# Expected Utility and Cognitive Consistency

Louis Lévy-Garboua

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Expected Utility and Cognitive Consistency

Louis LÉVY-GARBOUA

1999.104
EXPECTED UTILITY
AND
COGNITIVE CONSISTENCY

by

Louis Lévy-Garboua*

TEAM, Université Paris I (Sorbonne) and CNRS

This revision, December 1, 1999

Author's address:

Maison des Sciences Economiques, 106-112 boulevard de l'Hôpital 75647 Paris cedex 13

e-mail: llg@univ-paris1.fr
## RESUME

Avant de prendre une décision, un raisonnement peut survenir parce que l'on perçoit des objections potentielles à sa préférence-espérance d'utilité. On atteint la cohérence cognitive en utilisant toute l'information disponible, à savoir la préférence cohérente et ces raisons. Je montre que la coïncidence entre le choix rationnel et la préférence normative requiert une conscience parfaite, et j'indique des règles de décision maximisatrices conditionnelles à la préférence qui s'appliquent lorsque la conscience est imparfaite. En situation de risque, le choix rationnel converge vers l'espérance d'utilité par un apprentissage inconscient. La théorie résout de nombreux paradoxes et anomalies bien connus du choix, de l'évaluation et de l'information pour des préférences transitives et régulières.

**Mots-clés** : doute, cohérence cognitive, raisons, information, conscience, espérance d'utilité dépendante de l'objection.

**Classification JEL** : D0, D8.

## ABSTRACT

One may reason before making a decision on perceiving potential objections to expected utility-preference. Cognitive consistency is attained by making full use of available information, i.e. consistent preference and reasons. I show that coincidence between the rational choice and the normative preference requires perfect consciousness, and I provide maximizing rules of decision conditional on preference which are valid with imperfect consciousness. A necessary and sufficient condition for expected utility to be descriptively valid is given. Under risk, the rational choice converges towards expected utility through unconscious learning. Many well-known paradoxes and anomalies of choice, evaluation and information are solved for well-behaved preferences.

**Keywords:** doubt, cognitive consistency, reasons, information, consciousness, objection-dependent expected utility.

**JEL classification** : D0, D8.
"The action which follows upon an opinion depends as much upon the amount of confidence in that opinion as it does upon the favorableness of the opinion itself".

Frank Knight

# I. INTRODUCTION

The most comprehensive view of rationality is probably that a choice is rational if it is justified by a consistent set of reasons. However, the great generality of this definition has led various disciplines which purport to describe human behavior, like economics and cognitive psychology, and even various branches of economics, like theories of rational choice and of rational expectations, to follow divergent paths.

Modern economics describes rational behavior by a small set of axioms which ensure the _logical consistency_ of choices. Von Neumann and Morgenstern (1947) pioneered this approach for choices under risk, with objective probabilities, and Savage (1954) extended it to choices under uncertainty, with subjective probabilities. The main achievement of the axiomatic approach is to allow the derivation, for each individual, of a unique ordered set of preferences, the expected utility (EU), which can be defined _prior_ to knowing each particular choice set. Consequently, the decision-making process is seen by economists as irrelevant for determining rational choices.

By contrast, cognitive psychology has focused on decision-making procedures, information processing and the limitations of human mind. The definition of _cognitive consistency_ which is most often mentioned is that an individual suffers from holding two opposite cognitions and will thus seek ways of reducing his cognitive dissonance. For instance, Festinger (1957)'s famous theory of cognitive dissonance states that a person confronted to a bad experience after having made a choice will be looking for justifications of his past decision and tend to ignore the dissonant information. Two main criticisms might be addressed by a bear-headed economist to the approach followed by cognitive psychology. First, the key concepts of information, cognitive consistency, procedural rationality (Simon 1976), or bounded rationality are too loosely defined to bear fruit. And, furthermore, since revealed preferences depend upon the decision-making process, the existence and uniqueness of individual preferences may be lost.

One consequence of economics and cognitive psychology having taken so divergent paths is that, while these approaches have been opposed many times, few serious attempts have been made to bring them together into a common framework. This is rather unfortunate since the
tremendous growth of anomalies, paradoxes and puzzles to the conventional theory of choice discovered in the last two decades (Machina 1987, Thaler 1994, and Rabin 1998 give many examples) has led many researchers to introduce psychological assumptions into economic models with no check that the two approaches were behaviorally consistent. I demonstrate in this paper that it is possible to bring economics and cognitive psychology into a common framework, by an appropriate combination of cognitive consistency with logical consistency.

Once consistent preferences are defined, some reasoning should take place before coming up with a “rational” decision if reason is not to be “the slave of passions” (to paraphrase David Hume). Thus there is a logical possibility that rational choices may deviate from the normative preferences. While accepting the axioms underlying EU-preference, I relax the postulate that the rational choice must always coincide with the normative preference and I seek to rationalize under what circumstances such identification can be made. I argue that an individual will reason before making a decision whenever he doubts his preference, i.e. if he finds potential objections to it. Prior doubt is related to the expected value of perfect information and it indicates a latent demand for information. In contrast with the conventional view that the latter cannot be satisfied, it will be contended here that context, or choice set-dependent, reasons or arguments are a special kind of available information, not captured by the normative preference, which may be valuable by reducing doubt. Thus consistent reasoning overcomes doubt by making full use of available information. Since cognitive consistency is reached when doubt has been overcome, this new approach yields an economic theory of cognitive consistency (CC).

CC theory reconciles, perhaps for the first time so explicitly, the normative definition of rational behavior with informational efficiency which is also a widely used definition in economic analysis, at least since Muth's (1961) seminal paper on rational expectations. In two specific instances, as the paper shows, no further reasoning is needed and the normative approach is sufficient. Under certainty or dominance in a strong sense, first of all, choice and preference must coincide because there can be no objection to preference and no latent demand for information. Besides, when the decision-maker initially doubts his preference, he should still dismiss any potential objection to it if he is “perfectly conscious”. Consciousness is an individual's capacity to accept any new piece of accurate information, like the objective probability of an event, as usable knowledge. It will be described here by a positive parameter which may go to infinity. Following the subjectivist conception of probability initiated by Ramsey (1926), EU theory assumes perfect consciousness, that is an infinite value of the consciousness parameter. However, the perfect consciousness paradigm, which is needed to ensure that the rational choice always coincide with
the normative preference, has been abandoned some time ago by cognitive psychologists after they found repeated evidence of unconscious mechanisms of the mind. Indeed, unconscious behavior offers an intuitive explanation for many anomalies of choice under risk or uncertainty. For instance, it takes more than the acceptance of unfair gambles by perfectly-conscious risk-lovers, predicted by EU theory, to account for the popularity of lotteries and horsetrack betting. It can also be argued that moderate risk-averters will often bear risk out of unconsciousness. For rational but imperfectly-conscious individuals who treat any potential objection to their preference as unconscious information, the normative approach to rational choice needs to be completed by an explicit account of “unconscious reasoning”. The main achievement of the present paper is to provide such theory in a parsimonious, tractable, and testable form.

Accounting for unconscious reasoning and cognitive consistency unifies the so far divergent approaches of economics and cognitive psychology, at least to some extent. It leads to a generalization of EU and, at the same time, to a reformulation of the psychological theory of cognitive dissonance. It also seems to unify disparate arguments captured by various non-EU theories, including the distortion of probabilities(see, among others, Kahneman and Tversky 1979, Quiggin 1982, Schmeidler 1989, Tversky and Wakker 1996), the role of regret (Bell 1982, Loomes and Sugden 1982), bayesian behavior (Viscusi 1989), or similarity judgments (Rubinstein 1988, Leland 1994). The analysis of choices and judgments based on reasons and arguments is obviously an old one, but it has been recently revived by several researchers (see Boudon 1990, Hogarth and Kunreuther 1995, Shafir, Simonson and Tversky 1993, Simonson 1989) as an alternative to value maximizing behavior. I suggest here a simple way of incorporating reason-based choices into an economic framework.

CC is a descriptive theory of rational decision that describes how the latter eventually conforms to, or deviates from, EU-preference. It is not normative in nature because it yields context or choice- set dependent solutions ; but it has a prescriptive value precisely for this reason. In section 2, the normative preference, and specifically EU, is set as a prior to the rational choice. A theory of unconscious reasoning is then presented in section 3, which is applied to the important class of choices between true actions, with well-specified common states of the world. Choice-set dependent reasons and the latter's "weight" are characterized and it is shown in this case that all available information of potential value not captured by the normative preference is summarized by the set of pairwise-consistent objections to the prior preference. The objection set will not be empty in general and it may contain valuable information to an imperfectly-conscious person. If it does not, there is no valuable reason for rejecting the normative preference; and if it
does, one should choose the action which raises the most valuable objection to the prior. As a result, there always exists a rational choice conditional on the preference. At the moment of choice, doubt has been overcome by unconscious reasoning and cognitive consistency has been reached. It is shown in section 4 that a rational individual behaves as if he maximized a descriptive utility function (DU) conditional on his EU-preferred action. For pairwise choices, which have been extensively used in experiments, this simplifies into the maximization of an EU in which objection states are overweighted by the degree of unconsciousness (inverse of consciousness). The latter is called the objection-dependent EU (ODEU). These decision rules have clear psychological roots. Indeed, the rational choice coincides with EU-preference if and only if the consistent objection's "salience", which is a natural measure for cognitive dissonance, does not exceed the individual's level of consciousness. This looks very much like a simple rule of thumb which appeals to perception and can be implemented by unsophisticated agents. However, it is a rational decision rule.

Since it makes a distinction between unconscious and conscious reasoning, CC theory predicts that unconscious learning from experience may substitute for conscious learning. Thus experience may affect the rational choice without having to bring any conscious revision of the objective or subjective probabilities of outcomes. Unconscious learning, studied in section 5, is shown for instance to be responsible for making the rational choice under risk converge in probability, in a sequence of i.i.d. gambles, towards the EU-preference. However, the speed and the path of convergence critically depend on the individual's luck and on the objection's salience and weight. The rules of probability distortion are altered, and complicated, by unconscious learning. Nonetheless, it is rational for the researcher who cannot observe the past experience of individuals in similar gambles to adopt the single choice-rules with an apparent consciousness simply increased by experience. This will cause random error in his or her predictions. Section 6 provides an application of CC theory to pairwise choices under risk or uncertainty in an action-frame. The main lesson from this exercise is that we are able to explain many observed anomalies of choice with well-behaved (e.g. concave and differentiable everywhere) utility functions. It is also demonstrated that CC theory is inconsistent with the common consequence effect, which is one of the two versions of the Allais paradox, if gambles are presented in an action-frame. The essence of the Allais paradoxes seems to lie in the prospect-framing which is outside the scope of the present paper. CC theory is then applied in section 7 to bidding tasks. It is shown that risk and uncertainty do make a quantitative difference here although most qualitative conclusions would hold for both. For instance, we prove that, of two EU-indifferent bets yielding different positive
payoffs, an imperfectly-conscious individual attaches greater value to that yielding the higher prize. Although this qualitative prediction is equally valid under risk and uncertainty, the theory entails that a risky bet yielding a sure probability of a prize sells more than an uncertain bet yielding the same subjective probability of the same prize. The difference measures the opportunity cost of uncertainty relative to risk. Another goal of this section is to solve many observed pricing anomalies, like the endowment effect for lotteries (e.g. Thaler 1980, Knetsch and Sinden 1984), the utility evaluation effect (Karmarkar 1978, Machina 1987), the rejection of probabilistic insurance (Kahneman and Tversky 1979). Finally, CC theory solves the preference reversal phenomenon (Lichtenstein and Slovic 1971) which results from the combination of a comparison task and an evaluation task. The paradox is that many subjects choose the less risky beneficial bet but sell the riskier option at a higher price. This result is especially instructive because, although it is robust (e.g. Grether and Plott 1979), it is not naturally predicted by normative theories of choice. Finally, section 8 briefly revisits the demand for new information under uncertainty in the presence of available arguments. A major prediction of CC theory is that new valuable information will be discarded under imperfect consciousness whenever available arguments are more valuable. It is shown for illustration that this behavior may reinforce the “certainty effect” under uncertainty and is consistent with the puzzling observations made on the popular three-door game (eg. Nalebuff 1987, Friedman 1998). Concluding remarks are found in section 9.

## II. THE NORMATIVE PREFERENCE AS A PRIOR

At first sight, the broad definition of rationality found in the introduction is alien to the mathematical precision required by modern economics. But it turns out to be a surprisingly fertile exercise, because axioms of logical consistency appear to form a consistent set of reasons for justifying one's choice. A long time ago, Ramsey (1926) justified the *transitivity* of preferences by the immunization that it provides against the money pump danger. More recently, Sugden (1991) has provided a nice interpretation for the *completeness* of preferences, by saying
$$u_p^{-1}(p) = u^{-1}\left(p + \frac{1-p}{1+s^*}\right)$$
that it ensures that no choice will be made without having reasons for it.

For choices under uncertainty, the reduction of probabilities axiom, the independence axiom, or the sure thing principle may be viewed as elementary expressions of *description and procedure invariance*. The latter property is obviously a consistent reason for justifying one's choice without prior knowledge of the choice set. Under the veil of ignorance, rational individuals should
consider the context of their future decisions and the presentation of their future objects of choice
as irrelevant, by the principle of insufficient reason. I believe that this is what makes the expected
utility (EU) axioms of Von Neumann - Morgenstern (1947) and Savage (1954) so intuitive and
appealing. But our analysis also demonstrates that the axioms describing description and
procedure invariance essentially identify an individual's prior preference, i.e. the preference
judgment which can be made by him prior to being aware of the particular problems and
alternatives that he will face. Abstraction from the singularity of each decision characterizes the
normative perspective, whether it applies to individual or to social choices. It is thus more
accurate to speak of the individual preferences deriving from a set of axioms as *normative preferences*, and Savage (1954) himself recognized the normative character of his theory after his
controversy with Allais (1953).

Having described axioms of logical consistency as normative, i.e. context or choice set-
independent, reasons for justifying one's choices, it should now be obvious that axioms cannot
describe *all* the potential reasons that can be invoked to justify any single decision¹. The question
is: may the set of *consistent* reasons also contain context, or choice set-dependent, reasons? In
other words, should rational choices always coincide with normative preferences?

These questions are not trivial. While axioms consistently define preferences, i.e. stable
individual attitudes towards objects of choice, no preference per se can be said to be rational if
choice is not to be identified *a priori* with preference. The coincidence between choice and
preference should be rationalized rather than being postulated, and the way to do this is to
introduce reasoning in the decision-making process. By taking EU preference as a prior, it will be
shown in this paper that rational choice does coincide with preference under certainty, but may
systematically deviate from it under risk and uncertainty.

Cognitive psychologists (e.g. Nisbett and Ross 1980) observe that people often deviate from
their own prior judgments, and some philosophers contend that reasonable persons may
intentionally act in contradiction to their own normative preference (a contemporary figure of this
tradition is the American philosopher and former psychologist Donald Davidson 1982), a
phenomenon that Aristotle named “akrasia”. Even economists make distinctions between
individual values and collective choices, or between players' preferences and their actual choices
(e.g. the prisoner's dilemma), and Sen (1993, 1997) has recently advocated the distinction
between choices and preferences and the “menu-dependency” (i.e. context or choice-set
dependency) of rational choices. I contend that “akratic” behavior essentially applies to individual
decisions under risk and uncertainty. Perhaps the most rigorous testimony of such behavior is
provided by the numerous anomalies and paradoxes to EU theory displayed by the sharp experiments of Allais (1953), Ellsberg (1961), Lichtenstein and Slovic (1971), Kahneman and Tversky (1979), Grether and Plott (1979), Tversky and Kahneman (1981, 1986, 1992), and others. However, Allais and Ellsberg themselves made EU axioms responsible for the predictive failure of EU theory. Remarkable efforts were thus made to relax some of its postulates, mainly independence (e.g. Machina 1982), and probability additivity (e.g. Schmeidler 1989), or even transitivity (Fishburn 1991 is a good survey), while saving supposedly basic postulates like transitivity (for a majority of theorists) and dominance (e. g. Allais 1953: 518, Quiggin 1982, Tversky and Kahneman 1992). The result of these efforts has been mitigated. The non-expected utility (NEU) theories which have been suggested do not consistently organize observed choices, and recent tests even show that these do not systematically outperform EU (Camerer 1989, 1992; Battalio et alii 1990; Harless and Camerer 1994; Hey and Orme 1994). These somewhat negative results are perhaps not too surprising, given the normative nature of all theories of choice under risk and uncertainty which have been suggested in the literature (even when authors consider their theory to be descriptive).

Normative theories of choice overlook the possibility that there exist consistent choice set-
dependent reasons. This is a biased attitude which can explain their descriptive inadequacy under
risk and uncertainty. I hope to complete the description of rational behavior by showing that
choice set-dependent reasons do exist and that they may be valuable in overcoming doubt.

### III. REASONING IN CHOICES BETWEEN RISKY AND UNCERTAIN ACTIONS

**III.1. Doubt and dominance**

This paper presents a theory of rational choice which does not postulate that choice and preference must coincide and defines the preference as expected utility. Although much of the argument could be made with other normative preferences, EU is an especially attractive prior since it can be held under a partial veil of ignorance.

It is first assumed that there is a finite set of *n* risky or uncertain actions and *m* states of nature,

<figure><img src="image_2.png" /></figure>

with [ ] and [ ]. Any action A_k (k = 1,2,...,n) can be described by the
lottery:
$$A_k = (y_{k1}, \pi_1; \dots; y_{km}, \pi_m)$$

with $0 < \pi_1, \dots, \pi_m \le 1$ and $\pi_1 + \dots + \pi_m = 1$, where $\pi_i$ is the (objective or subjective) probability of state $i$ ($i = 1, \dots, m$) and $y_{ki}$ is the outcome of action $k$ when state $i$ occurs. Without loss of generality, consequences can be expressed in terms of final wealth if commodities are replaceable and can be exchanged on markets. Let $A_1$ denote EU-preference:

$$EU(A_1) \ge EU(A_k) \quad \text{for } k=(2,\dots,n) \iff \sum_{i=1}^{m} \pi_i U(y_{1i}) \ge \sum_{i=1}^{m} \pi_i U(y_{ki})$$

and assume that the marginal utility of wealth is positive ($U'(0) > 0$). Letting $A_1 \ge A_k$ denote that $A_1$ is preferred to (or indifferent with) $A_k$, EU theory of choice amounts to the statement: $A_1 \ge A_k \Leftrightarrow EU(A_1) \ge EU(A_k)$.

Although this normative preference is logically consistent, it may still raise doubt. Doubt is an unpleasant feeling which originates in the possibility of finding objections to own prior judgment. When the prior is the normative preference, doubt is further specified as prior doubt. In the words of cognitive psychology, doubt is the opposite of confidence and it is a state of pre-decisional cognitive dissonance.

Prior doubt arises if and only if $\bar{J} = \{j, j \in (1, \dots, m); \exists k \in (2, \dots, m) \text{ s.t. } y_{1j} < y_{kj} \} \ne \emptyset$. If $\bar{J} = \emptyset$, the preferred action is said to dominate all other actions. Equivalence of all actions $(y_{1j} \equiv y_{kj} \forall k, j)$ and certainty $(y_{ki} \equiv y_{kj}, \forall k, i, j)$ are special cases of dominance. Prior doubt can be related to the expected opportunity loss of choosing the EU-preferred alternative

$$ \begin{aligned} D(A_1) &= \sum_{j \in J} \pi_j \left[ \max_k (U(y_{kj})) - U(y_{1j}) \right] \\ &= \sum_{i=1}^{m} \pi_i \left[ \max_k (U(y_{ki})) - U(y_{1i}) \right] \end{aligned} \quad (1) $$

This magnitude is non-negative, and equal to zero if and only if $\bar{J} = \emptyset$, i.e. $A_1$ dominates all other actions. Furthermore, it coincides with the expected value (in utility terms) of perfect information (see Raiffa 1968, for instance). Thus $D(A_1)$ truly captures the magnitude of prior doubt. Doubt expresses a latent demand for information by the decision-maker. The latter wishes that some genie could tell him the gamble's outcome a few seconds before playing. He would then save $D(A_1)$ because, with the assurance of getting perfect information just before choosing, his expected utility would be $\sum_i \pi_i \max_k (U(y_{ki}))$ instead of the lower valued $\sum_i \pi_i U(y_{1i})$. Therefore, he would be ready to pay the compensating variation to the genie, if the latter ever existed.
Now, the genie does not exist in reality so that it is not clear whether the feeling of doubt should make actual decisions deviate from normative preferences. In the quotation heading this article, Frank Knight (1971: 227) gave a positive answer to this question, and so did Keynes (1921), but none of them explained how it could work out. Hirshleifer and Riley (1992: 11) take the opposite step by reasoning that the EU criterion is indispensable when the latent demand for information cannot be satisfied. The latter reasoning is logical, but a psychologist might say that it is cognitively inconsistent in the context of actual decisions. Doubt arouses an unpleasant feeling of pre-decisional cognitive dissonance that should trigger-off an appropriate response until it is eventually suppressed and cognitive consistency is found. Such conjecture makes sense of the survival value of feelings, but it is rather puzzling if we consider situations of risk, with stated probabilities, as it is then impossible to acquire new information from external sources. However, the economic and the psychological view can be reconciled by saying that reasons or arguments drawn from available data are commonly used to satisfy the latent demand for information and overcome doubt. Reasons are "internal information" since, like any other message, they serve the single purpose of making rational decisions, or, more precisely, of reducing doubt (or increasing confidence). If I am correct, the feeling of doubt reveals that the normative preference does not make full use of available information and that reasoning must take place. Decision-making can be described by the consistent set of informational actions aiming at the suppression of reasonable doubt, or attainment of cognitive consistency and confidence.

Opting for the EU prior minimizes prior doubt, since minimizing the expected opportunity loss is equivalent to maximizing the expected utility, by duality. This well-known result means that, in view of the choice to be made, own preference brings valuable information. However, it does not prove that there can be no other valuable, i.e. doubt reducing, reason unless there is no prior doubt.

**Proposition 1 (dominance):**

Under certainty and dominance, the rational choice must coincide with EU-preference.

Actions dominated by the prior can be safely eliminated from any further comparison of alternatives.

**Proof:** No objection to the prior can be found from a dominated action. ☐

This proposition validates the conventional theory of choice under certainty, but it is quite restrictive under risk or uncertainty. For instance it does not entail that actions which are
dominated by a non-preferred alternative without being dominated by the prior cannot be chosen. Nor is it sufficient to validate the widely held principle of first-order stochastic dominance because the latter property does not imply dominance (in the strong form used here), and thus does not eradicate prior doubt, unless the lotteries are *true actions* with well specified common states of the world. Therefore, I shall only deal with true actions in the rest of the paper in order to avoid the complications of framing which require further development.

### III.2. Choice-set dependent reasons

In this section, consistent choice set-dependent reasons will be characterized for risky or uncertain actions. When no confusion can be made, I use the shorter expression of reason or argument for choice set-dependent reason. When speaking of uncertainty, I assume until section 8 that no valuable new information can be acquired from an external source.

By the principle of independence of choice from irrelevant alternatives, potential arguments must be drawn from available information. By the assumption of common states, choice set-dependent arguments must be described by a joint subset of states associated with its probability share. For a pairwise comparison between two alternatives, $A_1$ and $A_k$ ($k \neq 1$), including the EU-preferred action $A_1$, they will be denoted ($a_{k1} \rightarrow a_{1k}; p_{k1}$), where $a_{1k}$ and $a_{k1}$ are vectors of state-contingent outcomes from $A_1$ and $A_k$ respectively, drawn from the same subset of states, and $p_{k1}$ is the total probability of these states. Since $A_1$ is given for all pairwise comparisons, this notation will be alleviated by ($a_k \rightarrow a_{1k}; p_k$). The total probability of states included in the argument, $p_k$, can be described as the *weight of the argument*.

Although reasons may be taken as internal information, the procedures that individuals use for processing information when they reason do not resemble the purchase of information services from external sources. The classical analysis of information (e.g. Hirshleifer and Riley 1992) makes the point that external sources are opaque so that the decision-maker is unable to choose the particular message he will receive. Instead, he must buy an information service. The situation is now totally different because the person *perceives* all the potential arguments which are simply drawn from available information. The arguments which make full use of available information form the set of consistent (choice set-dependent) arguments.

Choice set-dependent reasons either confirm or refute the decision-maker's normative preference. We call the former *justifications*, and the latter *objections*.

Since all potential justifications confirm the EU-preference, consistent justifications can only be found among the normatively equivalent presentations of the given lotteries that use all available
information. Consequently, the weight of a consistent justification must equal one. For choices between actions, this includes the prior presentation, and all those that can be obtained by permuting the order of states. Starting from an arbitrary ordering of states, we then turn to the consistent objection. For a pairwise comparison, it is simply formed by the complete subset of states pointing to an objection² because that composite objection makes full use of available relevant information. A consistent objection then exists (inclusive of the empty objection in the cases of certainty and dominance) and is unique. Since it does not depend on which particular ordering of states was adopted, we conclude that the ordering of states is decision-irrelevant. Thus, in choices between actions, justifications bring no valuable information and the only reasons must be objections.

Any action $A_k$ ($k \neq 1$) can be partitioned, conditional on the EU-preferred action $A_1$, into:

$$A_k = (a_k, p_k; b_k, 1 - p_k)$$

where $a_k$ is the vector of outcomes included in the consistent objection to $A_1$, $b_k$ is the complementary set of outcomes which describes the “counterargument”, and $p_k$ is the consistent objection's weight in the pairwise-comparison of $A_k$ with $A_1$. We showed for objections that $0 \le p_k < 1$, and $p_k > 0$ when certainty and dominance are ruled out. The latter partition is unique for each $k \neq 1$, so that the set of pairwise-consistent objections to $A_1$:

$$Q = \{Q_k = (a_k \to a_{1k}; p_k), k \neq 1\}$$

is also unique and well-defined. $Q$ may be simply called **the objection set**. When the prior dominates some or all other actions, $Q_k$ or $Q$ can be defined as the empty set $\emptyset$.

The foregoing discussion shows that, in choices between actions sharing common states of the world, consistent reasons are summarized by the set of pairwise-consistent objections to the prior preference. The prior preference is set as the EU-maximizing alternative. When the latter is not unique, one of them should be selected arbitrarily among the EU-indifferent candidates³. The objection set is unique and well-defined under risk and uncertainty and it is not empty, whenever all actions are not equivalent and no single action dominates others.

### III.3. The informational value of consistent objections

By now, it should be clear that normative theories of choice under risk and uncertainty do not make full use of available information. In the case of choices between true actions, the omitted information is summarized by the objection set. Whether the latter information is valuable, and thus decision-relevant, remains to be seen.
Let us first consider a pairwise-consistent objection $Q_k$. We assume here that $Q_k$ is not the empty set and that actions are not ambiguous. In making his decision, a rational individual should be aware of his own normative preference ($EU(A_1) \ge EU(A_k)$), each pairwise-consistent objection to it ($EU(a_k) > EU(a_{1k})$), and any prior relevant experience he might have ($I_0$). Since these three pieces of information each point out to an ordinal response (i.e. choose $A_1$ or choose $A_k$ in a pairwise comparison), they can be treated as being generated by an i.i.d. Bernoulli process with an unknown value of the parameter in the unit interval. The individual observes the objection's weight $p_k$, which is the mean of the prior distribution of that parameter, and $N+1$ realizations of the Bernoulli variable $\tilde{W}_k$, namely $w_{k0} = 1$ in the mental experience of perceiving the objection and $w_{ki} = 0$ or $1$ ($i = 1,...,N$) in the real experience of drawing $N$ prior i.i.d. gambles. If we suppose that the prior distribution of that unknown parameter is a Beta distribution with parameters $\mu_k > 0$ and $\nu_k > 0$, under bayesian behavior the posterior distribution of the Bernoulli parameter when $W_{ki} = w_{ki}$ ($i = 0,...,N$) is Beta with parameters:

$$ \mu_k + r_k + 1 > 0, \quad \nu_k + N + 1 - r_k - 1 = \nu_k + N - r_k > 0, \quad \text{with} \quad r_k = \sum_{i=1}^{N} w_{ki} $$

(see, for instance, De Groot 1970). The posterior mean is:

$$ p_k^* = \frac{\mu_k + r_k + 1}{\mu_k + \nu_k + N + 1} \quad (2) $$

which lies in the interior of the unit interval. By writing that the prior distribution has a known mean $p_k$, we get a relation between $\mu_k$, $\nu_k$ and $p_k$:

$$ p_k = \frac{\mu_k}{\mu_k + \nu_k} \quad (3) $$

By reporting (3) into (2), the posterior mean is found to be a weighted average of the prior ($p_k$) and the perceived (i.e. mentally or really observed) frequency ($\bar{p}_k = \frac{1+r_k}{1+N}$) of objections

$$ p_k^* = \frac{\mu_k + \nu_k}{\mu_k + \nu_k + 1 + N} p_k + \frac{1 + N}{\mu_k + \nu_k + 1 + N} \bar{p}_k \quad (4) $$

so that $\mu_k + \nu_k$ is the precision of the prior and $1+N$ is the precision of perceptual experience. Since the latter is simply measured by the number of perceived experiments that does not rest on which particular pairwise comparison is being made, the prior's precision captures an individual's general experience endowment that applies to any comparison task. Each person has his own
experience endowment $s^* > 0$, which I call his level of consciousness because it describes his own capacity of accepting given accurate information as “knowledge”. Conversely, $1/s^*$ is an indicator of unconsciousness because an unconscious person is one who has a vague, or imprecise, perception of given information. It is immediate that the objection is overweighted (underweighted) if and only if its perceived weight is greater (smaller) than its prior weight

$$p_k^* = \begin{cases} p_k & \text{iff } \bar{p}_k = \frac{1+r_k^*}{1+N} = p_k \end{cases}$$

The incidence of (real) experience on repeated choices will be examined at length in section 5. For the time being, I suppose that the individual makes a single choice, so that $I_0 = \emptyset$. In this important special case, the objection will always be overweighted because $N = r_k = 0 \Rightarrow \bar{p}_k = 1 > p_k$. Substituting $s^*$ for $\mu_k + \nu_k$ in (4), we get a simple expression for the posterior objection's weight:

$$p_k^* = \frac{s^*}{1+s^*} \cdot p_k + \frac{1}{1+s^*} \cdot 1 \quad (5)$$

The posterior weight is an average between one and the normative weight, with constant individual-specific coefficients. (5) recovers the intuition of a probability distortion underlying NEU theories of choice like prospect theory (Kahneman and Tversky 1979, Tversky and Kahneman 1992), rank-dependent expected utility under risk (e.g. Quiggin 1982) or uncertainty (e.g. Schmeidler 1989) and prospective reference theory (Viscusi 1989) among others. But the parallel should not be drawn too quickly in our case, given the distinctive conditionality of the objection's weight on prior preference. More fundamentally, (5) shows how consistent objections may induce a revision of prior beliefs and serve as valuable internal information. A necessary condition to be met is that the objection be overweighted, which will always occur under zero experience.

Letting $A_k^* = (a_k, p_k^*; b_k, 1-p_k^*)$ and $A_{1k}^* = (a_{1k}, p_k^*; b_{1k}, 1-p_k^*)$ denote the posterior representations of actions $A_k$ and $A_{1k}$ after using the objection $Q_k$, we derive from (5):

$$EU(A_k^*) = \frac{s^*}{1+s^*} EU(A_k) + \frac{1}{1+s^*} EU(a_k) \quad (6)$$

$$EU(A_{1k}^*) = \frac{s^*}{1+s^*} EU(A_{1k}) + \frac{1}{1+s^*} EU(a_{1k})$$
Notice that the prior may have several posterior representations $A_{1k}^*(k = 2,...,n)$ depending on the other actions with which it is compared. The informational value of objection $Q_k$ (in utility terms) is simply:

$$V(Q_k) = \max \left( (EU(A_k^*) - EU(A_{1k}^*)) , 0 \right) \quad (7)$$

and the informational value of the objection set $Q$ (in utility terms) is:

$$V(Q) = \max \left( EU(A_2^*) - EU(A_{12}^*) , \dots , EU(A_n^*) - EU(A_{1n}^*) , 0 \right) \quad (8)$$

If the level of consciousness were infinite, the rational choice would never deviate from EU-preference because the prior probabilities would not be distorted and objections would thus have no value. The postulate that rational choice under risk and uncertainty must always coincide with EU-preference can only be rationalized under the implicit assumption of perfect consciousness (i.e. $s^* = +\infty$). The latter seems an exceedingly strong assumption because a perfectly-conscious person would be able to learn without accumulating own experience, which is counterfactual (e.g think of children who trust their parents but do not always listen to them). *CC theory extends EU to rational but imperfectly conscious* (i.e. $s^* < +\infty$) individuals.

## IV. RATIONAL DECISION RULES WITH IMPERFECT CONSCIOUSNESS

Since all consistent objections are available information, the decision-maker takes each objection as a message and picks up the most valuable one, if any. Whatever the outcome of the search, and the resulting decision, he has now made full use of available information and can have no further reasonable doubt⁴. Doubt has been overcome and cognitive consistency has been reached in two steps, and the decision process comes to an end. If no consistent objection is valuable, i.e. $V(Q)=0$, there is no more reason for rejecting the normative preference which can now be revealed. Otherwise, the chosen action will not coincide with prior preference but be the action which raises the most valuable objection to the prior.

This rational decision rule under imperfect consciousness can be restated as a standard maximization problem *conditional* on the EU-preferred action.

**Proposition 2 (maximizing rules):**

The following maximizing rules are equivalent for describing the rational behavior of an imperfectly conscious individual making a single choice among a finite set of risky or uncertain actions, conditional on the EU-preference for action $A_1$.

(1) maximize the posterior EU difference ($\Delta PEU$)
$$ \max_{k \in (1, \dots, n)} \Delta PEU(A_k) = \frac{s^*}{1+s^*} [EU(A_k) - EU(A_1)] + \frac{1}{1+s^*} [EU(a_k) - EU(a_{1k})] \quad (9) $$

with $\Delta PEU(A_1) = 0$

(2) maximize the objection-dependent EU difference ($\Delta ODEU$)

$$ \max_{k \in (1, \dots, n)} \Delta ODEU(A_k) = \left( p_k + \frac{1}{s^*} \right) [EU(a_k) - EU(a_{1k})] + (1 - p_k) [EU(b_k) - EU(b_{1k})] \quad (10) $$

with $\Delta ODEU(A_1) = 0$

(3) maximize the descriptive utility function (DU)

$$ \max_{k \in (1, \dots, n)} DU(A_k) = EU(A_k) + \frac{1}{s^*} [EU(a_k) - EU(a_{1k})] \quad (11) $$

with $DU(A_1) = EU(A_1)$

In all these expressions, differences like $EU(a_k) - EU(a_{1k})$ are conventionally set to zero when $A_k$ is identical to, or dominated by $A_1$.

The prior $A_1$ is always chosen when it belongs to the set of maxima.

**Proof:** see appendix 1. □

The decision rules described by proposition 2 extend the EU rule by allowing for imperfect consciousness and they obviously boil down to EU when the consciousness parameter becomes infinite. But it would be particularly useful to know exactly when EU predictions are valid. Fortunately, it is possible to get a necessary and sufficient condition.

We first define the salience of the $k^{th}$ objection by

$$ s_k = \frac{EU(a_k) - EU(a_{1k})}{EU(A_1) - EU(A_k)} $$

This is a positive magnitude which can be conventionally equated to zero if and only if action $A_k$ is dominated by the prior. Because the objection's salience compares the intensities of the two conflicting “preferences”, the conscious and the unconscious, it truly captures the magnitude of cognitive dissonance before an informational action is taken. We then prove the following.

**Proposition 3 (coincidence of rational choice with EU-preference):**

The prior EU-preferred action is chosen by a rational individual making a single choice among a finite set of risky or uncertain non-equivalent actions if and only if cognitive dissonance is never greater than consciousness:
$$s_k \le s^* \quad \forall k \ne 1$$

**Proof:** Let us first assume a strict EU-preference for $A_1$. The k-th objection's value given by (7) may then be written

$$V(Q_k) = \frac{1}{1+s^*} \max([EU(A_1) - EU(A_k)][s_k - s^*], 0)$$

Since $EU(A_1) - EU(A_k) > 0 \forall k \ne 1$, $V(Q_k) = 0$ iff $s_k \le s^* \forall k \ne 1$. The rational choice must then coincide with EU-preference because the objection set has no value.

If the prior is EU-indifferent but not equivalent with action $A_k$ ($k \ne 1$), $s_k = +\infty$ and equations (9) or (11) immediately show that the objection $Q_k$ is valuable to an imperfectly conscious individual. In such case, we must have $s_k > s^*$. □

What proposition 3 says, in words, is that the EU decision rule will be valid if and only if it is cognitively consistent. This means that (ex ante) cognitive dissonance should not exceed consciousness, or that the objection's salience should not exceed the perceived precision of prior probabilities.

Proposition 3 demonstrates that EU theory cannot always be verified with imperfectly conscious individuals and that valuable objections can be found in systematic ways, at least when the decision-maker has no prior relevant experience. Perhaps the most general, albeit somewhat trivial, counterexample is that such individuals would never choose between EU-indifferent actions that are not equivalent simply by tossing a coin, irrespective of their level of consciousness. After having set a prior by tossing a coin, they would immediately change their mind because they would always find a valid objection to whatever prior they had. But the frequent rejection of strict EU-preference is also to be expected in choices between similar lotteries because the salience of objections tends to be great when the lotteries under comparison are nearly EU-indifferent. Indeed, many anomalies of choice which have been found experimentally involve alternatives of similar expected value and, even more so presumably, of similar expected utility⁵.

It is quite comforting that the constant emphasis put by psychologists on the role of salience in decision-making, borne by numerous empirical observations⁶, finds here direct theoretical support. Hence, given the perceptual nature of salience (of which a rigorous definition has been given), it should be no surprise that animals and laymen often behave roughly like sophisticated men or even trained scientists (e.g. Battalio, Kagel and MacDonald 1985). Indeed, propositions 1, 2 and 3 can be assembled to provide a four-step decision rule which can be implemented by
unsophisticated agents with limited computational powers because it appeals to perception and proceeds by successive eliminations. In the first step, the prior is set as an EU-preferred alternative. We then proceed to the second step and search for pairwise-consistent and valuable objections to the prior. We first eliminate dominated actions (proposition 1). If the remaining subset (not including the prior) $R(A_1)$ is empty, we choose the prior $A_1$. Otherwise, in a third step, we seek to eliminate non-dominated actions whose consistent objections are not salient enough (proposition 3). If the remaining subset (not including the prior) $S(A_1) \subseteq R(A_1)$ is empty, we choose $A_1$. Otherwise, in a final step, we discard the prior and choose within $S(A_1)$ the action whose pairwise-consistent objection $Q_k$ has the greatest informational value (proposition 2).

The analysis, which was carried so far with finite sets of actions, may extend to infinite sets. This is true, for instance, when DU (or other maximands appearing in proposition 2) is a piecewise-twice continuously differentiable function of one or several variables. Let $\mathbf{A}$ be such a set, and $\mathbf{A}_- = \mathbf{A} - \{A_1\}$ the subset of non-EU preferred actions. Further assume that the descriptive utility function DU: $\mathbf{A} \to \mathbb{R}$ has a maximum. The decision rule described by (11) has the important property of being *transitive conditional on the possibility of choosing the EU-preferred action*. For any given subset $P \subseteq \mathbf{A}_-$, actions which are non-DU preferred within $P+\{A_1\}$ must be non-DU preferred on $\mathbf{A}$ or on any subset of $\mathbf{A}$ containing $P+\{A_1\}$. The search for the optimal choice on $\mathbf{A}$ can thus be decomposed into the search for local optima on convenient subsets of the form $P+\{A_1\}$. Very convenient subsets of this form ($\alpha$) contain $A_1$ plus actions from $\mathbf{A}_-$ sharing the same objection states: i.e. $\forall A_k, A_l \in \mathbf{A} \text{ s.t. } k \neq l, a_{1k} = a_{1l} = a_1$ (hence $b_{1k} = b_{1l} = b_1$). It can be shown that the local optimum on $\alpha + \{A_1\}$ (which is assumed to exist) simply maximizes an expected utility in which objection states are overweighted by the degree of unconsciousness $\frac{1}{s^*}$.

I call this EU the *objection-dependent EU* (ODEU):

$$ODEU(A_k) = \left( p + \frac{1}{s^*} \right) EU(a_k) + (1-p) EU(b_k) \quad (12)$$

where $A_k \in \alpha + \{A_1\}$ and $p$ designates the common objection's weight for actions of $\alpha$. Of course, the ODEU can also be written in normalized form, with probabilities adding up to one, by dividing each probability by $1 + \frac{1}{s^*}$
$$ \text{normalized ODEU}(A_k) = \left( p + \frac{1-p}{1+s^*} \right) \text{EU}(a_k) + \left( 1-p - \frac{1-p}{1+s^*} \right) \text{EU}(b_k) \quad (12 \text{ bis}) $$

**Proposition 4 (objection-dependent expected utility)**

Let an imperfectly-conscious rational individual have an EU-preference $A_1$ on a set of actions $\mathbf{A}$ and let $\alpha$ be any non empty subset of actions from $\mathbf{A}_-$ that share the same objection states (i.e. $\forall A_k, A_l \in \alpha$ s.t. $k \neq l, a_{1k} = a_{1l} \equiv a_1$).

The rational choice over $\alpha + \{A_1\}$ is to maximize the objection-dependent EU:

$$ \max_{A_k \in \alpha + \{A_1\}} \text{ODEU}(A_k) = \left( p + \frac{1}{s^*} \right) \text{EU}(a_k) + (1-p) \text{EU}(b_k) $$

If there is a finite number $t$ of sets of objection states, let $\alpha_r (r=1,...,t)$ be the non-empty class of actions from $\mathbf{A}_-$ sharing the $r^{th}$ objection to $A_1$. Assume that local optima exist on all $\alpha_r + \{A_1\}$. The rational choice over $\mathbf{A}$ is to maximize the descriptive utility function on the finite set of locally optimal actions:

$$ \max_{r=(1,...,t)} \text{DU} \left( \left( \argmax_{k \in \alpha_r + \{A_1\}} \text{ODEU}(A_k) \right) \right) $$

A strictly non EU-preferred action may be chosen only if its descriptive utility is strictly greater than EU($A_1$).

**Proof:** See Appendix 2 □

An immediate corollary of proposition 4 is the following.

**Corollary to proposition 4**

A rational individual having an EU-preference always maximizes his objection-dependent EU in a pairwise choice.

The rational decision rule emerging from this section bears resemblance with other rules that previously appeared in the literature. First of all, it boils down to EU when the level of consciousness becomes infinite and under more general circumstances elicited by proposition 3. Hence, the present theory offers a nice generalization of EU theory to imperfectly-conscious individuals, which can be viewed as a specific form of bounded rationality or cognitive biases.
Second, the descriptive utility function looks a bit like the modified utility used by regret theory (e.g. Loomes and Sugden 1982). The crucial difference between the two is that rejoicing plays no role in the present theory, and that the new decision rule is conditional on the EU-preference⁷. Third, the objection-dependent EU is an expected utility in which probabilities are replaced by decision weights, and the same property characterizes prospect theory (Kahneman and Tversky 1979, Tversky and Kahneman 1992, Tversky and Wakker 1996) and rank-dependent EU (e.g. Quiggin 1982) among others. But the laws of probability distortion are different in these theories, and it is only in special cases like pairwise choices that the descriptive utility function introduced here has the simple form of an EU with distorted probabilities. Viscusi's (1989) prospective reference theory also assumes bayesian behavior, but specifies the informational action quite differently. Its prior is a given reference and what it counts as evidence is the external information contained in the objective probabilities. Finally, the present theory offers a justification for the observation that behavior departs from EU rule mainly when the alternatives are close in value (or, rather, in EU). Although the similarity hypothesis of Leland (1994), who generalizes Rubinstein (1988), is consistent, like ours, with people's possessing transitive preferences, it attributes choice deviations not to reasoning but to the limitations of human perception. Accounting for unconscious reasoning and cognitive consistency seems to unify disparate arguments captured by various NEU theories. It also unifies the treatment of risk and uncertainty in a single theory while, in the past, rank-dependent EU applied to risk, for instance, and Choquet-EU (e.g. Schmeidler 1989) applied to uncertainty. And, last but not least, it yields a new theory of cognitive dissonance that reconciles the economic theory of choice with cognitive psychology.

## V. UNCONSCIOUS LEARNING FROM EXPERIENCE

A surprising implication of unconscious reasoning is that people are able to learn unconsciously from experience, even under risk when they are aware that their own experience cannot alter the given probabilities of states. Thus experience may affect the rational choice without having to bring any conscious revision of the (objective or subjective) probabilities of outcomes. Unconscious learning is an unsystematic process of leaning by doing, where individuals seek to make the best of each single decision, given their own experience but without concern for the compound gamble induced by the future repetition of choices. We assume a sequence of i.i.d. gambles, with perfect memory of the experienced outcomes and non-revisable prior probabilities. Let us then calculate the informational value of a given objection by making use of
equation (4). Designating this value as a function of experience N by $V_N \equiv V_N(Q_k)$ and dropping all indexes in this section, we get the following expression (see the appendix 3.1):

$$ V_N = \begin{cases} 0 & \text{if } s + (r - Np) \cdot \frac{1+s}{1-p} \le s^* + N \\ \ge 0 & \text{otherwise} \end{cases} \tag{13} $$

The number of times (r) an objection to EU-preference was observed in the past conditional on experience and the objection's weight captures the individual's luck. In order to estimate the effect of experience on choice, we consider r as a binomial variable of mean Np and standard deviation $\sqrt{Np(1-p)}$. Hence, solving (13) for r,

$$ P_N \equiv \mathrm{Prob}(V_N = 0) = B\left( I\left( \frac{(1 + ps)N + (1 - p)(s^* - s)}{1 + s} \right) \right) \tag{14} $$

where $I(z)$ designates the integer part of z and $B(\cdot)$ is the cumulative density function of the binomial distribution.

**Proposition 5 (convergence of choice towards EU-preference):**

Imperfectly-conscious individuals unconsciously learn from experience under risk (with given probabilities) and, in a sequence of i.i.d. gambles, their rational choices converge in probability towards their EU-preference.

**Proof:** Convergence in probability derives from eq. (14). For a given small value of $\epsilon$ ($0 < \epsilon < 1$), define $r_e$ s.t. $B(r_e) \ge 1 - \epsilon$ and $N_e = I\left(\frac{(1+s)r_e - (1-p)(s^*-s)}{1+ps}\right)$. Since $P_N$ is non-decreasing in $N: \forall N \ge N_e, P_N \ge 1 - \epsilon$. $\square$

It is important to understand this conclusion. EU theory predicts that experience does not matter for rational behavior when the probabilities of states are given because it implicitly assumes that endowed experience is infinite. This theory confuses rationality with perfect consciousness by postulating that the rational choice always coincide with the normative preference. Our framework is more general and encompasses the behavior of imperfectly-conscious individuals. Obviously, the latter must accumulate own experience as a substitute for their limited endowed experience. Only in the long run do they converge in probability towards EU behavior; and even this convergence does not mean that they become “more rational” in the long run. They simply become more experienced and thus behave as if they had become more conscious. It should finally be noted that, in the problem under study, individuals have no vision of the future repetition of
choices and are, therefore, unaware that they will eventually converge towards their own EU-preference.

For great values of $N$, the normalized binomial converges in distribution towards the standard normal distribution, and

$$ P_N = \text{Prob} \left\{ \frac{r - Np}{\sqrt{Np(1-p)}} \le \sqrt{\frac{1-p}{Np} \cdot \frac{N+s^*-s}{1+s}} \right\} \quad (15) \\ \approx F \left( \sqrt{\frac{1-p}{Np} \cdot \frac{N+s^*-s}{1+s}} \right) $$

where $F(\cdot)$ is the cumulative density function of the standard normal distribution. The argument of $F(\cdot)$ in (15) behaves like $\sqrt{\frac{(1-p)N}{p}} \cdot \frac{1}{1+s}$ for great values of $N$ and given finite values of cognitive dissonance and consciousness. The values of $N$ for which $\sqrt{\frac{(1-p)N}{p}} \cdot \frac{1}{1+s}$ takes a given value $t$ are highly sensitive to the values of the objection's salience and weight. Table 1 indicates the result of these simple computations and the number of i.i.d. experiments needed for objection to be valueless with an error margin of less than 5% ($t = 1.645$).

### Insert table 1 here

While 25 shots are enough to converge for moderate values of salience ($s=2$) if $p=0.5$, it will take 2411 shots to reach the same point if $p=0.99$. Besides, convergence will be fast when options are widely different and very slow indeed when options are nearly EU-indifferent, causing cognitive dissonance to be very great. The slowness of convergence shown by table 1 when the objection's weight and salience are not small together is striking.

The rate of learning and convergence towards EU is also sensitive to luck, and history-dependent. A person experiencing more(less) objections than his lot in the past is more(less) likely to value a specific objection and to violate his EU-preference on making a new independent similar decision. In particular, an objection's value would always be nil irrespective of the objection's salience and consciousness, when the prior preference is strict$^8$, if $\bar{p} \le p$, i.e. $1+r \le (1+N)p$. But it may become positive when the objection is overweighted, and luck then interacts with the objection's salience. The latter conclusion is supported by the alternative expression of value:
$$ V_N = \begin{cases} 0 & \text{if } \frac{[1+r-(1+N)p]s-(N-r)}{1-p} \le s^* \\ 0 & \text{otherwise} \end{cases} \tag{16} $$

Expressions of the descriptive utility (DU) and the objection dependent-EU (ODEU) are provided in the appendix 3, as well as useful alternative expressions of the objection's value. By using one of these (eq. (3.8) in the appendix), it can be shown that an objection's value would be nil irrespective of the objection's weight if $(1+s)N \le s^*$.

I give now a closer look at the influence of the objection's weight and salience on the objection's value and the convergence path. I ran simulations of $P_N=\text{Prob}\{V_N=0\}$ for various values of the objection's salience ($s$), number of i.i.d. experiments ($N$), and the objection's weight ($p$). The chosen value for the level of consciousness, $s^*=4$, was kept constant. Preliminary tests of CC theory on a series of published experiments (Blondel 1997) showed this to be a reasonable average estimate that fits the data. Tables 2.1-2.5 describe the results for $p$ values of 0.01, 0.2, 0.5, 0.8, 0.99. When the objection's weight is very small (e.g. $p=0.01$, table 2.1), the following simple rule can be used as a good approximation for a wide range of values of $s$ and $N$ (see the proof in the appendix 3.3):

$$ P_N = \begin{cases} 1 & \text{if } s \le s^*+N \\ 0 & \text{otherwise} \end{cases} \tag{17} $$

For instance, when $p=0.01$ and $s=10$, convergence would be reached at the 10% level after just five i.i.d. gambles. However, in the meantime, imperfectly-conscious but rational persons would <u>never</u> respect their EU-preference. This strange behavior is greatly reduced when the objection's weight takes intermediate values like 0.10 or 0.20 (see table 2.2), but it may persist in an attenuated form as long as the objection's weight is less than one-half (see the proof in appendix 3.4).

On the other hand, when the objection's weight is close to one (e.g. $p=0.99$, table 2.5), $P_N$ is insensitive to variations of $s$ and $s^*$ in a wide range of experience values as long as $(1+N)s > s^*$ (see the proof in the appendix 3.3). In the first interval, $P_N=\text{Prob}\{r<N\}=1-p^N$. Hence, $P_N$ rises slowly with $N$, then falls discontinuously and slowly rises again from a lower value. For instance, when $p=0.99$ and $s=100$, $P_N$ rises continuously during the first 100 gambles, reaches a peak of 0.634, then falls abruptly to 0.268 after 101 gambles. In the second interval,

$$ P_N = \text{Prob}\{r<N-1\} = 1 - p^N - Np^{N-1}(1-p), $$
and so on. Intervals are of nearly equal length $\frac{1+s}{s}:(1-p)$, the first one being slightly longer than
all the others. The large magnitude of the oscillations of $P_N$ is strikingly illustrated by figure 1
($s=10, p=0.99$). The saw-shaped pattern of $P_N$ is also visible for descending values of the
objection's weight but with a decreasing amplitude. Vacillation is not uncommon when someone
has a strong doubt over his own preference.

An interesting prediction of the theory is that risk-averters might fail to purchase insurance
against very small probabilities of a disaster⁹ while they commonly buy coverage in the medium
range of probabilities. Notice that the interpretation of this observed phenomenon (see
Kunreuther et al. 1978) does not have to rest on the often alleged incapacity to perceive very
small probabilities. Another related intriguing phenomenon that CC theory may explain is why
people who seemed unconscious of low-probability dangers like an earthquake or some deadly
accident will often suddenly change their mind after being exposed to the disaster. Assume that no
loss has been observed during *N* periods and the disaster occurs for the first time at period *N*+1.
Hence, *r* = *N* for a risk-averter, and risk-seeking is entailed by (16) at period *N* if $(N+1)s > s*$.
However, after the loss occurs at the $(N+1)^{th}$ period, risk avoidance is now implied by (16) if
$$ \left(1 + N - \frac{p}{1-p}\right)s - \frac{1}{1-p} \le s* $$
The latter condition is likely to hold when the probability of loss
$(1-p)$ is very small, and is true irrespective of the objection's salience whenever $\frac{p}{1-p} \ge 1 + N$,
i.e. $1-p \le \frac{1}{N+2}$.

**Insert figure 1 here**

I summarize below the main conclusions concerning finite unsystematic sequences of similar
choices.

**Proposition 6 (finitely-repeated choices)**

Consider the rational choice of a given non-EU preferred action by an imperfectly-conscious
individual after a given finite number *N* ≥ 1 of i.i.d. experiments.
If the perceived weight of the objection $\bar{p}=\frac{1+r}{1+N}$ is smaller than the given (objective or subjective) value p of this weight, the action is not chosen. In case of a strict EU-preference relation, the same conclusion holds when the perceived weight equals the given weight.

If the given objection's weight is very small (e.g. $p=0.01$), the individual initially behaves as if he made a single choice with an apparent level of consciousness $s^*+N$. Hence, if the objection is salient enough, he never respects his EU-preference initially. After a sufficient number of experiments, however, he rapidly converges in probability towards his EU-preference.

It is possible to observe systematic and repeated violations of EU-preference initially iff $p<1/2$. At the opposite, it is impossible to observe any violation of EU preference as long as $(1+N)s \le s^*$.

If the given objection's weight is very big (e.g. $p=0.99$), most (but not all) individuals initially violate their EU-preference irrespective of their (finite) consciousness level and objection's salience as long as $(1+N)s > s^*$. The frequency of violations diminishes very slowly with experience, then suddenly jumps at some point and slowly diminishes again, and so on. It follows a characteristic saw-shaped pattern.

## Insert tables 2 here

I have assumed that the decision-maker knows his own luck. But the econometrician or the experimentalist, who often cannot observe luck, will minimize variance of error by setting r at the mean $Np$. Thus, for a pairwise choice, the researcher can base his inferences on the simple rule derived from (13):

$$ \begin{cases} \text{choose the prior if } s \le s^*+N \\ \text{reject the prior otherwise} \end{cases} \tag{18} $$

(18) extends the single choice rule by adding experience to the level of consciousness. Experience raises the apparent level of consciousness because consciousness is simply endowed experience. It is worth noticing that the simple rule given by (18) does not mention the objection's weight, and has undeniable computational virtues. However, an equivalent rational decision rule which is more familiar to economists would be to maximize the objection-dependent-EU (12) in which the objection's weight is simply overweighted by $\frac{1}{(s^*+N)}$.

The foregoing analysis of repeated choices under risk is consistent with the context effect exhibited by Hershey and Schoemaker (1980). Choices presented in an insurance context were
judged with greater risk aversion than mathematically identical choices presented as standard gambles. Mentioning the insurance context immediately recalls related experience and thus increases the likelihood that choice respect EU-preference.

## VI. RISK BEHAVIOR

The joint appearance of risk avoidance and risk seeking behavior in a single individual is probably the oldest puzzle raised by EU theory. Early studies by Friedman and Savage (1948) and Markowitz (1952) admitted that the EU utility function could be convex on some intervals and concave elsewhere. A similar property is retained by Kahneman and Tversky (1979) or Tversky and Kahneman (1992) for the value function, although a distinction is made between risk aversion and loss aversion. The answers given by such inductive approaches are contrived because they rationalize *post factum* a seemingly irrational fact.

The solution given by the present theory is more parsimonious, and allows the cardinal utility function to be concave (or convex) and differentiable everywhere in actual practice¹⁰. The Arrow-Pratt definition of risk aversion is also maintained but it is a property of the normative utility function which does not extend to the descriptive utility function.

Until the discussion of Allais paradox in an action-frame in subsection VI.4, I consider the choice between a sure outcome S and a risky prospect R, and rule out dominance. Without loss of generality, we write:

$$ \begin{cases} R = (y_1, 1-p-q; y_2, q; y_3, p) \\ S = (y_2) \end{cases} \qquad (19) $$

with $y_1 > y_2 > y_3$, $0 < p < 1$, $0 \le q < 1$. Some or all of these three outcomes may be vectors.

Letting $q = 0$, we get an important special case:

$$ \begin{cases} R = (y_1, 1-p; y_3, p) \\ S = (y_2) \end{cases} \qquad (20) $$

We set: $U(y_3) = 0, U(y_1) = 1$, hence $U(y_2) = x$ with $0 < x < 1$.

Notice that prospects can be assimilated to risky actions so far as a single lottery is compared to a sure outcome, irrespective of the number of consequences in the former.

Problems (19) and (20) are associated with two versions of the Allais paradox, known as the common consequence effect and the common ratio effect (Kahneman and Tversky 1979). There is only one objection to EU-preference. It will then be shown that certainty is commonly preferred
(S > R) in the “insurance problem” and risk is commonly preferred (R > S) in the “gambling problem”. An insurance problem is characterized by the fact that the probability of the worst outcome is small, while a gambling problem is found when the probability of the best outcome is small.

Going back to the general form of problem (19), CC theory predicts choice of the sure outcome (S > R) iff: (i) R is a strict prior preference which is not cognitively consistent; (ii) S is a strict prior preference which is cognitively consistent; and (iii) R and S are EU-indifferent, and R has been chosen as the prior (perhaps by tossing a coin).

Assuming that the researcher does not know the individual's luck, we use the simple decision rule associated with (18), which substitutes $\hat{s} \equiv s^* + N$ for s* in the zero experience-decision rule. We designate $\hat{s}$ as the apparent level of consciousness, since we are unable to identify s* and N separately. The first two conditions for S > R can be written¹¹:

$$ (i) \quad x < 1 - \frac{p}{1-q} \equiv x_0 \quad \text{and} \quad s_R > \hat{s} \qquad (21.1) $$

$$ \text{with } s_R = \frac{x}{1 - p - x - q(1-x)} $$

$$ (ii) \quad x > 1 - \frac{p}{1-q} \equiv x_0 \quad \text{and} \quad s_S \le \hat{s} \qquad (21.2) $$

$$ \text{with } s_S = \frac{1-x}{p - (1-x) + q(1-x)} $$

Subscripts R and S designate the EU-preferred gamble. $s_R$ is increasing and convex in x ($s'_R > 0 ; s''_R > 0$) in the interval $[0, +\infty[$; while $s_S$ is decreasing and convex in x ($s'_S < 0 ; s''_S > 0$) in the interval $]+\infty, 0]$. Hence, $s_R$ has one intersection with $\hat{s}$ at $x_1$ ($0 < x_1 < x_0$). Similarly, $s_S$ intersects $\hat{s}$ once at $x_2$ ($x_0 < x_2 < 1$).

Thus, cognitive consistency entails the “certainty effect” ($S > R$)¹² iff:

$$ x_1 < x \le x_0, \quad \text{or} \quad x_2 \le x \le 1 \qquad (22) $$

The lower bounds of the two disconnected intervals for which $S > R$ are easily computed:

$$ x_1 = \frac{(1-p-q)\hat{s}}{1+(1-q)\hat{s}} \quad x_2 = \frac{1+(1-p-q)\hat{s}}{1+(1-q)\hat{s}} \qquad (23) $$
For a given level of consciousness, the sure outcome is preferred to the risky alternative by those having a **strong** normative preference for the sure outcome and also by those having a **weak** normative preference for the risky alternative.

I shall now suggest a simple graphical analysis to visualize the frequency of choice S in a large sample by assuming that all individuals share a common level of consciousness. Figure 2 juxtaposes the distribution function of the risk aversion parameter g(x) upon the salience and consciousness level curves drawn separately for each of the R and S categories. The two bounds $x_1, x_2$ appear at the intersection of the $s_R$, $\hat{s}$ curves on one hand, and $s_S$, $\hat{s}$ curves on the other hand. Figure 2 describes the insurance problem, with a small probability of loss (p). The shaded areas under the given distribution function represent the total frequency of choice S. By contrast, EU prediction of the same choice, i.e. $x \ge x_0$, may be quite off the mark in quantitative terms and/or address qualitatively different individuals.

Insert figure 2 here

VI.1) The certainty effect:

A high frequency of choice S is predicted when the two intervals $]x_1, x_0]$ and $[x_2, 1]$ lie in the thick part of the distribution of x, i.e. the sure intermediate outcome's utility. Two instances of insurance problems (IP) are problems 3 and 14' from Kahneman and Tversky (1979) :

IP1 : $R = (4000, .80; 0, .20)$ ; $S = (3000)$ [80 %]

IP2 : $R = (0, .999; -5000, .001)$ ; $S = (-5)$ , [72 %]

where percentages of the modal response are shown in brackets. The former is known as the certainty effect, and the latter is the common prediction of full insurance at fair prices by EU theory. The probability of loss is equal to .20 in the first case, and to .001 in the second case. Taking $\hat{s} = 4$, we easily compute from (22) and (23):

IP1 : $S > R \text{ iff } x \in [.64, .80] \cup [.84, 1]$

IP2 : $S > R \text{ iff } x \in [.7992, .9990] \cup [.9992, 1]$

Noticing that risk neutrality coincides with values of .75 and .9990 for x respectively, it is quite plausible that the two intervals lie in the thick part of the distribution of x in both cases.
Allais (1953) and Kahneman and Tversky (1979 : problem 1) have replicated the certainty effect with three-outcome risky actions including rather small probabilities of the best outcome 1-p-q:

IP3 : $R = (5.10^6, .10; 10^6, .89; 0, .01)$ ; $S = (10^6)$

IP4 : $R = (2500; .33; 2400, .66; 0, .01)$ ; $S = (2400)$
[82 %]

In the Allais problem, for instance, $p = .01$ and $1-p-q = .10$. With these numerical values, the above calculations yield:

IP3 : $S > R \text{ iff } x \in [.278, .909] \cup [.972, 1]$

IP4 : $S > R \text{ iff } x \in [.559, .971] \cup [.983, 1]$

The certainty effect is enhanced in the Kahneman and Tversky example by the fact that a substantial group of subjects should be moderately risk averse or mildly risk loving for average gains (the risk neutral value of x is .960). Moreover, for a given distribution of x, the lower bounds of the intervals for which $S > R$ always decrease in q (see (23)). This shows, by making use of (22), that the certainty effect can be raised by increasing the share of the common consequence, which is consistent with the above data (e.g. $q = .89, .66$).

## VI.2) Gambling and the reflection effect:

The conditions for choosing the risky action ($R > S$) are simply derived from (21.1) and (21.2) by changing $>$ into $\le$ and vice-versa. By symmetry with (22), it is found that:

$$R > S \quad \text{iff} \quad 0 \le x \le x_1 ; 1 - \frac{p}{1-q} \le x < x_2 \qquad (24)$$

The typical gambling problem is characterized by a two-outcome risky action with a small probability of gain 1-p. Two instances of gambling problems (GP) are problems 3' and 14 from Kahneman and Tversky (1979):

GP1 : $R = (0, .20; -4000, .80)$ ; $S = (-3000)$
[92 %]

GP2 : $R = (5000, .001; 0, .999)$ ; $S = (5)$
[83 %]
These two experiments display a reflection effect, whereby the insurance problem IPi (i = 1,2) is converted into the gambling problem GPi by changing gains into losses and losses into gains of an equal amount. While the reflection effect was built in their theory by Kahneman and Tversky (1979), and implied that the value function be concave for gains and convex for losses, it is a simple consequence of cognitive consistency when the risky action has two outcomes (q = 0). It is enough to notice that the two inequalities (24.1) and (24.2) simply derive from each other in this case by converting > into ≤, p into 1-p, and x into 1-x. Changing gains into losses and losses into gains of an equal amount would have exactly the same effect under the assumption of risk neutrality so that an insurance problem would then perfectly reflect into an associated gambling problem. The reflection effect, though, is an approximate description of reality which might become misleading if the utility function were non-linear and outcomes rather extreme in comparison with endowed wealth.

People usually avoid symmetric bets of the form (y, .50 ; -y, .50). However, they may actually behave differently if outcomes are quite small by accepting a fair symmetric bet. This casual observation, which Markowitz (1952) had built in the shape of his utility function, was recently replicated by Battalio, Kagel and Jiranyakul (1990 : table 4) in the following experiments in which outcomes are expressed in dollars:

$$R = (10, .50; -10, .50) \qquad ; \qquad S = (0) \\ [60\%]$$

$$R = (20, .50; -20, .50) \qquad ; \qquad S = (0) \\ [57 \%]$$

While a majority of subjects rejected a fair symmetric bet when the stake reached $ 20, another majority 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, which violates prospect theory, fits exactly with cognitive consistency. In spite of risk aversion, the normative preference is refuted because the salience of objection to S goes to infinity when the stake approaches zero¹³. This exemplifies the case where, in figure 2, the x-distribution would be highly concentrated at a point slightly above 1/2, resulting in modal choice of the risky
action. It is worth noting that the described phenomenon is not limited to symmetric bets and applies to fair bets of any odds involving very small outcomes.

### VI.3) The fourfold pattern of risk behavior:

A major empirical regularity observed in several studies (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. Tversky and Kahneman (1992) claim that this fourfold pattern is implied by (cumulative) prospect theory. I show that it is also implied by cognitive consistency.

Assuming a gamble of form (20), equations (22) and (23) will be used to compare the predictions of CC theory about the preference for certainty against those of EU theory:

$$ \text{Prob}(S > R/CC) = \text{Prob}(S > R/EU) + [G(x_0) - G(x_1)] - [G(x_2) - G(x_0)] \quad (25) $$

where G( ) designates the cumulative density function of x, which is assumed to be continuous.

Reporting the corresponding values of x₁ and x₂ into (25), we get:

$$ \text{Prob}(S > R/CC) - \text{Prob}(S > R/EU) = \left[ G(1-p) - G\left(1-p-\frac{1-p}{1+\hat{s}}\right) \right] - \left[ G\left(1-p+\frac{p}{1+\hat{s}}\right) - G(1-p) \right] $$

The right-hand member is a difference between two terms shown in brackets. For most distributions G( ), this difference will be positive for small values of p (e.g. p = 0.2) and negative for big values of p (e.g. p = 0.8). Therefore, in comparison with EU, CC favors risk aversion for losses of low probability and risk seeking for losses of high probability. By reflection (see VI.2), it is possible to show that CC favors risk seeking for gains of low probability and risk aversion for gains of high probability.

The fourfold pattern of risk behavior is thus implied by CC under fairly general conditions.

### VI.4) Allais paradox in an action-frame:

In presenting his paradox, Allais (1953) did not frame the lotteries as risky actions with common states of the world. This choice of presentation is innocuous for the demonstration of the certainty effect, as I wrote in the introduction of this section, but it becomes crucial, according to the present theory, for exhibiting the common consequence effect. Remember that Savage (1954) judged that he had made a mistake in indulging to the latter paradox after reframing the lotteries as actions, without realizing himself that different framings may induce different choices. Going back to the choice between R and S described by (19), there are two conspicuous ways of withdrawing the common consequence (y₂, q) from these two lotteries and substituting the worst
outcome $(y_3, q)$ for it. The Allais solution, subsequently adopted by Kahneman and Tversky (1979) and many others, looks like the following:

**Insert table 3 here**

In this frame, the two states $(y_3, q)$ and $(y_3, p)$ yielding the same outcome are aggregated in $R'_{Allais}$ with the result that the transformed lotteries appear as two-state independent prospects with slightly ($p$ is small) different probabilities of occurrence. In contrast, the solution suggested by Savage is to transform the lotteries within an action-frame:

**Insert table 4 here**

I will not examine here the reasons why the prospect-frame adopted by Allais (1953) and Kahneman and Tversky (1979) gives rise to the common consequence effect, leaving a more comprehensive treatment of framing to a subsequent paper. I just demonstrate below that the common consequence effect vanishes but the common ratio effect may persist in the action-frame adopted by Savage (1954).

The substitution of the common consequence $y_3$ for $y_2$ if state 2 occurs (with probability $q$) alters neither the normative preference, nor the objection to it, nor the objection's salience. Consequently, the choice between $R$ and $S$ in (19) orients the choice between $R'_{Savage}$ and $S'_{Savage}$ in the same direction according to proposition 3.

The common ratio transformation is obtained in the action-frame by multiplying the probability of each state in the (20) pair 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.

**Insert table 5 here**

This transformation has no incidence on the normative preference and the objection state although it does reduce the objection's weight by a factor of $r$. Most importantly, the objection's salience is divided by the factor of $r$. This will have virtually no effect on the choice reversal if the common ratio is close to one (e.g. 0.90), but it may have a sizeable effect on the modal response when the common ratio is small (e.g. 0.25). The common ratio effect so described in the action-frame operates on individuals whose choices conformed to their preferences in the (20) gamble. Now, some $R$-preferring persons will be induced to shift to the safer option $S''_{Savage}$ while some $S$-preferring persons will change for the riskier option $R''_{Savage}$. For those individuals whose choices deviated from their preferences in the (20) gamble, the direction of choice towards the riskier or
safer alternative is unaffected by the common ratio transformation. The aggregate effect depends on the relative sizes of the four groups and on the distributions of the risk aversion and consciousness parameters in the population.

We conclude from the foregoing discussion of the Allais paradoxes in an action-frame that CC theory is inconsistent with the common consequence effect but is not inconsistent with the common ratio effect. However, the essence of the Allais paradoxes seems to lie in the prospect-framing¹⁴, which is outside the scope of the present paper.

## VII. THE VALUE OF A LOTTERY

In EU theory, the value of any lottery $B$ is given by its certainty-equivalent $v(B)$, defined by $u(v(B)) = EU(B)$. I shall only consider here bets of the form $B = (y, p; 0,1-p) \equiv (y, p)$ in order to simplify the exposition without great loss of generality¹⁵. For a bet yielding a prize, i.e. $y > 0$, $v(B)$ is the minimum selling price predicted by EU theory, and it would also be close to the EU prediction for the maximum buying price of the bet when income effects are small. However, Thaler (1980), and Knetsch and Sinden (1984), for instance, discovered a substantial wedge between selling prices and buying prices for lotteries that is inconsistent with EU theory. Thaler (1980) called this anomaly the endowment effect because owners of a good value it much more than non-owners. In another famous experiment, Slovic and Lichtenstein (1968) remarked that subjects who had to evaluate gambles yielding similar expected gains attached a greater value to that which yielded a higher payoff. Their observation contrasts with the EU prediction of equal prices for EU-indifferent lotteries. What made their result even more striking, and came to be known as the preference reversal phenomenon (Lichtenstein and Slovic, 1971), is that the tendency to attach a greater value to riskier beneficial bets is opposed to the certainty effect, which is the tendency to choose a sure outcome over a risky alternative yielding a similar expected gain. All of these and other findings can easily be accommodated by CC theory because the latter assumes that choices, not preferences, are influenced by the choice set and the choice sets are not the same when subjects compare lotteries, bid to sell or bid to buy. In the comparison ("choice") task, $A$ is compared to $B$. In the selling task, descending values of a sure outcome $v_s$ are compared to $A$ and $B$ separately until ODEU-indifference is reached. In the buying task, ascending values of $v_b$ are generated and $A - v_b$ or $B - v_b$ separately are compared to the status quo 0 until ODEU-indifference is reached. Previous sections were largely devoted to the basic comparison task. Bidding tasks are the main focus of this section.
## VII.1. Risk

We first determine the minimum selling price of a beneficial bet $B = (y, p)$ yielding a positive outcome y with a given positive probability $p (0 < p \le 1)$. When $p = 1$, B degenerates into a sure outcome and the CC predictions are identical to those made by the standard theory of choice. Let us then assume that $p < 1$. Descending values of $v_s$ are generated in the interval $[0, +\infty[$. The sure price $v_s$ strictly dominates B as long as $v_s \ge y$, and $u(v_s) > EU(B)$. The continuity of the utility function ensures that $u(v_s) \ge EU(B)$ still holds for descending values of $v_s < y$ in the vicinity of y. However, there is now an objection to selling the bet at such a price, which is $(y \to v_s; p)$. Thus, by applying here the general definition of ODEU given by (12)

$$ODEU(B) = \left( p + \frac{1}{s^*} \right) u(y) + (1-p) u(0) \quad \text{and} \quad ODEU(v_s) = \left( 1 + \frac{1}{s^*} \right) u(v_s) \qquad (26)$$

The minimum selling price $v_s(B)$ is the lowest value for which $u(v_s) \ge EU(B)$ and $ODEU(v_s) \ge ODEU(B)$. If values can be set continuously, we derive from (26)

$$\begin{aligned} u(v_s(B)) &= \frac{p + \frac{1}{s^*}}{1 + \frac{1}{s^*}} u(y) + \frac{1 - p}{1 + \frac{1}{s^*}} u(o) \\ &= \frac{s^*}{1 + s^*} EU(B) + \frac{1}{1 + s^*} u(y) \end{aligned} \qquad (27)$$

The evaluation method and equation (27) show that the minimum selling price is greater than the certainty-equivalent predicted by EU for an imperfectly-conscious individual

$$v_s(B) > v(B) \quad \forall s^* < +\infty \qquad (28)$$

Moreover, (27) helps us reformulate one of Slovic and Lichtenstein's (1968) observations.

**Proposition 8 (valuation of positive bets)**

Of two EU-indifferent bets yielding different positive payoffs, an imperfectly-conscious individual attaches greater value to that yielding the higher prize.

Similarly, the maximum buying price for the beneficial bet $B = (y, p)$ is the highest value $v_b(B)$ for which $EU(B-v_b) \ge u(0)$ and $ODEU(B-v_b) \ge ODEU(0)$. Since the objection to buying B at price $v_b \in ]0,y[$ is $(0 \to -v_b; 1-p)$,

$$ODEU(B-v_b) = p u(y-v_b) + \left(1 - p + \frac{1}{s^*}\right) u(-v_b) \quad \text{and} \quad ODEU(O) = \left(1 + \frac{1}{s^*}\right) u(o).$$
ODEU-indifference between $B-v_b$ and 0 determines the maximum buying price of the bet

$$ \frac{s^*}{1+s^*}EU(B-v_b(B)) + \frac{1}{1+s^*}u(-v_b(B)) = u(0) \quad (29) $$

(29) implies $v(B - v_b(B)) > 0$ for an imperfectly-conscious individual.

Since the buying price predicted by EU theory, $v_b^{EU}(B)$, is defined by $v(B - v_b^{EU}(B)) = 0$, we get

$$ v_b(B) < v_b^{EU}(B) \quad (30) $$

because the certainty-equivalent of a lottery increases when a sure positive outcome is added in all states of the world.

When income effects are small, $v_b^{EU}(B) \approx v(B)$, so that CC theory implies, by use of (28) and (30)

$$ v_b(B) > v(B) > v_s(B) \quad (31) $$

which is the endowment effect for lotteries. It is possible to be more specific about these three values if we assume that income effects are small or individuals are risk-neutral. The utility function is normalized so that $u(0) = 0$ and $v(B) = EU(B) = py$. Values for the selling price and the buying price of $B$ are then derived from (27) and (29) respectively.

$$ v_s(B) = \frac{p + \frac{1}{s^*}}{1 + \frac{1}{s^*}} y \quad ; \quad v_b(B) = \frac{p}{1 + \frac{1}{s^*}} y \quad (32) $$

The latter expressions and the endowment effect also apply to bets implying a fixed loss $y$ with a given probability $p$. $v_s(B)$ and $v_b(B)$ are then simply the insurance price that a seller requires and that a buyer accepts for exchanging the risk of loss. However, CC theory does not extend Slovic and Lichtenstein's (1968) observations and proposition 8 to negative bets $(-y, p) \equiv (0,1-p; -y, p)$ of the type involved in insurance purchasing decisions. The reason is simply that, in applying equation (27), the winning state should be replaced by the no-loss state whose payoff is invariably equal to zero. Hence

**Proposition 8 bis (valuation of negative bets)**

An individual will accept the same maximum insurance premium for covering two EU-indifferent negative bets involving different losses.
It is worth noticing from (32) that the wedge between the selling price and the buying price does not depend on the probability of the payoff when income effects are small. It remains when this probability gets very close to one as long as uncertainty remains. This may help understand why the endowment effect has also been observed with consumption goods (on rather inexperienced subjects), but not with induced-value tokens designed to eliminate uncertainty (Kahneman, Knetsch and Thaler 1990)¹⁶.

## VII.2. Uncertainty

CC theory makes a distinction between risk and uncertainty in evaluation tasks, which did not arise in comparison tasks. I will show this by determining the minimum selling price of a bet $\tilde{B} = (y, E)$ yielding a prize $y$ on the occurrence of an uncertain event $E$. The true probability of $E$ is unknown and follows a distribution with an unknown mean $Ep$, of which a prior estimate $\bar{p}$ is known. The latter is the Savagean subjective probability of the uncertain event. Hence, $EU(\tilde{B}) = \bar{p}u(y)$ if $u(0)$ is conventionally set at zero value. Since $Ep$ is unknown, the uncertain bet $\tilde{B}$ is truly perceived as a distribution of risky bets $B(p) = (y, p)$ associated with density $\varphi(p)$. Therefore, the assessment of a selling price $v_s \in ]0, y[$ such that $u(v_s) \ge EU(\tilde{B})$ no longer raises a single objection, as in the case of risk, but a whole range of objections. After $v_s$ has been chosen, the true bet may turn out to be any of the $B(p)$, where $p$ is such that $EU(B(p)) = p \ u(y) > u(v_s)$, that is, $p > \frac{u(v_s)}{u(y)} = w_s$. We write this composite objection as

$$ \left( (B(p); \varphi(p | p > w_s)) \rightarrow v_s ; 1 - I_{w_s} \equiv \int_{w_s}^{1} \varphi(p) dp \right) $$

Consequently,

$$ ODEU(\tilde{B}) = \left( 1 - I_{w_s} + \frac{1}{s^*} \right) \int_{w_s}^{1} EU(B(p)) \frac{\varphi(p)}{1 - I_{w_s}} dp + I_{w_s} \int_{0}^{w_s} EU(B(p)) \frac{\varphi(p)}{I_{w_s}} dp $$ 
and $ODEU(v_s) = \left( 1 + \frac{1}{s^*} \right) u(v_s)$.

The normalized minimum selling price $w_s(\tilde{B})$ is the value for which ODEU-indifference is reached between $\tilde{B}$ and $v_s$. It is the solution of the following equation

$$ w_s(\tilde{B}) = \frac{u(v_s(\tilde{B}))}{u(y)} = \frac{s^* \bar{p} + 1}{1 + s^*} E(p | p > w_s(\tilde{B})) \quad (33) $$

Since $w_s \ge \bar{p}$ by construction, (33) entails

$$ w_s(\tilde{B}) > \bar{p} \quad \forall s^* < + \infty $$
The latter condition parallels (28) in the risk case, and the endowment effect extends to uncertainty. However, (33) and (27) immediately show

$$w_s(\tilde{B}) \le w_s(B(\bar{p})) = \frac{s^*}{1+s^*} \bar{p} + \frac{1}{1+s^*} \cdot 1$$

A risky bet yielding a sure probability of a prize sells more than an uncertain bet yielding the same subjective probability of the same prize. The value difference is the opportunity cost of uncertainty.

By assuming again that p follows a Beta distribution with parameters a, b > 0, and recalling that: $\bar{p} = \frac{a}{a+b}$, we get (see the proof in the appendix 4):

$$E(p | p > w_s) = \bar{p} + w_s (1 - w_s) \frac{h(w_s)}{a + b} \quad (34)$$

where $h(w_s) = \frac{\varphi(w_s | a, b)}{1 - I_{w_s}(a, b)}$ is the hazard rate of the Beta (a,b) distribution. Then we report (34) into (33) and obtain:

$$w_s = \bar{p} + \frac{w_s (1 - w_s) h(w_s)}{1 + s^* (a + b)} \quad (35)$$

Since $a+b$ is positively related to the precision with which the probability of the uncertain event is estimated¹⁷, the overpricing of an uncertain beneficial bet declines as the individual accumulates knowledge about the uncertain event. But, even if he had little knowledge of the uncertain event, a perfectly conscious person would never overprice the bet, in accordance with EU theory. Equation (35) demonstrates that knowledge and consciousness have reinforcing effects.

When prizes are converted into losses, the same analysis shows that the maximum insurance premium is smaller for a risky loss than for an uncertain loss of equal size having the same subjective probability of occurrence. People are willing to pay an uncertainty premium in excess of the amount they would pay for covering themselves against a known harmful risk.

### VII.3. The preference reversal phenomenon

Tversky, Slovic and Kahneman (1990) concluded from a series of experiments that the primary cause of the preference reversal phenomenon is the failure of procedure invariance, especially the overpricing of low-probability high-payoff bets. Since the latter is predicted by our theory, we might expect that it also predicts the preference reversal phenomenon. It is shown here that it is well the case, at least when bets $B_1$ and $B_2$ are framed as actions¹⁸.
We assume here $B_i = (y_i, p_i)$, ($i=1,2$), with $y_1 > y_2 > 0$ and $0 < p_1 < p_2 \le 1$, and conventionally set: $u(y_1) = 1$, $u(o) = 0$, $u(y_2) = x$ ($0 < x < 1$).

For comparison purposes, we use the following frame

$$B_1=(y_1,p_1;0,p_2-p_1;0,1-p_2)$$

$$B_2=(y_2,p_1;y_2,p_2-p_1;0,1-p_2)$$

The preference reversal phenomenon normally occurs when $B_2$ is chosen against $B_1$ but $B_1$ would be sold at a higher price than $B_2$.

Two cases must be successively examined: (i) $EU(B_1) \ge EU(B_2)$; (ii) $EU(B_1) < EU(B_2)$.

$$(i) \quad EU(B_1) \ge EU(B_2), \text{ or } p_1 \ge p_2 x \qquad (34.1)$$

$B_2$ is chosen against $B_1$ if $ODEU(B_2) > ODEU(B_1)$, that is,

$$\left(p_2 + \frac{1}{s^*}\right) x > p_1 \qquad (35.1)$$

$B_1$ is sold at a higher price than $B_2$ if $u(v(B_1)) > u(v(B_2))$. Since $u(v(B_1)) = \frac{p_1 + \frac{1}{s^*}}{1 + \frac{1}{s^*}}$ and

$$u(v(B_2)) = \frac{p_2 + \frac{1}{s^*}}{1 + \frac{1}{s^*}} x, \text{ this implies}$$

$$p_1 + \frac{1}{s^*} > \left(p_2 + \frac{1}{s^*}\right) x \qquad (36)$$

Note that the latter condition is already implied by (34.1) if $s^* < +\infty$. Finally, (34.1) and (35.1) together imply

$$\frac{p_1}{p_2 + \frac{1}{s^*}} < x \le \frac{p_1}{p_2} \qquad (37)$$

To illustrate this possibility, we take the numerical values of the first monetary bet listed in Tversky et al (1990, table 1): $y_1 = 16, p_1 = \frac{11}{36}$; $y_2 = 4, p_2 = \frac{35}{36}$. We also assume $s^* = 4$. Conditions (37) require that $0.250 < x \le 0.314$, which covers reasonable values of risk aversion (the agent is risk-neutral when $x = 0.250$).

$$(ii) EU(B_1) < EU(B_2), \text{ or } p_1 < p_2 x \qquad (34.2)$$

$B_2$ is chosen against $B_1$ if $ODEU(B_2) \ge ODEU(B_1)$, or
$$ \left( p_2 + \frac{1}{s^*} \right) x \geq p_1 + \frac{1}{s^*} \qquad (35.3) $$

$B_1$ is sold at a higher price than $B_2$ if (36) holds. But this contradicts (35.3). Hence, CC theory circumscribes the preference reversal phenomenon to subjects whose choice of the less risky bet violates their EU-preference, that is, whose risk aversion parameter respects conditions (37). Contrary to what Tversky et al. (1990) conclude, the overpricing of low-probability high-payoff bets is not the primary cause of the preference reversal phenomenon because it is already implied by the violation of EU-preference. Thus the preference reversal phenomenon and the certainty effect have a single cause which is simply that the rational choice of imperfectly-conscious individuals may violate their EU-preference. This interpretation is consistent with the violation of procedure invariance emphasized by Tversky et al. (1990) because choice set-dependent choices obviously violate procedure invariance.

However, Lichtenstein and Slovic (1971) and Tversky et al. (1990) are partially right in emphasizing the overpricing of low-probability high-payoff bets because the preference reversal phenomenon is not commonly observed the other way round. Indeed, CC theory predicts that it is irrational to choose the riskier beneficial bet (i.e. $\binom{0}{-\frac{p}{2}, 1-p}$) and, at the same time, to underprice the latter $\binom{-\frac{p}{2}, 1-p}{-p, \frac{1}{2}, -x, \frac{1}{2}, p}$. The proof can be found in the appendix 5. In brief, CC theory circumscribes the preference reversal phenomenon to subjects whose choice of the less risky bet violates their EU-preference and rules it out the other way round. This conclusion is best described by figure 3, in which the dotted cells are ruled out and PR designates the locus of the preference reversal phenomenon.

Insert figure 3 here

## VII.4. Other pricing anomalies

### a) The utility evaluation effect

Under the EU model, certainty-equivalents can be used to recover the individual utility curves. Define the beneficial bet (y, p) as B(p) and adopt the normalization $u(0)=1$ and $A_k(k \neq 1)$. By asking a subject to state his or her minimum selling price for B(p) for various values of p, a single utility curve is determined by the relation : $u(v(p)) = p$. Interestingly, the same conclusion is implied by the pricing formula (27) under the CC model, once the individual's consciousness is known. However, the relation between the bet's value and the probability of its prize is modified:
$$u(v(p)) = \frac{p + \frac{1}{s}}{1 + \frac{1}{s}} = p + \frac{1 - p}{1 + s} \quad (38)$$

This new pricing formula suggests that, if we wanted to recover utility curves by the standard method of certainty-equivalents, we would seemingly get low “utility curves” at low values of p and high “utility curves” at high values of p. This phenomenon, found by Karmarkar (1978), is called the utility evaluation effect (Machina 1987). It appears in figure 4 where I have drawn the true utility curve u(y) and the “recovered” utility curves for probabilities 1/4, 1/2 and 3/4 under the assumption that $s^* = 4$.

### Insert figure 4 here

The inverse local utility curves which can be recovered from the evaluation task are:
$$u_p^{-1}(p) = u^{-1}\left(p + \frac{1-p}{1+s^*}\right).$$
They all lie below the true utility curve, and at greater distance for lower values of p. The correction terms $\frac{1-p}{1+s^*}$ depicted by figure 4 are 3/20 for p = 1/4, 2/20 for p = 1/2 and 1/20 for p = 3/4. It is worth noticing that the “recovered” utility curves are pure artefacts under CC theory. Machina (1987) interprets the utility evaluation effect¹⁹ as a sign that indifference curves in the Marshak-Machina triangle fan out instead of being parallel lines. But the apparent fanning out can result from the postulated coincidence between choice and preference under risk.

## b) Probabilistic insurance

Lastly, problem 9 from Kahneman and Tversky (1979) about probabilistic insurance will be solved here²⁰. In this problem, subjects are initially asked to consider the premium P at which they are indifferent between the options of purchasing full insurance against damage of their property X or leaving the property uninsured. Then, a new policy called “probabilistic insurance” is offered to subjects. Under this program, the latter pay half of the premium P. In case of damage, there is a 50 per cent chance that they pay the other half of the premium and the insurance company covers all the losses; and there is a 50 per cent chance that they get back their insurance payment and suffer all the losses. The puzzle raised by such probabilistic insurance is that most subjects reject it while EU theory predicts that it should be accepted by risk-averse individuals (proof of this last point is straightforward and can be found in Kahneman and Tversky 1979: 270).
In CC theory, the premium that makes one feel indifferent between full insurance and no insurance, that is, between the sure loss (-P) and the negative bet (-X, p) ≡ (0, 1-p; -X, p), is given by the pricing formula (27)

$$ 
\begin{aligned}
u(-P) &= \frac{s^*}{1+s^*} \cdot (1-p) + \frac{1}{1+s^*} \cdot I \\
&= \frac{1+(1-p)s^*}{1+s^*}
\end{aligned} 
\quad (39) $$

for which we assumed $u(-X)=0$ and $u(0)=1$. Now, the probabilistic insurance is described by the following lottery

$$ I = \left( -\frac{P}{2}, 1-p; \left( -P, \frac{1}{2}; -X, \frac{1}{2} \right), p \right) $$

Although $EU(I) > EU(-X, p)$ if $u$ is strictly concave, there is an objection to $I, \left( 0 \to -\frac{P}{2}; 1-p \right)$. Consequently, we show that

$$ \text{ODEU}(I) < \text{ODEU}(-X, p), \\ \text{or} \quad \left(1-p+\frac{1}{s^*}\right)u\left(-\frac{P}{2}\right) + p\left[\frac{1}{2}u(-P)+\frac{1}{2}u(-X)\right] < \left(1-p+\frac{1}{s^*}\right)u(0)+pu(-X) $$

Reporting the expression of $u(-P)$ given by (39) in this inequality yields

$$ 
\begin{aligned}
u\left(-\frac{P}{2}\right) &< 1 - \frac{\frac{p}{2}s^*}{1+(1-p)s^*} u(-P) \\
&= 1 - \frac{p}{2} \frac{s^*}{1+s^*}
\end{aligned} 
$$

In combination with (39), this entails

$$ 
\begin{aligned}
u\left(-\frac{P}{2}\right) - u(-P) &< \left(1 - \frac{p}{2} \frac{s^*}{1+s^*}\right) - \left(1 - p \frac{s^*}{1+s^*}\right) \\
&= \frac{p}{2} \frac{s^*}{1+s^*}
\end{aligned} 
\quad (40) $$

and (39) entails

$$ 1 - u(-P) = p \frac{s^*}{1 + s^*} \quad (41) $$

Comparing (40) to (41),

$$ u\left(-\frac{P}{2}\right) - u(-P) < \frac{u(0) - u(-P)}{2} $$

is implied by the concavity of the utility function (risk aversion). By adapting the same analysis to a convex utility function, it is straightforward to show that CC theory exactly predicts that risk lovers find probabilistic insurance attractive, while risk averters do not. Thus, the present theory
recovers the intuitive notion that probabilistic insurance is “riskier” than full insurance, which
could not be accommodated by EU theory.

VIII. THE DEMAND FOR NEW INFORMATION UNDER UNCERTAINTY AND
COGNITIVE DISSONANCE

The analysis which was carried so far assumed that no valuable new information can be acquired
from an external source. The latter assumption characterizes risk but it is obviously too strong in
the case of uncertainty. It is now relaxed.

Let us suppose that a new information service I may be acquired at a utility cost C(I). Both
negative costs (subsidies) and positive costs are being considered. Standard economic theory
predicts an effective demand for an information service which has a positive value net of cost:

$$
V(I) - C(I) > 0
$$

Under CC theory, the above condition is necessary but not sufficient, because available
arguments are potential competitors with new information. The necessary and sufficient condition
for an effective demand for new information is now:

$$
V(I) - C(I) > V(Q) \tag{42}
$$

If CC theory is to be accepted, EU theory certainly overstates the demand for new information.

**VIII.1. Rational ignorance of new valuable information**

We show below, by exhibiting a simple counter-example, that it may be rational for an
imperfectly-conscious individual to ignore new free perfect information.

Consider the choice between a risky bet $R = (y_1, p; y_3, 1-p)$ and the sure outcome $S = (y_2)$
that we examined in section 6 (eq. (20)) and assume that $R$ is EU-preferred to $S$. Hence, with the
same notations, $x \le p$. The expected value of perfect information (in utility terms) is then

$$
VPI = u(v_s(T)) - u(v_s(R))
$$

in which $T = (y_1, p; y_2, 1-p)$, because purchasing a perfect information service is like
obtaining the right to win in all states of nature instead of only winning under one state. By
application of (27), we get

$$
u(v_s(R)) = \frac{s^*}{1+s^*} \cdot p + \frac{1}{1+s^*}
$$

$$
u(v_s(T)) = \frac{s^*}{1+s^*} [p + (1-p)x] + \frac{1}{1+s^*}
$$

Hence
$$VPI = \frac{s^*}{1+s^*}(1-p)x \qquad (42)$$

VPI must be compared to the value of raising the objection $Q = (y_2 \rightarrow y_3; 1-p)$ to the preference $R$. Therefore, according to (7),

$$V(Q) = \max(EU(S^*) - EU(R^*), 0)$$

By using (6) to derive the posterior EU of $S$ and $R$, we get

$$V(Q) = \max\left(x - \frac{ps^*}{1+s^*}, 0\right) \qquad (43)$$

and, by comparison of (42) with (43)

$$V(Q) \geq VPI / x \leq p \Leftrightarrow \frac{ps^*}{1+ps^*} \leq x \leq p \qquad (44)$$

which further requires $(1-p)s^* \leq 1$.

**Proposition 9 (rational ignorance of new valuable information):**

It may be rational for an imperfectly-conscious individual to ignore new valuable information.

A necessary condition is that there exists at least one valuable objection to the EU-preference.

**Proof:** (44) exhibits a counter-example to the contrary assertion when $s^* < +\infty$. In the latter, new information is assumed to be free and perfect. (41) shows that costly information will always be discarded if the same information is discarded when freely available. It also shows that it is necessary that $V(Q) > 0$ for discarding new information $I$ when $V(I) - C(I) > 0$. $\square$

CC theory does not presume that new information may have a negative value, but merely that it has a positive reservation value in the presence of a valuable objection to the EU-preference. Since (42) makes no requirement on past experience, our theory is consistent with the psychological theory of cognitive dissonance (Festinger 1957)²¹. A person may ignore a bad experience after a choice of his and invoke justifications for his previous decision. Although a bad experience is, ex post, a valuable and free message, it would be rational to ignore the latter if it were less valuable than the reasons which led to the decision and can now justify it. However, the cognitive dissonance phenomenon has limits. It should not be observed when there was no valuable reason for the previous decision or when the bad experience is catastrophic and thus has sufficient value; and it should eventually disappear after a number of repeatedly bad experiences.
## VIII.2. More on the certainty effect under uncertainty

The simple example treated above can be further extended to establish the robustness of the certainty effect under uncertainty and endogenous information. We initially assumed that R was EU-preferred to S and showed then that sufficiently unconscious gamblers would stick to their choice of S by discarding new valuable information which might have pulled them back to R. If we now make that opposite assumption that S is EU-preferred to R (ie. $x \ge p$), we can recalculate

$$ 
\begin{align*}
VPI &= u(v_s(T)) - u(v_s(S)) \\
&= \frac{s^*}{1+s^*} [p + (1-p)x] + \frac{1}{1+s^*} - x \\
&= \frac{1+ps^*}{1+s^*} + \frac{s^*}{1+s^*} (1-p)x - x
\end{align*}
$$

as well as the value of objection $Q' = (y_1 \rightarrow y_2; p)$ to the preference S

$$ 
\begin{align*}
V(Q') &= \max(EU(R^{**}) - EU(S^{**}), 0) \\
&= \max\left(\frac{1 + ps^*}{1 + s^*} - x, 0\right)
\end{align*}
$$

The comparison of VPI and $V(Q')$ immediately shows that $VPI > V(Q')$. Therefore, even imperfectly conscious individuals would accept new information if it were valuable and cheap enough, with the risk of shifting back to their preference S. Put briefly, more decision paths lead to S than to R when additional information can be acquired and this reinforces the certainty effect under uncertainty. These conclusions can be easily extended to the more general choice problem described by eq. (19) in section 6.

## VIII.3. The three-door game

CC theory is now used to solve the popular three-door game which has been recently replicated by David Friedman (1998) in a controlled experiment. Following Nalebuff's (1987) account, host Monty Hall of the once-popular TV game show "Let's Make A Deal" asked his final guest of the day to choose one of three doors. One door led to the grand prize and the other two doors led to worthless prizes. After the guest chose a door, Monty Hall opened one of the other two doors to reveal a worthless prize and offered the guest the opportunity to switch his choice to the remaining unopened door. The fact that very many guests ignored the opportunity to switch suggests that they did not make use of new valuable information since the probability of winning is actually increased from 1/3 for non switchers to 2/3 for switchers.

Prior to choosing one door, the guest faces three EU-indifferent lotteries
$$L_1 = \left( 1, \frac{1}{3}; 0, \frac{1}{3}; 0, \frac{1}{3} \right)$$

$$L_2 = \left( 0, \frac{1}{3}; 1, \frac{1}{3}; 0, \frac{1}{3} \right)$$

$$L_3 = \left( 0, \frac{1}{3}; 0, \frac{1}{3}; 1, \frac{1}{3} \right)$$

where $L_i$ ($i = 1,2,3$) designates the gamble played if the $i^{th}$ door is chosen, and the utilities of the grand prize and worthless prizes have been normalized to 1 and 0 respectively. Remember that the guest will never choose his prior, under CC theory, when the three gambles are EU-indifferent. Assume that $L_2$ is the prior and $L_1$ is the choice before Monty Hall reveals additional information. Since the objection to $L_2$ is that the grand prize hides behind the first door with a probability of one-third, the objection's value (in utility terms) is

$$ 
\begin{align} 
V(Q) &= EU(L_1^*) - EU(L_2^*) \tag{45} \\ 
&= \frac{\frac{1}{3} + \frac{1}{\hat{s}}}{1 + \frac{1}{\hat{s}}} - \frac{\frac{1}{3}}{1 + \frac{1}{\hat{s}}} \nonumber \\ 
&= \frac{1}{1 + \hat{s}} \nonumber 
\end{align} 
$$

where $\hat{s}$ denotes the apparent consciousness of the guest at the time he decides whether he wants to switch or not.

This is to be compared with the expected value (in utility terms) of the information displayed by Monty Hall, which amounts to $u(v_s(M)) - u(v_s(L_2))$, where $M = (0, \frac{1}{3}; 1, \frac{2}{3})$ is the would-be-preferred gamble if the guest accepts the new valuable information$^{22}$. We have

$$u(v_s(L_2)) = \frac{\hat{s}}{1+\hat{s}}\frac{1}{3} + \frac{1}{1+\hat{s}}$$

$$u(v_s(M)) = \frac{\hat{s}}{1+\hat{s}}\frac{2}{3} + \frac{1}{1+\hat{s}}$$

Hence,

$$u(v_s(M)) - u(v_s(L_2)) = \frac{\hat{s}}{1+\hat{s}}\frac{1}{3} \qquad (46)$$

By comparing (45) and (46), we conclude that the guest ignores the additional information, although it is valuable, if
$$ \frac{\hat{s}}{1 + \hat{s}} \frac{1}{3} \le \frac{1}{1 + \hat{s}} $$

or

$$ \hat{s} \le 3 \qquad (47) $$

Thus sufficiently unconscious guests ignore the new valuable information. But there is an additional trick in this game. Remember that Monty Hall provides partial experience to his guest by opening one of the two unchosen doors and recall that the posterior objection's weight, given by appendix 3.2, is $p^* = p + \frac{1+r-(1+N)p}{(1-p)s^*+N-r} = p + \frac{1}{\hat{s}}$ with $p = \frac{1}{3}$ and $N = 1$ in the present context.

But the experience does not convey a complete information on $r$. If the second door is opened, the guest knows for sure that his prior is refuted; hence $r = 1$ and $\hat{s} = \frac{s^*}{2}$. Conversely, if the third door is opened, the guest cannot be sure that his prior is confirmed. This would be true if the winning gamble were $L_1$, but false if the winning gamble were $L_2$. Hence, $r = 0$ with probability

$$ \frac{1}{3}, \quad r = 1 \quad \text{with probability} \quad 2/3, \quad \text{and} \quad \frac{1}{\hat{s}} = \frac{1}{3} \frac{1}{\frac{1}{3} + \frac{2}{9}s^*} + \frac{2}{3}s^*. \quad \text{Since } s^* > 0, \quad (47) $$

yields: $s^* \le \frac{7+\sqrt{73}}{2} = 7.77$. On average, half of the guests ignore the valuable information provided by Monty Hall if $s^* \le 6$ and the other half behaves similarly if $s^* \le 7.77$.

### VIII.4. Conscious and unconscious learning under uncertainty

Under risk, unconscious learning caused rational choice to converge in probability towards EU-preference. This process is likely to be inhibited under real-world uncertainty by the need for conscious learning whenever little is known about the environment. If the uncertainty is great, the decision-maker is likely to consciously update his beliefs by making use of each new experience. This can explain the experimental evidence collected by Fox, Rogers and Tversky (1996), showing that experienced options traders behave differently under risk and uncertainty. The latter were apparently perfectly conscious under risk (and risk-neutral), but not under uncertainty. Our interpretation for this result is that unconscious learning from experience can exert its full effects under risk but is partly inhibited by the need for conscious learning under real-world uncertainty. This brings relevance to the Knightean distinction between risk and uncertainty even though I have assumed that individuals are able to make prior subjective probability assessments.
## IX. CONCLUSION

A built-in assumption of expected utility theory is that all human knowledge and beliefs are the output of conscious reasoning. To postulate the coincidence of the rational choice with the normative preference is another way of putting it. In light of widely accepted psychological research about unconscious mechanisms of the mind, this approach is too restrictive. Accounting for unconscious reasoning and learning has enabled us to solve many anomalies and paradoxes to EU theory both in choice and evaluation tasks, both under risk and uncertainty. For instance, we were able to predict the subtle combination of risk avoidance and risk seeking described in the classical works of Friedman and Savage (1948), Markowitz (1952), Kahneman and Tversky (1979), and the fourfold pattern of risk behavior evidenced more recently by Cohen et al. (1987), Tversky and Kahneman (1992) or Tversky and Wakker (1996). We also predicted that risk averters might fail to purchase insurance at very small probabilities of a disaster while they commonly buy coverage in the medium range of probabilities (Kunreuther et al. 1978). In applying our theory to evaluation tasks, we predicted inter alia the endowment effect for lotteries (Thaler 1980, Knetsch and Sinden 1984), the preference reversal phenomenon (Lichtenstein and Slovic 1971, Grether and Plott 1979), and the uncertainty premium that people are willing to pay in excess of the amount they would pay for covering themselves against a known harmful risk. CC theory appears to be no less successful (and perhaps more) than other leading NEU theories like prospect theory or rank-dependent EU in solving many anomalies and paradoxes to EU theory. Furthermore, the new theory provides a parsimonious solution that assumes well-behaved (everywhere concave and differentiable, for instance) utility functions, fully specifies the weighting function in a simple way, and just adds one parameter (consciousness) to EU.

As CC theory makes a distinction between choice and preference, it does not impose the same consistency requirements on both. Transitivity and procedure and description invariance, assumed by EU theory, seem to be proper requirements for defining a choice-set independent preference which can be held as a prior in all kinds of choice problems. But these axioms are too strong for defining rational choices which, by necessity, are always conditional on preferences. The weaker requirements of dominance and transitivity conditional on the preferred action remaining in the choice set seem sufficient for the latter purpose. Finally, it is worth mentioning that our analysis might be accommodated to other normative theories of preference making other consistency requirements for preference, because it does no claim to be a theory of consistent preference but a theory of rational choice conditional on a given preference.
Although the goal of decision-making under uncertainty is to overcome doubt, not to maximize any given utility function, a desirable property of our model is that rational behavior can be described as if individuals maximized a descriptive utility function which easily derives from EU with just one additional parameter. The task of defining local optima on the subset of actions sharing common objection states is even simpler because the maximand is then an objection-dependent EU which is linear in the (distorted) probabilities. On several grounds, our theory is more tractable than other NEU models but the conditionality of choice obviously adds a complication in that a single choice can originate from several priors. This last feature is the price to be paid for predicting, say, that a large majority of people, which is not a unanimity though, will exhibit the certainty effect, despite the great diversity of individual attitudes.

The ultimate reason for believing in CC theory is that it unifies so many insights and approaches from economics and cognitive psychology. Three widely used concepts of rational behavior, i.e. normative rationality, informational efficiency, and cognitive consistency have been reconciled and fitted in the common broad definition introducing the paper. The first omits that rational choices under uncertainty may be conditioned by choice-set dependent reasons along with choice-set independent preferences. The second is quite general but it has been so far restricted in economic applications to a conscious treatment of information. Cognitive consistency presumes unconscious reasoning, as shown here, and amounts to full informational efficiency including both conscious and unconscious information. EU theory makes a synthesis of the first two concepts of rational behavior on the premises that beliefs are separated from outcomes, which is akin to saying that all reasoning is conscious. Cognitive consistency theory generalizes EU by allowing for unconscious reasoning and learning. Whenever a valuable objection is being raised, beliefs are no longer separated from outcomes. This occurs if and only if cognitive dissonance (i.e. the objection's salience) exceeds consciousness.

CC theory makes some novel testable predictions. One striking implication is that unconscious learning from experience will occur under risk with objective probabilities and result in the stochastic convergence of rational choice towards the normative preference. Another one is that rational ignorance of new valuable information will occur whenever available arguments are more valuable. These new insights shed light on the Knightean distinction between risk and uncertainty. Our first conclusion is that it is warranted because risky and uncertain actions are treated differently in evaluation tasks and they give rise to different learning patterns. However, the intuitions of Keynes (1921) and Knight (1921) are far from being fully supported by our analysis. First, people can make prior subjective probability assessments out of conscious reasoning.
Second, doubt or confidence plays a role on the rational choice (as recalled by the quotation from Knight (1971: 227) heading this article) as much under risk as under uncertainty. Third, since new information may compete with available arguments only under uncertainty, unconscious learning from experience can exert its full effects under risk while it is partly inhibited by the need for conscious learning under real-world uncertainty. Fourth, it was shown that risk and uncertainty make no difference in comparison tasks (choices) between true actions with common states of the world, but they do make a difference in evaluation tasks (pricing).

An important shortcoming of the present analysis is that it has been restricted to choices presented in an action-frame, since there can be no valuable justification in this case and the objection set summarizes all consistent reasons. We showed that the Allais paradox thus loses a great deal of its power of illusion. However, essentially the same theory provides the foundations of an economic theory of framing, since the latter consists in studying valuable justifications. Uncertainty, ambiguity and probability judgments is another subject that surely deserves more attention. Because cognitive consistency brings unconscious reasoning and learning into economic theory, and thus extends EU theory substantially, it is likely to provide many new insights in the fields of information, dynamic and strategic choices, insurance and portfolio behavior to which the latter has been applied. This is a vast agenda for future research.
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<table><caption>Table 1 - Minimum level of experience N₀ s.t. P_N ≥ .95 u(0) = 1</caption><thead><tr><th>p =</th><th>.01</th><th>.20</th><th>.50</th><th>.80</th><th>.99</th></tr></thead><tbody><tr><th>s =</th><td></td><td></td><td></td><td></td><td></td></tr><tr><th>2</th><td>na</td><td>na</td><td>25</td><td>98</td><td>2411</td></tr><tr><th>4</th><td>na</td><td>na</td><td>68</td><td>271</td><td>6698</td></tr><tr><th>10</th><td>na</td><td>82</td><td>328</td><td>1310</td><td>32416</td></tr><tr><th>50</th><td>71</td><td>1760</td><td>7039</td><td>28154</td><td>696799</td></tr><tr><th>100</th><td>279</td><td>6901</td><td>27605</td><td>110417</td><td>2732812</td></tr></tbody></table>

na : not available. The binomial distribution of r cannot be approximated by a normal distribution when convergence is obtained for small values of N.

Tables 2 - Values of P_N for various levels of salience

<table><caption>Table 2.1 : p = 0.01</caption><thead><tr><th>N =</th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>...</th><th>50</th><th>...</th><th>100</th><th>...</th><th>1000</th></tr></thead><tbody><tr><th>s =</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>2</th><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>...</td><td>1</td><td>...</td><td>1</td><td>...</td><td>1</td></tr><tr><th>4</th><td>1</td><td>.990</td><td>.980</td><td>.970</td><td>.961</td><td>.999</td><td>.999</td><td>.998</td><td>...</td><td>1</td><td>...</td><td>1</td><td>...</td><td>1</td></tr><tr><th>10</th><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>.941</td><td>.932</td><td>...</td><td>1</td><td>...</td><td>1</td><td>...</td><td>1</td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>50</th><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>...</td><td>.605</td><td>...</td><td>.921</td><td>...</td><td>1</td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>100</th><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>...</td><td>.605</td><td>...</td><td>.736</td><td>...</td><td>.993</td></tr></tbody></table>

<table><caption>Table 2.2: p = 0.2</caption><thead><tr><th>N =</th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>...</th><th>50</th><th>...</th><th>100</th><th>...</th><th>1000</th></tr></thead><tbody><tr><th>s =</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>2</th><td>1</td><td>1</td><td>.960</td><td>.896</td><td>.973</td><td>.942</td><td>.983</td><td>.967</td><td>...</td><td>1</td><td>...</td><td>1</td><td>...</td><td>1</td></tr><tr><th>4</th><td>1</td><td>.800</td><td>.640</td><td>.896</td><td>.819</td><td>.737</td><td>.901</td><td>.852</td><td>...</td><td>.997</td><td>...</td><td>1</td><td>...</td><td>1</td></tr><tr><th>10</th><td>0</td><td>0</td><td>.64</td><td>.512</td><td>.410</td><td>.328</td><td>.655</td><td>.577</td><td>...</td><td>.889</td><td>...</td><td>.944</td><td>...</td><td>1</td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>50</th><td>0</td><td>0</td><td>0</td><td>0</td><td>.410</td><td>.328</td><td>.262</td><td>.210</td><td>...</td><td>.584</td><td>...</td><td>.559</td><td>...</td><td>.874</td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>100</th><td>0</td><td>0</td><td>0</td><td>0</td><td>.410</td><td>.328</td><td>.262</td><td>.210</td><td>...</td><td>.444</td><td>...</td><td>.559</td><td>...</td><td>.725</td></tr></tbody></table>
<table><caption>Table 2.3: p = 0.5</caption><thead><tr><th>N =</th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>...</th><th>50</th><th>...</th><th>100</th><th>...</th><th>1000</th></tr></thead><tbody><tr><th>s =</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>2</th><td>1</td><td>1</td><td>.750</td><td>.875</td><td>.938</td><td>.813</td><td>.891</td><td>.938</td><td>...</td><td>.992</td><td>...</td><td>1</td><td>...</td><td>1</td></tr><tr><th>4</th><td>1</td><td>.500</td><td>.750</td><td>.500</td><td>.688</td><td>.813</td><td>.656</td><td>.773</td><td>...</td><td>.941</td><td>...</td><td>.982</td><td>...</td><td>1</td></tr><tr><th>10</th><td>0</td><td>.500</td><td>.250</td><td>.500</td><td>.313</td><td>.500</td><td>.656</td><td>.500</td><td>...</td><td>.760</td><td>...</td><td>.816</td><td>...</td><td>.998</td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>50</th><td>0</td><td>.500</td><td>.250</td><td>.500</td><td>.313</td><td>.500</td><td>.344</td><td>.500</td><td>...</td><td>.556</td><td>...</td><td>.540</td><td>...</td><td>.726</td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>100</th><td>0</td><td>.500</td><td>.250</td><td>.500</td><td>.313</td><td>.500</td><td>.344</td><td>.500</td><td>...</td><td>.444</td><td>...</td><td>.540</td><td>...</td><td>.612</td></tr></tbody></table>

<table><caption>Table 2.4: p = 0.8</caption><thead><tr><th>N =</th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>...</th><th>50</th><th>...</th><th>100</th><th>...</th><th>1000</th></tr></thead><tbody><tr><th>s =</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>2</th><td>1</td><td>1</td><td>.360</td><td>.488</td><td>.590</td><td>.672</td><td>.738</td><td>.790</td><td>...</td><td>.897</td><td>...</td><td>.953</td><td>...</td><td>1</td></tr><tr><th>4</th><td>1</td><td>.200</td><td>.360</td><td>.488</td><td>.590</td><td>.672</td><td>.738</td><td>.423</td><td>...</td><td>.810</td><td>...</td><td>.871</td><td>...</td><td>.999</td></tr><tr><th>10</th><td>0</td><td>.200</td><td>.360</td><td>.488</td><td>.590</td><td>.263</td><td>.345</td><td>.423</td><td>...</td><td>.556</td><td>...</td><td>.638</td><td>...</td><td>.930</td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>50</th><td>0</td><td>.200</td><td>.360</td><td>.488</td><td>.590</td><td>.263</td><td>.345</td><td>.423</td><td>...</td><td>.556</td><td>...</td><td>.540</td><td>...</td><td>.606</td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>100</th><td>0</td><td>.200</td><td>.360</td><td>.488</td><td>.590</td><td>.263</td><td>.345</td><td>.423</td><td>...</td><td>.416</td><td>...</td><td>.540</td><td>...</td><td>.544</td></tr></tbody></table>

<table><caption>Table 2.5: p = 0.99</caption><thead><tr><th>N =</th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>...</th><th>50</th><th>...</th><th>100</th><th>101</th><th>...</th><th>1000</th></tr></thead><tbody><tr><th>s =</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>2</th><td>1</td><td>1</td><td>.0199</td><td>.0297</td><td>.0394</td><td>.0490</td><td>.0585</td><td>.0679</td><td>...</td><td>.395</td><td>...</td><td>.634</td><td>.638</td><td>...</td><td>.871</td></tr><tr><th>4</th><td>1</td><td>.0100</td><td>.0199</td><td>.0297</td><td>.0394</td><td>.0490</td><td>.0585</td><td>.0679</td><td>...</td><td>.395</td><td>...</td><td>.634</td><td>.638</td><td>...</td><td>.781</td></tr><tr><th>10</th><td>0</td><td>.0100</td><td>.0199</td><td>.0297</td><td>.0394</td><td>.0490</td><td>.0585</td><td>.0679</td><td>...</td><td>.395</td><td>...</td><td>.634</td><td>.638</td><td>...</td><td>.543</td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>50</th><td>0</td><td>.0100</td><td>.0199</td><td>.0297</td><td>.0394</td><td>.0490</td><td>.0585</td><td>.0679</td><td>...</td><td>.395</td><td>...</td><td>.634</td><td>.638</td><td>...</td><td>.543</td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>.</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>100</th><td>0</td><td>.0100</td><td>.0199</td><td>.0297</td><td>.0394</td><td>.0490</td><td>.0585</td><td>.0679</td><td>...</td><td>.395</td><td>...</td><td>.634</td><td>.268</td><td>...</td><td>.543</td></tr></tbody></table>
<table><caption>**Table 3:**</caption><thead><tr><th>R'<sub>Allais</sub>:</th><th>Gain</th><th>Probability</th></tr></thead><tbody><tr><td></td><td>y<sub>1</sub></td><td>1 - p - q</td></tr><tr><td></td><td>y<sub>3</sub></td><td>p + q</td></tr></tbody></table><table><thead><tr><th>S'<sub>Allais</sub>:</th><th>Gain</th><th>Probability</th></tr></thead><tbody><tr><td></td><td>y<sub>2</sub></td><td>1 - q</td></tr><tr><td></td><td>y<sub>2</sub></td><td>q</td></tr></tbody></table>

<table><caption>**Table 4:**</caption><tbody><tr><td>States of the world</td><td>1</td><td>2</td><td>3</td></tr><tr><td>Probability</td><td>1 - p - q</td><td>q</td><td>p</td></tr><tr><td>R'<sub>Savage</sub>:</td><td>y<sub>1</sub></td><td>y<sub>3</sub></td><td>y<sub>3</sub></td></tr><tr><td>S'<sub>Savage</sub>:</td><td>y<sub>2</sub></td><td>y<sub>2</sub></td><td>y<sub>2</sub></td></tr></tbody></table>

<table><caption>**Table 5:**</caption><tbody><tr><td>States of the world</td><td>1</td><td>2</td><td>3</td></tr><tr><td>Probability</td><td>r(1 - p)</td><td>rp</td><td>1 - r</td></tr><tr><td>R''<sub>Savage</sub>:</td><td>y<sub>1</sub></td><td>y<sub>3</sub></td><td>y<sub>3</sub></td></tr><tr><td>S''<sub>Savage</sub>:</td><td>y<sub>2</sub></td><td>y<sub>2</sub></td><td>y<sub>3</sub></td></tr></tbody></table>
$$s^* = 4, s = 10, p = 0.99$$

<figure><img src="image_3.png" /><figcaption>Figure 1. Probability that choice respect EU-preference after N i.i.d. gambles</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 2 - Modal choice of the sure outcome in the insurance problem
(small probability of loss)</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 3</figcaption></figure>

Preferences, choices and values of beneficial bets: relations of exclusion and the preference reversal phenomenon.

<figure><img src="image_6.png" /><figcaption>Figure 4</figcaption></figure>

The true and the “recovered” utility curves for probabilities 1/4, 1/2 and 3/4
# APPENDIX

## 1. Proof of proposition 2

By conventionally setting $EU(A_1^*) - EU(A_{11}^*) = 0$, we can rewrite $V(Q)$ in equation (8) as the value function for the posterior EU difference, with $\Delta PEU(A_1) = 0$. If $V(Q) = 0$, $A_1$ is chosen and maximizes $\Delta PEU(A_1)$; and if $V(Q) > 0$, the chosen action still maximizes the posterior EU difference. By using (6), this takes the form of equation (9).

Report $EU(A_k) = p_k EU(a_k) + (1-p_k)EU(b_k)$; and $EU(A_1) = p_k EU(a_{1k}) + (1-p_k)EU(b_{1k})$ into (9). Rearranging terms and dividing by $\frac{s^*}{1+s^*} > 0$, we get expression (10). The latter is an EU difference in which the objection states are globally overweighted by merely adding the degree of unconsciousness to the objection's weight. It may thus be called the objection-dependent EU difference. Although the sum of weights adds up to $1 + \frac{1}{s^*}$ in (10), it is easy to normalize this expression in order that the normalized weights add up to one.

Rewrite $\Delta ODEU(A_k) = EU(A_k) - EU(A_1) + \frac{1}{s^*}[EU(a_k) - EU(a_{1k})]$ with $\Delta ODEU(A_1) = 0$. Defining $DU(A_k)$ by (11), we get

$$\Delta ODEU(A_k) = DU(A_k) - EU(A_1)$$

Since $EU(A_1)$ is a common term, it is equivalent to maximize $\Delta ODEU$ and to maximize $DU$, with $DU(A_1) = EU(A_1)$. Therefore, $DU$ is a descriptive utility function conditional on the normative preference. ☐

## 2. Proof of proposition 4

Assume that the descriptive utility function $DU: A \to \mathbb{R}$ has a maximum and rewrite (11) as:

$$DU(A_k) = \left( p_k + \frac{1}{s^*} \right) EU(a_k) + (1-p_k) EU(b_k) - \frac{1}{s^*} EU(a_{1k})$$

If $\alpha$ is a non-empty subset of actions from $A_-$ sharing the same objection states, $p_k = p$ and $a_{1k} = a_1 \forall A_k \in \alpha$. Hence, $\forall A_k, A_\iota \in \alpha \text{ s.t. } k \neq 1$, $DU(A_k) \geq DU(A_\iota)$ iff $ODEU(A_k) \geq ODEU(A_\iota)$. Assume that a local optimum exists on $\alpha + \{A_1\}$. If it belongs to $\alpha$, it surely maximizes the ODEU. But it might as well be $A_1$. In order to identify whether the local optimum is $A_1$ or an alternative action in $\alpha$, $DU(A_k)$ is to be compared with $EU(A_1)$, for all $A_k \in \alpha$, since $DU(A_1) = EU(A_1)$. Taking the above expression for $DU(A_k)$ and writing $EU(A_1) = p EU(a_1)$
+ (1-p) EU(b₁), it is shown that DU(A_k) ≥ EU(A₁) iff ODEU(A_k) ≥ ODEU(A₁) by a simple rearrangement of terms. Hence, the optimal action on α + {A₁} maximizes ODEU on this subset. Thanks to the conditional transitivity of DU (defined in the text), the latter local optimum can be defined irrespective of the set of alternative actions ∉ α.

If there is a finite number t of sets of objection states, let α_r (r=1,...,t) be the non empty class of actions from A_ sharing the r^th objection to A_1. The α_r's form a partition of A_. By the conditional transitivity of DU, the global optimum on A must be one of the t local optima on α_r + {A_1} for r=(1,...,t), and it maximizes DU on this finite set by proposition 2.

### 3. Extension of results to repeated choices

1. The posterior objection's weight after the repetition of N choices from a set of i.i.d. non-dominated actions A_k (∈ A_) is given by equation (4) in the text:

$$p_k^* = \frac{s^*}{s^*+1+N} p_k + \frac{1+N}{s^*+1+N} \frac{1+r_k}{1+N}$$

$$1 - p_k^* = \frac{s^*}{s^*+1+N} (1 - p_k) + \frac{1+N}{s^*+1+N} \left(1 - \frac{1+r_k}{1+N}\right)$$

From these expressions, we derive:

$$\text{EU}(A_k^*) = \frac{s^*}{s^*+1+N} \text{EU}(A_k) + \frac{1+N}{s^*+1+N} \left[ \frac{1+r_k}{1+N} \text{EU}(a_k) + \left(1-\frac{1+r_k}{1+N}\right) \text{EU}(b_k) \right]$$

$$\text{EU}(A_{1k}^*) = \frac{s^*}{s^*+1+N} \text{EU}(A_1) + \frac{1+N}{s^*+1+N} \left[ \frac{1+r_k}{1+N} \text{EU}(a_{1k}) + \left(1-\frac{1+r_k}{1+N}\right) \text{EU}(b_{1k}) \right]$$

We eliminate EU(b_k) and EU(b_{1k}) from these two expressions by remembering that

$$\text{EU}(A_k) = p_k \text{EU}(a_k) + (1-p_k) \text{EU}(b_k)$$

$$\text{and: } \text{EU}(A_1) = p_k \text{EU}(a_{1k}) + (1-p_k) \text{EU}(b_{1k})$$

and get after some calculations:

$$\text{EU}(A_k^*) - \text{EU}(A_{1k}^*) = \frac{1}{(1-p)(s^*+1+N)} \left\{ \begin{aligned} & (1+r_k - (1+N)p) (\text{EU}(a_k) - \text{EU}(a_{1k})) \\ & - (s^*(1-p) + N - r_k) (\text{EU}(A_1) - \text{EU}(A_k)) \end{aligned} \right\} \quad (3.1)$$

Let V_N(Q_k) designate the informational value of objection Q_k after N choices. By (7) and the definition of the objection's salience s_k, we get if EU(A_1) ≠ EU(A_k):

$$V_N(Q_k) = \frac{1}{(1-p)(s^*+1+N)} \max(0, (\text{EU}(A_1) - \text{EU}(A_k))[(1+r_k - (1+N)p_k)s_k - (s^*(1-p_k) + N - r_k)]$$
Since EU(A_1) - EU(A_k) > 0,
$$ V_N(Q_k) = \begin{cases} 0 & \text{if } \frac{(1+r_k - (1+N)p_k) s_k - (N-r_k)}{1-p_k} \le s^* \\ > 0 & \text{if } \frac{(1+r_k - (1+N)p_k) s_k - (N-r_k)}{1-p_k} > s^* \end{cases} \quad (3.2) $$

If $EU(A_k) = EU(A_1)$, $V_N(Q_k)$ is simply derived from (3.1) and (7), and

$$ V_N(Q_k) = \begin{cases} 0 & \text{if } 1+r_k - (1+N) p_k \le 0 \\ > 0 & \text{if } 1+r_k - (1+N) p_k > 0 \end{cases} \quad (3.3) $$

Since (3.3) entails (3.2) when salience tends towards $+\infty$, (3.2) may be used as a general formulation.

An alternative expression of (3.2) is:

$$ V_N(Q_k) = \begin{cases} 0 & \text{if } s_k + (r_k - Np_k) \cdot \frac{1+s_k}{1-p_k} \le s^*+N \\ > 0 & \text{if } s_k + (r_k - Np_k) \cdot \frac{1+s_k}{1-p_k} > s^*+N \end{cases} \quad (3.4) $$

2. Omitting the constant (for a given prior) term in $EU(A_1)$ from (3.1), we get the descriptive utility function:

$$ DU(A_k) = \begin{cases} EU(A_1) & , \text{ if } k=1 \\ EU(A_k) + \frac{1+r_k - (1+N)p_k}{s^*(1-p_k) + N - r_k} (EU(a_k) - EU(a_{1k})) & , \text{ if } k \neq 1 \end{cases} \quad (3.5) $$

(3.5) reduces to (11) when $N=0$ because $r_k$ must then equal zero as well. It is worth noticing that the coefficient of $EU(a_k)$ - $EU(a_{1k})$ in (3.5) may be negative when $N > 0$, and this occurs when $\bar{p}_k < p_k$. However, the objection-dependent EU must have positive weights since:

$$ ODEU(A_k) = \frac{(1-p_k)(1+r_k+p_k s^*)}{s^*(1-p_k) + N - r_k} EU(a_k) + (1-p_k) EU(b_k) \quad (3.6) $$

for $A_k \in \alpha + \{A_1\}$.

The coefficient of $EU(a_k)$ is always positive, but it is smaller than $p_k$ when $\bar{p}_k < p_k$. It can be written as $p_k + \frac{1+r_k - (1+N)p_k}{s^*(1-p_k) + N - r_k}$.

3. The following alternative expressions of (3.2) and (3.4) are also useful:
$$ V_N(Q_k) = \begin{cases} 0 & \text{if } r_k \le \frac{s^*+N-s_k}{1+s_k} + p_k \cdot \frac{(1+N)s_k-s^*}{1+s_k} \\ > 0 & \text{if } r_k > \frac{s^*+N-s_k}{1+s_k} + p_k \cdot \frac{(1+N)s_k-s^*}{1+s_k} \end{cases} \quad (3.7) $$

$$ V_N(Q_k) = \begin{cases} 0 & \text{if } r_k \le N - (1-p_k) \frac{(1+N)s_k - s^*}{1+s_k} \\ > 0 & \text{if } r_k > N - (1-p_k) \frac{(1+N)s_k - s^*}{1+s_k} \end{cases} \quad (3.8) $$

If $p_k$ is small enough, (3.7) shows that the right member of the inequality is negative when $s_k > s^*+N$ so that $V_N > 0$. When $s_k \le s^*+N$, $B(0) = (1-p_k)^N \approx 1$ for moderate values of N, so that $\text{Prob}\{V_N = 0\} \approx 1$ for such values.

(3.8) shows that $V_N = 0$ if $(1+N) s_k \le s^*$ for any value of $p_k$.

If $p_k$ is close to one and $(1+N) s_k > s^*$, (3.8) shows that the right member of the inequality is close to N, and insensitive to the variability of $s_k$ and $s^*$ unless the latter become very large.

4. I finally examine whether it is possible to have $\text{Prob}\{V_N = 0\} > 0$, $\forall N \ge 1$ and $\forall s, s^* > 0$. Starting from (3.4) for instance, this requires that

$$ N(1+p_k s_k)+(1-p_k)(s^*-s_k) \ge 0 \quad \forall N \ge 1 $$

Hence, the same inequality must hold for $N = 1$

$$ 1+p_k s_k+(1-p_k)(s^*-s_k) \ge 0, $$

$$ \text{or} \qquad s^* \ge \frac{s_k(1-2p_k)-1}{1-p_k}, $$

which is always true iff $p_k \ge 1/2$. Conversely, there always exist a finite level of cognitive dissonance and consciousness at which experienced individuals will not respect their EU-preference when the objection's weight is less than ½.

**4. Proof of equation (37)**

It is assumed here that $p$ follows a Beta distribution with parameters $a, b > 0$. Hence, the density function is

$$ \varphi(p) = \frac{p^{a-1}(1-p)^{b-1}}{B(a,b)}, \quad \text{with} \quad B(a,b) = \int_0^1 p^{a-1}(1-p)^{b-1} dp = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} \quad (4.1) $$

We wish to calculate for this distribution
$$
\begin{align*}
E(p/p > w_s) &= \frac{1}{1 - I_{w_s}(a,b)} \int_{w_s}^{1} p \varphi(p) dp \tag{4.2} \\
&= \frac{1}{1 - I_{w_s}(a,b)} \int_{w_s}^{1} \frac{p^a (1-p)^{b-1}}{B(a,b)} dp \\
&= \frac{B(a+1,b) (1 - I_{w_s}(a+1,b))}{B(a,b) (1 - I_{w_s}(a,b))} \\
&= \frac{1 - I_{w_s}(a+1,b)}{1 - I_{w_s}(a,b)}
\end{align*}
$$

Using the recurrence relation

$$
I_{w_s}(a+1,b) = I_{w_s}(a,b) - \frac{\Gamma(a+b)}{\Gamma(a+1)\Gamma(b)} w_s^a (1-w_s)^b,
$$

and reporting it into (4.2), we get

$$
\begin{align}
E(p/p > w_s) &= \bar{p} + \frac{\bar{p} w_s^a (1-w_s)^b}{a B(a,b) (1 - I_{w_s}(a,b))} \tag{4.3} \\
&= \bar{p} + \frac{w_s (1-w_s)}{a+b} h(w_s/a, b) \nonumber
\end{align}
$$

where $h(w_s / a, b) = \varphi(w_s / a, b) / (1 - I_{w_s}(a, b))$ is the hazard rate of Beta $(a, b)$ distribution. $\square$

5. Proof that the preference reversal phenomenon cannot occur when the riskier
beneficial bet is chosen

Assume here that $B_1$ is chosen against $B_2$ but $B_2$ would be sold at a higher price than $B_1$.
Thus we should have: $ODEU(B_1) > ODEU(B_2)$ and $v(B_2) > v(B_1)$. But this is impossible,
whether $EU(B_1) \ge EU(B_2)$ or $EU(B_1) < EU(B_2)$. In the former case, $p_1 > \left(p_2 + \frac{1}{s^*}\right)x$ and
$p_1 + \frac{1}{s^*} < \left(p_2 + \frac{1}{s^*}\right)x$ should be observed together, which is impossible. In the latter case, both
$p_1 + \frac{1}{s^*} > \left(p_2 + \frac{1}{s^*}\right)x$ and $p_1 + \frac{1}{s^*} < \left(p_2 + \frac{1}{s^*}\right)x$ should obtain at the same time, which is also
contradictory. $\square$
Notes

* I am grateful to Serge Blondel with whom I had many discussions over years and to Fabrice Etilé for assisting me in the computations. I received helpful comments on various drafts from Alban Bouvier, Marc-Arthur Diaye, Marc Le Ménestrel, Bertrand Munier, Pierre Romelaer, Bernard Sinclair-Desgagné, and participants at the Journées de Microéconomie Appliquée, European Econometric Society meeting, European Public Choice Society meeting, FURVIII meeting, and at seminars in the Ecole Normale Supérieure de Cachan, University of Chicago, Humboldt University, La Sapienza Roma, Université de Montréal, Universités de Paris I, Paris VI and Paris IX. Financial support from the Commissariat Général du Plan, Fédération Française des Sociétés d'Assurances, and Groupement d'Intérêt Scientifique "Sciences de la Cognition" is gratefully acknowledged.

$$ 
\begin{aligned}
U(-y) &\equiv 0 \approx x - y U'(0) \left[ 1 + \frac{y}{2} \cdot \text{ARA} \right], \\
U(y) &\equiv 1 \approx x + y U'(0) \left[ 1 - \frac{y}{2} \cdot \text{ARA} \right]
\end{aligned}
$$

where ARA is the absolute risk aversion coefficient - U''(0)/U'(0). From these two equations, we eliminate

$U'(0) = \frac{1}{2y}$ and get: $x \approx \frac{1}{2} + \frac{y}{4} \text{ARA}$. Thus, assuming ARA > 0: $s_s = \frac{1-x}{x-1/2}$ behaves like $\frac{2}{y \cdot \text{ARA}}$ and goes to infinity if $y \to 0$.

<footnote n="1">¹ Shafir et al. (1993: 34) have made a similar remark by suggesting that "the axioms of rational choice act as compelling arguments, or reasons, for making decision when their applicability has been detected, not as universal laws that constrain people's choices".</footnote>

<footnote n="2">² The states pointing to an objection are well determined if lotteries are not ambiguous. The analysis can be extended to ambiguity but, for brevity, this will not be done here. However, see subsection VIII.2.</footnote>

<footnote n="3">³ It is assumed that EU-indifferent actions are not equivalent. When two actions or more are equivalent, it is safe to treat them as identical and keep just one for further comparison of alternatives (see proposition 1).</footnote>

<footnote n="4">⁴ Notice that the expected opportunity loss, which measured prior doubt, does not capture posterior doubt anymore.</footnote>

<footnote n="5">⁵ In Allais's (1953) experiments, for instance, huge differences in prizes result in fairly small differences in EU (see, e.g. problem IP3 in section 6 of the present paper).</footnote>

<footnote n="6">⁶ The psychologists' view can be summarized by a sentence drawn from the classical work of Nisbett and Ross (1980: 8): "We argue that people effectively assign inferential weight to physical and social data in proportion to the data's salience and vividness. Information is heeded, processed, stored, and retrieved in proportion to its sensory, cognitive, and affective salience".</footnote>

<footnote n="7">⁷ The analogy with regret theory is far less clear when choices can be repeated because the coefficient of regret may then become negative (see eq. (3.5) in the appendix).</footnote>

<footnote n="8">⁸ In the case of EU-indifference, salience becomes infinite and the product [1+r-(1+N)p]s in equation (16) is an indeterminate form when $\bar{p} = p$. Direct calculation of value shows that the latter is positive if $\bar{p} = p$, and equals zero if $\bar{p} < p$.</footnote>

<footnote n="9">⁹ The objection to the purchase of insurance is that, without insurance, one can save an insurance premium. Even if the latter is not very substantial, the objection's weight gets so large, when the probability of disaster is very small, that imperfectly conscious risk-averters might durably fail to purchase insurance as illustrated by figure 1.</footnote>

<footnote n="10">¹⁰ I am not saying that the cardinal utility function is actually concave (or convex) everywhere. If utility were bounded from above and from below and if final wealth could take any positive or negative value, there should be at least one change of curvature. Consequently, an S-shaped utility function is a plausible assumption. But the inflection point has no reason to lie at some reference wealth, and it may even be out of reach if there are limited liability rules.</footnote>

<footnote n="11">¹¹ The objection to the risky action in case (i) is that the worst outcome may occur with probability p; and the objection to the sure outcome in case (ii) is that the best outcome may occur with probability (1-p-q).</footnote>

<footnote n="12">¹² When $x = x_0$, i.e. R and S are EU-indifferent, S(R) is chosen with a probability of one-half.</footnote>

<footnote n="13">¹³ To show this formally, we define: U(-y) = 0, U(y) = 1, U(0) = x; and develop U(-y) and U(y) in Taylor series around 0 at the second order:</footnote>

<footnote n="14">¹⁴ It should be noticed that true actions might be presented in a prospect-frame (e.g. Tversky and Kahneman 1992: 303-305) and, alternatively, independent prospects might be framed as actions (e.g. Loomes and Sudgen 1982).</footnote>
$$v(C(p,r)) < v(pr); \text{ and } v(D(p,r)) > v(1-r+rp).$$

<footnote n="15">¹⁵ For a lottery $B = (y_1, p_1; y_2, p_2; \dots; y_n, p_n)$ with $n \ge 3$ discrete outcomes, the main additional insight given by the theory would consist in the endogenous definition of “winning states” in the context of gambling decisions, or “no-loss states” in the context of insurance decisions. These have obvious definitions in the case of bets.</footnote>

<footnote n="16">¹⁶ Lévy-Garboua and Montmarquette (1996) provide an interpretation of the endowment effect and other observed anomalies of choice in a “seemingly riskless” environment.</footnote>

<footnote n="17">¹⁷ The variance of a Beta (a,b) distribution is: $\text{Var} = \frac{ab}{(a+b)^2 (a+b+1)}$. Since $a = \bar{p}(a+b)$ and</footnote>

<footnote>$b = (1 - \bar{p})(a + b)$, we see that $a + b + 1 = \frac{\bar{p}(1 - \bar{p})}{\text{Var}}$. Precision, i.e. $\frac{1}{\text{Var}}$, is thus proportional to $a+b+1$, for a given value of $\bar{p}$.</footnote>

<footnote n="18">¹⁸ In many experiments, outcomes are determined by dice or roulette wheels which are simple ways of setting common states of the world.</footnote>

<footnote n="19">¹⁹ Machina (1987: figure 6a) defines the recovered utility functions differently. He considers three types of bets: $B(p)=(1,p)$; $C(p,r)=(v(p),r)$; $D(p,r)=(1, 1-r)$; $v(p),r$ with $0 < p < 1$ and $0 < r < 1$. Then he exploits the property that, under the EU model: $v(C(p,r))=v(pr)$; and $v(D(p,r))=v(1-r+rp)$.</footnote>

<footnote>The utility evaluation effect consists in finding:</footnote>

<footnote>These two inequalities can be derived from CC theory under imperfect consciousness. They both reflect the overweighting of high payoffs in lotteries' evaluation which is implied by equation (30). The proof is available upon request from the author.</footnote>

<footnote>Of course, all definitions of the recovered utility functions are equivalent under EU theory. For simplicity, I only used bets of the first type $B(p)$ in the above discussion, but that does not alter the main conclusion.</footnote>

<footnote n="20">²⁰ CC theory is also consistent with a different type of probabilistic insurance recently introduced by Wakker, Thaler and Tversky (1997). For brevity, the latter problem will not be treated here.</footnote>

<footnote n="21">²¹ Akerlof and Dickens (1982) examine the economic consequences of cognitive dissonance. However, they do not offer a rational explanation for this phenomenon.</footnote>

<footnote n="22">²² Whether Monty Hall opens the second or the third door is immaterial at this stage.</footnote>