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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]

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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]

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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.

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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

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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

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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>

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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

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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

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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

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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

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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

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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

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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

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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

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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.

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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>

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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

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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

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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 &

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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.

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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.

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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

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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>

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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.

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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

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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.

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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

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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

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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

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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

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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

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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.

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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

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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

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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).

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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).

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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

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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

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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

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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.

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The Anna Karenina Effect in Risky Choice 40

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The Anna Karenina Effect in Risky Choice 45

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 .

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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 .

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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

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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 .

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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>

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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:

[email protected].

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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

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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

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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.

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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

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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

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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.

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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.

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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.

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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.

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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

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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.

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The Anna Karenina Effect in Risky Choice 62

FIG. 1

x

v(x)

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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

)

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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))

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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

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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

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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

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The Anna Karenina Effect in Risky Choice 68

FIG. 5

.1 .5 .9

p

w(p

) M, N

A

C

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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

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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

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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

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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

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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

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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

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The Anna Karenina Effect in Risky Choice 75

FIG. 9

x

v(x)

All

Non-Markowitzians

Markowitzians

Cautious

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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

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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

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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.

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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).