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Executive
Summary
The Microeconomics of Choice under
Risk and Uncertainty: Where Are We?
WDA Bryant
Financial decision-making is not straightforward, in part, because such decisions
generally involve comparing financial assets the payoffs from which are subject to risk
and uncertainty.
Given that situation, two questions naturally arise: How do economic agents go about
the business of making choices in the face of risk and uncertainty? And, how should
economic agents make choices in the face of risk and uncertainty?
This paper concentrates on the first of these questions and discusses some of the main
attempts made by economic theory to understand how economic agents go about the
business decision-making under conditions of risk and uncertainty.
Theoretical possibilities considered in the context of decisions under conditions of
risk include: Expected value maximization, Expected utility maximization, Rank
dependent utility maximization, Prospect theory, and the Topology of fear approach
to decision-making in the face of catastrophic risk. This paper also considers empirical
tests of these theoretical possibilities and some of the anomalies and responses thrown
up by those tests such as: Allais Paradox, Discovered Preference Hypothesis, and the
choice behaviour of CEOs when faced with risk.
KEY WORDS
Decision-making
Uncertainty
Utility Theory
Expected Value
Maximization Hypothesis
Expected Utility
Maximization Hypothesis
The paper concludes with a brief excursion into choice under uncertainty where,
unlike in risky choice situation, the existence of objective probabilities over states of
the world cannot be relied on. In that context, the author briefly canvases the Subjective
Expected Utility approach – which is unable in general to account for ambiguity
aversion – Choquet utility, Wald’s Multiple Priors, and the Case Based approach
This paper highlights the fact that the rich and fascinating field of decision-making
under risk and uncertainty is characterized by a constant interplay between theoretical
conjecture, empirical testing, and theoretical refinement. Such interplay is mirrored by
this paper and contributions in the Colloquium Section of this Issue, where the thoughts
of practitioners and academics interact.
Allais Paradox
Catastrophic Risk
VIKALPA • VOLUME 39 • NO 1 • JANUARY - MARCH 2014
21
“One of the most intriguing areas of research in finance is that which examines and seeks to explain how and
why individuals make decisions. This research is aimed at predicting accurately decisions that will be made
given certain conditions and circumstances. The most commonly accepted model of rational choice today is the
theory of utility of wealth developed by von Newmann and Morgenstern (1953). Questions have arisen recently,
however, concerning the completeness of this theory.”
Edwards (1996; p. 19)
“There can be few areas in economics that could claim to have sustained such a rich interaction between theory
and evidence in an ongoing effort to develop theories in closer conformity with the facts. Considered together,
the accumulated theory and evidence present an opportunity to reflect on what has been achieved.”
Chris Starmer (2000; p. 332)
T
he valuation of financial assets is a primary focus
of Finance Theory because it is asset values that
guide asset allocation, (see the discussion in
Wang, 2008). The valuation of assets is not a straightforward exercise, however, because asset values typically
derive from future payoffs. Such payoffs are subject to
risk and are uncertain in both timing and extent. Risk,
uncertainty, the attitudes of economic agents to them, and
the way that agents make their demand and supply decisions under conditions of risk and uncertainty are therefore crucial in determining asset values.
Consequently, this paper discusses some of the attempts
made by economic theory to understand choice and decision-making under conditions of risk and uncertainty.
This is a rich and fascinating field of endeavour where
the constant interplay between empirical testing and theoretical refinement is clearly visible – an interplay interestingly mirrored in the Colloquium Section of this Issue,
where contributions by practitioners and acade-mics interact.
SETTING THE SCENE
The paper begins with a comparison of choice under certainty and choice under uncertainty as a way of coming
to conceptual grips with the choice under uncertainty
situation of a decision-maker.
more highly by the decision-maker than is c*– although
there may be ties.
Choice under uncertainty enriches the choice under certainty picture by supposing that decision-makers now
have only indirect access to the set, C, now referred to as
the set of consequences. Decision-makers have direct access only to a set A of actions. It is in the set of actions A
that agents have to operate, and over which they somehow have to define a preference ordering, çA.
Connecting A and C via the World
The set A is connected to the set, C by the world, W. The
idea is that if a decision-maker takes an action a ∈ A, then,
depending on the state or mood of the world, that action
may lead to any one of a number of consequences. This is
expressed by saying that W (a) ⊆ C, meaning that a given
action may be associated with a subset, or even the whole
set, C.
This formulation illustrates a central aspect of decisionmaking under uncertainty, namely, a decision-maker does
not know in advance, the consequences of a given action.
Contingent on the state of the world, a given action may
be mapped to any one of a number of consequences – see
Figure 1 for an illustration.
Figure 1: The World Maps Actions into Consequences
Choices under Certainty vs Uncertainty
The standard model of choice under certainty involves
the idea of: (i) a choice set, C, to which the decision-maker
has direct access; (ii) an ordering çC defined over the choice
set that captures a decision-maker’s preferences; and (iii)
a behavioural hypothesis that if the decision-maker choses
option c* out of C, then there is no other option in C ranked
22
THE MICROECONOMICS OF CHOICE UNDER RISK AND UNCERTAINTY: WHERE ARE WE?
In this way of looking at things, uncertainty is thought of
as meaning: (i) that decision-makers have only indirect
access to the space they are really interested in, the space
of consequences, C; (ii) their access is through the space
of actions A; (iii) the mapping W:A → C is multivalued,
i.e. it is a correspondence and not a function1.
Remark: The elements in the spaces of actions, A and
consequences, C can be very general. The case where the
consequences and/or actions are monetary is dealing
with financial decisions.
ing; so, C = {0, —1, +2, —3, +4, —5, +6}.
p(–1|ar)=p(+2|ar)=p(–3|ar)=p(+4|ar)=p(–5|ar)=p(+6|ar) = 6-1
(1)
p(0|aw) = 1
The decision-maker can then make a probabilistic analysis of the world they are in, as in equation which gives
them a picture of the map W: A→ C, as in Figure 2.
→ C of Example 1
Figure 2: The Map W: A→
The Decision-maker’s Knowledge:
Objective Probabilities
An interesting aspect of choice under uncertainty concerns the decision-maker’s knowledge of the world in
which they have to operate. In this, there are two leading
cases. Firstly, it may be that the decision-maker has access to objective probabilities about the likelihood of a particular state of the world, W1, ..., Wn, occurring. This then
provides connections between a given action and elements in the set of consequences.
Such a situation is common in, for example, casino games.
It can be expressed as the objective probability, p(c|a), of a
particular consequence c ∈ C being experienced when a
particular action a ∈ A is taken. By letting c range over all
of C and a range over all of A, the decision-maker can get
a probabilistic picture of the uncertainty they face and
hence information about the map W : A → C.
Situations in which a decision-maker has objective
probabilistic information about the way actions are connected to consequences will be referred to as Risk. In such
circumstances, it can also be useful to think of actions as
probability distributions over consequences. It may be written
as ak = (c1, c2, ..., cn; p1|k, p2|k, ..., pn|k) where the notation p1|k
indicates the probability that consequence ci will be experienced if action ak is chosen for i = 1, ..., n and k = 1, ..., m.
Example 1: Suppose an individual is given the option to
roll a fair die, (ar), or to walk away (aw), the space of actions A is then, {ar, aw}. If they decide to roll the die, they
are promised a loss of $1m if 1 comes up; a gain of $2m if
2 comes up; a loss of $3m if 3 comes up; a gain of $4m if 4
comes up; a loss of $5m if 5 comes up and a gain of $6m if
6 comes up. If they walk, they gain nothing and lose noth1
If W is a function then for each a ∈ A, ∃!c ∈ C such that W(a) = c,
and we have choice under certainty.
VIKALPA • VOLUME 39 • NO 1 • JANUARY - MARCH 2014
The Decision-maker’s Knowledge:
No Objective Probabilities
A second leading case involves situations where the decision-maker does not have complete – or any – objective
information about the nature of the world they are in.
They have no idea, even in probability terms, how actions and consequences are connected, i.e. they are uncertain about the nature of the map, W: A → C. Such a
decision context is referred to as Uncertainty.
As is traditional, the author begins with trying to understand the selection of actions by a decision-maker in the
special case of Risk. Equipped with some important
insights gained by studying that case, he moves to consider decisions under Uncertainty. Collecting the thoughts
so far gives the following:
Definition 1 (Decision under risk and uncertainty): Decisions under risk or uncertainty involve making choices between actions that yield consequences contingent on
realizations of a priori unknown states of the world. In the
case of decisions under Risk, agents have complete knowledge of the objective likelihood of each state. Under conditions of uncertainty, that is not the case.
23
The author follows the tradition in the literature of considering this problem firstly in the context of decisions
under risk and then taking what we learn there to the
general case of decisions under uncertainty.
The Basic Research Question
Given this setting, the central, fascinating, theoretical problem in the choice under risk and uncertainty literature is:
How do agents order the set of actions available to them? In
thinking about this question, it is important to be aware
of the mindset that a decision-maker is imagined to have,
namely that actions are not inherently interesting; they
are viewed merely as means to consequential ends. It is
the consequences that are inherently interesting.
Thus the mindset brought to the task of theorizing about
how people go about ordering a space of risky actions is
to suppose that the actions themselves are not inherently
interesting. So, for example, if a property developer has to
appear every night for the next month on a very tedious
television programme in order to drum up business for
their latest real estate development, they will take that
action – even though they would much prefer to take
the action of relaxing in a spa bath with a bottle of Champagne.
This contrasts with a person who just loves being on TV
and will take any opportunity to appear on any programme for free, and no matter what the consequence.
The latter person has his focus on the space of actions,
with no regard for the space of consequences. The former
person is intently focused on the space of consequences
and is trying to select from the space of actions in such a
way as to get themselves to a good consequence2.
TWO FUNDAMENTAL BEHAVIOURAL
HYPOTHESES: EVMH AND EUMH
Given an agent’s ordering of the space of consequences,
çc along with their probabilistic understanding of the
world, W, how do they go about making a preference order, çA over the space of risky actions available to them3?
The Expected Value Maximization Hypothesis
(EVMH)
Let a decision-maker be in a risk situation in which i = 1,
..., n possible consequences c1, ..., cn make up the space of
consequences C. Let the space of actions A consist of
k = 1, ..., m elements a1, ..., ak. One possibility is that decision-makers order the space of actions by calculating the
expected value of each action.
Definition 2 (Expected value of an action): In the case
where space of consequences is a non-empty finite set,
the expected value of action ak ∈ A is EV(ak) = p1|kc1 + ... +
pn|kcn with pi|k being the probability of consequence i being enjoyed, when action is taken4.
The Expected Value Maximization Hypothesis: In a risk
situation, if a* ∈ A is chosen, then EV(a*) ≥ ΕV(a), ≤a ∈ Α,
so that a decision-maker will choose an action a* that is
not dominated, in expected value terms, by any other action in A.
Is the EVMH Theoretically Viable?
The answer is ‘yes’ if: (i) there are conditions under which
expected values exist; and (ii) there are conditions under
which maximum expected value exists. There are such
conditions but they are not perfectly general.
As far as (i) is concerned, the existence of an expected
value is equivalent, in the case of a discrete consequence
space to the absolute convergence of the corresponding series. A series converges absolutely if the sum of the series
remains finite when the terms in the series are replaced
by their absolute values. Well-known cases where expected values can’t be defined, include the case where an
action is defined by a Cauchy distribution over consequences, e.g. p(ci|a) = 1/π (1 + c2i).
As for (ii), if the set of actions is assumed to be non-empty
and compact, then the expected value function when it
exists being affine and continuous will take a maximum
value.
Is the EVMH Empirically Viable?
2
This may be thought of as the ‘consequentialist point of view. See
the discussion of this approach in Hammond (1988).
3
We refer to the elements of the set as risky actions. Other terms
found in the literature include ‘lotteries’, ‘gambles’ and sometimes,
‘prospects’.
24
There are well-known empirical objections to this hypothesis, and they will be discussed momentarily. Before
4
We stick with the finite case in this paper, however, many arguments can be generalized e.g. if C is a separable metric space and
p(c|a) probability density function on C then EV(a) = ¶c x p(c|a) dc.
THE MICROECONOMICS OF CHOICE UNDER RISK AND UNCERTAINTY: WHERE ARE WE?
doing so, it is perhaps worth noting the work of Fan
(2002)5 who finds that, in the context of a small-payoff
experiment and irrespective of whether payoffs were hypothetical or real, that:
“Most experimental choices were consistent with simple
expected value maximization [and] [t]he consistency was
slightly more pronounced when payoffs were real.” Fan
(2002; p. 419).
Counter-examples to EVMH were known to Bernoulli
(1738), and they may be illustrated as follows.
{
}
Let A ≡ {a0 = [$2m;p1 = 1]; a1 = $5m, $0; p1 = 1-2 , p2 = 2-1 }.
Calculating expected values, we have EV(a1)=12- x $5m+
1- x $0 = $2.5 and EV (a ) = 1 x $2m = $2m, so EV (a ) > EV
2
0
1
(a0). However, people regularly report that they would
take action a0 which delivers $2m for sure, ahead of action a1 which, on the toss of a fair coin, delivers $5m or $0.
On the basis of this sort of thinking, Bernoulli (1738) suggested attention be paid to the utility of consequences, not
just their numerical value, and that a probability weighted
sum of the utility of consequences be used to order the
space of actions. Thus was born the influential idea of
expected utility.
As Machina (2008) notes, EUMH is, one of the most influential ideas in decision theory:
“The expected utility hypothesis is the predominant descriptive and prescriptive theory of individual choice
under conditions of risk or uncertainty… [The] approach
can be applied to a tremendous variety of situations, and
most theoretical research in the economics of uncertainty,
as well as virtually all applied work in the field (for example, insurance or investment decisions) is underta-ken
in the expected utility framework.” (Machina, 2008; p.
130).
Similarly, Schoemaker (1982) and Camerer (1989) argue
that expected utility theory is the foundation of the economics of uncertainty and the focus of much research
and application in decision theory, finance, and psychology. Apart from directly addressing the question of how
agents order the set A, another attractive aspect of the
expected utility formulation is that it allows us to capture
a decision-makers attitude to risk, through the concavity
(or convexity) of their Bernoulli utility function uB.
Given the centrality of EUMH and the massive stimulus
to the literature it has provided, this paper discusses
EUMH along with some of its tests and alternatives.
The Expected Utility Maximization
Hypothesis (EUMH)
Is EUMH Theoretically Viable?
In a risk environment in which the decision-maker has a
real valued Bernoulli utility function uB defined on the space
of consequences so that uB:C → R, it is theoretically possible that agents order the space of actions by calculating
the expected utility of each action.
EUMH is theoretically viable if: (i) it can be shown that
there are circumstances under which an expected utility
function exists which represents an agent’s preference
ordering çA on the space of risky actions A; and (ii) the
function takes a maximum in the space of actions.
Definition 3 (Expected utility of an action): When C is a
non-empty finite set, the expected utility of action ak ∈ A is
EU (ak) = p1|kuB(c1) + ... + pn|kuB(cn). The function EU (.)
over the space of actions is an expected utility function for
the decision-maker, (also known as a von Neuman –
Morgenstern utility).
As far as (i) is concerned, Proposition 1 will show that
under the conditions about to be listed6, an expected utility representation of çA on A is possible.
The expected utility maximization hypothesis, then supposes that agents pick an action having the greatest expected utility.
The Expected Utility Maximization Hypothesis: In a
choice under risk situation, a decision-maker chooses an
action, a* ∈ Α, such that EU (a*) ≥ EU(a), ≤a ∈ A.
Axiom 1: çA is a binary relation on A
Axiom 2: çA is complete on A
Axiom 3: çA is transitive on A
Axiom 4: çA continuous on A so for any a, a’, a’’ ∈ A the sets {0
≤ t ≤ 1|a’’ çA ta + (1 — t)a’} and {0 ≤ t ≤ 1|ta + (1 - t)a’ çA a’’}
are closed
Axiom 5: çA satisfies independence on A so that ≤a, a’, a’’ ∈ A,
and 0 < t < 1 we have a çA a’ if and only if [ta + (1 - t)a’’] çA [ta’
+ (1 - t)a’’].
6
5
See also the results in Cokely and Kelley (2009).
VIKALPA • VOLUME 39 • NO 1 • JANUARY - MARCH 2014
Axioms 1-5 are often referred to as ‘the von Neumann-Morgenstern
axioms’.
25
Remark: Each of these conditions on preferences over
risky actions has been subject to scrutiny. Of particular
interest from our point of view are Axioms 4 and 5. Consider first the requirement that çA is continuous on A. As
Mas-Colell, Whinston, and Green (1995) note, continuity
of çA on A means that sufficiently small changes in probabilities do not change the ordering of actions
Example (Mas-Colell et al. (1995; p. 171): Suppose in C,
there are three possible consequences: c1 ≡ death by car
accident, c2 ≡ staring at the wall at home, c3 ≡ a beautiful
and uneventful car trip. Suppose the agent’s utility ordering on C be such that uB(c1) < uB(c2) < uB(c3).
Let A = {a0, a1} where a0 ≡ stay at home, a1 ≡ go driving.
Writing out the actions gives us that: a0 = (c1, c2, c3; p1|0 =
0, p2|0 = 1, p3|0 = 0), a1 = (c1, c2, c3; p1|1 = t, p2|1= 0, p3|1 = 1
- t). Continuity of çA on A requires that if a0 \A a1 when t =
0 it will also be the case that a0 \A a1 for sufficiently small
positive values of t.
This clearly rules out possible preference orders where
‘safety first’ is prioritized and any non-zero chance of
death would reverse the ranking of a0 and a1. Another
way to put this is that continuity makes the preference
ordering insensitive to small variations in the probability
of catastrophic consequences. We return to this condition below.
As far as the independence axiom is concerned, the following observation by Oliver (2003) is pertinent:
“Independence implies that the intrinsic value that an
individual places on any particular outcome in a gamble
will not be influenced by the other possible outcomes (either within that gamble or within other gambles to which
the gamble is being compared), or by the size of the probability of the outcome occurring. The axiom requires that,
when comparing gambles, all common outcomes that
have the same probability of occurring will be viewed by
the individual as irrelevant.” (Oliver, 2003; p. 36).
Proposition 1: If a preference relation çA on A satisfies the
conditions in axioms (1)–(5) and the space of consequences C is
a non-empty finite set, then çA admits representation by a function that has an expected utility form, then ak çA ak’ g EU(ak)
= p1|kuB(c1) + ... + pn|kuB(cn) ≤ EU(ak’) = p1|k’uB(c1) + ... +
pn|k’uB(cn).
As for (ii), since expected utility functions are affine and
hence continuous, if A is non-empty and compact, then
there will be an expected utility maximum in A.
This is given formal expression as follows:
Proposition 2: If a preference relation çA on A admits an expected utility representation where A is non-empty and compact,
then there is an a* ∈A such that EU(a*) ≥ EU(a), ≤a ∈ A.
Proof : As an expected utility function is linear, it is continuous. By a theorem of Weierstrass7, such a function
has a maximum on non-empty, compact sets.
Hence, there is a non-empty set of conditions under which
EUMH is theoretically viable.
Is the EUMH Empirically Viable?
As noted above, the theoretical foundation of the EUMH
rests on the existence of a function that makes complete,
transitive, mixture-continuous and independent preferences over risky actions representable by the expected utilities of those actions.
There is a vast amount of empirical work aimed at scrutinizing the EUMH – and the individual axioms on which
it relies. A brief list of such work includes stretching from
Mosteller and Nogee (1951), Allais (1953), Ellsberg (1961),
Kahneman and Tversky (1979), Schoemaker (1982),
Harrison (1994), Hey and Orme (1994), Camerer (1995) to
Blondel (2002), Fan (2002), Oliver (2003), Schmidt and
Neugebauer (2007), Harrison and Rutström (2009),
Harrison, Humphrey and Verschoor (2010), Roger (2011),
Birnbaum and Bahara (2012), Blavatsky (2012a), Huck
and Müller (2012) and Cavagnaro et al. (2013). In the space
available, only a very selective treatment of the literature
is possible, focusing on some classic studies, but mostly
more recent work and especially studies testing EUMH
in a financial markets. As a prelude to doing that however, it might be useful to note the general negative implications of empirical findings for the empirical viability of
EUMH up to about the year 2000:
“There is substantial evidence that EUMH is likely to be
descriptively misleading in at least some important contexts and, given the accumulating evidence supporting,
in particular, probability weighting and loss aversion,
we have at least some well-grounded hypotheses about
Proof : See Mas-Colell et al (1995; pp. 176-178).
7
26
See E.A. Ok (2007).
THE MICROECONOMICS OF CHOICE UNDER RISK AND UNCERTAINTY: WHERE ARE WE?
important factors generating departures from the standard theory” (Starmer, 2000; p. 376).
0.10 x uB(500) + 0.9 x uB(0) < 0.11 x uB(100) + 0.89 x uB(0)
(3)
In principle, the failure of any of the conditions (1)–(5)
given above could be responsible for an empirical failure
of EUMH . In reality, it seems that one of the most potent
sources of failure of EUMH seems to be inapplicability of
the ‘independence axiom’, condition (5) which says: ≤a,
a’ ∈ A, a çA a’ if and only if [ta + (1 - t)a’’] çA [ta’ + (1 - t)a’’]
for any a’’ ∈ A and all 0 < t ≤ 1. The earliest and one of the
most potent demonstrations of the failure of independence is due to an experimental design known as the ‘Allais
paradox’.
But if (3) holds, then a’2 \A a2. This contradicts the ‘empirical fact’ from the Allais experiment which revealed that
a2 \A a’2. The author is therefore forced to drop the idea
that an expected utility function represents çA and that
describes decision-making under risk.
The Allais Paradox
As a canonical example of behaviour inconsistent with
EUMH, and an example that has stimulated an enormous
amount of subsequent work, consider the example of
Allias (1953):
The Allais Paradox: Let {c1, c2, c3} with c1 = €500m, c2 =
€100 m and c3 = €0. Agents’ are given two sets of actions
A1 = {a1, a’1} and A2 = {a2, a’2} which they are asked to
order, where:
a1 = (c1, c2, c3; p1|1 = 0, p2|1 = 1, p3|1 = 0);
a’1 = (c1, c2, c3; p1|1’ = 0.10, p2|1’ = 0.89, p3|1’ = 0.01)
a2 = (c1, c2, c3; p1|2 = 0, p2|2 = 0.11, p3|1 = 0.89)
a’2 = (c1, c2, c3; p1|2’ = 0.10, p2|2’ = 0, p3|2’ = 0.09)
Allais (1953) reported that in situations like this a majority percentage of respondents revealed the following preference orderings: a’1 \A a1 and a2\A a’2. Such behaviour is
inconsistent with the supposition that A ≡ {a1, a’1, a2, a’2} is
being ordered using an expected utility function.
To see why, suppose that the ordering a’1 \A a1 and a2 \ a’2
is consistent with expected utility theory – and derive a
contradiction as follows. Let uB (500), uB(100) and uB(0)
be the Bernoulli utility of the three consequences. The preference ordering a’1 \A a1 means, supposing an EU representation of \A that:
0.10 x uB(500) + 0.89 x uB(100) + 0.01 x uB(0) < 1 x uB(100)
(2)
Add 0.89 x uB(0) - 0.89 x uB(100) to both sides of equation
(2) and we have:
VIKALPA • VOLUME 39 • NO 1 • JANUARY - MARCH 2014
Given the implications of the Allais paradox for such an
appealing notion as EUMH it is not surprising that the
paradox has been subjected to scrutiny along the following lines.
Real vs Hypothetical Payoffs
Harrison (1994; p. 223) argues that ‘the experimental evidence against expected utility theory is, on balance, either uninformative or unconvincing’ and that
‘pronouncements of the passing of expected utility theory
are premature’. Burke et al. (1996) suggest that the reason
for Harrison’s (1994) position is that his review of numerous influential experiments that find against EUMH
lead him to conclude that many of the experiments are
invalid because they fail to meet the requirements of saliency or dominance suggested by Smith (1982)8. This typically happens, because ‘rewards are hypothetical, fixed,
or insufficiently sensitive to alternative choices’. Consequently,
“Based on this critique and several new experiments of
his own, Harrison concludes that expected utility theory
is still alive and well.” (Burke et al., 1996; p. 617).
Camerer (1995) notes9 that Harrison’s Allais-paradox
experiment provides the only evidence that real, as opposed to hypothetical payoffs, influence behaviour in directions consistent with EUMH. He cites Camerer (1989)
and Battalio, Kagel, and Jiranyakul (1990) who investigated the impact of financial incentives in Allais experiments, without finding much qualitative difference in
behaviour between hypothetical and real lotteries.
Given the apparent uniqueness of Harrison’s contribution, Burke et al. (1996) set out to test whether viola8
Saliency means that rewards in an experiment are systematically
related to actual choices. Dominance means that rewards outweigh other motivations that might arise from an individual’s subjective costs or benefits of participation in an experiment.
9
“…Harrison’s Allais-paradox experiments provide the only evidence that actually playing a gamble substantially reduce[s] the
rate of EU violations.” (Camerer, 1995, p. 634).
27
tions of expected utility are significantly diminished in
Allais-type set-ups when payoffs are real rather than hypothetical. The upshot of their work is that:
“We share Harrison’s (1994) concern that the empirical
case against expected utility theory might be overstated.
Lottery-choice experiments often fail to satisfy the precepts of saliency and/or dominance. This is necessarily
true when lottery choices are hypothetical.” (Burke et al.,
1996; p. 623-624).
“We share Harrison’s (1994) concern that the empirical
case against expected utility theory might be overstated.
Lottery-choice experiments often fail to satisfy the precepts
of saliency and/or dominance. This is necessarily true
when lottery choices are hypothetical.” (Burke et al., 1996;
p. 623–624).
The Discovered Preference Hypothesis
Another line of deference of comes in the form of discovered
preference hypothesis (DPH). The idea behind this
hypothesis is that individuals have true underlying
preferences that are EUMH consistent, but which
experiments fail to reveal for a variety of reasons. If this is
the case then, as Cubitt, Starmer, and Sugden (2001; p.
385) note, ‘the discovered preference hypothesis may seem
to insulate expected utility theory from disconfirming
experimental evidence’.
Consequently, Cubitt et al. (2001) identify the confounding effects to be expected in subject responses when DPH
is true, and then consider how they might be controlled
for. They argue for a design in which each subject faces
just one distinct choice task, and for real. The authors then
review the results of some tests of EUMH in which this
design has been employed and find that such experiments
‘reveal the same violations of the independence axiom as
other studies’. Cubitt et al. (2001) therefore argue that does
not justify scepticism about recoded rejections of EUMH.
Interestingly, Cubitt et al. (2001) regard as an open
question the issue of whether violations of EU become
less frequent as subjects gain experience. This question is
addressed by van de Kuilen and Wakker (2006) who aim
to show that ‘learning can reduce violations of expected
utility’. On the basis of a study which allows learning in
an Allais paradox context, they find that:
“… choices converge to expected utility maximization if
subjects are given the opportunity to learn by both thought
28
and experience …” (van de Kuilen & Wakker, 2006; p.
155).
Modified Probability Similarity
Blavatskyy (2013c) notes an important feature of the Allais
experiment which is an apparent similarity of probabilities
in the second Allais question. Recall in the second choice
situation:
a2 = (c1, c2, c3; p1|2 = 0, p2|2 = 0.11, p3|1 = 0.89)
a’2 = (c1, c2, c3; p1|2’ = 0.10, p2|2’ = 0, p3|2’ = 0.09)
So, in the second decision context, individuals face a tradeoff between the intermediate consequence c2 achieved
with probability 0.11 if they take action a2 and the most
desirable consequence c1 which they will arrive at with
probability 0.10, if they select action a’2. The argument is
that since probability 0.11 { probability 0.10, this similarity
in probabilities may contribute to the Allais paradox
outcome by inducing a kind of ‘optical illusion’ in
experimental subjects. That opens the possibility of the
Allais paradox disappearing in experiments which do
not employ such a similarity of probabilities.
Blavatskyy (2013c) points to studies which modify the
probability similarity feature of the original Allais design
and find amelioration – and in some cases a reversal – of
the Allais paradox as a result10. He suggests, given the
experimental situation surveyed by him that:
“… the existing experimental evidence tentatively
suggests that the Allais paradox may be reversed in a
design that decreases the similarity of probabilities in the
second Allais question.” (Blavatskyy, 2013c; p. 61).
Blavatskyy (2013c) presents results from a new
experiment11 to support his conjecture that it is probability
similarity that leads to Allais paradox behaviour. His
experiment does indeed reveal interesting modifications
of the Allais paradox – and indeed some cases of paradox
10
These studies include Camerer (1989, pairs 5 and 12, p. 79), Starmer
(1992, pairs 1 and 4, p. 815), Loomes and Sugden (1998), Humphrey
and Verschoor (2004). Studies which modify other aspects of the
Allais design include Bierman (1989), Conlisk (1989) and Carlin
(1990) report a reduction in Allis paradox behaviour if lotteries are
presented in a compound form or in a frequency format as in
Carlin (1990); when very large hypothetical outcomes are replaced with small (real or hypothetical) monetary amounts as in
Conlisk (1989, Appendix IV), Camerer (1989, Table 7, p. 92), Huck
and Müller, 2012).
11
See Blavatskyy (2013; pp. 61–62) for the details of the experimental design.
THE MICROECONOMICS OF CHOICE UNDER RISK AND UNCERTAINTY: WHERE ARE WE?
reversal. However, from the point of view of the bigger
picture of the consistency of agent behaviour with the
EUMH, his findings are that:
“Our experimental results reveal systematic violations
not only of expected utility theory but also of a more
general class of non-expected utility theories based on
the betweenness axiom.” (Blavatskyy, 2013; p. 63).
“… offers a further challenge to the descriptive validity of
EUMH”. (Oliver, 2003; p. 35).
In light of the Allais paradox, the empirical work it has
stimulated indicates that there are interesting circumstances in which EUMH is not a generally adequate theory
of choice under risk; it is of considerable interest to review
some of the alternative theories that have been formulated
to account for choice under risk.
Some Field Evidence
Huck and Müller (2012) take testing of the EUMH out of
the laboratory and into general real world populations.
They summarize their findings in doing this as follows:
“Earlier laboratory experiments (see Camerer, 1995 or
Starmer, 2000 for surveys) have documented the Allais
paradox in student samples. Our paper highlights that, if
anything, these studies underestimate the true prevalence
of the paradox in general populations ...” (Huck and
Müller, 2012; p. 277, emphasis added).
Interestingly, and also as a result of artefactual field
experiments12, using data generated by very poor (rural)
individuals in Ethiopia, India, and Uganda, Harrison et
al. (2010) report roughly equal support for two major
models of choice under risk and uncertainty, namely
Expected Utility Theory and Prospect Theory (discussed
below).
“…there is roughly equal support for the two major models
of choice under uncertainty considered here. It is not the
case that EUT or PT wins but that the data is consistent
with each playing some roughly equal role, as in Harrison
and Rutström (2009) for comparable laboratory tasks with
university students.” (Harrison et al., 2010; p. 101).
The Allais Paradox with Non-monetary Payoffs
As Oliver (2003) notes, there have been numerous tests of
EUMH using monetary outcomes. It is therefore interesting
to round out this Section by noting a test of the Allais
paradox in the context of gambles over health outcomes.
Using a sample of 38 subjects, he found that the choice
behaviour of 17 subjects strictly violated the independence
axiom. He concluded that ‘the violations were thus significant and systematic’. He concludes that the evidence
found in his investigation:
12
Harrison et al. (2010; p. 80) call field experiments artefactual if
they involve taking procedures from the laboratory and applying
them in the field.
VIKALPA • VOLUME 39 • NO 1 • JANUARY - MARCH 2014
CEOs and Expected Utility Maximization
List and Mason (2011) begin by noting that the issue of
whether individuals are expected utility maximizers is of
more than academic interest because at stake in the answer are the underpinnings of a large number of models
of choice under risk and uncertainty. List and Mason
(2011) also note that the almost ubiquitous use of benefit–
cost analysis in government agencies ‘renders the expected utility maximization paradigm literally the only
game in town’. In order to investigate the empirical usefulness of EUMH, they study CEO preferences over small
probability, high loss lotteries and using undergraduate
students as our experimental control group, find:
“… that both our CEO and student subject pools exhibit
frequent and large departures from expected utility theory.
In addition, as the extreme payoffs become more likely
CEOs exhibit greater aversion to risk.” (List & Mason,
2011; p. 114)
This is a particularly interesting study given the model of
decision-making with ‘catastrophic risks’ that is considered later.
MODELS ALTERNATIVE TO EU FOR çA
The empirical realization that the expected utility
maximization hypothesis is not a perfectly general theory
about how human beings go about making choices among
risky actions lead to a flood of alternatives vying to either
replace completely – or to plug some of its significant
gaps. At a general level, it is clear what is needed. Since
an expected utility function is of the form EU(a) = p1|auB
(c1)+ ... +pn|auB(cn) playing around with the way the
probabilities enter this expression and/or the nature of
the utility function used to evaluate consequences is the
obvious strategy if one wants to augment and generalize
EUMH.
29
Approaches which have done one or both these things
include: Allais’ theory of random choice (see Allais, 1988;
Anticipated or Rank dependent utility (see Quiggin, 1982;
Quiggin & Wakker, 1994); Cumulative Prospect theory (see
Tversky & Kahneman, 1992); Disappointment aversion
theory (see, Gul, 1991; Grant & Kajii, 1998); Implicit (or
linear) expected utility theory (see Yu, 2008); Implicit rank
linear utility theory (see Luce & Fishburn, 1991; Carlier,
2008); Implicit weighed utility theory (see Cheung, 1992 for
theory and Hu et al., 2012) for an application to transport
choice); Generalised expected utility theory (see Machina,
1987; Quiggin, 1995; Chu and Halpern, 2008 for theory
and Sopher and Gigliotti, 1997 for some tests); Perspective
theory (see Hu, 2013); Prospect theory (see Kahneman &
Tversky, 1979; Barberis, 2013); Prospective reference theory
(see Viscusi, 1989 for theory and Harless, 1993 for some
tests); Regret theory (see Bell, 1982; Loomes & Sugden, 1982
for the theory, Egozcue, 2012 for an application to portfolio diversification and Humphrey, 2004 for some empirical work); SSB theory (see Fishburn, 1984a; 1984b,
Alcuntud, 2002 for theory and Cigola, 1996 for an application to financial markets and bank management);
Yaari’s dual theory (see Yaari, 1987; Maccheroni, 2004;
Galichon and Henry, 2012 for theory, Hadar and Seo,
1995 for an application to asset diversification and Wang
& Young, 1998 for an application to insurance; Realization utility (see Barberis & Wei, 2012; Ingersoll & Jin, 2013
for theory and applications to stock market behaviour);
Similarity explanations of EUMH failure (see Buschena &
Atwood, 2012 for theory and some empirical results).
It is clearly beyond the scope of this paper to explain each
of these approaches and to evaluate their empirical plausibility. We therefore concentrate on just two approaches
that seem to have enduring value – Rank Dependent Utility Theory and Prospect Theory – and then present an
interesting new approach, that was developed and presented in a series of papers by Chichilnisky (1996, 2000,
2002, 2009, 2010). All three approaches appear to have
particular relevance in a financial market context.
Rank Dependent (or Anticipated) Utility Theory
Introduced in Quiggin (1982) and axiomatized by Quiggin
and Wakker (1994), this way of dealing with violations of
expected utility theory – particularly violations of the independence axiom shown up by the Allais paradox. As
Perpiñán and Prades (2001) note, the key idea of rank
dependent expected utility theory is that people substi-
30
tute decision weights for probabilities – while respecting
rank dependence.
Definition 4 (Rank dependent utility): The rank-dependent
utility of an action a that yields consequence ci with probability pi|a for i = 1, ..., n when consequences are rankordered from least to most preferred c1 çc c2 çc ...çc cn is
given by RDU(a) = ∑ni=1φiuB(ci) where the decision weights
are φi = w (∑ni=j pi|a) - w(∑ni=j+1 pi|a).
Example (Perpiñán & Prades, 2001; pp. 180-82): Suppose
that a person is diagnosed with an illness that causes
migraine at a frequency of 8 days per month. They are
offered a treatment a ≡ (c1 = pain free; c2 = 6 days of migraine; c3 = no change: p1|a = 0.1, p2|a = 0.7, p3|a = 0.2).
To work out the RDU(a), the following steps need to be
gone through.
1. Rank-order the consequences from least to most preferred. For sake of illustration, suppose: (no change)
çc (6 days of migraine) çc (pain free), where çc means
‘at best as good as’.
2. Calculate both the cumulative probability of reaching
each consequence or any other more preferred outcome
and the cumulative probability of reaching all other
more preferred consequences. Here, the probability of
achieving at least no improvement = 1; the probability
of achieving all other outcomes better than no improvement = 0.8; the probability of enjoying at least 6 days
of migraine per month = 0.8; the probability of gaining
all other outcomes better than 6 days of migraine per
month 0.1; the probability of achieving at least complete recovery = 0.1; the probability of achieving all
outcomes better than complete recovery = 0.
3. Transform by a weighting function the cumulative probabilities calculated in Step 2, where w is the one that
behaves as follows: 1 → w(1) = 1, 0.8 → w(0.8), 0.1 → w
(0.1) and 0 → w(0) = 0.
4. The decision weight φ associated with each consequence
is calculated by finding the difference between the
transformed cumulative probability of achieving at
least the outcome and the transformed cumulative
probability of achieving all other more preferred consequences. Then φ (no change) = 1 - w(0.8); φ (6 days
migraine) = w(0.8) - w(0.1) and φ (pain free) = w(0.1).
5. Then RDU(a) = [1 – w(0.8)]uB(c3) + [w(0.8) - w(0.1)]uB(c2)
+ w(0.1)uB(c1).
THE MICROECONOMICS OF CHOICE UNDER RISK AND UNCERTAINTY: WHERE ARE WE?
Comparing the expression EU(a) = ∑ni=1pi|auB(ci) with the
expression for the rank-dependent expected utility of an
action RDU(a) = ∑ni=1φiuB(ci) with φi = w(∑ni=j pi|a) - w(∑ni=j+1
pi|a), it is clear how decision weights get substituted for
probabilities.
While empirical support for RDEU has been found (see,
for instance, Quiggin, (1993) – perhaps the most important contribution is the idea of decision weights, that were
incorporated into the (Cumulative) Prospect Theory. The
Cumulative Prospect Theory of Tversky and Kahneman
(1992) discussed in the next Section is more general than
RDEU since it allows probability weighting functions to
differ in gains and in losses.
Prospect Theory
As Edwards (1996) notes, Prospect Theory was proposed
by Kahneman and Tversky (1979) in response to their
observation that the choices made by individuals in risky
situations have several characteristics that are inconsistent with EUMH . The first is that individuals underweigh
probable outcomes in comparison with outcomes that are
certain – a phenomena they named as the certainty effect.
They also observed that this effect can lead to risk-aversion in choices involving certain gains and risk-seeking
in choices involving certain losses – for details, see
Kahneman and Tversky (1979’ p.265). Secondly, individuals facing choices among different prospects disregard
components that are common to all prospects under consideration – a phenomenon they termed as the isolation effect.
This effect can cause the framing of a prospect to change
choices (see, Kahneman & Tversky, 1979; p.271 for details). Thirdly, individuals display a reflection effect in
which choices involving negative prospects and positive
prospects are treated equivalently (see, Kahneman &
Tversky, 1979, p.268 for details).
To account for these phenomena, Kahneman and Tversky
(1979) supposed that in contrast to the EUMH where
choices over risky actions are driven by considerations of
final wealth and probabilities, decisions are made based
on values assigned to gains and losses with respect to a
reference point and decision weights. Schmidt and Zank
(2012) rate reference dependence as the central innovation in Prospect Theory. They observe that it is consistent
with at least three things: observed sign dependence (i.e.
attitudes to risk and uncertainty depend on the sign of
outcomes); diminishing sensitivity for outcomes (i.e. peoVIKALPA • VOLUME 39 • NO 1 • JANUARY - MARCH 2014
ple are more sensitive to outcome changes near the reference point than to changes a long way from it) which
shows up in an S – shaped utility which is convex over
losses and concave over gains; and loss aversion, that is, a
negative deviation from the reference point has a higher
impact on an individual than does a positive deviation of
equal size.
At the empirical level, Schmidt and Zank (2012) cite many
studies which support Prospect Theory in general and
reference dependence in particular. For instance, studies
they cite which support sign dependence include
Edwards (1953; 1954), Hogarth and Einhorn (1990),
Tversky and Kahneman (1992), Abdellaoui (2000),
Bleichrodt (2001), Etchart-Vincent (2004), Payne (2005),
Abdellaoui et al. (2005, 2010). Diminishing sensitivity is
reported in Kahneman and Tversky (1979), Tversky and
Kahneman (1991; 1992), Hershey and Schoemaker (1982),
Budescu and Weiss (1987), Camerer (1989), Fennema and
van Assen (1998), Luce and Marley (2000), Abdellaoui
(2000) and Abdellaoui et al. (2005; 2007; 2008). Empirical
support for loss aversion is found in Battalio et al. (1990),
Tversky and Kahneman (1992), Camerer (1998),
Kahneman and Tversky (2000), Schmidt and Traub
(2002), Camerer et al. (2004), Brooks and Zank (2005);
and Abdellaoui et al. (2007, 2008). Given this extensive
empirical support, we follow Schmidt, Starmer and
Sugden (2008) and present a brief account of 3rd generation Prospect Theory.
There is a finite state space, S, consisting of the states of the
world si for i = 1, ..., n and a set of consequences C given by
an interval of the real line (to be thought of as wealth levels). In decisions under risk, each state si has an objective
probability pi of occurring with ∑ni=1pi=1. A is the set of all
actions and a particular action a ∈ A is a function from S
to A so that an action a specifies for each state of the world
si the resulting consequence a(si) ∈ C. As in other versions
of prospect theory, preferences over actions are reference
dependent; so:
For any three acts, a, a’, a’’ ∈ A to say that a is weakly
preferred to a’ when viewed from a’’, where a’’ is the reference act (the status quo position, say), we write a’ ça’’ a.
Given this idea, it is possible to define a relative value
function.
Definition 5 (Relative value function): The desirability of
the consequence of action a in state si relative to the conse-
31
quence of a reference act a’’ in the same state is given by a
relative value function v(a(si), a’’ (si)).
Remark: The reference act a’’ is not required to be a constant act, i.e. an act which gives the same consequence in
every state. This represents a significant generalization
of standard Prospect Theory.
Assumption 3P.1 The function v(a(si), a’’ (si)) is strictly
increasing in its first argument.
Assumption 3P.2 v(a(si), a’’ (si)) = 0 when a(si) = a’’(si).
We now define the idea of a function that assigns a real
number to any act a ∈ A viewed from any reference action
a’’. Such a function will be called an expected relative value
function.
Definition 5 (Expected relative value): A function V:A x A
→ R is an expected relative value function if it is defined by
V(a, a’’) = ∑iv(a(si),a’’(si)) x pi.
Remark: The function V(a,a’’) represents the preference
order ça’’ in the sense for all a, a’, a’’ ∈ A we have a’ ça’’ a g
V(a’, a’’) ≤ V(a, a’’).
As Schmidt et al. (2008) note:
“This approach is based on a state-contingent conception
of reference-dependence. That is, gain/loss comparisons
are made separately for each state of the world; thus, the
pattern of gains and losses associated with any act [a]
viewed from any reference act [a’’], depends on the statecontingent juxtaposition of consequences in [a] and [a’’].
(Schmidt et al., 2008, p. 207).
Notice that the function V(a, a’’) = ∑iv(a(si)) x pi is linear in
probabilities. The third generation Prospect Theory relaxes that restriction by generalizing this form of V (a, a’’),
using decision weights to V (a, a’’) = ∑iv(a(si), a’’(si)) x w(si; a,
a’’) where w(si; a, a’’) is the decision weight assigned to
state si when a is being evaluated from the reference point
a’’.
Assumption 3P.3 The decision weights of an agent are
determined cumulatively using a rank-dependent transformation of the sort introduced by Quiggin (1982) and
discussed above.
In order to construct cumulative decision weights for a
given pair of actions a, a’’, states must be ordered accord-
32
ing to the attractiveness of a’s consequences in each state.
The position of each state si is determined by the values of
v(a(si), a’’ (si)). Consequently, as Schmidt et al. (2008; p.
208) argue, the ordering of states must be constructed separately for each a, a’’ pair. Also, in the spirit of cumulative
prospect theory, there must be separate rank-dependent
transformations of probability for gains and losses that
are mirror-images of one another.
Illustration (Schmidt et al., 2008; p.208-209): Consider any
pair (a, a’’) ∈ A X A. Relative to that pair, there is a weak
gain in a state si if a’’ (si) ≤ a(si) and a strict loss if a’’ (si) >
a(si). Partition the total number of states n into m+ ≡ the
number of states in which there are weak gains and m– =
n - m+ the number of states in which there are strict losses.
Subscripts on states are assigned so that, ≤i, j = 1, ..., n, i
comes after j (so j < i ), if and only if it is the case that
v(a(sj), a’’(sj)) ≤ v(a(si), a’’(si)). Then, states with weak
gains are indexed by m+, ..., 1 while the states with strict
losses are indexed by -1, ..., -m-. Cumulative decision
weights can be defined by letting:
⎛
w+(pi) if i = m+
+
⎜ w (∑j≥ipj) – w+ (∑j>ipj) if 1≤ i ≤ m+ - 1
w(si; a, a’’) =
⎜ w- (∑j≤ipj) - w- (∑j<ipj) if —m- + 1 ≤ i ≤ –1
⎝
w- (pi) if i = –mHere both w+ and w– are strictly increasing maps of [0, 1]
onto [0, 1]. They are interpreted, respectively as the probability weighting functions for the gain and loss domains.
The third generation Prospect Theory is closed by the following:
Assumption 3P.4 When making choices in risky environments, agents behave so as to maximize V(a, a’’) =
∑iv(a(si), a’’(si)) x w(si; a, a’’).
Special cases: 1. RDEU is the special case in which decision weights are untransformed state probabilities so that
w+(pi) = w- (pi) = pi; 2. Cumulative prospect theory emerges
when the relative value function has the form v (a(si), a’’
(si)) = u(a(si), a’’(si)) where u(.) is some value function and
where the reference actions are certainties so that a’’ (si) =
a’’ (sj) = a’’ (sj), ≤i, j = 1, ..., n; 3. Expected utility theory is the
special case in which decision weights are untransformed
state probabilities and relative value is independent of
the reference outcome, so w (si; a, a’’) = pi and v(a(si), a’’
(si)) = EU (a(si)) where EU (.) is an expected utility function.
THE MICROECONOMICS OF CHOICE UNDER RISK AND UNCERTAINTY: WHERE ARE WE?
In summary, the third generation Prospect Theory as presented by Schmidt et al. (2008) retains all the empirically
inspired features of the earlier versions of Prospect Theory
such as loss aversion, diminishing sensitivity, and non-linear probability weightings, and adds in the generalization
of allowing reference points to be uncertain.
How well does it perform empirically? According to
Schmidt et al. (2008), the third generation Prospect Theory:
“… retains all the predictive power of those previous variants, but in addition provides a framework for determining the money valuation that an agent places on a lottery
… More surprisingly, when [the theory] is made operational by using simple functional forms with parameter
values derived from existing experimental evidence, it
predicts observed patterns of preference reversal across a
wide range of specifications … consistent with the range
in which that phenomenon has in fact been observed.
…The result is a flexible and parsimonious model of
choice under uncertainty which organises a large body
of experimental evidence.” (Schmidt et al., 2008; pp. 220221).
In a particularly interesting study of the behaviour of finance professionals, Abdellaoui et al. (2013) begin by
noting that much of the empirical support for Prospect
Theory, of the sort reported in Schmidt et al. (2008), for
example, comes from laboratory experiments using student subjects. The authors are interested to know if this
empirical support extends to the ‘real world’ – in particular to the behaviour of hard-nosed finance professionals.
To that end, Abdellaoui, Bleichrodt, and Kammoun (2013)
assess the descriptive potency of Prospect Theory using a
sample of private bankers and fund managers and in the
event find ‘clear support for Prospect Theory. In particular the authors found that:
“Our financial professionals behaved according to prospect theory and violated expected utility maximization.
They were risk averse for gains and risk seeking for losses
and their utility was concave for gains and (slightly) convex for losses. They were also averse to losses, but less so
than commonly observed in laboratory studies … A substantial minority focused on gains and largely ignored
losses, behavior reminiscent of what caused the current
financial crisis.” (Abdellaoui et al., 2013; p.411).
This is an interesting finding, especially when read in
VIKALPA • VOLUME 39 • NO 1 • JANUARY - MARCH 2014
conjunction with the self-reports of finance professionals
in the Colloquium Section of this issue.
Catastrophic Risk and the Topology of Fear
In a series of papers, Chichilnisky (1996, 2000, 2002, 2009,
2010), revisited the von Neumann – Morgenstern axioms
that underpin EUMH, in particular, the continuity axiom
for preferences over risky alternatives. Its implication,
discussed earlier, that preference rankings are insensitive to small changes in probabilities, even for catastrophic
outcomes, is the focus here. Chichilnisky’s motivation is
nicely captured in the following remark:
“For many years experimental observations have raised
questions about the rationality of economic agents – for
example, the Allais Paradox or the Equity Premium Puzzle. The problem is a narrow notion of rationality that
disregards fear [and rare but catastrophic events]. This
article extends the notion of rationality with new axioms
of choice under uncertainty and the decision criteria they
imply … In the absence of catastrophes, the old and the
new approach coincide, and both lead to standard expected utility. A sharp difference emerges when facing
rare events with important consequences, or catastrophes.” (Chichilnisky, 2009, p. 807).
Drawing on results in experimental psychology showing that human brains behave quite differently in potentially catastrophic situations that invoke extreme fear, to
the way they operate ‘normally’13, Chichilnisky (2009)
argues that these experimental observations would seem
‘to be relevant to the issue of rationality in decision-making under uncertainty’ – and to the general applicability
of EUMH.
The particular shortcoming of an expected utility representation of preferences over risky alternatives that
Chichilnisky (1996, 2000, 2002, 2009, 2010) focuses on, is
that it underestimates (reasonable and rational) responses
to rare events no matter how catastrophic their consequences may be.
She argues that:
“… the insensitivity of expected utility to rare events and
the attendant inability to explain responses to events that
invoke fear, are the source of many of the experimental
13
See Le Doux (1996) and Hertwig et al. (2003).
33
paradoxes and failures of that have been found over the
years.” (Chichilnisky, 2009, p. 808).
The observation from which Chichilnisky’s work departs
is contained in the following example due to Arrow (1971,
pp. 48–49), an example identical in spirit to the example
in Mas-Colell et al. (1995; p. 171) considered by us earlier.
Arrow’s Example: Let a1 be an action that involves receiving one cent, a2 an action that involves receiving zero
cents, and a3 an action involving receiving one cent and
facing a small probability of death. Suppose that uB (zero
cents) < uB (one cent), then the continuity axiom underlying the EUMH (Axiom 4 above), requires that the third
action involving death and one cent should be preferred
to the second involving zero cents when the probability of
death is small enough (but still positive).
Supposing that death is regarded by an individual as a
catastrophe, then the insensitivity to rare events forced on
decision-makers by the EUMH is a potential source of
failure of the hypothesis – and one that Chichilnisky (1996,
2000, 2009) addresses by replacing the continuity requirement in Axiom 4 above with a condition requiring sensitivity to rare events. The details of how all this works are
quite technical; however, the gist is as follows. Replace
the Axioms underlying Proposition 1 above with the following:
Axiom 1’: The ranking functional FR: A → R is linear and
continuous on the space of risky actions
Axiom 2’: The ranking functional FR: A → R is sensitive
to rare events
Axiom 3’: The ranking functional FR: A → R is sensitive
to frequent events.
Example (Chichilnisky, 2009; p. 811): An example of
behaviour that is consistent with these three axioms is,
when a portfolio is chosen, it is structured so as to maximize expected utility while seeking to minimize total value
losses in the event of a catastrophe.
In a set-up governed by these axioms, the following can
be shown to be generally true:
(Chichilnisky, 2009; Theorem 1]: A ranking of risky actions satisfies the continuity requirement in Axiom 4, if
and only if it is insensitive to rare events. Therefore, Axiom
4 g ¬Axiom 2’.
34
(Chichilnisky, 2009; Corollary 1]: In the absence of rare
events, a ranking of the space of risky actions that satisfies Axioms 1’, 2’, and 3’ is consistent with Expected Utility Theory.
One reason for being interested in the approach of
Chichilnisky is that it is consistent with the behaviour of
agents who face ‘heavy tails’ in the distributions of, say,
returns in financial markets. As she points out:
“… [this] new understanding of rationality consistent
with previously unexplained observations about decisions involving rare and catastrophic events, decisions
involving fear, the Equity Premium Puzzle, ‘jump diffusion’ processes and ‘heavy tails’ …” (Chichilnisky, 2009;
p. 807).
This is an interesting collection of ideas at a theoretical
level, but does this approach have any empirical support? Chanel and Chichilnsiky (2009) addressed this
question experimentally – in a set-up the details of which
are described in Chanel and Chichilnsiky (2009; pp. 27478). In brief, they study decisions made under conditions
of fear, when a catastrophic outcome is introduced in a
risky alternative or lottery. The experimental results reported by Chanel and Chichilnsiky (2009):
“… provide evidence that fear influences the cognitive
process of decision-making by leading some subjects to
focus excessively on catastrophic events. Such heterogeneity in subjects’ behavior, while not consistent with EUbased functions, is fully consistent with the new type of
utility function implied by the new axioms [i.e. Axioms
1’, 2’, and 3’].”
For studies specifically focused on the effects of fear in
financial markets see Lo et al. (2005), Chang and Tu (2011),
Hu and McInish (2013).
A Methodological Suggestion
There is an interesting methodological suggestion in
Harrison and Rutström (2009) to abandon the search for
a unique theory to explain all choices under risk (and uncertainty). Harrison and Rutström (2009) suggest that we
should aim instead for a combination of hypotheses, the
weights on which correspond to the percentage of the
population exhibiting expected utility behaviour, prospect theory behaviour and so on. As Harrison and
Rutström (2009) put it:
THE MICROECONOMICS OF CHOICE UNDER RISK AND UNCERTAINTY: WHERE ARE WE?
“Debates over the validity of … models have often been
framed as a horse race, with the winning theory being
declared on the basis of some statistical test in which the
theory is represented as a latent process explaining the
data. In other words, we seem to pick the best theory by
‘majority rule’. If one theory explains more of the data
than another theory, we declare it the better theory and
discard the other one. In effect, after the race is over, we
view the horse that ‘wins by a nose’ as if it was the only
horse in the race. The problem with this approach is that
it does not recognize the possibility that several behavioral latent processes may coexist in a population.”
(Harrison & Rutström, 2009, p. 134).
As Harrison et al. (2010) put it:
“… substituting PT for EUT would be tantamount to replacing one ‘half wrong’ assumption with another. [Consequently] policies should not be designed under the
assumption that one or other theory explains all behaviour. Policymakers are therefore faced with the substantial challenge of examining the sensitivity of the predicted
impact of policy interventions to the replacement of single
key assumptions with a probability distribution over competing assumptions …” (Harrison et al., 2010; p. 101-102).
This is an interesting suggestion – and one that has been
tried in other areas of economics – such as macroeconomics (e.g., Hansen & Sargent, 2001). It remains to be seen
the benefits it might yield in the current context.
DECISION-MAKING UNDER UNCERTAINTY
Leaving the relatively safe harbour of decision-making
under risk – where agents at least have access to objective
probabilities over states of the world – the paper now
heads into the ocean of choice under uncertainty where
objective probabilities over states are not available (hopefully without having to jettison everything we have
learned about choice in a risky environment). The reason
for needing to make this adjustment in our thinking is
that, outside the context of a casino, it is rare for agents to
have access to objective probabilities about events and
states of the world – a point emphasized by Keynes (1921)
and Knight (1921). Decision-makers are therefore typically forced to make subjective judgements about the likelihood of various events and states in order to select an
action.
Despite this significant change in context from risk to
VIKALPA • VOLUME 39 • NO 1 • JANUARY - MARCH 2014
uncertainty, the central question that we are trying to
answer remains essentially the same, namely: How are we
to suppose agents order the space of actions available to them –
and then make a selection? We now look at some possible
answers to this question.
In order to get started and following Wakker (2008), we
introduce the following ideas and notation. Decision-making under uncertainty involves choices between actions generically written as (c1; e1; c2; e2 ... cn: en) yielding a monetary
consequence ci if event ei happens with i = 1, ..., n. The events
e1, e2, ..., en are mutually exclusive and exhaustive and the
agent does not know for sure whether they will occur –
nor do they have any objective probabilities about the
likelihood of any event happening. Because agents are
unsure about which event will happen, they are also uncertain about which consequence they will enjoy/suffer
as a result of taking an action. They are therefore making
decisions under uncertainty.
Subjective Expected Utility Theory
An ‘obvious’ approach to getting a theory of choice under uncertainty is to use the machinery developed for
choice under risk, but instead of employing objective probabilities, we suppose instead that agents are able to come
up with subjective probabilities psi for i = 1, ..., n over the
space of events. Agents can then be imagined to order the
space of uncertain actions using a subjective expected
utility function. This is essentially the approach suggested
by de Finetti (1931), Ramsey (1931), and Savage (1954).
One problem that this approach runs into is the so-called
Ellsberg paradox – which plays a role in choice under
uncertainty to that played by the Allais paradox in risky
choice situations. As it is such an important empirical
result, we spend a little time exploring it.
Ellsberg Paradox
Imagine a situation with two urns, K and A. Urn K has a
known composition of 50 red balls and 50 black balls,
while the nature of urn A is ambiguous because it has 100
red and black balls in total, but in unknown proportion.
One ball is drawn from each urn and let:
rK denote the event that a red ball is drawn from the known
urn K
bK denote the event that a black ball is drawn from the
known urn K
35
rA denote the event that a red ball is drawn from the ambiguous urn A
bA denote the event that a black ball is drawn from the
known urn K
Experimental evidence shows that a common preference
pattern is:
(bA: $100; rA: $0) \ (bK: $100; rK: $0) and (bA: $0; rA: S100)
\ (bK: $0: rK: $100)
Now, if people are revealing subjective probabilities by
these preference rankings, then what they are saying is
that ps (bA) < ps (bK) and ps (rA) < ps (rK). However, if these
quantities are to be probabilities then ps (bK) + ps (rK) = 1 =
ps (bA)+ ps (rA) = 1. But ps (bA) + ε1 = ps (bK) and ps (rA) + ε2
= ps (rK); so, 1 = ps (bK) + ps (rK) = ps (bA) + ε1 + ps (rA) + ε2 =
1 + (ε1 + ε2) > 1, which is a contradiction. Consequently,
the behaviour exhibited in the Ellsberg’s paradox violates the subjective expected utility hypothesis.
As Zhang (2002) notes, the Ellsberg study stimulated a
great deal of empirical work which reinforced the early
Ellsberg findings that the Subjective Expected Utility (SEU)
model, axiomatized by Savage (1954), is not generally able
to account for aversion to uncertainty or aversion to ambiguity. The leading contenders in this regard are Choquet utility, Wald’s (1950) multiple priors model and the Case Based
Approach of Gilboa and Schmeidler (2001). As work on
these topics is taken up by others in the Colloquium, we
stop at this point and offer a brief conclusion.
CONCLUSION
The aim of this paper has been to present something of
the microeconomics of uncertainty and financial decision
making. This is a fascinating and lively area of Economics – particularly at points where it interacts with cognate disciplines such as Finance. While as Giboa (2009)
recently remarked, no clean, unified and general theory
of behaviour under risk and uncertainty is around the
corner, our view is certainly clearer now than it has ever
before been about the obstacles in the path of obtaining
such a theory.
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WDA (Tony) Bryant is an Associate Professor of Economics
in the faculty of Business and Economics at Macquarie University, Australia. He has for many years worked in the areas
of General Equilibrium Theory and Game Theory, widely
regarded as centrepieces of theoretical economics. He also
has active research interests in the general field of
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Microeconomics, especially Choice under Conditions of Risk
and Uncertainty and in Applied Macroeconomics as well as
Economics and Mathematics Education.
e-mail: [email protected]
THE MICROECONOMICS OF CHOICE UNDER RISK AND UNCERTAINTY: WHERE ARE WE?