A theory of framing, probability transformation, and decision under risk
LOUIS LEVY-GARBOUA*
June 2016
Preliminary draft
Abstract
This paper presents a descriptive theory of expected utility which explains how lotteries are framed and how
the frame impacts the decision process and the revealed preference. I consider decisions under risk among two
independent prospects by intuitively-rational decision makers who rely on evidence to decide and have a
sequential perception of the objects of choice. If decision makers perceive sequentially dissonant evidence,
they may be led to reverse their preference within the decision process. I describe here the process of decision
of intuitively-rational decision makers who start with a context-free, expected utility (EU) prior.
Whenever dominance cannot be observed, choosing the EU-preferred option always raises an objection, i.e.
doubt. The role of framing is to reveal dominance or make the objection visible in order to reach a decision that
one will feel “reasonably sure” of. However, objections are frame-dependent and so will be the revealed
preference. I develop a Bayesian representation of the decision process based on the sequential perception of
EU preference (the prior) and a visible objection to the latter. In a pairwise comparison, individuals with doubt
maximize an objection-dependent expected utility (ODU) conditional on their prior EU preference.
All pairwise comparisons without dominance can be categorized in just two patterns: nested lotteries and
overlapping lotteries. Nested lotteries exacerbate the dilemma between risk-taking and certainty by making
the nested lottery appear relatively safe and by highlighting the objection to any prior preference, whereas
overlapping lotteries may bring certainty in decisions that are intrinsically risky by suggesting stochastic
dominance or “quasi-stochastic dominance” if lotteries are similar. With this simple theory of framing under
risk, all the anomalies of decision under risk exhibited by Kahneman and Tversky (1979) and a few more can be
predicted with a single parameter in excess of risk aversion. It is shown in particular that the Allais paradox may
result from the use of a nested-lotteries frame in presence of a sure outcome and an overlapping-lotteries
frame exhibiting quasi-stochastic dominance in presence of two similar risky lotteries. Finally, a crucial test of
this analysis and of the relevance of framing in the decision process is presented.
Key words: Decision under risk, intuitive rationality, doubt, framing, anomalies of choice,
objection-dependent expected utility.
JEL codes: C11, C91, D03, D81, D83
* Université Paris 1 Panthéon-Sorbonne, Paris School of Economics, Centre d’ Economie de la
Sorbonne, 106-112 Bd de l’Hôpital 75647 Paris Cedex 13, France; and Cirano, Montreal,
Canada ([email protected]).
1
Kahneman and Tversky (KT 1979) built prospect theory to predict a number of salient
features of single decisions under risk (with objective probabilities) among simple
independent prospects including anomalies to expected utility (EU) theory1: Certainty effect
and Allais paradox (common consequence and common ratio effect), insurance and
gambling, reflection effect, risk seeking for losses and aversion to symmetric bets, possibility
effect, isolation effect. Their paper starts with some considerations on framing which have
been overlooked in further developments of the theory (Quiggin 1982, Gilboa 1987,
Schmeidler 1989, Tversky and Kahneman 1992, Wakker 2010). The goal of my paper is to
revive KT’s emphasis on the role of framing while saving essential parts of expected utility
theory. I present a descriptive theory of expected utility which explains how lotteries are
framed and how the frame impacts the revealed preference. This theory gives a
parsimonious solution to all the anomalies discussed in KT (1979) and more.
I assume that decision-makers are intuitively-rational in the sense that they are logical but
have a sequential perception of the objects of choice and rely on evidence to decide.
Framing and evidence are endogenously perceived with the aim of helping individuals to
make decisions that they feel “reasonably sure” of2. If decision makers perceive sequentially
dissonant evidence, they may be led to reverse their preference within the decision process.
At the beginning of the decision process, they stand at a distance under a veil of ignorance
such that the presentation of objects and procedure of decision are still unknown to them.
Thus, their prior preference must be invariant to presentation and procedure and meet
normative requirements (Tversky and Kahneman 1986). In this paper, presentation and
procedure invariance will be summarized by the EU axioms and the well-defined EUpreference will be adopted as the prior. We describe here the subsequent process of
decision.
1
Classic reviews of non-expected utility theories are Machina (1987) and Starmer (2000).
I started formulating this basic idea as a major criticism of the economic postulate of known
preferences almost forty years ago in my paper “Perception and the Formation of Choice” (LévyGarboua 1979). In this early paper, I developed a theory of selective perception resting on the two
hypotheses mentioned about the decision process. This paper, however, did not deal with simple
lottery choices but with complex consumer decisions. Later on, over many years and many
unpublished drafts, I began to apply this approach to decisions under risk. See, for instance, my
working paper “Expected utility and cognitive consistency” (Lévy-Garboua 1999).
2
2
Under certainty, dominance is the only observable pattern whereas, under risk or
uncertainty, dominance is just a possibility. Whenever dominance cannot be observed,
choosing the EU-preferred option always raises an objection that triggers doubt. After
perceiving a frame-dependent objection, intuitively-rational decision makers update their
prior in a Bayesian fashion, which may lead to preference reversals within the decision
process and cause the so-called anomalies to EU theory. The model of objection-dependent
expected utility that I develop here precisely formulates the probability transformation
induced by doubt which is a cornerstone of prospect theory, and it is consistent with
important psychological insights found in other existing economic theories of decision under
risk like regret (Bell 1982, Loomes and Sugden 1982), imprecision (Butler and Loomes 2007),
similarity judgments (Rubinstein 1988, Leland 1994), salience (Bordalo et al. 2012), and in
the reward prediction error model of neurosciences (e.g. Schultz et al. 1997). This model
gives a parsimonious account of framing, the Allais paradox and many other anomalies of
pairwise choice among simple lotteries.
I. EU Preference and the decision process under risk
A. Intuitive rationality
We derive a specific form of bounded rationality from two basic requirements of perception:
(i) all decisions rely on characteristics of the objects of choice recalled or perceived by the
senses at the moment a decision is made, which we call “evidence”; (ii) perception and recall
occur sequentially through successive cognitions. For our purpose, each cognition can be
associated with a specific belief or preference and the sequence of preferences defines the
decision process. Only final preferences are revealed by observable choices. We designate
rational individuals who are constrained by their imperfect perception as “intuitive” (see
Lévy-Garboua et al. 2015 for an application to confidence). Intuitive individuals make up
their mind on the sole basis of what they perceive and ignore forgotten or imperceptible
information. We consider the normative EU preference as prior evidence for a logical
decision maker because, when the decision process begins, she cannot yet perceive the
singular context of her forthcoming decision. However, such prior evidence may be
completed by contextual information perceived in a later stage of the decision process.
3
Evidence and information overlap but do not juxtapose: for instance, contextual evidence is
irrelevant information for normative rationality, while hidden dominance would make no
evidence. If risky options are described by their gains and losses relative to a reference point
while prior wealth is not recalled, gains and losses are evidence and initial wealth is not. An
intuitive individual would then treat reference-dependent gains and losses as carriers of
utility or value3 and respect at least this basic tenet of prospect theory in the context of
choices which are not liquidity-constrained or do not participate to wealth management.
Frames convert information into evidence but they may also hide information which
becomes unavailable to an intuitive decision-maker. Therefore, intuitively-rational
individuals may be sensitive to framing and driven by frames into normatively irrational
decisions.
B. Dominance and doubt in lottery comparison
We consider pairwise choices among prospects presented in the common form adopted by
Kahneman and Tversky (1979):
= 1 − with 0 ≤ ≤ 1
(1)
Gains and losses and are defined with respect to an unspecified reference state. We can
easily extend this notation to more than two sure outcomes and to contingent outcomes ,
presented under the same form (1). Lottery degenerates into a sure outcome if =
, implying = 1.
Let us compare two lotteries A and B presented in form (1). Without loss of generality, we
set: ≥ . Since the theory of decision under risk extends the theory of decision under
certainty, it is illuminating to assume first that B is a sure outcome.
= 1 − = with ≥ , and 0 < ≤ 1.
(2)
From the perspective of decision-making, two very different patterns may arise4: (i)
≤ or ≥ ; (ii) > > . The first pattern exhibits dominance of one
lottery over the other (either A or B) whereas the second pattern does not. The essential
3
However, prior wealth remains background information which would condition her risk aversion.
4 In choices between a risky lottery and a sure outcome, these patterns immediately extend to any
finite number of outcomes in the lottery.
4
difference between them lies in the fact that no objection to the prior preference can be
found under dominance whereas there always exists an objection otherwise. This is what
makes the acceptance of dominance a universal criterion of rationality. When the
dominance pattern is perceived, the decision raises no doubt ex ante and no regret ex post:
the decision-maker opts for the dominant lottery which is her EU-preference and feels sure
of making the right decision. Under certainty, i.e. = , dominance (including
equivalence) is the only observable pattern and, indeed, preferences are known and raise no
doubt. Under risk or uncertainty, however, dominance is just a possibility, not a necessity.
Whenever dominance cannot be observed, risk does not only affect the objects of choice; it
plagues the decision itself. Indeed, choosing the EU-preferred option under the second
pattern always raises a visible objection. When the risky lottery A is EU-preferred to B,
choosing A exposes to the downside risk of drawing the low outcome - with probability
1 − p - instead of surely getting if one had opted for B. And, if the sure outcome is EU-
preferred to A, choosing exposes to the upside risk of missing the higher outcome -
with probability - that might have been won by opting for A. We write conventionally these
two objections to choosing the EU-preferred option as:
= y → y ; 1 − p = → y ; p
Existence of an objection to prior preference-i.e. doubt- is equivalent to the expectation of
an opportunity loss:
= 1 − !" − " # = !" − " # ,
which signals the value of perfect information in utility terms (Raiffa 1968). Under the
dominance pattern, no objection can be found to the prior preference and the expected
opportunity loss has zero value. In contrast, under the objection pattern, the expected
opportunity loss is positive, the risk of decision exists and the prior EU preference is no
longer certain. This means that the decision process has not come to an end. An intuitivelyrational decision maker will feel unsure of her prior preference after perceiving the visible
objection and revise upward the probability of the objection’s state in the risky lottery. This
cognitive process generates an “availability bias” in a form anticipated by Tversky and
Kahneman (1983)5. We assume that the decision process stops after perceiving this
5
Tversky and Kahneman end their conclusion (1983: 178) by these words:
5
objection to the EU prior. Our theory departs from decision field theory (Busemeyer and
Townsend 1993) in this important respect for the simple lotteries that we consider. The
reason why we rule out vacillation in a pairwise comparison –i.e. objecting to the prior’s
objection, then objecting to the objection of the prior’s objection, and so forth- is that all
objections of a higher order are invisible to an intuitively-rational individual because they are
hidden by the first perception. No further evidence can be found and the decision process
must come to an end6. At this point, the intuitively-rational decision maker has gone
“beyond reasonable doubt” and reached cognitive consistency. Eliminating the unpleasant
feeling of doubt has been obtained at a cost since the logical consistency of the prior EU
preference does not necessarily convey to the revealed preference; but a single decision
cannot provide conclusive evidence of that cost ex ante.
C. Doubt and probability transformation
Since the perception of an objection to the prior causes preference uncertainty, we shall
now develop a Bayesian representation of the decision process7 based on the sequential
perception of EU preference and a visible objection to the prior. This model precisely
formulates the probability transformation induced by doubt8 and predicts choices of
cognitively consistent individuals between pairs of A and B bets and valuations of such bets
(Valuations and the preference reversal phenomenon are studied in a companion paper
(Blondel and Lévy-Garboua 2016)). The decision maker sequentially perceives, first her EU
preference, then the available objection to the latter. The prior cognition is a choice under
risk between A and B while the second cognition, focused on the objection, is another
mental choice between two sure outcomes: and if the prior preference was A, and
“Continued preoccupation with an outcome may increase its availability, and hence its perceived likelihood.
People are preoccupied by highly desirable outcomes, such as winning the sweepstakes, or with highly
undesirable outcomes, such as an airplane crash. Consequently, availability provides a mechanism by which
occurrences of extreme utility (or disutility) may appear more likely than they actually are...”
6 Further objections to objections would become evidence with experience but they remain
invisible in single decisions. Lévy-Garboua et al. (2016) examine repeated decisions under risk.
7 Our analysis is not inconsistent with the experimental rejection of Bayesian calculus because the
latter assumes a prior that most subjects would not perceive in the formulation of the problem.
Cognitively consistent individuals are logical so that they are Bayesian conditional on their
perception of information. Indeed, Bayesian models are currently used for the prediction of
saccadic eye movements (see, for instance, Rehder and Hoffman 2005).
8
For brevity, we skip the adjective “reasonable” when no confusion can be made.
6
if the prior preference was B. The two cognitions sequentially perceived dictate dissonant
preferences within the decision process, that is, before the final preference is revealed. Since
no logical mind can hold two different preferences with certainty, the next step of the
argument is to recognize that a logical decision-maker who experiences doubt while
choosing among risky or uncertain options must feel uncertain of her true beliefs (LévyGarboua and Blondel 2002). Indeed, the decision-maker observes two values for the
objection’s likelihood, first the objective probability of the objection state $ (i.e., 1 − if A is
the prior, or if B is the prior), then probability 1. Assume that this unknown probability
follows a Beta distribution on the unit interval. If the decision maker has no experience or
prior information about the gambles, and makes a single choice, the perceived likelihood of
the objection is a weighted average of the two beliefs (DeGroot 1970, chap. 9):
$ ∗ = &$ + 1 − &1
for all $(#0,1!
(3)
The weights are the relative precisions of the prior probability and its objection. We
designate µ as the prior’s weight and 1 − & as the objection’s weight, with 0 < & ≤ 1. EU
theory assumes perfect rationality: & ≡ 1 and rules out probability transformation.
However, even under risk, intuitively-rational decision-makers (i.e., with & < 1) transform
objective probabilities into objection-dependent subjective ones. The weight of an objection
depends on the visibility conferred by its framing but not on its probability. Our model
shares this distinctive property with salience theory of choice under risk (Bordalo et al. 2012)
because both theories rely on the sensory mechanisms of perception.
By writing (3) differently:
π * = π + (1 − µ )(1 − π )
for all $(#0,1! ,
(3’)
it appears that the objection state is always overweighted iff & < 1, and that the smaller is $
the more over-weighted is the perceived probability $ ∗ . From (3), we can derive the
probability transformation curve that describes the variation of the perceived probability of
winning the high prize, noted ∗ , as a function of its objective probability . It is obtained by
noticing that $ = if the sure outcome is EU-preferred (i.e. 0 < < *) whereas $ = 1 − if the risky lottery is EU-preferred (i.e. * < < 1). Hence,
7
if p = 0
0
µp + 1 − µ if 0 < p < x

p* = 
if x < p < 1
µp
1
if p = 1
(4)
For a given value of *, the perceived probability of winning a high prize against a sure
smaller gain is a two-piece
piece affine transformation of the given probability with a
downward jump at = *,, when the EU preference for B (at low probabilities of winning)
converts into A (at higher probabilities of winning). The probability transformation curve
conditional on the comparison value * is described by figure 1.
+−,
Figure 1 – A probability transformation curve for µ = 0.7
As assumed by prospect theory, low probabilities of winning are overweighted and high
probabilities of winning are underweighted9 by intuitively-rational
rational decision makers.
m
However, the present curve is dependent on the comparison value,, has three points of
9
The same probability transformation curve holds by reflection for negative bets against a sure loss.
8
discontinuity in 0, x,, and 1 and it is composed of two linear, increasing and parallel,
segments on the open intervals ]0,x[ and ]x,1[. Since the probability transformation
transf
curve
has commonly been elicited from a large sample of comparison outcomes, the present
analysis is consistent with an inverse-S
inverse S shape for the “aggregate” probability transformation
curve (figure 2). Remarkably, this is the shape postulated by KT (1979) or Tversky and
Kahneman (1992) and elicited from careful empirical studies (Abdellaoui 2000). However, a
within-subject
subject curve would still be discontinuous in general at the two extreme values 0 and
1, with ∗ 0- = 1 − & and ∗ 14 = &.
Figure 2 – An aggregate probability transformation curve for an individual
Proposition 1 (0DU rule): In the pairwise comparison between a risky lottery and a sure
outcome, intuitively-rational
rational individuals with doubt maximize an objection
bjection-dependent
expected utility (0DU) conditional on their prior EU preference:
preference
− &"56 where 56 is the outcome in L’s
. "|0120 = &3" + 1
’s objection state.
state
If their prior preference is A,, they compare:
. "|
"
= &3" + 1 − &" = &" + 1 − &" ,
with:
(5)
. "|
"
= " .
And, if their prior preference is , they compare:
9
0 "| = &3" + 1 − &" = 71 − &1 − 8" + &1 − " . "| = " .
with:
(6)
It is easy to demonstrate from expressions (5) and (6) that, whichever be their prior
preference, intuitively rational decision makers may reverse it during the decision process
under specific conditions10. The latter are examined below.
It may seem strange that rational decision makers perceive their normatively rational
preference and eventually deviate from the latter. However, intuitively rational individuals
who perceive further evidence feel unsure of their true preference and thus receive this
information.
Camerer,
Loewenstein
and
Weber
(1989)
conducted
experiments
demonstrating that participants were not able to ignore previously received information
when subsequently making a decision and ended up making worse decisions as a result,
even though obtained information was accurate.
II. Certainty effect, insurance and gambling
A. EU violations
In this section, we examine under which conditions on probabilities and outcomes the prior
preference will or will not be reversed during the decision process under the maintained
assumption that a nondegenerate lottery is compared with a sure outcome. We discard the
trivial cases of dominance and certainty in which the EU-preference is always respected.
Thus, we assume: > > and 0 < < 1. As we have only three outcomes, we
normalize the utility function and set: 9 = 0; 9 = 1; 9 = * with 0 < * < 1.
Since the objection depends on the prior, we need to consider two cases:
(i)
3" ≥ 3" , or ≥ * :
Since = y → y ; 1 − p, the revealed preference violates the prior according to (5)
iff & < * ≤ . Thus, our theory predicts the certainty effect when & < 1 and the expected
Proof: With (5) and "7: 8 < " , it is possible to have if & < 1: 3" ≥ 3" and &3" +
1 − &" < " ; hence, . "⁄ < . " ⁄.
With (6) and "7< 8 > " , it is possible to have if & < 1: 3" ≤ 3" and &3" + 1 −
&"1>"; hence, . ">. ".
10
10
utilities of the two lotteries are close enough. If the lotteries are EU-indifferent, the prior is
always violated. In general, the condition for violating the EU prior may be written:
0 ≤ − * < 1 − &
(5’)
This condition is more easily met when the objection’s weight, the probability of winning,
and the potential opportunity loss * are relatively high.
(ii)
3" ≤ 3" ,or ≤ *:
Since = → y ; p, the revealed preference violates the prior according to (6) iff
+ 1 − &1 − > *, that is, when & < 1 and the expected utilities of the two lotteries
are close enough. This will always occur if the lotteries are EU-indifferent. In general, the
condition for violating the prior preference for the sure outcome is the following:
0 ≤ * − < 1 − &1 − (6’)
This condition is met more easily when the objection’s weight and the potential opportunity
loss 1 − * are high while the probability of winning is low. A number of implications can be
derived from this analysis.
Since EU-preference and the revealed preference may either coincide or differ, the sequence
of perceived preferences for either A or B can follow four decision paths. Using expressions
(5’) and (6’), these are summarized by AA, AB, BA,BB, in which the first symbol indicates the
prior preference (EU) and the second symbol the posterior preference (0DU). The four
decision paths are summarized by the following positions of * on the open unit interval:
AA: x ≤ µp
AB: µp < x < p
BA: p < x < µp + 1 − µ
BB: x ≥ µp + 1 − µ
(7)
Notice that the 0DU function is not continuous at the point of indifference (i.e. p = x ) as a
result of the change in the prior and objection across this point. The discontinuity of the 0DU
function at the point of EU-indifference is responsible for the apparent inconsistency of an
alternation of the revealed preference when x moves along the unit interval. Preference
consistency is preserved when x moves at a distance from , either below (AA) or above
(BB). We shall say that preferences are “strong” in such case because the posterior
preference confirms the prior. However, preferences are “weak” and alternate between A
and B close to the point of indifference because the posterior preference then violates the
prior. The length of this interval around the indifference point is 1 − &, which measures the
11
relative imprecision of the prior preference. Such interpretation is in line with Butler and
Loomes’s (2007) ‘imprecision interval’ surrounding lottery evaluations, although I differ from
their analysis by banning indecision and eliciting well-defined preferences on this interval
(with the exception of the indifference point, see sub-section (a) below).
B. Applications
(a) Indifference:
If A and B are EU-indifferent, the prior preference cannot be selected by the EU criterion
alone and will be determined by an arbitrary criterion. 0DU theory predicts in this special
case that the final choice will systematically violate this arbitrary prior. However, insofar the
criterion used for selecting the prior is unknown to the observer, EU-indifference transposes
into 0DU-indifference.
(b) The certainty effect:
Allais (1953) first noted the tendency to prefer a huge sure gain to a lottery of higher
expected value that offers a small probability of no gain. Kahneman and Tversky (1979)
replicated this finding with reasonably-valued amounts and labeled it the certainty effect.
We indicate below their examples (percentages of modal choice are indicated in brackets):
2500.33
= =2400.66D
0.01
[82%]
= 2400, 1.00
′ = 4000.80
0.20
′ = 30001.00
[80%]
Notice first that a small generalization of the previous analysis is in order because lottery A1
on the left has three outcomes with the middle one equal to the sure gain. Since ( )=
2400 → 0; .01, the objection-state is to receive no gain with a 1% probability when A1 is
the prior. Hence, “winning” must be considered lato sensu (i.e., as not losing) in this three-
outcome lottery. Therefore, is equal to 99% in the first example and to 80% in the second
example. The previous analysis predicts the likely occurrence of a certainty effect (apparent
risk aversion) when the probability of not losing with the risky lottery is high, as these
examples show. We replicated these findings in two incentivized experiments with small €
gains:
12
12.33
= =11.66D
0.01
(N=109)
(N=126)
= 111.00
[72%]
[72%]
′ = 16.80
0.20
′ = 121.00
[75%]
[73%]
The consensus11 in favor of the sure gain is obtained by the addition of people who had a
prior preference for the risky lottery but reversed their preference (AB), and those who had
a prior preference for the sure gain and confirmed this preference (BB). Moreover, there are
a higher proportion of subjects switching from A to B than from B to A because, thanks to
the high probability of winning, the potential opportunity loss must be much higher if
subjects stick to A than if they stick to B when the expected gains are close. In the first
example, switching from A to B may cause a loss of 2400 in case of bad luck (strong
objection) whereas switching from B to A would cause a rather small loss of 100 under the
same circumstances (weak objection).
(c) Insurance, gambling, and the reflection effect in presence of a sure outcome:
An insurance problem is characterized by the fact that the probability of the bad outcome in
the risky bet is small, while a gambling problem is found when the probability of the good
5000.001
−5000.001
= 51.00 ′ ≡ − = ′ ≡ − = −51.00
0.999 0.999
[83%]
[72%]
outcome is small. KT (1979) give one example of each:
= The bad outcome is a large loss (-5000) for the A’2 lottery on the right while the good
outcome is a large gain of equal absolute value (5000) for the lottery on the left. A
consensus emerges for avoiding risk in the insurance problem and for seeking risk in the
gambling problem. The insurance problem offers a particular instance of certainty effect
with a high probability of winning and the gambling problem offers an instance of risk-taking
with a small probability of winning.
The insurance and gambling decisions above display a reflection effect (i.e., ′ ≡
− ;′ ≡ −), whereby the insurance problem on the right is converted into the
gambling problem on the left and vice-versa by changing gains into losses and losses into
gains of an equal amount. The reflection effect is an anomaly for EU theory if the utility
11
I use this word to designate the decision of a large majority like three-fifths, two-thirds or more.
13
function remains concave or convex everywhere. For instance, a risk-averse subject would
prefer the sure outcome in KT’s insurance and gambling problems alike. The early solutions
proposed by Friedman and Savage (1948) and Markowitz (1952) to explain the joint
appearance of risk avoidance and risk seeking behavior in a single individual assumed that
the utility function could be convex on some intervals and concave elsewhere. A similar
property is retained by prospect theory where the value function is concave for gains and
convex for losses.
In the 0DU model, the reflection effect in presence of a sure outcome is strictly implied
under risk neutral EU-preferences. Indeed, the conditions for observing a certainty effect,
i.e. AB or BB in (7), simply convert into the conditions for observing a gambling effect, i.e. BA
or AA, by changing into 1 − , and * into 1 − *. Changing gains into losses and losses into
gains of an equal amount has exactly the same effect under the assumption of risk neutrality
so that an insurance problem then perfectly reflects into a gambling problem and vice-versa.
The reflection effect can be extended to nonlinear utility functions in the 0DU model under
certain conditions. Insofar the EU preference is weak and remains unchanged by reflection
(which is true under risk neutrality), the reflection effect may still be observed in our model
thanks to the preference reversal caused by the prior’s objection. However, if the EUpreference is unchanged, the observation of a reflection effect requires the prior’s objection
to be weak on one side and strong on the other. For instance, the objection to the prior
preference for certainty = 0 → −5; .999 is weak in the insurance problem; and
= 5000 → 5; .001 is strong in the gambling problem12. It is worth noticing that the
reflection effect is perfectly consistent with an everywhere concave, linear or convex utility
function in this theory.
(N=109) ′ = 15.50
−15.50
′ = 7.501.00 −′ = −′ = −7.501.00
0.50
0.50
[84%]
[64%]
(N=126) ′ = 15.50
−15.50
′ = 71.00 −′ = −′ = −71.00
0.50
0.50
[75%]
[49%]
We replicated the reflection effect with incentives and small gains and losses below:
12
Remember that the probability of the objection’s state has no incidence on the objection’s
strength in our model.
14
12.33
= =11.66D
0.01
′ = 16.80
0.20
(N=109) = = 111.00
′ = 121.00
!72%#
[73%]
100.01
= 11.00
0.99 [70%]
(N=126) = −12.33
− = =−11.66D − = −111.00
0.01
[74%]
−16.80
0.20
[56%]
−′ = = −′ = −121.00
−100.01
= −11.00
0.99 [50%]
20.01
−20.01
= 0.501.00 − = − = −0.501.00
0.99 0.99
[51%]
[60%]
Although some reflection is visible, the attraction to certainty also operates on those data.
When gains and losses are small, the difference in magnitude between the objections to the
prior on the gains and on the loss side is obviously reduced, which favors the alignment of
preferences by reflection.
(d) Symmetric bets, risk aversion and loss aversion:
People usually avoid symmetric bets of the form (y,.50 ; -y,.50) and their aversion increases
when the stakes get higher. This fact is well predicted by EU theory if the utility function is
everywhere concave but it is not unambiguously supported otherwise. Indeed, loss aversion
was introduced by KT (1979: 279) so as to make prospect theory compatible with the
rejection of symmetric bets. The rejection of symmetric bets is predicted in the 0DU model
as soon as the EU-preference for the status quo becomes a strong preference, and the
increasing aversion towards symmetric bets when the stakes get higher is a mere
consequence under 0DU of the same phenomenon under EU.
However, 0DU theory predicts that decision makers with concave utility functions will
actually accept a fair symmetric bet when the outcomes become small enough and the EU
preference for the status quo gets vulnerable to an objection. This fact was anticipated by
Markowitz (1952) and built in the shape of his utility function, then experimentally
15
replicated by Battalio, Kagel and Jiranyakul (1990 : table 4). In their experiment, 57% of
subjects rejected a fair symmetric bet when the stake reached $ 20, but 60% accepted it
when the stake was only $ 10. Moreover, the same sort of risk-seeking attitude was
observed by these authors (questions 12 and 13) if the latter outcomes were uniformly
increased by $ 15 or $ 20, transforming all losses into small positive gains. Such evidence
violates prospect theory but it is consistent with 0DU theory.
Rabin (2000) shows that, within the EU model, turning down a small favorable bet like (11,
.50; -10, .50) at all levels of wealth (within a certain range) implies implausibly high risk
aversion over large stakes. Loss aversion is one way of reconciling the frequently observed
rejection of small favorable bets with small rates of decline of the marginal utility of money
for small changes in wealth. 0DU theory suggests another solution to this problem. While
maintaining risk aversion in the concavity of the utility function, it introduces another kind of
loss aversion in the probability weighting. Turning down a small favorable bet like (11, .50; 10, .50) is consistent with virtual risk neutrality over modest stakes because the risk of losing
10 is a visible objection to accepting this bet which magnifies the perceived probability of
loss.
(e) The fourfold pattern of risk behavior:
A major empirical regularity (Cohen, Jaffray and Saïd 1987; Tversky and Kahneman 1992;
Tversky and Wakker 1996) is a distinctive fourfold pattern of risk behavior: risk aversion for
gains and risk seeking for losses of high probability; risk seeking for gains and risk aversion
for losses of low probability. This pattern is implied by (cumulative) prospect theory (Tversky
and Kahneman 1992). I show that it is also implied by the 0DU model for choices between a
risky lottery and a sure outcome.
Since the 0DU model shares the EU predictions when EU preferences are strong, the
difference in the two models’ predictions comes entirely from weak EU preferences. By
using conditions (7), we write13:
JK ≽ |. "M − JK ≽ |3"M = JKM − JKM
(8)
= JK − 1 − & < * < M − JK < * < + 1 − &1 − M.
13
I assume in conditions (7) that B is nested in A. If A was nested in B, the same conditions would
hold by permuting A and B and their respective probabilities of winning. This transformation of
conditions (7) is used in appendix 2 (case b).
16
Varying * uniformly on the open interval #0,1! for any value of , the probability difference
in (8) is equal to the difference in the lengths of the and intervals for *, that is,
1 − &2 − 1. If & < 1, this difference is positive for > 1/2, negative for < 1/2, and
null for = 1/2. Hence,
Proposition 2: (0DU and the fourfold pattern of risk behavior): In comparison with EU
predictions, the 0DU model predicts at the individual level more preference for certainty for
probabilities of winning (or not losing) higher than one-half and more preference for risk
taking for probabilities of winning (or not losing) lower than one-half when a risky lottery is
compared with a sure outcome.
III. Framing: theory and applications
A. Theory:
Framing under risk occurs because there is an infinite number of ways of describing a pair of
nondegenerate lotteries. Whereas all potential presentations are normatively equivalent
under EU theory, they are not descriptively equivalent for intuitive decision makers if the
characteristics describing the lotteries under comparison are unequally and sequentially
perceived. The particular form in which the prospects are perceived may actually be
conditioned by visible patterns working as persuasive heuristics that simplify decisions and
help overcome reasonable doubt. Framing matters to intuitively rational individuals because
their perception of dominance or objections essentially depends on the selected frame.
Let us now compare two nondegenerate independent lotteries presented in the same form:
O
= 1 − = 1 − O (9)
with > , > , and 0 < , O < 1. Without loss of generality, we further
assume that is not dominated by , i.e. : > .
Loomes and Sugden (1982) proposed in their theory of regret to convert independent
prospects into actions separating states of regret from states of rejoicing; and Bordalo et
al. (2012) used the same decomposition in their theory of salience. In the present case,
the lotteries would be decomposed into:
17
O
1 − O
PP = Q
R
1 − O
1 − 1 − O
O
1 − O
PP = Q
R
1 − O
1 − 1 − O
This presentation elicits the objection states that would cause regret if they occurred, but an
intuitive decision maker would not perceive it because it is not directly visible. From the
perspective of intuitive decision making, three general patterns14 may arise:
(a) ≤ ; (b) > > > or > > > ; (c) ≥ >
≥ or ≥ > ≥ .
We designate them as: (a) dominance (of A over B); (b) nested lotteries (B (A) nested in A
(B)); (c) overlapping lotteries (A (B) overlaps B (A)). This typology can be extended to lotteries
with more than two outcomes by grouping adjacent sure outcomes to form contingent
aggregated outcomes insofar the number of final outcomes does not exceed two per lottery.
(a) Dominance:
If A dominates B, A is EU-preferred to B and no objection can be found to this prior
preference. The decision maker feels sure of her choice because she incurs no risk of
decision: whatever happens with A is better (no worse) than its counterfactual in B.
(b) Nested lotteries:
The nested-lottery pattern (b) generalizes the sure-outcome case studied in previous
sections. Choosing the EU-preferred option under this pattern always raises a visible
objection, assigned either to the occurrence of the low outcome if A is the prior or to the
high outcome if B is the prior. All the analyses done in previous sections immediately
extend to nested lotteries by a mere substitution of for , and 3" ≡ * for " .
Proposition 3: The fourfold pattern of risk behavior (proposition 2) extends to nested
lotteries.
For example, the “safe” lottery = =
14
20.50
is nested in the two riskier lotteries
10.50
30.50
60.25
and P = . By proposition 3, risk-averse individuals should opt for
0.50
0.75
To be general, a theory of framing must be concerned with general patterns. However, I cannot
rule out at this stage the perception of familiar particular patterns. The simplest example I can
think of is a sure outcome facing a fair coin of equal expected value in combination with a common
outcome. This case is further discussed in note 23.
18
safety against but moderately risk-averse individuals would opt for risk with P . Since
expected gains of the three lotteries are equal, the fourfold pattern of risk behaviour refutes
second-order stochastic dominance.
(c) Overlapping lotteries and the action frame:
Overlapping lotteries are non-nested independent prospects that evoke stochastic
dominance of A over B if ≥ > ≥ because, for all outcomes facing each other
in a row, the outcome in lottery A is not lower than the outcome in lottery B and strictly
higher for one outcome at least15. However, first-order stochastic dominance truly obtains if
and only if ≥ O. An extension of stochastic dominance, called quasi-stochastic dominance
(QSD), is provided when < O but these two probabilities remain close. When the
difference between the win probabilities widens, the impression of dominance of A over B
fades out, and B is eventually preferred.
(i)
First-order stochastic dominance:
If and only if ≥ O, pattern (c) exhibits first-order stochastic dominance. Lottery A is then
always EU-preferred to B. However, choosing the first- order stochastically dominating
lottery A does not eliminate the risk of decision when lotteries are independent because
and may be drawn simultaneously with positive probability 1 − O. Tversky and
Kahneman (1986) showed that subjects invariably respect first-order stochastic dominance
when it is transparent but will frequently make choices that refute it when dominance is
hidden or not transparent, i.e. when it is no evidence. Their observation suggests that the
visibility of dominance contributes to the suppression of doubt. In our present setting, firstorder stochastic dominance is transparent if ≥ O and should thus be invariably respected.
To see why, let us first assume that = O. The two risky prospects appear as:
= 1 − = 1 − (10)
This presentation evokes alternative actions with true states of the world, such that if
occurs in A, then must occur too in B, and if occurs in A, must occur too in B.
We call this an action frame. The latter has the virtue of making objections visible for each
state separately. In this particular instance, no objection to the stochastically dominating
Further developments apply symmetrically to overlapping iff If ≥ > ≥ and
O ≥ . Throughout this section, it will be assumed that overlaps .
15
19
lottery can be found and the intuitive decision maker feels reasonably sure of her EU
preference for A16. More generally, if ≥ O, dominance is made apparent by the following
action frame because ≥ and the random outcome ,
the sure outcome :
(ii)
O
S4T
W 4T
\
′ = V
X
Y 1 − O [
4S
4T
U
Z
S4T
4T
; ,
4S
4T
O
= 1 − O dominates
(10’)
Searching for dominance:
Under pattern (c), there is no visible objection to the prior if < O. The decision maker faces
a dissonance between the search for an objection to her prior and the perception of
dominance of A over B suggested by the overlapping lotteries pattern. The pair of prospects
would then take one of the following forms highlighting the dominance pattern:
4S
= 1 − ′ = Υ 1 − , with Υ = X
4T Y; (11)
4S
T4S
T
O
Υ O
_ = 1 − O , with Υ = X
or P = ^ T4SY.
1 − O
T
S
(11’)
Which form is selected will depend on which characteristics of lotteries the decision maker
considers to be “decisive”, that is, to be susceptible to reverse the choice if perceived (LévyGarboua 1979). The decisiveness of one characteristic depends on how different and
important the latter appears in the lotteries. The absolute differences |" − " |
and |" − " | dictate the perceptual attraction of high versus low outcomes in the
decision process (Lévy-Garboua 1979, Bordalo et al. 2012, Közsegi and Szeidl 2013). The
perception that lottery A overlaps B and seems to dominate it is seen either as a justification
of prior preference for A or as an objection to prior preference for B. Conversely, the doubt
raised by the fact that lottery A does not stochastically dominate B triggers the search for an
objection to A. However, whichever form they perceive, intuitively-rational decision makers
16
In the special case of equivalence ( = , = ; = O, the two lotteries are indifferent.
20
are likely to view overlapping lotteries through an action frame and perceive dominance or
the visible objections induced by this frame.
Let’s assume initially that the high outcomes differ by a wider margin than the low outcomes
and form (11) is adopted. Then, consider first that A is EU-preferred to B. The action frame
(11) elicits a potential objection to A: = Υ → y ; 1 − p with3"Υ =
T4S
4S
" +
4T
4S
" . If the two probabilities of winning are so close that 3"Υ ≤
" , the impression of dominance persists because the objection is masked by the
frame17. As diverges from O, an objection to A becomes eventually visible. However, the
AA decision path is still taken and the prior remains by the 0DU rule (5) iff:
AA:
Or:
&3" + 1 − &" ≥ &3" + 1 − &3"Υ &3" − 3" ≥ 1 − &!3"Υ − " #
(12)
(12’)
Note that (12’) still holds when the impression of dominance persists with 3" ≥ 3"
as the right-hand term in brackets would then be non-positive. The expression (12’) of the
0DU rule (12) may be given the psychological interpretation that the perception of
dominance prevails insofar the objection is less “salient” than the prior preference18. We call
this quasi-stochastic dominance (QSD). Developing the expected utilities appearing in (12’),
we get another formulation of this condition:
&!" − " # + 1 − &!" − " # ≥ 1 − & 4S !" − " # (12’’)
T4S
The left-hand term is the difference in the expected utilities of lotteries A and B in which the
objective probability has been converted into &. This modified EU difference is always
positive and reflects a virtual preference for the seemingly dominating prior A. The righthand term is also positive if < O and proportional to the difference between the win
probabilities O − . Thus, the lower is this probability difference, the more likely is lottery A
to be perceived as quasi-stochastically dominating lottery B and to be chosen by an
T4S
4S
!" − " # ≤ " −
" . A necessary condition is:" > " .
The natural definition of the objection’s salience deriving from 0DU theory is the product of the
potential opportunity loss and the objection’s weight. Symmetrically, the EU-preference salience is
the product of the EU-difference and the prior’s weight. The relative salience of the objection,
which is the ratio of these two values, bears some analogy with Bordalo et al.’s (2012) definition of
salience. The present definition of relative salience appeared with a slight modification in my
earlier working paper (Lévy-Garboua 1999).
17
The condition for the suppression of doubt can be rewritten as:
18
21
intuitively-rational decision maker. However, as the difference between and O widens, the
illusion of dominance fades out, the objection’s salience increases, and the prior preference
is eventually violated in favor of B.
Consider now that B is the prior. In the action frame, the objection to B is now19: =
→ y ; and A is eventually preferred (BA path) by 0DU rule (6) iff:
&3" + 1 − &" > &3" + 1 − &" BA:
&3" − 3" < 1 − &!" − " #
Or
(13)
(13’)
The perception of dominance prevails now iff the prior preference for B is less salient than
its objection. By developing the latter, we get the equivalent formulation:
71 − &1 − 8!" − " # + &1 − !" − " # >
&1 − 4S !" − " # (13”)
T4S
Again, the left-hand term of (13’’) is a modified expected utility difference in favor of the
seemingly dominating lottery A and the right-hand term is proportional to the difference
between the win probabilitiesO − . When the latter probability difference is small
enough, the EU-preference for the seemingly dominated lottery B is violated and the QSD of
lottery A prevails.
The same analysis can be replicated if the form (11’) is adopted for comparison because low
outcomes differ by a wider margin than high outcomes. The conclusions are the same (see
proofs in Appendix 1).
Proposition 4 (QSD): An illusion of stochastic dominance is created by adoption of the QSD
frame irrespective of the risk attitude and EU preference when two risky lotteries overlap
with > and close probabilities of winning of intermediate value20.
B. Applications:
Since 3" ≥ 3", we must have: 3"Υ > " if y < y .
In what follows, O − is kept constant. Assume that Min (" − " , " − " = 0
which is true in particular for the Allais paradox and its reflection. When " − " >
" − " ≡ 0 and form (11) is adopted, a high value of (in the interval (0,1/2)) is favorable
to QSD. When 0 ≡ " − " < " − " and form (11’) is adopted, the predictions are
reversed for O, and a low value of O (in the interval (1/2,1)) is favorable to QSD. Since 1 − O is
transformed into by reflection, it will be possible to observe QSD with the Allais paradox and its
reflection if and O take close intermediate values on the interval (0,1/2).
19
20
22
(a) Allais paradox and quasi-stochastic dominance:
The Allais paradox (Allais 1953, Kahneman and Tversky 1979) consists in the joint
observation of a certainty effect and a risky decision which contradicts the latter within EU
theory. Starting from KT’s (1979) examples of a certainty effect studied in sub-section II.B.b,
the common consequence (2400, .66) in the left-hand lotteries is converted into (0, .66) and
the probability of gains in the right-hand lotteries is divided by a common ratio of four.
These transformations preserve the EU preference thanks to the independence axiom.
However, as shown below, the modal revealed preference is reversed and this holds too for
the individual preferences of a majority of subjects. The left-hand version of the Allais
paradox is known as the common consequence effect and the right-hand version as the
common ratio effect.
` = 2500.33
0.67
` = 2400, .34
0, .66
′` = ^4000.20_
0.80
′` = 3000.25
0.75
The two pairs of lotteries obtained by those transformations overlap, with = .33.20 and
[83%]
[65%]
O = .34.25, and they share a common outcome (0). Although lotteries A don’t
stochastically dominate lotteries B since < O, we assume that intuitively-rational decision
makers can use the action frame (11) for overlapping lotteries in search for dominance.
We demonstrated in III.B (ii) that a consensus should emerge by quasi-stochastic dominance
in favor of both lotteries A if O − is low and takes an intermediate value. Indeed,
O − = .01and = .33 for the left-hand pair whereas O − = .05 and = .20 for the
right-hand pair. Note that the proportion of subjects choosing lottery A is smaller (65%
instead of 83%) in the right-hand pair (common ratio example) as the probability difference
is larger.
Proposition 5: (Allais paradoxes) The common consequence and common ratio effect on
independent prospects are both predicted in 0DU theory by the change of framing: the
nested-lotteries frame is adopted when B is a sure outcome whereas quasi-stochastic
dominance is most likely perceived when B overlaps with A with close probabilities of winning
of intermediate value.
We give below a closer comparison of the certainty effect in favor of B1 (B’1) and the quasistochastic dominance effect in favor of A3 (A’3).
23
Considering the common ratio effect first where the probability of winning with A J < 1 is
divided by a factor a > 1 (i.e., = J⁄a), we must compare respectively conditions (5) and
(12) that describe an AA path and conditions (6) and (13) that describe a BA path. To make
them easily comparable, we develop these conditions and rearrange terms as in (12’’) and
(13’’):
" − " ≥
AA|B sure:
" − " ≥
AA|B risky and QSD:
4bc
bc
!" − " #
4bS T4S
bS 4S
!" − " #
(14)
" − " > 4b4c !" − " #
BA|B sure:
b4c
" − " > 4b4S 4S !" − " #
b4S T4S
BA|B risky and QSD:
Since = = 0 in the KT’s examples, the utility differences appearing on the left or on
the right of these four conditions are identical, so that what makes the difference between
the certainty effect and quasi-stochastic dominance lies in the proportionality factors.
Dividing the probability of winning P by a factor n increases the ratio, but introducing the
multiplicative factor
T4S
4S
in the QSD diminishes the ratio. The magnitudes of these two
effects are in favor of a diminution in the Allais paradox. For example, if J = .80, a = 4; O −
= .05; & = .721, the ratio is divided by 2.05 along the AA path and by 20.48 along the BA
path when we move from the certainty effect to the common ratio effect. As a result, the
modal preference for the sure outcome converts into a modal preference for the riskier
lottery by quasi-stochastic dominance.
The analysis of the common consequence effect uses the same reasoning and the conditions
for choosing A are similar:
AA|B sure:
AA|B risky and QSD:
BA|B sure:
21
" − " ≥
" − " ≥
4b4T4S
bS
4bS T4S
bS 4S
!" − " #
!" − " #
(15)
" − " > 4b4S !" − " #
b4S
.7 is the estimated mean value of µ in my companion paper (Blondel and Lévy-Garboua 2016).
24
BA|B risky and QSD:
" − " > 4b4S 4e !" − " #
b4S d4e
The main factor which is responsible for the two Allais paradoxes is the multiplicative factor
T4S
4S
in the QSD. In the numerical example of KT reproduced here for the common
consequence effect, the difference between the certainty effect and quasi-stochastic
dominance is considerably amplified along the AA path by a factor of 8.04, and along the BA
path by a factor of 20.12. Thus, both the common consequence effect and the common ratio
effect are essentially produced by the illusion of persistence of stochastic dominance even
when the seemingly dominant lottery is not the prior.
We replicated these findings in two incentivized experiments with small € gains. The
certainty effect was clearly visible on the comparisons of with , and ′ with ′ . We
compare below the two risky lotteries of the Allais paradox on these two independent
samples:
` = (N=109)
12.33
0.67
!66%#
(N=126) !68%#
` = 11.34
0.66
′` = 16.20
0.80
[55%]
!51%#
′` = 12.25
0.75
Again, with the chosen numerical values, the common consequence effect is more neatly
replicated than the common ratio effect. This result is consistent with conditions (14) and
(15) for QSD being formally identical for the two effects because the chosen numerical
values are different between the two examples. For instance, assuming risk-neutral EUpreferences and a QSD frame, the AA path should be followed if & = .7 in the comparison
between ` and ` but (marginally) not in the comparison between ′` and ′` 22.
The critical role of framing in the Allais paradox is also illustrated by the disappearance of
the common consequence effect when lotteries A and B are framed as true actions (with
three states of the world), in conformity with Savage’s (1954) intuition in his controversy
with Allais (1953). Cancelling the common consequence is then harmless because it neither
changes the prior EU-preference nor its objection and their relative salience. Experimental
results of Conlisk (1989) and Wakker, Erev and Weber (1994) confirmed this prediction.
22
Condition (14) for the QSD of ′` over′` : (16-12)≥
4.f∗.g .gh
.f∗.g .ig
12 − 0, i.e. 4≥ 4.605, marginally
doesn’t hold. In contrast, condition (15) for the QSD of ` over ` holds: (12-11)≥
i.e. 1≥ .547.
4.f∗.`` .g
.f∗.`` .jf
11 − 0,
25
However, the common ratio effect can still be found with an action frame. The common
ratio transformation is obtained in the action-frame by multiplying the probability of each
state by a common ratio r (0 < r < 1) , and finally adding a common third state yielding the
worst outcome with probability 1 − r in order that the three probabilities add up to one. By
doing so, the objection doesn’t change but the salience of the prior preference is multiplied
by r and the relative salience of the objection is multiplied by 1⁄0 > 1.
(c)Apparent risk aversion and the possibility effect:
KT (1979) show another violation of the independence axiom with the following example.
When there is a substantial probability of winning, the consensus favors the lottery with the
larger probability of gain (apparent risk aversion). But, when winning is possible but has a
very low probability, most people choose the lottery offering the more unlikely but larger
6000.001
3000.90
3000.002
6000.45
k = ′k = ^
_ ′k = 0.999
0.10
0.998
0.55
[86%]
[73%]
We replicated these findings in two incentivized experiments with small € gains and a less
extreme probability of winning:
prize (risk-taking). This last effect is called the possibility effect:
k = (N=109)
(N=126)
k = 30.45
0.55
k = 20.45
0.55
k = 30, .25
0.75
30.01
15.90
′k = 0.99
0.10
[83%]
[68%]
8.90
20.01
k = ′k = 0.10
0.99
[59%]
[63%]
20.25
k = =10.25D
0.50
[81%]
k = ′k = ′k = 15.02
0.98
10.02
0.98
Suppose now that the majority of people are risk averse and the EU-preference is B4 (′k ).
Since A4 (′k ) overlaps23 B4 (′k ), we apply here the 0DU rule (13’) to characterize the
preference for A4 (′k ):
23 It is worth noticing that lotteries and demonstrate a preference for pseudo-certainty, evoking the
k
k
preference for certainty against a risky lottery with close expected gains and a probability of winning of onehalf. For instance, the first pair may be written:
30.50
15.90
.90
n "k = . Since most people are familiar with tossing fair coins, they might
"k = m 0.50
0.10
0.10
perceive this familiar pattern and, from there, adopt an action frame and cancel the common outcome 0. They
26
&3" − 3" < 1 − &!" − 3" # (13’)
When we move from the left-hand to the right-hand pair of lotteries, the gains remain
identical but the probabilities of winning are divided by 450 in KT’s examples and by 45 in
ours. Consequently, the salience of the prior preference for B4 is divided by 450 or 45
whereas the objection’s salience is the same. Thus, condition (13’) demonstrates that it is
most likely that the modal preference shifts from B4 to A’4. Note that, in contrast with
proposition 5, QSD is obtained here for the right-hand pair of lotteries although win
probabilities are very small because their difference gets very small too.
As for the minority of risk lovers, the choice of A’4 is predicted by (12’):
&7. 001"6000 − .002"30008 ≥ 1 − &!.001"3000#
"6000 ≥
1+&
"3000
&
Only will the more risk-loving individuals confirm their EU-preference for A’4. For instance, if
& = .7, the possibility effect requires from risk-loving subjects "6000⁄"3000 ≥ 2.43.
In these examples, the possibility effect results from the attraction of QSD for practically the
whole majority of risk-averse subjects and a minority of highly risk-loving subjects.
Note that our analyses of Allais paradox and the possibility effect elicit the decision
processes underlying Rubinstein’s (1988) similarity judgments.
(d) The reflection effect with two risky lotteries:
The Allais paradox and the possibility effect are conserved by reflection on the reversed
preferences, as shown by the KT’s (1979) examples reproduced below.
−4000.80
−4000.20
−3000.25
−A’1 = − B’1 = −30001.00; −’3 = −’3 = 0.20
0.80
0.75
[92%]
[58%]
30.50
of equal expectation. Such
0.50
transparent familiar pattern may mask the overlapping lotteries frame. However, even though the same
pattern is present in choices between ′k and ′k , a consensus obtains for the riskier lottery, suggesting that
the QSD frame is now perceived.
Notice too that the last lottery k has three outcomes but the two positive ones lie strictly
20.50
between 30 and 0 and can thus be grouped as a contingent outcome with probability .50.
10.50
The pseudo-certainty pattern then becomes apparent.
would then be left with a choice between the sure gain 15 and the lottery 27
−3000.90
−6000.001
−3000.002
−6000.45
−k = −k = ;−′k = −′k = 0.10
0.999
0.998
0.55
[92%]
[70%]
The 0DU model can predict the reflection effect with two risky lotteries. The overlapping
lottery’s frame is conserved by reflection with a reversal of the dominance relation. QSD is
now obtained when subjects take paths –B-B or –A-B. Since 0 is now the common high
outcome within each pair, intuitively-rational decision makers will use the action frame (11’)
in search for dominance. Condition (16’) characterizes the –B-B path and condition (17’’)
characterizes the –A-B path. However, if and O designate the win probabilities for A and B
respectively, these two conditions apply here, due to the reversal of the dominance relation,
by reflection of into 1 − O and of O into 1 − . After these transformations, conditions
(16’) and (17”) for QSD become respectively:
"− − "− ≥
"− − "− >
4bS T4S
bS 4S
!"− − "− #
b4S T4S
4b4S 4S
(16’reflect.)
!"− − "− #(17’’reflect.)
Conditions (16’reflect.) and (17’’reflect.) are the exact replications of (14’) and (15’’)
respectively with the reflected outcomes. Thus, the reflection effect is strictly implied for
QSD under risk neutral EU-preferences with two risky lotteries, which extends to QSD the
result previously obtained for the certainty effect. Similarly, the reflection effect can be
extended for QSD to nonlinear utility functions in the 0DU model insofar the curvature of the
utility function does not push too strongly in the opposite direction. Actually, the range of
QSD may be extended in the loss domain with a concave utility function and reduced with a
convex utility function. An S-shaped utility function of the type postulated by prospect
theory would approximately maintain the same range of QSD in the gain and in the loss
domain. However, loss aversion would enhance QSD for small losses relative to small gains.
The reflection effect for QSD is replicated below for small losses in incentivized experiments
and extended to the common consequence effect:
−12.33
−11.34
(N=126) −` = −` = ;
0.67
0.66
!60%#
−20.45
(N=126) −k = 0.55
−8.90
−k = ;
0.10
[60%]
−16.20
−12.25
−′` = −′` = 0.80
0.75
[60%]
−20.01
−10.02
−′k = −′k = 0.99
0.98
[62%]
28
−30.25
−k = 0.75
−20.25
−k = = −10.25 D
0.50
!68%#
It can be observed with small outcomes that the preferences reflect better –at least in the
aggregate- when QSD holds than if it doesn’t, including the case where one option is a sure
outcome. This may be due to the fact that the objection’s salience has little relevance when
QSD holds (proposition 4) and a lot more when lotteries are nested or QSD doesn’t hold.
(e) The isolation effect:
The isolation effect (KT 1979) is a violation of the reduction of compound lotteries axiom.
This axiom of EU theory states that a decision maker is indifferent between a compound
lottery and the simple lottery in which the probabilities of the two-stages of the compound
lottery have been multiplied out. Lotteries h and h below describe a two-stage gamble in
which there is a probability of .75 to end the gamble with nothing and a probability of .25 to
move to the second stage. If you reach the second stage, you have a choice between the
simple lotteries A’1 and B’1. The choice must be made before the first stage. If the axiom of
compound lotteries holds, the compound lottery below should be equivalent to A’3, and B5
to B’3, the two lotteries that illustrate the common ratio effect. Therefore, A5 should be
′ .25
0.75
′ .25
0.75
[78%]
preferred to B5. However, 78% of KT’s sample chose lottery B5.
h = h = 4000.80
′ = 3000
0.20
Naturally, lotteries h and h appear in the action frame since ‘ending the gamble after the
With ′ = first stage’ and ‘reaching the second stage’ are two states of the world with probabilities .75
and .25 respectively. In an action frame, the common state 0.75 can be cancelled
because it has no incidence on the prior and on its objection. Once the common state has
been removed, the problem is reduced to a choice between ′ and ′, and the certainty
effect is observed. The action frame has allowed the decision maker to isolate the second
stage of the compound lotteries and, consequently, to change her perception by converting
initially overlapping prospects into nested ones.
(f) Violation of the continuity axiom:
29
An example, due to Prelec (1990), is the following. Most people prefer A6 to B6 but C6 to A6:
30000.01
20000.34
30000.17
20000.34
j = j = j = rj = =20000.32D
0.66
0.83
0.66
0.67
[94%]
[82%]
However, rj =
j
f
j +
f
j and, by the continuity (or, betweenness) axiom, if A6 is
preferred to B6, it should still be preferred to any lottery lying between A6 and B6, like rj .
This paradox of EU theory may be seen as another consequence of the critical effect of
framing on preferences under risk. Although j is likely to be strongly preferred to j, when
j is compared with rj the nested lotteries frame becomes apparent and many subjects will
view this as a comparison between j and r′j = ^
dominating sj = ^
30000.01
_ , with j stochastically
sj .99
20000 . 32⁄. 99
_. The objection to the nested prior j , (j )=
0.67⁄. 99
30000 → Aj ; .01, is considerably more salient than the EU-preference, and r′j is now
0DU-preferred to j .
(g) Violation of first-order stochastic dominance:
The fact that first-order stochastic dominance does not eliminate the objective decision risk
suggests that it may be violated if the action frame does not reveal the dominance relation.
This doesn’t occur with two-outcome prospects but Birnbaum (2005) has shown a number of
such violations with three-outcome prospects. His benchmark example is the following:
96.90
f = =14.05D
12.05
96.85
f = =90.05D
12.10
[74%]
Although A7 first-order stochastically dominates B7 because J* > t|f ≥ J* > t|f for
all t, the visible action frame obtained by regrouping the two common outcomes 96 and 12
no longer exhibits the dominance relation:
′f = Q
96 .uh
.ug
X
.ghY.95 R
12 .uh
14.05
96
.uh
X
Y.95
′f = Q 12 .g
R
.uh
.ih
90.05
Instead, this frame reveals an objection to EU-preference: ′ = 90 → 14; 05. By using
equation (5), we can compare the 0DU of the two pseudo-actions:
30
. "′f |f = &3"f + 1 − &"14
. "′f |f = &3"f + 1 − &"90
The size of the potential opportunity loss incurred in choosing the EU-preferred (in fact,
stochastically dominating) lottery "90 − "14 is so large that a preference reversal is
highly plausible. It is worth noticing that dominance might have been elicited by grouping
the first and second outcome (96 and 90) in′f and the second and third outcome (14 and
12) in ′f ; but this is no evidence to an intuitive decision maker if the second row of f and
f is perceived as a “common state”24, the consequences of which cannot be felt
independently in alternative actions.
IV. Framing: a test of the theory
A. A comparison of overlapping versus nested lotteries:
If framing matters, preferences must be shaped in part by the type of frame being used on a
given pair of lotteries. I showed however that each pair of lotteries without dominance leads
to a particular frame, essentially nested or overlapping lotteries. My strategy for testing the
effect of framing is to compare the probabilities of preferring A to B conditional on the frame
around the point where the substitution between these two frames takes place. Since the
individual parameters and their distribution are invariant when the frame is changed, the
two probabilities of preferring A to B conditional on the frame should not differ if frames do
not matter.
I compare the frequency of choice of lottery A over B in the following sequence:
O
O
v1 − ≻ 01 − ≽ v1 − O ≻ 01 − O Or, with obvious notations:
x ≻ ≽ x ≻ where > > 0, 0 < < O < 1,and v is a small gain. The reflection of this sequence
of lotteries is also examined. The operator ≻indicates a transparent relation of first-order
stochastic dominance and ≽ signals a potential preference relation (including indifference).
We compare A with x , A with , and x with . Since vis a small positive value, x is
nested in A but B and x overlap. Moreover, the frames are conserved by reflection. If
24
Viewing the second row of the pair of prospects, resp. (90,.05) and (14,.05), as a common state is
based on the evidence that these two outcomes face each other and have the same probability.
31
framing does not matter, the frequency of preference for A should vary little whether A is
compared with or with x , the preference for should also vary little whether it is
compared with A or with x .
Proposition 5 : (A proof that framing matters) In the vicinity of Allais-type risky lotteries, the
frequency of risky choices when probabilities of winning are close and at some distance from
1 is normally higher for overlapping than for nested lotteries if & < 1.
The reverse proposition is true by reflection in the domain of losses.
The proof is relegated to appendix 2. In the KT’s (1979) examples and our own, O takes an
intermediate value like .34 or .25. It is worth noticing that this proposition is not restricted to
lotteries that yield only gains or losses and it is potentially applicable to mixed lotteries.
B. Results of the test:
We collected experimental data on lottery choices with gains and losses similar to those
found in KT(1979). Two incentivized experiments were realized in 2013 and 2014 at the LEEP
(Laboratoire d’économie expérimentale de Paris) using the random lottery incentive
procedure25. Participants received a show-up fee of 5€ and whatever they earned from the
random draw of their selected decision. For choices involving losses, they received an
endowment prior to their choice and their loss was eventually deducted from this
endowment before payment. A Holt and Laury (2002) test of risk aversion was included in
the design prior to lottery choices to separate the sample in two groups: risk-averse (N=75)
and risk-tolerant (N=51).
I chose for , O values of .33, .34 and (.20, .25) which have proved to be appropriate for
quasi-stochastic dominance. Table 1 shows the frequency (in %) of choice of the riskier
lottery when x is compared with B (col. 1) while A is compared with x (col. 2) and with B
(col. 3). This frequency has been computed for the whole sample (in bold letters), and for
12.33
11.34
and` = with v =10 cents,
0.67
0.66
the groups of risk-averse subjects on the left and risk-tolerant subjects on the right. For
example, in the comparison between ` = 79% preferred x to B, 61% of subjects preferred tox , and 68% preferred A to B. The
difference between the probabilities of preferring A to B and x to B is significant at the 5%
25
Notice that the isolation effect which guarantees the effectiveness of the random lottery
incentive procedure is consistent with ODU theory, as demonstrated in section III.B.e.
32
level (p=.0449). When we take the reflection of these lotteries, 78% preferred -B to−x ,
44% of subjects preferred−x to−, and 60% preferred -B to -A. The difference between
the probabilities of preferring −x to−and -B to -A is significant at the 5% level ( = .017),
and the difference between the probabilities of preferring -B to −x and -B to -A is
significant at the 1% level ( = .0018). Table 1 demonstrates proposition 5: the overlapping-
lotteries frame enhances the choice of the lottery yielding the higher gain, or the lower loss
by reflection, relative to the nested-lotteries frame for similar Allais-type lotteries. This
framing effect is maximal under strict QSD (compare col. 1 with col. 2) but remains when the
lotteries include the common component26 0 (compare col. 3 with col. 2). Moreover, the
prediction by proposition 4 that the illusion of quasi-stochastic dominance operates almost
equally irrespective of the risk attitude for Allais-type lotteries is not refuted by the data.
Whereas the risk attitude partly conditions the preference if lotteries are nested, its
influence appears negligible under QSD.
Table 1- Frequency of choice for the higher-gain (lower-loss by reflection) lottery, and its
variation between risk-averse and risk-tolerant subjects
12.33
11.34
≽ 0.10.67
0.66
16.20
12.25
≽
0.10.80
0.75
77 79 82
−16.20
−12.25
≽
−0.10.80
0.75
75 74 73
16.20
12.25
≽
0.80
0.10.75
−11.34
−12.33
≽
−0.10.66
0.67
61 52 39
16.20
12.25
≽
0.80
0.75
44 51 61
45 44 43
−16.20
−12.25
≽
0.80
−0.10.75
12.33
11.34
≽
0.67
0.66
68 68 69
41 53 71
−11.34
−12.33
≽
0.66
−0.10.67
77 78 78
55 61 71
61 63 65
12.33
11.34
≽
0.67
0.10.66
−11.34
−12.33
≽
0.66
0.67
61 60 57
−16.20
−12.25
≽
0.80
0.75
65 60 51
26
Notice that, for lotteries including a common outcome, QSD is less clearcut on the gain side than
on the loss side. Indeed, on the gain side, the quasi-stochastically dominated lottery may be
perceived alternatively as nested if (and only if) it was EU-preferred. A number of subjects may
then opt for the nested-lotteries frame instead of QSD, particularly when the win probabilities are
not “very” close as for the second pair of lotteries (row 2). This phenomenon does not occur on the
loss side since the high outcome (0) is now common to both lotteries and thus cannot serve as a
visible objection.
33
V. Conclusion
I presented a descriptive expected utility theory of decision under risk that predicts all the
anomalies of EU theory exhibited by KT (1979) with greater parsimony than prospect
theory27. This theory can predict choices among two independent prospects including four
(sure or contingent) outcomes with a single parameter in addition to risk aversion. It
generalizes expected utility under risk (Von Neumann and Morgenstern 1947) by
transforming objective into well-defined subjective probabilities conditional on their prior
stated values. The probability transformation curve is consistent with the inverse-S shape
postulated by prospect theory, thus overestimating low probabilities and underestimating
high probabilities. As another resemblance with prospect theory, the utility function is
reference-dependent if the initial wealth remains implicit and not binding before the
decision. However, it is a regular Von Neumann-Morgenstern utility curve which may be
concave, linear or convex everywhere, a property that preserves the definition of (prior) risk
aversion in EU theory. The revealed risk aversion will in general deviate from this prior value
and also depend upon the transformation of probabilities. The gain in parsimony with
respect to prospect theory is obtained by our simple account of the decision process under
risk or uncertainty and our systematic inclusion of framing in the latter. A simple account of
the decision process is made possible by the assumption that decision makers are intuitivelyrational in the sense that they are logical but have a sequential perception of the objects of
choice and rely only on evidence to decide. Under risk or uncertainty, the preference is no
longer in general a relation of dominance and it will then raise an objection. Since the
perception of an objection to the prior triggers doubt (epistemic uncertainty), intuitivelyrational decision makers update their prior in a Bayesian fashion, which may lead to
preference reversals within the decision process and cause the so-called anomalies to EU
theory. The role of framing is to help individuals decide beyond reasonable doubt either by
revealing a dominance relation or by making the prior’s objection visible. In the latter case,
an objection-dependent expected utility (ODU) will be maximized. Framing, in this theory,
depends unambiguously on easily identifiable patterns of each pair of lotteries. Three
patterns only are sufficient to describe behavior with simple independent prospects:
27
The rejection of probabilistic insurance discussed by KT (1979) is not included in the present
paper because it involves the valuation of lotteries. However, the resolution of this problem can be
found in my earlier working paper (Lévy-Garboua 1999).
34
dominance, nested lotteries and overlapping lotteries. Nested lotteries exacerbate the
dilemma between risk-taking and certainty by making the nested lottery appear relatively
safe and by highlighting the objection to any prior preference, whereas overlapping lotteries
may bring certainty in decisions that are intrinsically risky by suggesting stochastic
dominance or “quasi-stochastic dominance” if lotteries are similar.
ODU theory elicits cognitive mechanisms which people use for overcoming doubt. These
mechanisms allowed us to deduct from the general assumption of intuitively-rational
behavior the laws of probability transformation that prospect theory sets out by induction.
These “laws” are summarized by the inverse S-shape of the probability transformation curve,
the Allais paradox and the possibility effect. The same cognitive mechanisms are sufficient to
predict the reflection effect with no additional assumption about the shape of the value
function, and to suggest a general theory of framing for pairwise choices among simple
prospects. Therefore, this theory may offer a micro-foundation and simplification of
prospect theory.
The failure of non-EU models of decision under risk to provide a superior and parsimonious
alternative to EU on a wide range of outcomes inside and beyond the Marshak-Machina
triangle (e.g., Camerer 1989, Harless and Camerer 1994, Hey and Orme 1994, Blondel 2002)
has given rise to an increasing reliance on noise and errors to explain observed departures
from EU predictions (e.g., Hey and Orme 1994, Loomes and Sugden 1995, Ballinger and
Wilcox 1997, Blavatskyy 2010). This research is increasingly linked with psychological models
of drift diffusion that optimize the speed of the choice-response for a given level of accuracy
(e.g., Ratcliff 1978, Busemeyer and Townsend 1993, Ratcliff and McKoon 2008). The two
assumptions on which I rely –accumulation of evidence and sequential perception- are
consistent with those of drift-diffusion models. For simplicity, I don’t assume noisy evidence
since the focus here is on observed modal choice among simple prospects28. Instead, I
consider epistemic uncertainty (doubt) to be a major cause of systematic preference
inconsistencies and instability which are not errors in the eyes of intuitively-rational decision
makers.
28 A major concern of drift diffusion models is to predict the response time for making a decision.
Although this goal is not pursued here, ODU theory may have something to say about decision
times, like predicting a faster decision when (stochastic) dominance is apparent than when it is not
and when preference is strong than when it is weak.
35
ODU theory also predicts anomalies not considered by KT’s original paper. Violations of
betweenness (Prelec 1990) and of first-order stochastic dominance (Tversky and Kahneman
1986, Birnbaum 2005) offered good examples of framing effects in the present study. The
resolution of lottery evaluation problems like the gap between the willingness to accept and
the willingness to pay, the endowment effect, and the preference reversal phenomenon is
another important application examined in a companion paper (Blondel and Lévy-Garboua
2016).
ODU theory is consistent with basic psychological mechanisms. It describes decision under
risk as the output of a decision process which makes it context-dependent. Thanks to the
assumption of intuitively-rational decision makers, it provides a tractable Bayesian theory of
information treatment by the brain that is consistent with the reward prediction error model
in neurosciences (Schultz et al. 1997) axiomatized for economics by Caplin and Dean (2008).
It integrates psychological insights borrowed by other existing economic theories of decision
under risk like regret (Bell 1982, Loomes and Sugden 1982), imprecision (Butler and Loomes
2007), similarity judgments (Rubinstein 1988, Leland 1994) and salience (Bordalo et al.
2012). Doubt triggers the perception of an objection to the prior preference which might
cause regret if the latter were implemented. However, doubt truly signals the presence of a
decision risk ex ante and the need for more information before choosing whereas regret and
rejoicing are negative and positive ex post feelings that can only reinforce future identical
decisions (Camille et al. 2004). The illusion of quasi-stochastic dominance provides a
rationale for similarity judgments and the ODU rule may be interpreted as a comparison
between the salience of the prior preference and its objection.
ODU theory sets a bridge between the normative and descriptive theory of expected utility.
The context-free EU-preference enters the decision process as a prior and thus has a genuine
behavioral existence, though a transient one. This prior is confirmed by ODU rule if it is a
strong preference and refuted if it is a weak preference. Only strong preferences are fully
rational and internally consistent. Weak preferences and the violation of normative
principles are a consequence of the imprecision of the prior EU-preference to intuitivelyrational decision makers.
In its present state, ODU theory is restricted to single pairwise choices among simple
lotteries with stated probabilities. The theory is extended to general correlated actions and
evaluation problems, and to repeated choices in companion papers. Extensions to decisions
36
under ambiguity and decisions from experience are potential applications of the present
framework. The extension of the present theory of framing of independent lotteries to more
than four outcomes must await further developments.
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APPENDIX 1
We replicate here the analysis of quasi-stochastic dominance when the form (11’) is adopted for
comparison because low outcomes differ by a wider margin than high outcomes. If lottery A is the
prior, its potential objection is: = y → Υ ; q. If the two probabilities of winning are so
close and/or O is so large that3"Υ ≥ " , the impression of dominance persists because the
objection is masked by the frame29. Otherwise, an objection to A is visible in the action frame.
However, A quasi-stochastically dominates B iff:
AA:
&3" + 1 − &3"Υ ≥ μEUB + 1 − μUy Or 71 − &1 − O8!" − " # + &1 − O!" − " # ≥
1 − &1 − O
T4S
!" −
T
(16)
" # (16’)
We derive from (16’) that, if lottery A is the prior, A is most likely to quasi-stochastically dominate B
with the action frame (11’) if O ≈ and O ≫ 0.
If lottery B is the prior, we must have 3"Υ < " and the objection to B is: = →
y ; 1 − q. Lottery A is eventually preferred to B iff:
BA:
Or
&3" + 1 − &"y > &3"B + 1 − μUy &73" − 3"8 < 1 − &!" − 3"y #
(17)
(17’)
Even though lottery B has a higher EU than A, A would be preferred if the objection to B in the
overlapping lotteries frame was more salient than the EU preference for B.
(17’) may be written otherwise:
&O!" − " # + 1 − &O!" − " # > &O
" . A necessary condition is:" > " .
29
T4S
!" T
The condition for the suppression of doubt can be rewritten as:
T4S
T
− " # (17’’)
!" − " # ≤ " −
40
We derive from (17’’) that, if lottery B is the prior, A is most likely to quasi-stochastically dominate B
with the action frame (11’) if O ≈ and O ≫ 0.
APPENDIX 2
Considering pairs of lotteries of general form (9) with < O and their reflection, let us calculate
separately for the nested and overlapping lotteries frames the probability of choosing the lottery
with the higher prize (lower loss) in excess of what the EU model predicts. In all the problems studied
by KT(1079) and here, the frame being used is (11) for the lotteries implying gains, or (11’) for the
reflected lotteries implying losses. These two cases will be examined separately.
(a) Frame (11) is used for lotteries implying gains, s.t. − ≥ | − | ≥ 0 with one strict
inequality at least. We compare Δ ≡ JK ≽ |. "M − JK ≽ |3"M = JK|0s€11M −
JK|0s€11M between the nested frame and the QSD frame. We use the following notations:
" = 1; " = 90 < 9 < 1. For the nested lotteries frame, " = 0 and " =
‚ƒ 0 ≤ ‚ƒ ≤ 1 − 9; and, for the overlapping lotteries frame, " = 0 and " = ‚6 0 ≤
‚6 ≤ 1 − 9.
Conditions (7) yield:
∆ |a…t† = JK < O9 + 1 − O‚ƒ < & + 1 − &M − JK& < O9 + 1 − O‚ƒ < M ;
and conditions (12) and (13) yield:
∆ |2‡0 = J ˆ + 1 − ‚6 < O9 < + 1 − ‚6 +
4b
‚g
b
− 4S 9 < O9 < + 1 − ‚6 ‰
T4S
4b
1
b
− 9‰ − J ˆ + 1 − ‚6 +
Solving in 9 and taking the limit of these functions when ‚ƒ , ‚6 → 0, we get: Š1€∆ |a…t† =
Jˆ < 9 <
S
T
bS-4b
S
‰ − J ˆ& T
T
<9< ‰
T
S
Š1€∆ |2‡0 = J ˆT < 9 < bT-4b‰ − J ‹
S
bS-4b
T-
S
<ŒŽŒ
 <Œ
< 9 < T
S
Noting ‘t = J9 < t for all t∈ 0,1, we can write these two limits more concisely:
Š1€∆ |a…t† = ‘ bS-4b
+
T
‘ & T − 2‘T Š1€∆ |2‡0 = ‘ bT-4b + ‘ m
bS-4b
S
S
<ŒŽŒ
T <Œ
S
n − 2‘T ,
S
and express the difference ∆∆ ≡ Š1€∆ |2‡0 − Š1€∆ |a…t† as the sum of two differences in
brackets:
41
& + 1 − &
& + 1 − &
∆∆ = “‘ ^
_−‘^
_” + •‘ Q
R − ‘ ^& _–
1
−
&
O
−
&O + 1 − &
O
O
O+ & 1−
If 1 − & > 0, the first difference in brackets is negative whereas the second difference in brackets is
positive. Can one of these effects dominate, with the first being smaller than the second for
instance? The first difference will be small if & < 1 and ≈ O, because the first ratio will then be
marginally below 1 and the second ratio will be above 1; whereas the second difference will be large
if 1: 1 +
4b T4S
b T4S
≫ &, that is, if 1 − & T4S ≫ 0. These two conditions are reasonably met if
S4T
& < 1 and ≈ O. For example, if & = .7, = .33, O = .34, ∆∆ = −!1 − ‘.987# + !‘. 880 −
‘.679# which is normally30 positive. Likewise, if & = .7, = .20, O = .25, ∆∆ = −!1 − ‘.964# +
!‘. 784 − ‘.560# is normally positive.
(b)
strict
Frame (11’) is used for lotteries implying losses, i.e. − ≥ | − | ≥ 0 with one
inequality
at
least.
We
compare
Δ4 ≡ JK− ≽ −|. "M − JK− ≽ −|3"M =
JK− − |0s€11P M − JK− − |0s€11P M between the nested frame and the QSD
frame. We use the following notations: "4 = 0; "4 = ‡0 < ‡ < 1. For the nested
lotteries frame, "4 = 1 and "4 = 1 − ™ƒ 0 ≤ ™ƒ ≤ ‡; and, for the overlapping
lotteries frame, "4 = 1 − ™g and "4 = 10 ≤ ™6 ≤ ‡.
The adaptation of conditions (7) for nested lotteries to this problem is obtained by changing A (B)
into -B (-A) and into 1 − O (see footnote 12). This yields: Δ4 |a…t† = JK&1 − <
1−O1−™a+O‡<1−−J1−<1−O1−™a+O‡<&1−+1−&. On the other hand, conditions
(12) and (13) for overlapping lotteries yield: Δ4 |2‡0 = JK!&O‡ + 1 − O + 1 − &‡ >
&1−1−™0∩!O‡+1−O<1−1−™0#−J&O‡+1−O+1−&O−1−‡+1−O1−<&1−1−™0+1−&
1−™0∩!O‡+1−O>1−1−™0#. Solving in ‡ and taking the limit of these functions when
™ƒ , ™6 → 0, we get: O − & 1 − &
O−
O−
O − &
−
<‡<
<‡<
œ−J›
œ
O
O
O
O
O
O−
&O − + 1 − & 1 − &O − O−
O−
Š1€∆4 |2‡0 = J ‹
<‡<
<‡<
−J
O− ž
&O + 1 − &
O
O
&O + 1 − & 1 − Š1€∆4 |a…t† = J ›
More concisely, Š1€∆4 |a…t† = 2‘ T4S
T4bS
−‘ T −
T
‘
T4bS
T
−
4b
T
For these two examples, ∆∆ < 0 requires almost lexicographic preferences s.t. most decision
makers are only satisfied if they get the highest gain.
30
42
O−
&O − + 1 − & 1 − O−
&O − Š1€∆4 |2‡0 = 2‘ ^
_−‘m
n−‘X
O− Y
O
&O + 1 − &
&O + 1 − &
1−
Hence, ∆∆4 ≡ Š1€∆4 |2‡0 − Š1€∆4 |a…t† may be expressed as the sum of two differences in
brackets:
∆∆4 =
T4bS
Ÿ‘ T −
bT4S
‘
+
bT-4b
T4bS
¡‘ T
−
4b
−
T
‘m
ŽŒ
<Œ
ŽŒ
bT-4b
<Œ
bT4S-4b
n¢ .
The first difference in brackets is positive if & < 1, because the second ratio has a lower numerator
and a higher denominator than the first ratio. The second difference in brackets is negative if & < 1,
b4S44T
T
≈ O, and ≪ 1. Indeed, the first term may be rewritten as ‘ assumptions; and the second term is equal to ‘ m1 −
bS
bT-4b
ŽŒ
<Œ
= 0 under these
n ≅ 0 under the same assumptions.
Hence, ∆∆4 , being the sum of a positive term which can be substantial if & is significantly lower
than 1, and a negative term which is small in absolute value if the win probabilities are close and
stand sufficiently below 1, is unambiguously positive. With our numerical examples that illustrate the
reflection effect, we have: ∆∆4 = !‘. 321 − ‘. 0130# + !0 − ‘. 0473# with & = .7, =
.33, O = .34; and ∆∆4 = !‘. 44 − ‘. 0737# + !0 − ‘. 378# with & = .7, = .20, O = .25.
43