Table Of ContentRisky Curves
On the empirical failure of expected utility
Daniel Friedman, R. Mark Isaac,
Duncan James, and Shyam Sunder
Routledge
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© 2014 Daniel Friedman, R. Mark Isaac, Duncan James, and Shyam Sunder
The right of Daniel Friedman, R. Mark Isaac, Duncan James, and Shyam
Sunder to be identified as authors of this work has been asserted by them
in accordance with the Copyright, Designs and Patent Act 1988. _
All rights reserved. No part of this book may be reprinted or reproduced or
utilized in any form or by any electronic, mechanical, or other means, now List of figures vil
known or hereafter invented, including photocopying and recording, or in
any information storage or retrieval system, without permission in writing List of tables
from the publishers. Acknowledgments and permissions Xl
Trademark notice: Product or corporate names may be trademarks or List of abbreviations xiii
registered trademarks, and are used only for identification and explanation
without intent to infringe.
British Library Cataloguing in Publication Data The challenge of understanding choice under risk
A catalogue récord for this book is available from the British Library
Library of Congress Cataloguing in Publication Data Historical review of research through 1960
Friedman, Daniel, 1947-
Risky curves : on the empirical failure of expected utility /
Daniel Friedman, R. Mark Isaac, Duncan James, and Shyam Sunder. Measuring individual risk preferences 20
pages cm
ISBN 978-0-415-63610-0 (hardback) — ISBN 978-1-315-81989-1 (e-book)
Aggregate-level evidence from the field 54
1. Risk. 2. Utility theory. 3. Decision making. I. Title.
HB615.F55 2014
338.5—-de23 What are risk preferences? 81
2013030113
ISBN 13: 978-1-138-09646-2 (pbk)
Risky opportunities 96
ISBN 13: 978-0-415-63610-0 (hbk)
Typeset in Times New Roman
by Out of House Publishing Possible ways forward 115
Index 133
Figures
2.1 A Friedman-Savage Bernoulli function
2.2 A Markowitz Bernoulli function
3.1 Alternative lotteries presented as pie charts 28
3.2 Data from Isaac and James (2000) 35
3.3 Dorsey and Razzolini decision interface with
42
probabilities of winning displayed
43
3.4 Clock format Dutch auction
44
3.5 Tree format Dutch auction
57
4.1 Friedman—Savage Bernoulli function and the optimal gamble
5.1 Scatterplot of o (Stdev) versus X7 (Exp(loss)) for 121 lotteries
(each with a Uniform distribution), with fitted regression line 91
5.2 Scatterplot of o (Stdev) against X7 (Exp(loss)), with fitted
regression line, for 100 lotteries on [-0.5, 0.5] governed by
the beta distribution with differing parameters 92
6.1 White cup/black faces in profile 97
6.2 Estimates of Bernoulli functions for nine oil men 99
6.3 Smith and Stulz (1985) figure 1 101
6.4 Smith and Stulz (1985) figure 2 102
6.5 Net payoff functions (y = net payoffs; x = gross payoffs) 106
7A Neuron activity over time (binned and averaged) across
the five different probability of reinforcement conditions 123
7.2 Mean coefficient estimates as a function of posterior
winning probability, plotted in separate panels for
left and right side of brain 125
Tables
3.1 The lottery menu of Binswanger (1980) 22
3.2 Lottery choice frequencies in Binswanger (1980) 22
3.3 Choice table proposed in Holt and Laury (2002) 27
3.4 Treatments from Cox, Sadiraj, and Schmidt (2011) 33
3.5 Small stakes binary choices from Cox, Sadiraj, Vogt, and
Dasgupta (in press) Table 1 37
3.6 Binary choices conducted in Calcutta from Cox, Sadiraj,
Vogt, and Dasgupta (in press) Table 5 37
3.7 Calibrations for probability weighting functions reported in
Cox, Sadiraj, Vogt, and Dasgupta (in press) Table 7 37
3.8 Lottery treatment payoffs from Binswanger (1981) 40
Acknowledgments and permissions
We gratefully acknowledge the following people who have given their time
to read earlier drafts of this book: Paul Beaumont, Sean Collins, James
Cox, Shane Frederick, Dave Freeman, Antonio Guarino, Gary Hendrix,
Ryan Oprea, and Mattias Sutter. Likewise, we are thankful for the helpful
correspondence of Peter Bossaerts. The four authors entered into this project
after turning anew to the questions raised in two previous working papers:
James and Isaac and Friedman and Sunder (referenced in Chapter 6). We
would like to thank all those who provided comments on that research in
both earlier and more recent incarnations. Our sincere appreciation goes to
Susan Isaac who copy-edited the manuscript prior to submission and to Tom
Campbell and Qin Tan who prepared the bibliographies. The usual disclaimer
applies; we alone are responsible for any and all remaining errors.
Permission to reproduce Figure 3.1 (from Abdellaoui, M., Barrios, C.,
and Wakker, P. P. (2007) “Reconciling Introspective Utility with Revealed
Preference: Experimental Arguments Based on Prospect Theory,” Journal of
Econometrics 138(1): 356-378, p. 370, Figure 5. Online. Available http://peo-
ple.few.eur.nl/wakker/pdfspubld/07.1mocawa.pdf (accessed June 19, 2013)) is
gratefully acknowledged from Elsevier Limited.
Permission to reproduce Figure 3.2 (from Isaac, R. M., and James, D. (2000)
“Just Who Are You Calling Risk Averse?” Journal of Risk and Uncertainty
20(2): 177-187) is gratefully acknowledged from Springer Publishing.
Permission to reproduce Figure 3.3 (from Dorsey, R., and Razzolini, L.
(2003) “Explaining Overbidding in First Price Auctions Using Controlled
Lotteries,” Experimental Economics 6(2): 123-140) is gratefully acknowledged
from Springer Publishing.
Permission to reproduce Figure 6.2 (from Grayson, C. J. (1960) Decisions
Under Uncertainty: Drilling Decisions by Oil and Gas Operators. Cambridge,
MA: Harvard University Press) is gratefully acknowledged from Harvard
Business Publishers.
Permission to reproduce Figures 6.3 and 6.4 (from Smith, C. W., and Stuiz,
R. M. (1985) “The Determinants of Firms’ Hedging Policies,” Journal of
Financial and Quantitative Analysis 20(4): 391405) is gratefully acknowl-
edged from Cambridge University Press.
xii Acknowledgments and permissions
Permission to reproduce Figure 7.1 (from Fiorillo, C. D,, ‘Tobler, P.N., and
List of abbreviations
Schultz, W. (2003) “Discrete Coding of Reward Probability and Uncertainty
by Dopamine Neurons,” Science 299: 1898-1902) is gratefully acknowledged
from The American Association for the Advancement of Science.
Permission to reproduce Figure 7.2 (from Preuschoff, K., Quartz, S., and
Bossaerts, P. (2008) “Markowitz in the Brain?” Revue d’Economie Politique
118(1): 75-95) is gratefully acknowledged from Elsevier Limited.
ARA absolute risk aversion
AREC Acme Resource Exploration Corporation (hypothetical entity)
BDM Becker, DeGroot, and Marschak
CAPM capital asset pricing model
CARA constant absolute risk aversion
CPT Cumulative Prospect Theory
CRRA constant relative risk aversion
CRRAM constant relative risk aversion model
CRT Cognitive Reflection Test
DM decision maker
DMU diminishing marginal utility
EUT Expected Utility Theory
EWA Experience Weighted Attraction
FPSB first-price sealed-bid (auction)
fMRI functional Magnetic Resonance Imaging
HL Holt and Laury
Lid. independent identically distributed
PAI pay all independently
PAC pay all correlated
PAS pay all sequentially
POR pay one randomly
RRA relative risk aversion
UIP Uncovered interest parity
VNM Von Neumann and Morgenstern
1 The challenge of understanding
choice under risk
Life is uncertain. We hardly know what will happen tomorrow; our best-laid
plans go awry with unsettling frequency. Even the recent past is often a matter
of conjecture and controversy. Everyday decisions, small and large, are made
without certainty as to what will happen next.
It would therefore be comforting to have a well-grounded theory that orga-
nizes our observations, guides our decisions, and predicts what others might
do in this uncertain world. Since the 1940s most economists have believed
they have had such a theory in hand, or nearly so with only a few more tweaks
needed to tie up loose ends. That is, most economists have come to accept that
Expected Utility Theory (EUT), or one of its many younger cousins such as
Cumulative Prospect Theory (CPT), is a useful guide to behavior in a world in
which we must often act without being certain of the consequences.
The purpose of this book is to raise doubt, and to create some unease with
the current state of knowledge. We do not dispute that the conclusions of
EUT follow logically from its premises. Nor do we dispute that, in a suffi-
ciently simple world, EUT would offer good prescriptions on how to make
choices in risky situations. Our doubts concern descriptive validity and pre-
dictive power. We will argue that EUT (and its cousins) fail to offer useful
predictions as to what actual people end up doing.
Under the received theory, it is considered scientifically useful to model
choices under risk (or uncertainty) as maximizing the expectation of some
curved function of wealth, income, or other outcomes. Indeed, many social
scientists have the impression that by applying some elicitation instrument
to collect data, a researcher can estimate some permanent aspect of an indi-
vidual’s attitudes or personality (e.g., a coefficient of risk aversion) that gov-
erns the individual’s choice behavior. This belief is not supported by evidence
accumulated over many decades of. observations. A careful examination of
empirical and theoretical foundations of the theory of choice under uncer-
tainty is therefore overdue.
To begin with the basics: What do we mean by “uncertainty” and “risk”?
Economists, starting with, and sometimes following, Frank Knight (1921),
have redefined both words away from their original meaning.’
2 The challenge of understanding risky choice The challenge of understanding risky choice 3
In the standard dictionary definition, risk simply refers to the possibility enough free parameters to a given finite sample. That exercise is called “over-
of harm, injury, or loss. This popular concept of risk applies to many spe- fitting,” and it has no scientific value unless the fitted model can predict out-
cialized domains including medicine, engineering, sports, credit, and insur- side the given sample. Any additional parameters in a theory must pay their
ance. However, in the second half of the twentieth century, a very different way in commensurate extra explanatory power, in order to protect against
definition of risk took hold among economists. This new technical definition needless complexity.
refers not to the possibility of harm but rather to the dispersion of outcomes We shall see in Chapter 3 that the Expected Utility Theory and its many
inherent in a probability distribution. It is typically measured as variance or generalizations have not yet passed this simple test in either controlled labo-
a similar statistic. Throughout this book we will be careful to distinguish the ratory or field settings. These theories arrive endowed with a surfeit of free
possibility-of-harm meaning of risk from the dispersion meaning. parameters, and sometimes provide close ex post fits to some specific sam-
Although the notion of risk as dispersion seems peculiar to laymen, ple of choice data. The problem is that the estimated parameters, e.g., risk-
economists acclimated to it easily because it dovetails nicely with EUT. For aversion coefficients, exhibit remarkably little stability outside the context in
centuries, economists have used utility theory to represent how individuals which they are fitted. Their power to predict out-of-sample is in the poor-
construct value. In the 1700s Daniel Bernoulli (1738) first applied the notion to-nonexistent range, and we have seen no convincing victories over naive
to an intriguing gamble, and since the 1940s the uses of expected utility have alternatives.
expanded to applications in a variety of fields, seemingly filling a void. Other ways of judging a scientific model include whether it provides new
At the heart of Expected Utility Theory is the proposition that we each, insights or consilience across domains. Chapter 4 presents extensive failures
individually or as members of a defined class, have some particular knowable and disappointments on this score. Outside the laboratory, EUT and its gen-
attitudes towards uncertain prospects, and that those attitudes can be cap- eralizations have provided surprisingly little insight into economic phenom-
tured, at least approximately, in a mathematical function. In various contexts, ena such as securities markets, insurance, gambling, or business cycles.
it has been referred to as a value function (in Prospect Theory), or a utility After almost seven decades of intensive attempts to generate and validate
of income function, or a utility of wealth function. Following the standard estimates of parameters for standard decision theories, it is perhaps time to
textbook (Mas-Colell, Whinston, and Green [1995]) we shall often refer to it ask whether the failure to find stable results is the result. Chapter 5 pursues
as a Bernoulli function. Such a function maps all possible outcomes into a this thought while reconsidering the meaning and measures of risk and of
single-dimensional cardinal scale representing their desirability, or “utility.” risk aversion.
Different individuals may make different choices when facing the same risky But does it really matter? What is at stake when empirical support for a the-
prospects (often referred to as “lotteries”), and such differences are attributed © ory is much weaker than its users routinely assume? We write this book because
to differences in their Bernoulli functions. the widespread belief in the explanatory usefulness of curved Bernoulli func-
In particular, the curvature of an individual’s Bernoulli function determines tions has harmful consequences.
how an individual reacts to the dispersion of outcomes, the second definition
of risk. Because the curvature of the Bernoulli function helps govern how 1. It can mislead economists, especially. graduate students. Excessively lit-
much an individual would pay to avoid a given degree of dispersion, econo- eral belief in EUT, or CPT, or some other such model as a robust charac-
mists routinely refer to curvature measures as measures of “risk aversion.” terization of decision making can lead to a failed first research program,
Chapter 2 explains the evolution and current form of EUT, and the which could easily end a research career before it gets off the ground. We
Appendix to Chapter 2 lays out the mathematical definitions for the inter- hope that our book will help current and future graduate students be bet-
ested reader. Presently, we simply point out that in its first and original mean- ter informed and avoid this pitfall.
ing, aversion to risk follows logically from the definition. How can one not be 2. It encourages applied researchers to accept a facile explanation for devia-
averse to the possibility of a loss? If a person somehow prefers the prospect tions, positive or negative, that they might observe from the default pre-
of a loss over that of a gain, or of a greater loss over a smaller loss, in what diction, e.g., of equilibrium with risk-neutral agents. Because preferences
sense can the worse outcome be labeled a “loss” in the first place? By contrast, are not observable, explaining deviations as arising from risk aversion
under the second definition of risk as dispersion of outcomes, aversion to risk (or risk seeking) tends to cut off further inquiry that may yield more
is not inevitable; aversion to, indifference to, and affinity for risk remain open useful explanations. For example, we will see in Chapter 3 that, besides
possibilities. risk aversion, there are several intriguing explanations for overbidding in
It is a truism that to deserve attention, a scientific theory must be able to first-price sealed-bid auctions.
predict and explain better than known alternatives. True predictions must, 3. It impedes decision theorists’ search for a better descriptive theory of
of course, be out-of-sample, because it is always possible to fit a model with choice. Given the unwarranted belief that there are only a few remaining
4 The challenge of understanding risky choice
gaps in the empirical support for curved Bernoulli functions, many deci-
2 Historical review of research
sion theorists invest their time and talent into tweaking them further, e.g.,
by including a probability weighting function or making the weighting through 1960
function cumulative. As we shall argue, these variants add complexity
without removing or reducing the defects of the basic EUT, and the new
free parameters buy us little additional out-of-sample predictive power.
The question remains, what is to be done? Science shouldn’t jettison a bad
theory until a better one is at hand. Bernoulli functions and their cousins have
dominated the field for decades, but unfortunately we know of no full-fledged
alternative theory to replace them.
The best we can do is to offer an interim approach. In Chapter 6 we show
how orthodox economics offers some explanatory power that has not yet
This is an illustration of the ephemeral nature of utility curves.
been exploited. Instead of explaining choice by unobservable preferences
C. Jackson Grayson, 1960, 309
(represented, ¢.g., by estimated Bernoulli functions), we recommend look-
ing for explanatory power in the potentially observable opportunity sets that
decision makers face. These often involve indirect consequences (e.g., of fric-
The theory of choice under uncertainty is marked by two seminal works
tions, bankruptcy, or higher marginal taxes), and some of them can be ana- separated by more than two centuries, with remarkably little in the interim.
_ lyzed using the theory of real options. Beyond that, we recommend taking a These two contributions are D. Bernoulli's “Exposition of a New Theory
broader view of risk, more sensitive to its first meaning as the possibility of
on the Measurement of Risk” (1738) and John Von Neumann and Oskar
loss or harm.
Morgenstern’s Theory of Games and Economic Behavior (1943 [1953]). A
The interim approach in Chapter 6 has its uses, but we do not believe that
metaphor of “bookends” can be used to describe seminal works that bracket
it is the final answer. In Chapter 7 we discuss process-based understanding of
a large intervening literature. Because these two works stand almost alone, a
choice. We speculate on where, eventually, a satisfactory theory might arise.
more appropriate metaphor might be that they are “virtual bookends” across
Will neurological data supply an answer? What about heuristics / rule-of-
200 years.
thumb decision making? Can insights from these latter approaches be inte-
In his “mean.utility [moral expectation]’ Bernoulli introduced many of the
grated with the modeling structure outlined in Chapter 6? We are cautiously
critical elements of what is now called Expected Utility Theory. Although
optimistic that patient work along these lines ultimately will yield genuine
formal expositions of theories of risk attitudes came later, the core of his
advances.
argument is a parallel assertion about an individual’s nonlinear “utility”
for money: specifically that individuals have diminishing marginal utility of
income. Further, Bernoulli asserted a “moral expectation” that is a log func-
Notes
tion of income: The motivating factor for Bernoulli was the infamous St.
1 Knight said that a decision maker faced risk when probabilities over all possible
Petersburg Paradox. This was a gamble that pays 2” ducats with probability
future states were truly “known” or “measurable,” and faced uncertainty when these
2* for every n = 1, 2, ... 0%, and so has an expected value 1 +1+1+... =.
probabilities (or some of the possible outcomes) were not known. Knight himself
noted that this distinction is different than that of popular discourse. Apparently, neither Bernoulli nor his contemporaries could imagine trading
anything greater than a modest amount of ducats for this gamble. Bernoulli
“solved” the St. Petersburg Paradox by asserting that what mattered was not
Bibliography
the mathematical expectation of the monetary returns of the St. Petersburg
Bernoulli, D. (1738) “Exposition of a New Theory on the Measurement of Risk,” gamble, but rather the mathematical expectation of the utility of each of the
trans. Louise Sommer (1964) Econometrica 22: 23-26. outcomes, which implied only a modest monetary value of the gamble for
Knight, F H. (1921) Risk, Uncertainty and Profit. Boston: Hart, Schaffner most individuals.
and Marx. Even as Bernoulli’s logarithmic model became the foundation for the more
Mas-Colell, A., Whinston, M. D., and Green, J. R. (1995) Microeconomic Theory. general concave structure of the “utility function” that later writers used
Oxford: Oxford University Press.
to axiomatize choice under uncertainty and to characterize risk attitudes,
6 Historical review of research through 1960 Historical review of research through 19607
Bernoulli recognized situations in which a utility-of-income function would But the enthusiasm for Bernoulli, in spite of this rediscovery, was attenu-
not be concave. However, he saw these counterexamples only as “exceedingly ated by several factors. First, there was extensive discussion of how to deal
rare exceptions” (25). with the apparent.inconsistency between declining marginal utility of income
Over two hundred years later, Von Neumann and Morgenstern developed and the widespread phenomenon of gambling. According to Blaug (1968),
an axiomatized structure of expected utility over “lotteries” that referenced Marshall also accepted the general idea of a diminishing marginal utility of
Bernoulli’s logarithm function as a special case. This concept of expected util- income and begged the question of building an economic theory of choice
ity was central to their exposition of a theory of games. They went on to under uncertainty by simply attributing gaming at less than fair odds directly
describe a process by which such a utility function could be estimated from to the “love of gambling.” Specifically, in a footnote on page 843 of the eighth
data on an individual’s choices between a series of pairs of “prospects” with edition of his Principles (1890 [1920])? Marshall wrote:
certain versus risky outcomes.
In hindsight, it is astonishing that Bernoulli’s idea of expected utility The argument that fair gambling is an economic blunder is generally
received little, largely negative, attention between 1738 and 1943. On the based on Bernoulli’s or some other definite hypothesis. But it requires
other hand, the publication of The Theory of Games and Economic Behavior no further assumption than that, firstly, the pleasures of gambling may
ignited a veritable explosion of interest in expected utility and theories of atti- be neglected; and secondly 9”(x) is negative, where 9(x) is the pleasure
tudes towards risk. By the mid-1970s, after the publication of Pratt’s “Risk derived from wealth equal to x.
Aversion in the Small and the Large” (1964) and the series of papers in the
Journal of Economic Theory by permutations of Diamond, Rothschild, and Furthermore, in a footnote on page 842, Marshall rederives Bernoulli’s log-
Stiglitz (1970, 1971, 1974) neo-cardinalist theories of expected utility and risk arithmic utility function (in today’s familiar notation) of:
preferences took the driver’s seat.
Of course, taking a commanding place in economic orthodoxy does not U,(y) = K log (y! a)
necessarily mean that the theory has also created a useful empirical structure
for explaining or predicting “Economic Behavior” (as the second part of the (where y is actual income and a is a subsistence level of income). Contra
Von Neumann and Morgenstern title suggests). That is the point of our book. Bernoulli, Marshall suggests “Of course, both K and a vary with the
However, before turning to that larger theme, it may be useful to contrast why temperament, the health, the habits, and the social surroundings of each
Bernoulli’s foundational exposition received so little attention for more than individual.” Marshall’s treatment of Bernoulli is pretty much limited to this
two hundred years, and why Von Neumann—Morgenstern had such a revolu- and a discussion of the implications for progressive income taxation. Schlee, in
tionizing impact on this aspect of economics. a different and extended discussion of Bernoulli, Jevons, and Marshall, argues
To begin with, it seems commonplace to think of Bernoulli as modeling . that Marshall implicitly accepted the idea of some kind of expected utility
some concave utility function. In fact, he explicitly suggested only a specific — calculations in valuations of durable goods, although he (Marshall) claimed
logarithmic — concave function. Even with this more general (concave) inter- that “this result belongs to Hedonics, and not properly to Economics.”?
pretation, there were (and remain) empirical problems with this model in Second, even as Bernoulli was being cited by Jevons and Marshall, the
terms of such things as the existence of gambling and the form of insurance movement to ordinal utilities soon gained traction, and indifference curves
contracts. In the case of gambling, Bernoulli argues (29) that “indeed this is were described by utility functions defined only up to an increasing mono-
Nature’s admonition to avoid the dice altogether.” tonic transformation. Changes of marginal utility per se were undefined in
Schlee (1992) argues that until the arrival of the early marginal-utility the new ordinal paradigm.
theorists such as Jevons in the late nineteenth century, “few economists Moreover, there was. Menger’s demonstration (1934) that Bernoulli did not
appear to have noticed” Bernoulli’s theory. Jevons (1871 [1931], 160)! cited actually solve the St. Petersburg Paradox, he merely solved one member of the
Bernoulli’s work and asserted, “It is almost self-evident that the utility of family of such paradoxes. Even with Bernoulli’s logarithmic function, there
money decreases as a person’s total wealth increases.” But Jevons argued that exist other St. Petersburg gambles with appropriately increasing payments
along the way to diagnosing a reasonable answer to “many important ques- that would lead to the same conundrum that Bernoulli sought to resolve. A
tions” (presumably including the St. Petersburg Paradox), “having no means full solution required even more restrictive assumptions on the utility func-
of ascertaining numerically the variation in utility, [emphasis added] Bernoulli tion (boundedness would be sufficient).*
had to make assumptions of an arbitrary kind” (again, we presume Jevons Finally, Bernoulli’s “solution” of a logarithmic utility function was not the
means Bernoulli’s explicit adoption of the logarithmic form from the class of only one put forth to the St. Petersburg Paradox. As Bernoulli himself noted,
concave utility functions). Cramer offered one approach from an entirely different direction, suggesting
