so is „unsafe sex,‟ in
which a relatively small enjoyment invites a large potential cost. The latter example is an
interesting one, because, in gambling, incurring a small loss opens the possibility of a large gain,
whereas, in unsafe sex, obtaining a small gain opens the possibility of a large loss. Alternative
forms of playing safe and taking a chance may receive closer attention in future analyses.
Our modification of prospect theory essentially involves assumptions about the elevation
of the probability-weighing function (depending on the person, it is more or less elevated), and
assumptions about the elasticity of the value function (depending on the person and the domain,
its elasticity decreases or increases with outcome magnitude). Stake-dependent elasticity has not
previously been associated with prospect theory. After Tversky and Kahneman (1992), users of
prospect theory routinely resort to a power value function, the elasticity of which is constant in
outcome magnitude. We found decreasing elasticity in gains. In losses, however, some seem to
combine diminishing sensitivity with decreasing elasticity, while others seem to combine it with
increasing elasticity. Thus, decreasing elasticity is not a universal property of the value function
over losses, but constant elasticity is certainly disconfirmed by our findings.
While the preference heterogeneity observed in our experiments can be accommodated by
prospect theory, we are still left with the question why preference uncertainty is greater for losses
than for gains, as indicated, in our experiments, by greater choice inconsistency and longer choice
times (see also Arkoff, 1957; Minor, Miller, & Ditrichs, 1968; Murray, 1975; Schill, 1966), with
choice times being especially long among those who exhibited choice inconsistency. To account
for this, it may be necessary to go beyond the deterministic, static, and psychophysical approach
taken by prospect theory. One candidate theory that does this is decision field theory (Busemeyer
& Townsend, 1993). This is a probabilistic, dynamic, and motivational theory, which predicts
choice times and choice inconsistency as well as the choices made. It casts the contrast between
The Anna Karenina Effect in Risky Choice 39
gains and losses in dynamic terms, and it can predict that losses lead to longer choice times and
greater choice inconsistency than gains. In Appendix B, we show how prospect theory can be
incorporated into decision field theory, and how this preserves the above reconstruction of our
evidence while permitting us to explain why preference uncertainty is greater for losses than for
gains.
In sum, we examined Markowitz‟s hypothesis about decision under risk, and, instead of
the hypothesized pattern, we obtained the Anna Karenina effect. This effect indicates that loss
aversion does not fully capture the asymmetry between gains and losses. Losses not only loom
larger than gains, they also engender more diversity, with different people exhibiting different
preference patterns (preference heterogeneity), and many people exhibiting preferences that do
not conform to a definite pattern (preference uncertainty; also revealed by longer choice times).
Although our analysis was restricted to the Markowitzian choice task, it covered three central
issues in decision under risk: How risk preferences depend on the magnitude and sign of the
outcomes, and on the level of risk involved. It is still an open issue how pervasive the Anna
Karenina effect is in other choice tasks, but it appears that the analysis of modal preference
patterns as governed by representative agents is misleading: At least mid-range representative
agents must be postulated to understand decision under risk.
The Anna Karenina Effect in Risky Choice 40
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Appendix A
In this Appendix, we discuss the concept of elasticity. Following Loewenstein and Prelec
(1992), elasticity is defined as
)(
)(
/1
)(/)(
)log(
))(log()(
xv
xvx
x
xvxv
x
xvxv
.
Thus, elasticity is the rate at which outcome value v(x) changes relative to the rate at which
outcome x changes. Diminishing sensitivity, which is commonly stated as 0)( xv for x > 0 and
0)( xv for x < 0, can also be stated as 1)(0 xv . This means, for instance, that a 10%
change in the magnitude of x corresponds to less than a 10% change in the magnitude of v(x).
Elasticity can be constant for all x, or it can vary with x. A power value function v(x) = x
has constant elasticity, because xx-1
/x = . The elasticity of the value function is reflected in
the ratios between outcome values, such as the value ratio R(x, p) = v(x) / v(x / p) discussed in the
paper. The value ratio is inversely related to elasticity. When 1)( xv , meaning that the value
function is linear, we have R(x, p) = p. When 0)( xv , meaning that the value function is
constant, we have R(x, p) = 1. When 1)(0 xv , the value ratio is a concave increasing
function of p, i.e., R(x, p) = p.
An example of a value function with increasing elasticity is an additive combination of
two power value functions, one characterized by a lower elasticity, or more strongly diminishing
sensitivity, than the other:
1)( xxxv ,
where 0 < < ½ and > 0. The elasticity of this value function is
1
1)1()(
xx
xxxv .
The Anna Karenina Effect in Risky Choice 46
For this value function, elasticity approaches as x 0, and 1- as x . Intuitively, when x is
very small, most of the curvature is determined by the first term of the value function, x, with
elasticity. As x becomes larger and larger, however, most of the value, and, therefore, most of
the curvature, is determined by the second term of the value function, x1-
, with elasticity 1-.
Diminishing sensitivity is easier to combine with decreasing elasticity than with
increasing elasticity (see Scholten & Read, 2010). For instance, the value function v(x) = log(x)
has as its elasticity )log(/1)( xxv , which is clearly decreasing in x. A log function like this one
was proposed by Bernoulli (1738/1954) to describe utility from final states of wealth. However, it
has several shortcomings when used to describe value from changes in wealth relative to a neutral
reference point. An alternative log function, proposed by Scholten and Read (2010) in the domain
of intertemporal choice, is the following:
)1log(1
)( xxv
.
The elasticity of this value function is
)1log()1()(
xx
xxv
,
which approaches 1 as x 0, and 0 as x .
The Anna Karenina Effect in Risky Choice 47
Appendix B
In this Appendix, we discuss how prospect theory can be incorporated into decision field
theory, and we derive a Markowitzian choice pattern in which choice inconsistency is greater for
losses than for gains.
In decision field theory, choice is the end point of a stochastic-dynamic process. At each
point of the choice process, valences, or motivational values, are assigned to two actions (e.g.,
choice of the sure thing and choice of the gamble) as a function of the valences and the weights
assigned to the consequences of the actions (e.g., $2.50 and $25). The weights, in turn, are a
function of the attention weights assigned to the events that determine the consequences (e.g.,
heads or tails if the gamble is a coin flip) and the so-called „goal gradient weights‟ (discussed
below). At each point of the choice process, the decision maker‟s preference state is updated by
the difference between the valences of the actions at that point. Across the choice process, the
valence differences fluctuate because of fluctuations of the attention weights and the goal
gradient weights. The choice process ends when the preference state reaches a threshold. The
action that is preferred at that point is taken.
Prospect theory can be incorporated into decision field theory by setting the attention
weights equal to the weights from the probability-weighing function, and setting the valences
equal to the values from the value function. This allows us to explain why preference uncertainty
is greater for losses than for gains.
The core difference between gains and losses lies, according to decision field theory, in
the goal gradient weights. The consequences of an action become more salient as the preference
state for that action approaches its threshold. If there is little or no possibility of taking an action
(a great distance between the preference state and threshold), virtually no attention is given to its
consequences, but if the possibility of taking an action increases (a smaller distance between the
The Anna Karenina Effect in Risky Choice 48
preference state and the threshold), attention to its consequences increases. The goal gradient is
steeper for losses than for gains, meaning that the goal gradient weights are a more steeply
negative function of the distance between the preference state and the threshold. Metaphorically
speaking, decision making is hill-climbing, and the hill is steeper for losses than for gains. This
leads to longer choice times in losses than in gains. For further details on this, see Busemeyer and
Townsend (1993). The steeper goal gradient for losses than for gains also leads to greater choice
inconsistency in losses than in gains, as illustrated below.
In decision field theory, the probability of choosing the sure thing instead of the gamble
can be given as )/()/(2 dF , where F is the standard logistic cumulative distribution
function, d is the mean valence difference between the sure thing and the gamble, , i.e., V(x) -
V(10x, .1), is the standard deviation of the valence difference, and is the threshold.
We consider a simplified formulation of decision field theory, in which the valence of the
sure thing is V(x) = (1 - g)v(x), and the valence of the gamble is V(10x, .1) = (1 - g)w(.1)v(10x),
where 0 < g < 1 is the slope of the goal gradient. The outcome valences are given by Scholten and
Read‟s (2010) value function v(x) = (1 / ) log(1 + x) for x > 0 and v(x) = ( / ) log(1 + x)
for x < 0, where , > 0 are diminishing sensitivity and > 1 is constant loss aversion. This
describes the case of the Markowitzians and the Audacious since the value function is
decreasingly elastic in both gains and losses. The attention weights are given by Prelec‟s (1998)
probability-weighing function w(p) = exp(-b (-log(p))a), where 0 < a < 1 is diminishing
sensitivity and 0 < b < 1 is elevation.
The standard deviation of the valence difference between a sure thing and a gamble is
equal to the standard deviation of the gamble:
22)1,.10(0)9(.)1()1,.10()10()1(.)1( xVwgxVxvwg .
The Anna Karenina Effect in Risky Choice 49
Figure B1 exhibits the case where g = .1 for x > 0 and g = .9 for x < 0 (i.e., a steeper goal
gradient for losses than for gains), = = 0.25, = 2, a = 0.5, b = 0.6, and = . A steeper goal
gradient reduces both the mean valence difference d and the standard deviation of the valence
difference , but reduces d by a greater proportion than , leading to choice probabilities that are
closer to chance level. Constant loss aversion increases d and by the same factor , and thus has
no effect on choice probabilities. However, if we drop the assumption that is identical to , or,
more generally, proportional to (see Busemeyer & Townsend, 1993), and if we assume instead
that is constant (Busemeyer, personal communication), then constant loss aversion decreases
the ratio / by a factor 1 / , and, therefore, also leads to choice probabilities that are closer to
chance level.
<Insert Figure B1 about here>
The Anna Karenina Effect in Risky Choice 50
Authors‟ Note
Marc Scholten, associate professor with habilitation at the ISPA University Institute,
Lisbon, Portugal. Daniel Read, professor of behavioral economics at Durham Business School,
Durham University, Durham, United Kingdom, and visiting professor at Yale School of
Management, Yale University. We acknowledge the financial support received from the
Fundação para a Ciência e Tecnologia (FCT), program POCI 2010, and project PTDC/PSI-
PCO/101447/2008. This paper has greatly benefited from discussions with Jerome Busemeyer
and Peter Wakker. We are also very grateful to Andrew Meyer for his assistance in conducting
the experiments.
Correspondence concerning this paper should be addressed to Marc Scholten, ISPA
University Institute, Rua Jardim do Tabaco 34, 1149-041 Lisboa, Portugal. E-Mail:
The Anna Karenina Effect in Risky Choice 51
TABLE 1
Markowitz’s hypothesis.
Sure outcome
Small Large
Gains
Risk seeking
($1, .1) $0.10
Risk aversion
$1m ($10m, .1)
Losses
Risk aversion
-$0.10 (-$1, .1)
Risk seeking
(-$10m, .1) -$1m
The Anna Karenina Effect in Risky Choice 52
TABLE 2
Predictions from prospect theory for choices between x and (10x, .1): One fourfold
pattern, one twofold pattern, and two threefold patterns of risk preferences.
Assumption about the elasticity
of the value function
Sure outcome
Small Large
Any Gains Risk seeking Risk aversion
Losses Risk aversion Risk seeking
Any Gains Risk aversion
Losses Risk seeking
Loss amplification Gains Risk aversion
Losses Risk aversion Risk seeking
Gain amplification Gains Risk seeing Risk aversion
Losses Risk seeking
The Anna Karenina Effect in Risky Choice 53
TABLE 3
Disaggregate results of Experiment 1: Preference groups.a
Designation Gains Losses N nb
Cautious Risk seeking – risk aversion Risk aversion 49 53
Wavering Risk seeking – risk aversion Inconsistent 43 14
Markowitzians Risk seeking – risk aversion Risk aversion – risk seeking 23 37
Non-Markowitzians Risk seeking – risk aversion Risk seeking – risk aversion 22 31
– Inconsistent Inconsistent 10 –
– Inconsistent Risk aversion – risk seeking 8 –
– Risk aversion Risk aversion 6 –
Audacious Risk seeking – risk aversion Risk seeking 5 7
– Inconsistent Risk aversion 5 –
– Inconsistent Risk seeking – risk aversion 5 –
– Risk aversion Inconsistent 4 –
PT compatible Risk aversion Risk seeking 3 –
PT compatible Risk aversion Risk aversion – risk seeking 2 –
– Risk seeking Risk aversion – risk seeking 2 –
– Risk seeking Risk aversion 1 –
– Risk aversion Risk seeking – risk aversion 1 –
aNine of the 25 possible preference patterns did not occur, and are not listed.
bReclassification of the Wavering.
The Anna Karenina Effect in Risky Choice 54
TABLE 4
Predictions from prospect theory and Markowitz for choices between px and (x, p).
Prospect theory
Assumption about the elasticity
of the value function
Outcome probability
Low High
Any Gains Risk seeking Risk aversion
Losses Risk aversion Risk seeking
Any Gains Risk aversion
Losses Risk seeking
Loss amplification Gains Risk aversion
Losses Risk aversion Risk seeking
Gain amplification
Gains Risk seeing Risk aversion
Losses Risk seeking
Markowitz
Assumption about the location of
inflection points
Outcome probability
Low High
Any Gains Risk seeking
Losses Risk aversion
Any Gains Risk aversion
Losses Risk seeking
Further removed from the origin
in losses
Gains Risk aversion
Losses Risk aversion
Further removed from the origin
in gains
Gains Risk seeking
Losses Risk seeking
The Anna Karenina Effect in Risky Choice 55
TABLE 5
Disaggregate results of Experiment 2: Preference groups in choices between px and (x, p).
Designation Gains Losses N
PT compatible Risk seeking – risk aversion Risk aversion – risk seeking 71
PT compatible Risk aversion Risk aversion – risk seeking 42
M compatible Risk aversion Risk aversion 25
PT compatible Risk seeking – risk aversion Risk seeking 16
– Risk seeking – risk aversion Risk aversion 16
PT / M compatible Risk aversion Risk seeking 12
– Risk seeking – risk aversion Risk seeking – risk aversion 11
– Risk seeking Risk aversion – risk seeking 11
M compatible Risk seeking Risk seeking 11
– Risk aversion- risk seeking Risk aversion- risk seeking 9
– Risk aversion Risk seeking – risk aversion 8
– Risk aversion- risk seeking Risk seeking – risk aversion 6
– Risk seeking Risk seeking – risk aversion 5
M compatible Risk seeking Risk aversion 5
– Risk aversion – risk seeking Risk aversion 5
– Risk aversion – risk seeking Risk seeking 2
The Anna Karenina Effect in Risky Choice 56
TABLE 6
Disaggregate results of Experiment 2: Preference groups in choices between x and (2x, .5).a
Designation Gains Losses N nb
Wavering Risk seeking – risk aversion Inconsistent 63 17
Non-Markowitzians Risk seeking – risk aversion Risk seeking – risk aversion 55 80
– Inconsistent Inconsistent 17 –
Cautious Risk seeking – risk aversion Risk aversion 16 18
– Risk aversion Inconsistent 16 –
– Risk aversion Risk aversion 14 –
– Inconsistent Risk seeking – risk aversion 14 –
Audacious Risk seeking – risk aversion Risk seeking 11 20
Markowitzians Risk seeking – risk aversion Risk aversion – risk seeking 9 19
– Inconsistent Risk aversion – risk seeking 8 –
– Risk aversion Risk seeking – risk aversion 8 –
– Risk seeking Risk seeking 5 –
– Risk aversion Risk aversion – risk seeking 4 –
– Inconsistent Risk aversion – risk seeking 4 –
– Inconsistent Risk aversion 4 –
– Inconsistent Risk seeking 4 –
– Risk seeking Inconsistent 3 –
– Risk seeking Risk aversion – risk seeking 2 –
PT compatible Risk aversion Risk seeking 2 –
– Risk aversion- risk seeking Risk aversion- risk seeking 1 –
– Risk aversion- risk seeking Risk aversion 1 –
– Risk aversion- risk seeking Inconsistent 1 –
– Risk seeking Risk aversion 1 –
aTwo of the 25 possible preference patterns did not occur, and are not listed.
bReclassification of the Wavering.
The Anna Karenina Effect in Risky Choice 57
TABLE 7
Experiments 1 and 2: Relative size (%) of preference groups in choices between x and (10x, .1) and choices between x and
(2x, .5), and difference of proportions tests with separate variance estimates.a
Designation
Preference pattern
x vs. (10x, .1) x vs. (2x, .5) t p Gains Losses
Markowitzians RS – RA RA – RS 20
(12)
7
(4)
-3.65
(-3.27)
.00
(.00)
Audacious RS – RA RS 4
(3)
8
(4)
1.91
(0.97)
.06
(.33)
Non-Markowitzians RS – RA RS – RA 16
(12)
31
(22)
3.78
(2.86)
.00
(.00)
Cautious RS – RA RA 28
(26)
7
(6)
-5.76
(-5.57)
.00
(.00)
Wavering RS – RA Inconsistent 7
(23)
7
(25)
-0.30
(0.48)
.76
(.63)
Miscellaneous 25 40 3.36 .00
aNumbers in parentheses are those before reclassification of the Wavering.
The Anna Karenina Effect in Risky Choice 58
TABLE 8
Experiment 3: Relative size (%) of preference groups in choices between x and (10x, .1) and choices between x and (2x,
.5), and McNemar tests.a
Designation
Preference pattern
x vs. (10x, .1) x vs. (2x, .5) 2 p Gains Losses
Markowitzians RS – RA RA – RS 21
(13)
5
(2)
62.04
(42.96)
.00
(.00)
Audacious RS – RA RS 6
(4)
9
(6)
4.66
(2.75)
.03
(.10)
Non-Markowitzians RS – RA RS – RA 16
(9)
25
(18)
19.06
(25.25)
.00
(.00)
Cautious RS – RA RA 12
(10)
6
(5)
17.28
(13.14)
.00
(.00)
Wavering RS – RA Inconsistent 7
(25)
6
(19)
0.13
(7.01)
.72
(.01)
Miscellaneous 38 49 19.67 .00
aNumbers in parentheses are those before reclassification of the Wavering.
The Anna Karenina Effect in Risky Choice 59
TABLE 9
Experiment 3: Observed minus expected number of transitions between preference groups across choices between x and (10x,
.1) and choices between x and (2x, .5), upon reclassification of the Wavering.a
Preference group in 10x choices
Preference group in 2x choices
Markowitzians Audacious Non-Markowitzians Cautious Wavering Miscellaneous
Markowitzians 6.78 1.63 -3.03 -2.84 8.53 -11.07
Audacious 0.31 7.19 0.13 -0.86 -2.02 -4.75
Non-Markowitzians -4.75 0.09 29.86 -1.22 -2.69 -21.29
Cautious -1.69 -4.15 7.78 9.94 -0.43 -11.45
Wavering 0.89 2.49 -2.84 0.68 0.47 -1.68
Miscellaneous -1.55 -7.24 -31.88 -5.70 -3.86 50.23
a2(25) = 187.58, p = .00.
The Anna Karenina Effect in Risky Choice 60
Figure Captions
Figure 1. Markowitz‟s asymmetric, triply inflected value function.
Figure 2. Constant and decreasing elasticity. Panel 1 shows the power value function v(x) = x
with 0 < < 1 (dashed curve) and the logarithmic value function v(x) = (1 / ) log(1 + x) with
> 0 (solid curve). Panel 2 shows the value functions in log-log coordinates: The power value
function is a straight line (constant elasticity), while the logarithmic value function is concave
(decreasing elasticity). These are the value functions for gains. The value functions for losses
would be v(x) = x with 0 < < 1 and > 1, and v(x) = ( / ) log(1 + x), with > 0 and > 1.
Figure 3. Aggregate results of Experiment 1 (N = 189): Probability of choosing sure thing x over
the gamble (10x, .1), and 95% confidence intervals, as a function of outcome magnitude and sign.
Figure 4. Aggregate results of Experiment 2 (N = 255). Panel 1 shows the probability of choosing
the sure thing px over the gamble (x, p), and 95% confidence intervals, as a function of outcome
probability and sign. Panel 2 shows the probability of choosing sure thing x over the gamble (2x,
.5), and 95% confidence intervals, as a function of outcome magnitude and sign.
Figure 5. The probability-weighing functions of group C (solid curve), groups M and N (dashed
curve), and group A (dotted curve).
Figure 6. Rw-mapping plots, combining probability weights, w(p), and value ratios, R(x, p) =
v(x) / v(x / p). Panel 1 shows the case of gains for all evaluation groups. Panel 2 shows the case of
losses for groups and M and A. Panel 3 shows the case of losses for groups N and C.
Figure 7. Aggregate results of Experiment 3 (N = 569). Panel 1 shows the probability of choosing
the sure thing px over the gamble (x, p), and 95% confidence intervals, as a function of outcome
probability and sign. Panel 2 shows, on the left, the probability of choosing sure thing x over the
The Anna Karenina Effect in Risky Choice 61
gamble (10x, .1), and, on the right, the probability of choosing sure thing x over the gamble (2x,
.5), and 95% confidence intervals, as a function of outcome magnitude and sign.
Figure 8. Difference (d) in incidence rates of the Markowitzians, the non-Markowitzians, the
Cautious, and the Audacious between 2x and 10x choices, separately for the randomized order
condition and the non-randomized order conditions.
Figure 9. The Markowitzians, the non-Markowitzians, and the Cautious from a Markowitzian
perspective: An S-shaped, inverse S-shaped, and concave value function in losses, respectively.
Figure B1. A simulated set of predictions from decision field theory for the Markowitzians: A
greater choice inconsistency for losses than for gains.
The Anna Karenina Effect in Risky Choice 62
FIG. 1
x
v(x)
The Anna Karenina Effect in Risky Choice 63
FIG. 2.1
0 75 150 225 300 375 450 525 600 675 750
x
0
5
10
15
20
25
30v(x
)
The Anna Karenina Effect in Risky Choice 64
FIG. 2.2
0.25 0.50 0.75 2.50 5 7.50 25 50 75 250 500 750
log(x)
0.1
0.2
0.3
0.4
0.5
0.60.70.80.9
1
2
3
4
5
6789
10
20
30lo
g(v
(x))
The Anna Karenina Effect in Risky Choice 65
FIG. 3
Gains Losses
$0.25 $2.50 $25 $250 $2,500 $25,000
x
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1P
rob
ab
ility o
f ch
oo
sin
g x
The Anna Karenina Effect in Risky Choice 66
FIG. 4.1
Gains Losses
.05 .95
p
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1P
rob
ab
ility o
f ch
oo
sin
g p
x
The Anna Karenina Effect in Risky Choice 67
FIG. 4.2
Gains Losses
$0.25 $2.50 $25 $250 $2,500 $25,000 $250,000
x
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1P
rob
ab
ility o
f ch
oo
sin
g x
The Anna Karenina Effect in Risky Choice 68
FIG. 5
.1 .5 .9
p
w(p
) M, N
A
C
The Anna Karenina Effect in Risky Choice 69
FIG. 6.1
.1 .5 .9
p
w(p
) a
nd
R(x
, p
)
Small x
Large x
M, NC
A
The Anna Karenina Effect in Risky Choice 70
FIG. 6.2
.1 .5 .9
p
w(p
) a
nd
R(x
, p
)
Small x
Large x
M
A
The Anna Karenina Effect in Risky Choice 71
FIG. 6.3
.1 .5 .9
p
w(p
) a
nd
R(x
, p
)Small x
Large x
C
N
The Anna Karenina Effect in Risky Choice 72
FIG. 7.1
Gains Losses
.05 .95
p
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1P
rob
ab
ility o
f ch
oo
sin
g p
x
The Anna Karenina Effect in Risky Choice 73
FIG. 7.2
Gains Losses
10x-type choices
$0.2
5
$2.5
0
$25
$25
0
$2,5
00
$25
,00
0
$25
0,0
00
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1P
rob
ab
ility o
f ch
oo
sin
g x
2x-type choices
$0.2
5
$2.5
0
$25
$25
0
$2,5
00
$25
,00
0
$25
0,0
00
The Anna Karenina Effect in Risky Choice 74
FIG. 8
Randomization
No
Yes
Markowitzians Non-Markowitzians Cautious Audacious-.25
-.20
-.15
-.10
-.05
.00
.05
.10
.15
.20
.25d
The Anna Karenina Effect in Risky Choice 75
FIG. 9
x
v(x)
All
Non-Markowitzians
Markowitzians
Cautious
The Anna Karenina Effect in Risky Choice 76
FIG. B1
Gains Losses
$0.25 $2.50 $25 $250 $2,500 $25,000
x
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1
Pro
ba
bility o
f ch
oo
sin
g x
The Anna Karenina Effect in Risky Choice 77
Footnotes
1The Constance Garnett translation of Leo Tolstoy‟s novel, with “choice” substituted for
“family.”
2We use the term „preference reversal‟ to denote a reversal of risk preferences as revealed by a
person‟s choices. This should be distinguished from the „preference reversals‟ between choice
and pricing discovered by Lichtenstein and Slovic (1971; Slovic & Lichtenstein, 1983), in which
a person chooses a safer gamble but assigns a higher cash equivalent to a riskier one.
3Another possibility is that the probability-weighing function always outweighs the value
function:
)10(
)(
)10(
)()10/1(
xv
xv
mxv
mxvw .
This, however, is not a realistic possibility in prospect theory. The probability weight w(1/10)
generally remains far removed from its upper limit of 1, but, given a decreasingly elastic value
function v, the value ratio v(x) / v(10x) moves toward 1 as the magnitude of x increases. It will
thus become increasingly unlikely that w(1/10) exceeds v(x) / v(10x).
4Weber and Chapman (2005, Table 5) report results that are consistent with those of Hershey and
Schoemaker‟s (1980) study. However, they did not examine choices between a sure thing and a
gamble, but rather choices between two gambles. In their study, aggregate choices were globally
and increasingly risk averse in gains and globally and increasingly risk seeking in losses. That is,
there were no reversals, but larger gains resulted in more risk aversion and larger losses resulted
in more risk seeking. As in Hershey and Schoemaker‟s (1980) study, choice probabilities were
closer to chance level for losses than for gains
The Anna Karenina Effect in Risky Choice 78
5The regression analysis included a contrast between Wavering and other participants as
independent variable. The intercept tested whether the logarithms of the loss/gain ratios departed
reliably from zero.
6The elevation of the probability-weighing function has been interpreted has the motivational
aspect of probability weighing (Wakker, 2004, 2010): More overweighing means that people are
more „optimistic‟ about the delivery of a risky gain, and more „pessimistic‟ about the delivery of
a risky loss.
7The conventional formulation, in which low probabilities are overweighed and moderate to high
probabilities are underweighted, has seen confirmation in other experimental arrangements than
ours (e.g., see Abdellaoui, 2000, Figure 5; Bleichrodt & Pinto, 2000, Figure 2; Tversky & Fox,
1995, Figure 9; Tversky & Kahneman, 1992, Figure 1). However, an overweighing of low to
moderate probabilities has been found in decisions of whether or not to purchase insurance
(Kusve, van Schaik, Ayton, Dent, & Chater, 2009). We discuss the purchase of insurance and
other decisions under risk in the General Discussion.
8Comparison of Figures 7.2 and 7.3 shows that R(x, p) is less elevated for small x in groups N and
C than for large x in groups M and A, meaning that v is more elastic in the former case than in the
latter. We do not include this among the Characteristics, because value functions differ not only
in whether elasticity increases or decreases, but also in the range within which, and the rate at
which, it increases or decreases. For instance, in Appendix A, the elasticity of the increasingly
elastic value function ranges from to 1-, whereas the elasticity of the decreasingly elastic
value function ranges from 1 to 0. Therefore, elasticity lies in a higher and narrower range in the
former case than in the latter.
The Anna Karenina Effect in Risky Choice 79
9A situation where the probability-weighing function always outweighs the value function is a
realistic possibility under increasing elasticity, not under decreasing elasticity (see Footnote 3).