Self-Deception and Choice∗

Igor Kopylov Jawwad Noor

March 17, 2010

Abstract

The temptation literature has provided models where desires at-tain satisfaction by commanding the agent’s attention. We considera model of self-deception, where desires command the agent’s reason-ing, leading her to rationalize and justify actions that eventually leadinto temptation. Formally, we write axioms for a three-period exten-sion of Gul and Pesendorfer’s (2001) framework and obtain a specialfunctional form for the temptation utility at the interim stage. Therepresentation portrays an agent who is tempted (i) to relax her nor-mative attitude towards future indulgence and (ii) to turn a blind eyeto any possibility of temptation altogether. Welfare implications ofself-deception are discussed.

1 Introduction

“This self-deceit, this fatal weakness of mankind, is the source of half thedisorders of human life.”

∗Kopylov is at the Dept of Economics, University of California, Irvine; E-mail: ikopy-lov@uci.edu. Noor is at the Dept of Economics, Boston University; Email: jnoor@bu.edu.We thank Larry Epstein for very useful discussions, and seminar participants at BostonUniversity, U of Munich, Bayreuth U., U of Zurich, NYU, Michigan and Harvard-MITand participants at the Canadian Economic Theory Conference 2009, Econometric Soci-ety Summer Meeting 2009 and RUD Conference 2009 for helpful comments. The usualdisclaimer applies.

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– Adam Smith, The Theory of Moral Sentiments.

The literature on temptation arising from Gul and Pesendorfer [13] (hence-forth GP) describe agents who are hit with desires that command their atten-tion and clamor for immediate satisfaction. Refer to this as overt temptation.In this paper we propose that temptation has another side. Desires may affectone’s reasoning when by doing so they can achieve satisfaction. For example,a person can be motivated to follow suspect reasoning like “well, one cookiewon’t harm me” (Baumeister, Heatherton, and Tice [3, p. 139]) to get off arestraining diet. Here the tempting desires are not overt, but rather impactthe agent in a more sophisticated and covert way: they command the agent’sreasoning rather than her attention. The agent’s thinking is motivated andself-deceived, leading her to find ways of rationalizing actions she hithertobelieved she should not take.

The psychology literature highlights the importance and prevalence ofsuch motivated reasoning for behavior. Based on an extensive survey ofpsychology research, Kunda [18, pg 482] proposes that people who are “mo-tivated to arrive at a particular conclusion attempt to be rational and toconstruct a justification of their desired conclusion that would persuade adispassionate observer. They draw the desired conclusion only if they canmuster up the evidence necessary to support it” (italics added). Thus it isplausible that tempting desires can distort an agent’s reasoning – withoutnecessarily ever becoming overt – and thus also distort her consumption.

In this paper, we model self-deception in a three period extension of GP’sframework.1 Going in reverse chronological order, the choices faced by theagent are as follows. In the ‘ex post’ period, the agent faces overt temptationas she makes a choice from a menu – say she is in a bar and chooses howmuch to drink. In the ‘interim’ period, she selects a menu – she selects whatbar or restaurant to go to. Self-deception occurs here, as she constructsrationalizations for choosing a menu that contains temptation. In the ‘exante’ period, she is in a cold emotional state and her thinking is determinedby her ‘normative perspective’, that is, her view of how she should behavein subsequent periods. This underlies her preferences over menu of menus– she evaluates neighborhoods (containing bars and restaurants) keeping inmind her tendency to rationalize her way into bars and drink heavily. This

1On a technical level our results exploit Kopylov’s [15, 17] extensions of GP’s model todynamic settings with more than two periods and to representations with finitely manyadditive components.

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ex ante preference is our primitive.The following example illustrates the behavioral meaning of self-deception

in our model. An agent can either drink moderately xm or drink heavily xh

at social events. Let a = xm and b = xm, xh. For instance, the menua may be obtained on Friday night at a restaurant dinner with friends, andthe menu b at a bar. The agent has a normative preference to drink less, butshe cannot resist the temptation to drink heavy at the bar. This would leadto her to commit to a rather than to b ex ante (say a few days in advance)because the former menu leads to less drinking. At the same time, if she waitsuntil the interim period (say 4pm on Friday), then she may be prompted bytempting desires to tell herself things like “I have the will-power not to drinkmoderately if I go to the bar” or even “I’m going to live only once, so I shouldallow myself to have some fun”. Anticipating such motivated reasoning, shemay actually prefer to commit to a ex ante rather than to wait and choosebetween a and b at the interim stage, that is, she may exhibit a ≻ a, b.By contrast, imagine that this person can resist the temptation to drinkheavy at the bar. Then self-deception has no strategic value and no reasonto exist. Hence, the ranking a ∼ a, b seems plausible.

This behavior captures a basic aspect of self-deception, namely, that it isassociated with motivated reasoning. However, the philosophy and psychol-ogy literature also associates with self-deception the presence of an internaltension – these are recurring and nagging doubts that arise from maintaininga belief in a proposition in the presence of some awareness that the proposi-tion is false, and which may even give rise to a divergence between belief andaction. In our model, this internal tension aspect of self-deception is behav-iorally expressed in the form of restraint in action. The choice of a menu inthe interim period may not be optimal given the agent’s self-deceived views.For instance, the hero in the above example may choose a in the menu a, beven if her self-deceptive beliefs are that drinking is not a vice on Fridaynight.2

With the possibility of self-deception thus permitted, we write two axiomsthat solidify the interpretation of the model as one of self-deception andseparate it from other stories. The axioms express the idea that self-deceptionis motivated and restricted : (i) it arises only if it ultimately leads to a greater

2This is the analog of ‘self-control’ in GP’s model, whereby the agent’s choice deviatesfrom what temptation dictates. Note also that motivated reasoning in the interim stageis the analog of ‘temptation’ in GP’s model.

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satisfaction of ex post desires, and (ii) not every menu can be rationalized,since self-deception can exist only to the extent that a facade of rationalitycan be sustained.

The corresponding representation for preference is obtained in Theorem2. This representation portrays an agent who relaxes her ex ante normativeperspective at the interim stage in the direction of anticipated ex post de-sires, and possibly also turn a blind eye to future temptations. Note thatrationalizations arise subjectively in this paper, as part of the representationfor our axioms. An unexpected corollary of the Theorem is that GP’s keyaxioms – Independence and Set-Betweenness – rule out some interesting ra-tionalizations that are in spirit consistent with the two additional axioms wespecify. For instance, the agent does not distort her prediction about herex post choices in a way that make her look more virtuous. We concludethat while the menus framework can indeed by used to model self-deception,standard axioms in the temptation literature need to be relaxed in order toobtain a rich model of self-deception.

The remainder of this paper proceeds as follows. The introduction con-cludes with a mention of related literature. Section 2 describes the phe-nomenon of self-deception and differentiates it from other closely relatedphenomena. Section 3 introduces the primitives of the model and presentsa benchmark case. Section 4 presents our model of self-deception. Section 5extends this model to permit a virtuous self-perception and derives some ofits implications for welfare policy. Section 6 concludes. Proofs are relegatedto appendices.

1.1 Related Literature

To our knowledge, the idea of temptation-driven self-deception that we con-sider has not been studied in economics. The behavioral economics litera-ture discusses positive illusions (Benabou and Tirole [4]) and wishful think-ing (Brunnermeier and Parker [6]). The internal tension that defines self-deception is absent in these models – the agent’s choices are always optimalgiven her beliefs. The decision-theoretic literature on temptation has mod-elled temptation only as arising in its naked, unstrategic form and beingrelevant mainly at the moment of choice. Cherepanov, Feddersen and San-droni [7] axiomatize a model of motivated reasoning – an agent maximizespreferences only over the set of alternatives that can be justified accordingto some reason. An interpretation of the model in terms of self-deception is

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limited by the fact that the choice domain (choice from menus) is not richenough to express the existence of any internal tension. Indeed, the authorsshow that their model either subsumes or is observationally equivalent toothers that involve stories orthogonal to motivated reasoning.

The idea that an agent may be tempted to change her view of world(specifically, beliefs over a state space) is considered in Epstein [10] and Ep-stein and Kopylov [11]. The temptation to change beliefs is unexplained, andin particular, its existence is not motivated in the sense of serving a strategicfunction. In our paper, a temptation to change the ex ante perspective (abouther propensity for self-control, etc) serves the purpose of, and owes its exis-tence to, the desire for tempting final consumption. There is no temptationby final consumption in Epstein [10] and Epstein and Kopylov [11].

The idea that temptation may influence the choice of menu is present inNoor [22, 23] and Noor and Ren [24], but the mode in which temptation actsthere is presumed to be direct and overt. Our paper maintains that an agentmay be tempted by menus, but it hypothesizes that the agent is only temptedby menus that she can justify choosing. Indeed, we find that the models oftempting menus in [22, 24] cannot be regarded as one of self-deception.

A version of our self-deception model also shows up in Noor [23]. Thereare two important differences, however. First, the model appears in [23]mainly as a means to axiomatically unify other temptation models in theliterature. In contrast, the current paper starts by behaviorally defining anecessary condition for self-deception, and it turns out that the representa-tion theorem delivers the same model. A novel interpretation of that modelis obtained here as a result. Second, our choice domain differs substantiallyfrom [23] and so do our axioms. Intuitively, our axioms are imposed on theagent’s perspective in a cold state – in a special period 0 – where she antici-pates all temptation (by self-deception or otherwise) but is not subject to ityet. In contrast, axioms in [23] are imposed directly on choice in a hot state,when the agent is subject to all kinds of temptation. A counterpart of ourmain axiom does not appear in [23].

2 Defining Self-Deception

Motivated reasoning is the process of arriving at a conclusion on the basis ofmotivationally biased information processing. Thus, an agent who desires tobelieve p may interpret, misinterpret, seek and recall evidence in a way that

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supports p, even if the stock of available evidence supports ¬p. For instance,consider an older boy who wants to believe that he is a natural leader butlacks the respect of his peers. He may find the company of younger teenagersmore congenial, and the consequent selective exposure to data leads him tosupport the hypothesis that he is indeed a natural leader.

When motivated reasoning takes place without the agent being aware ofit, we refer to it as simple motivated reasoning. Self-deception is a specialtype of motivated reasoning. Losonsky [20] writes:

[S]elf-deceivers who come to recognize that they were self-deceivers often report that although they had rejected the belief,say, that they were anorexic (or that their children were usingdrugs), “on another level I knew I was anorexic,” or, “I knewall along she was abusing drugs, but I refused to accept it.”...Self-deception involves some kind of recognition of the fact thatthe available evidence warrants the undesirable proposition morethan the desirable one.

That is, self-deception involves some awareness of the motivated nature ofone’s reasoning – Sartre [25] writes that ‘I must know the truth very exactlyin order to conceal it’. Philosophers differ on the nature of this prereflectiveawareness and much debate has originated from the idea that self-deceptionseemingly involves simultaneously believing in p while also believing in ¬p(Sartre [25], Fingarette [12]).3

The support for the existence of an underlying awareness comes fromthe observation that, unlike simple motivated reasoning, self-deception ischaracterized by an internal tension. Losonsky [20] continues:

This can be manifested in various ways. One way is in a recur-ring or nagging doubt that typically does not occur when subjectsfix their beliefs. Similarly, self-deceivers can find themselves re-peatedly and obsessively entertaining the undesirable proposition

3The paradoxes arising from the idea of co-existing contradictory beliefs has led morerecent philosophers to weaken the requirement of holding contradictory beliefs to inten-tionally bringing about a belief despite an initial recognition that the evidence may notwarrant this belief (Talbott [28], Bermudez [5]). The intentionality is understood to existat a prereflective level. Without necessarily denying the possibility of self-deception, crit-ics emphasize that most cases of self-deception are observationally equivalent to cases ofsimple motivated reasoning (Barnes [2], Mele [21]).

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and going over the same line of reasoning that supports the de-sirable proposition.

That is, internal tension is argued to manifests in such ways.4 More to thepoint, it has been argued that nagging doubt can express itself in actionsthat are restrained relative to beliefs. For instance, the boy who believesthat he is a natural leader may not act as such when he is in the presence ofhis peers, or the cancer patient who honestly feels that she will survive maynever be found to make serious long term plans.

Our interpretation of the philosophy and psychology literature suggeststhe following definition: self-deception is a state where an agent engages inmotivated reasoning while also suspecting that she is engaged in motivatedreasoning. That is, the agent forms a belief while also experiencing naggingdoubt about the soundness and validity of the reasons underlying the belief.The suspicion induces restraint in action, giving rise to choices that are notnecessarily optimal given the belief. Note also that the existence of suspicionpresumes that the agent’s motives are sufficiently masked behind a facadeof rationality – transparency will give rise to certainty about motivated rea-soning. To take an analogy with interpersonal deception: a deceiver cannotsucceed in his deception if the agent is aware of the deceiver’s motives, buthe can nevertheless succeed if his motives are sufficiently masked and theagent is merely suspicious about them.

We conclude by clarifying that our interest is in what may more accu-rately be referred to as temptation-driven self-deception. In the psychologyliterature, self-deception has also been discussed in the context of achievinglong term goals or the need to adjust and function in a social environment.This accommodates the case of positive illusions (Taylor [29], Taylor andBrown [30]) that take the form of self-aggrandizing self-perceptions, exagger-ated perception of control over surroundings and unrealistic optimism, whichare viewed as serving an important adaptive function. We also mention thatwishful thinking is related but distinct from self-deception because the lattersurvives in the ‘teeth of evidence’ (by explaining away or reinterpreting theevidence) while the former does not (Szabados [26]).

4Other expressions of internal tension discussed in the literature are opacity and indi-rection (Levy [19]), emotional resistance to evidence (Talbott [28]) and hypersensitivityto criticism, confrontation or opposing evidence (Gur and Sackeim [14]).

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3 Preliminaries

To aid exposition in subsequent sections, we first present a basic three-periodextension of GP’s model.

Let X = x, y, z, . . . be the set ∆(Z) of all Borel probability measureson a compact metric space Z of deterministic consumptions. Let d be theProhorov metric of the weak convergence topology on X. More generally, letX be the class of all Anscombe–Aumann acts f that map a finite state spaceΩ into ∆(Z), and let d be the corresponding product metric in X.5

Suppose that choices are made in three stages—ex ante, interim, and expost—and these choices determine the decision maker’s consumption in Xafter the ex post stage. Let M1 = a, b, c, . . . be the set of all interimmenus—non-empty compact subsets a ⊂ X. Interpret any menu a ∈ M1

as a course of action that, if taken at the interim stage, restricts the ex postchoice to the set a ⊂ X. Endow the space M1 with the Hausdorff metric µ1

and define mixtures

αa+ (1− α)b = αx+ (1− α)y : x ∈ a, y ∈ b

for all α ∈ [0, 1] and menus a, b ∈ M1.Similarly, let M0 = A,B,C, . . . be the set of all ex ante menus—non-

empty compact subsets A ⊂ M1. Interpret any menu A ∈ M0 as a course ofaction that, if taken ex ante, restricts the interim choice to the set A ⊂ M1.Endow the space M0 with the Hausdorff metric µ0 and define mixtures

αA+ (1− α)B = αa+ (1− α)b : a ∈ A, b ∈ B

for all α ∈ [0, 1] and menus A,B ∈ M0. Then both M0 and M1 are compact(see Theorem 3.71 in Aliprantis and Border [1]) and the mixture operationsin these spaces are continuous.

Let a binary relation ≽ on M0 be the decision maker’s weak preferenceover ex ante menus. Write the symmetric and asymmetric parts of thisrelation as ∼ and ≻ respectively. Note that our model does not take thedecision maker’s interim and ex post choices as primitive, but instead derivesher anticipation of these choices from her ex ante preference.

Adapt GP’s list of axioms for the preference ≽.

5Menus of lotteries are first used by Gul and Pesendorfer [13] and—for finite Z—byDekel, Lipman, and Rustichini [8]. Menus of acts are proposed by Epstein [10].

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Axiom 1 (Order). ≽ is complete and transitive.

Axiom 2 (Continuity). For all menus A ∈ M0, the sets B ∈ M0 : B ≽ Aand B ∈ M0 : B ≼ A are closed.

Axiom 3 (Independence). For all α ∈ [0, 1] and menus A,B,C ∈ M0,

A ≽ B ⇒ αA+ (1− α)C ≽ αB + (1− α)C.

Axiom 4 (Set-Betweenness). For all menus a, b ∈ M1 and A,B ∈ M0,

a ≽ b ⇒ a ≽ a ∪ b ≽ b, (1)

A ≽ B ⇒ A ≽ A ∪B ≽ B. (2)

Order and Continuity are standard conditions of rationality. To motivateIndependence, interpret any mixture αA+ (1− α)C as a lottery that yieldsthe menus A and C with probabilities α and 1−α respectively and is resolvedafter the ex post stage. In this interpretation, the decision maker’s interimchoice αa+ (1− α)c in αA+ (1− α)C and her ex post choice αx+ (1− α)yin αa + (1 − α)c determine her consumptions x ∈ a ∈ A and y ∈ c ∈ Ccontingent on the resolution of the lottery between the menus A and C. Ifthe timing of the resolution of this objective uncertainty is irrelevant forpreference, then the decision maker should be indifferent between the menuαA + (1 − α)C and a hypothetical lottery α A + (1 − α) C that yieldsthe menus A or C with probabilities α and 1−α respectively, but is resolvedimmediately after the ex ante stage. (Here the preference ≽ is extendedfrom the original domain M0 to lotteries over menus.) Then the standardseparability argument suggests that

A ≽ B ⇒ α A+ (1− α) C ≽ α B + (1− α) C

because the possibility of getting the menu C with probability 1− α shouldnot affect the decision maker’s comparison of A and B. Independence follows.

Set-Betweenness is imposed separately on the preference ≽ over the entireM0 and on the restriction of ≽ to singleton menus a that provide a strictcommitment to the menu a ∈ M1 at the interim stage, but may still requireself-control ex post when choice in a has to be made. It is assumed that thedecision maker’s ex ante evaluation of any such menu a is based on heranticipation of two factors:

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• the consumption xa ∈ a that she will choose if a is feasible ex post,

• the self-control that she will use to resist the strongest temptationya ∈ a in this menu.

This informal assumption suggests that for all menus a, b ∈ M1,

xb ∈ a, ya ∈ b ⇒ a ≽ b. (3)

Indeed, if xb ∈ a and ya ∈ b, then the decision maker should expect thatif she chooses xa from the menu a at the ex post stage, then she will (i)obtain the same consumption that she plans to choose from b and (ii) resistthe temptation ya, which belongs to b and hence, should not be harder toresist than the strongest temptation yb in b. Therefore, the ranking a ≽b is intuitive because the menu a offers a weakly better combination ofconsumption benefits and self-control costs than b does. Condition (3) implies(1).6 Analogously, condition (2) assumes that the decision maker shouldevaluate any menu A ∈ M0 based on her anticipated interim choice aA ∈A and the most tempting alternative bA ∈ A in this menu. Note that iftemptations are cumulative or uncertain (as in Dekel, Lipman, Rustichini[9]), then both parts of Set-Betweenness can be violated.

The following condition is used to obtain uniqueness in representationresults below.

Axiom 5 (Regularity). There are x, y, x′, y′ ∈ X such that

x ≻ x, y ≻ yx′ ≻ x′, y′ ≻ y′.

The two rankings in this axiom are intuitive if the agent plans to resistthe tempting consumption y in the menu x, y at the ex post stage, and toresist the tempting menu y′ in x′, y′ at the interim stage.

Say that a function u : X → R is linear if for all α ∈ [0, 1] and x, y ∈ X,

u(αx+ (1− α)y) = αu(x) + (1− α)u(y).

Let U be the set of all continuous linear functions u : X → R. Similarly,define linearity for functions on M1 and let U1 be the set of all continuouslinear functions V : M1 → R.

6To show this claim, take any menus a ≽ b. Let c = a ∪ b. Then xa, xb, ya, yb ∈ c.By (3), if xc ∈ a, then a ≽ c; if xc ∈ b, then b ≽ c. In either case, a ≽ c. By(3), if yc ∈ a, then c ≽ a; if yc ∈ b, then c ≽ b. In either case, c ≽ b.

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Theorem 1. The preference ≽ satisfies Axioms 1–4 if and only if ≽ isrepresented by a utility function U0 such that for all A ∈ M0 and a ∈ M1,

U0(A) = maxa∈A

[U(a)−max

b∈A(V (b)− V (a))

](4)

U(a) = maxx∈a

[u(x)−maxy∈a

(v(y)− v(x))], (5)

where u, v ∈ U and V ∈ U1.Moreover, if ≽ satisfies Regularity, then it has another representation

(4) with components u′, v′ ∈ U and V ′ ∈ U1 if and only if u′ = αu + βu,v′ = αv + βv, and V ′ = αV + βV for some α > 0 and βu, βv, βV ∈ R.

This theorem provides a joint characterization for GP’s utility represen-tations (4) and (5) over M0 and M1 respectively. The restriction of U toM1 can be interpreted as the decision maker’s ex ante normative perspec-tive on what menu should be chosen at the interim stage. Temptations thatimpact her at this stage are captured by V , and the nonnegative componentmaxb∈A(V (b)−V (a)) is interpreted as the self-control cost of choosing a fromA. Similarly, the function u can be interpreted as the ex ante normative per-spective on what should be chosen at the ex post stage, and the nonnegativeterm maxy∈a(v(y) − v(x)) as the mental cost of ex post self-control. Theseinterpretations suggest that in order to balance her normative perspectivewith the costs of self-control, the decision maker should plan to maximizeU + V and u+ v respectively at the interim and ex post stages.

Before turning to our models of distorted self-perception, note the bench-mark case with V = 0 when the decision maker does not expect to have anytemptations at the interim stage. In this case, she obeys strategic rationalityso that for all A,B ∈ M0

A ≽ B ⇒ A ∼ A ∪B.

Note that she may still anticipate costly temptations at the ex post stage, asshe may exhibit a preference for commitment a ≻ a ∪ b for some menusa, b ∈ M1.

4 A Model of Self-Deception

We model self-deception in the form of interim temptation in GP’s setup. Itshould be noted, however, that self-deception and temptation are different

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phenomena in general (Szabados [27]). Knowledge of a craving is typical incases of temptation, and indeed is presupposed in the notion of self-control.But self-deception cannot survive knowledge of self-deception, just as a de-ceiver cannot deceive an agent who knows the deceiver’s intention to deceive.Since GP’s model is tailored for overt temptation and as such involves a de-gree of sophistication and self-awareness, using their model for self-deceptiondemands cautious reinterpretation.7

Take a preference ≽ over M0 that satisfies Axioms 1-4. For the pur-poses of illustration, consider an agent who is tempted to drink excessivelywhen in a bar and who also tends to rationalize her way into bars. Here,alcohol consumption levels are final consumption (elements on X), bars andrestaurants are menus (elements of M1), and neighborhoods consisting ofbars/restaurants are menus of menus (elements of M0). As indicated in thetimeline below, temptation is experienced in the ex post period when decid-ing how much to drink at a bar/restaurant. We suppose that this temptationis overt, though the reader is free to interpret it as covert – our model willnot draw a distinction. Self-deception arises in the interim period, when theagent is deciding which bar/restaurant to go to. We describe this furthershortly. The agent is clear-headed in the ex ante period and recognizes herself-control problems and her tendency to deceive herself.8 In light of this,she formulates her normative perspective on what neighborhood to go to (re-flected in U0), what bar/restaurant she should go to (reflected in U) in anygiven neighborhood, and what she should consume (reflected in u) in anygiven bar/restaurant. In the ex ante period, she chooses a neighborhood inlight of her normative perspective and her anticipated behavior.

t=0•choose menu of menus A

↑purely normative

↑anticipates on basis of memory

————t=1•

choose menu a∈A↑

self-deception↑

selective recall, reinterpretation,

but nagging suspicion remains.

————t=2•

choose x∈a↑

temptation

We now describe the interim period in more detail. Having already chosen

7An empirical distinction between overt temptation and self-deception is also made inthe context of the Rationalizable Self-Deception axiom below.

8Empirical foundations for such a ‘special period 0’ perspective are provided in Noor[23] on the basis of the hypothesis that distant temptations have smaller impact on currentchoices than do immediate temptations. Thus, our ex ante stage may be interpreted as atime sufficiently distant from the interim stage.

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a neighborhood, she enters it in the interim period with an ex ante norma-tive perspective on what bar/restaurant to go to – she goes in thinking sheshould maximize U . But her desires subconsciously prompt her to reevaluateher normative perspective by constructing rationalizations. These rational-izations draw on selective recollection, interpretation and misinterpretationof her available ‘evidence’, and affect her normative standards, her predic-tions on how susceptible she is to temptation, how much self-control she has,etc.9 She not only rationalizes away her ex ante normative views and herpredictions about the ex post stage, but also her recognition of her tendencyto rationalize, such as by arguing that “at that time I was not appreciatingthe fact that ...”. Her new interim perspective is reflected in V . She findsher new perspective compelling and rational and wishes to maximize V , butat the same time, however, she has nagging doubts about her reasoning.These leads to restraints in action – her choices put some weight on her exante perspective, although this involves a psychological cost of underweight-ing her compelling interim reasoning. This story about the agent’s interimexperience is reflected in the functional form (5): her nagging doubts leadher to put weight on U , and thus her choice maximizes U + V while shesimultaneously feels compelled by her reasoning to simply maximize V .

Observe that in this story, ex ante sophistication does not destroy thepossibility of interim self-deception because the latter rationalizes the for-mer. Also, the internal tension underlying interim self-deception gives riseto features that are analogous to the notion of self-control in GP’s model:self-control means placing weight on the ex ante preference despite currentdesires, and in the current context this describes the self-restraint that isprompted by nagging doubts that characteristically accompany self-deception.

While Axioms 1-4 permit such a self-deception interpretation, they donot necessitate it. The agent may well be experiencing overt temptation inthe interim period. We proceed by imposing two axioms on ≽ that expressproperties that are germane to self-deception and rule out the overt tempta-tion story. Axioms 1-4 imply minimal structure on V , but our representationtheorem will uncover the implications of the new axioms for the structure ofV .

9Rationalizations are unmodelled, and will appear only as part of our representationtheorem.

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4.1 Axioms

The following interpretation of behavior are needed to interpret the axiomsand subsequent discussions. The reader should keep track of whether each isa statement about temptation or choice in the interim or ex post period.

• a ≻ a, b reveals an ex ante preference for avoiding menu b in theinterim stage. The interpretation is that the agent anticipates engaging inself-deceiving rationalizations for selecting b over a in the interim stage.

• a ≻ a ∪ b reveals a preference for facing menu a rather than a ∪ bin the ex post stage, which in turn suggests that something in b is a sourceof temptation in menu a ∪ b. Observe that interim choice is trivial in botha and a ∪ b, and so interim considerations (choice and self-deception)are irrelevant in the comparison of the two.

• a ∪ b ≻ b reveals that ex post choice from a ∪ b belongs to a. Tosee this, note that if the anticipated choice from a ∪ b lay in b, then therewould be no reason to strictly prefer committing to the larger menu a∪ b. Inthe GP model, the ranking a ∪ b ≻ b implies more: it implies that theex post choice from a and a ∪ b is the same.10

Our first axiom expresses the fact that self-deception is directed: it mustbe motivated by expected ex post consumption.

Axiom 6 (Motivated Self-Deception). For all a, b ∈ M1,

a ∪ b ≻ b ⇒ a ∼ a, a ∪ b.

Motivated Self-Deception (MSD for short) states that if the ex post choicefrom a and a ∪ b are identical, then the agent does not deceive herself intochoosing a ∪ b over a. When both menus a and a ∪ b lead to the same finalchoice, they also yield the same degree of satisfaction of desires. Conse-quently, there can simply be no motivation for the agent to deceive herselfinto choosing a∪b over a. Self-deception has no strategic value in such cases.

While MSD is a natural requirement for self-deception, it may also besatisfied by models of overt temptation where interim desires are sensitiveto ex post choice. (For instance, see Noor and Ren [24] where the agentis tempted by guiltless indulgence.) Our second axiom delineates cases ofself-deception that are behaviorally distinct from those of overt temptation.We illustrate with the example stated earlier. An agent can either not drink

10This is because ex post choice satisfies WARP in GP’s model.

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x0, drink moderately xm or drink heavily xh. Let a = xm, b = x0, xh,and a ∪ b = x0, xm, xh. For instance, the menu a may be obtained at aformal social event, the menu b at a bar where any drinking is heavy, anda ∪ b at a home party. The agent has a normative preference to drink less,but she cannot resist the temptation to drink heavy at the bar. Then shewould commit to a rather than to b ex ante because the former menu leadsto less drinking ex post, and by Set-Betweenness, she exhibits the rankinga ≽ a ∪ b ≽ b.

We claim that a temptation to go to the bar rather than stay at home,

a ∪ b ≻ a ∪ b, b,

should be generic in cases of overt temptation, especially when the sophisti-cation implicit in MSD is present: if the agent anticipates drinking heavily inthe bar and only moderately at home, then nothing keeps her from desiringthe bar in a story of overt temptation. Therefore in order to rule out anovert temptation story, it suffices to impose:

Axiom 7 (Rationalizable Self-Deception). For all menus a, b ∈ M1,

a ∪ b, b ≽ a ∪ b.

It remains to check, however, that there exist cases of self-deception thatare consistent with this axiom. Before presenting confirmatory examples,we note that Rationalizable Self-Deception (RSD for short) is very muchconsistent with the spirit of self-deception. RSD asserts that the agent neverdeceives herself into choosing a menu b over a larger menu a∪b.11 When a∪boffers greater temptation than b, such deception is not motivated to beginwith. But when, as in the case of the bar vs home, it offers as much temp-tation as b, then any rationalization for choosing b would have to overturnthe fact that a∪ b offers strictly more options without more temptation. Theaxiom rules out such rationalizations, and this is well justified if such ratio-nalizations are too transparent to generate the facade of rationality neededfor self-deception.

To see that RSD accommodates intuitive examples of self-deception, con-sider the following three possible rationalizations the agent can use to justifygoing to the bar.

11It is enough to state the axiom for the hypothesis a∪ b ≽ b, since if b ≻ a∪ b,then a ∪ b, b ≽ a ∪ b follows from Set-Betweenness.

15

• Overestimation of propensity for self-control: “If I go to the bar I knowI will be tempted to drink, but I have the will-power to abstain.”

• Underestimation of susceptibility to temptation: “I am not even goingto be tempted to drink in the bar”.

• Relaxation in normative standards: “I am only going to live once, so Ireally should allow myself to enjoy life a little.”.

Observe how the first two rationalizations exploit the fact that the barcontains the normatively superior alternative x0. Indeed, under the exagger-ated self-control or underestimated temptation, the bar leads to a norma-tively better outcome than attending the social event, where some drinkingis unavoidable. The third rationalization leads the agent to view the bar in apositive light, without necessarily invoking a distorted view of her self-controlability or susceptibility to temptation.

Each of these rationalizations is consistent with RSD because none ofthem can help the agent justify going to the bar over staying home. This isimmediately evident (given that the bar offers only a subset of the optionsavailable at home) except perhaps in the following case:12 Suppose the agenttells herself that she has substantial will-power, and that she will choose x0 inthe bar but xm at home. This would seem to make the bar look normativelybetter in terms of final consumption. However, if xm is optimally chosen whenheavy drinking is possible then it must be better (perhaps after accountingfor self-control costs) than choosing x0 when heavy drinking is possible. Butthe latter describes precisely what she expects in the bar.

We refrain from claiming that there cannot exist cases of self-deceptionthat violate RSD. A rationalization for choosing the bar must make thepresence of xm look bad. If the agent chooses xm at home, then the agent mayview her ex post choice as her temptation, or she may completely reverse hernormative and temptation perspectives. Whether or not this is characteristicof agents who delicately try to get past their normative defenses by appealingto reason, we note that it is readily attributable to an agent who is overtlyconsumed by her temptation, and thus needs to be ruled out in order toseparate our model from one of overt temptation.

12Note that the following relies on the idea that WARP is satisfied ex apost.

16

4.2 Representation Result

Say that functions u, v ∈ U are independent if for all α, β, γ ∈ R,

αu+ βv + γ = 0 ⇒ α = β = γ = 0.

Note that u and v are independent if and only if the functions u, v, u + vrepresent three different rankings on X.

Theorem 2. ≽ satisfies Axioms 1–7 if and only if ≽ has a utility represen-tation (4)–(5) such that for all a ∈ M1,

V (a) = κU(a) + λmaxy∈a

v(y), (6)

where κ ≥ λ > 0, and u, v ∈ U are independent.Moreover, ≽ has another representation (4), (5), (6) with parameters

κ′, λ′ ∈ R and functions u′, v′ ∈ U if and only if κ′ = κ, λ′ = λ, u′ = αu+βu,and v′ = αv + βv for some α > 0 and βu, βv ∈ R.

Here the temptation utility V can be interpreted as a distortion of theinterim normative utility U in the direction of the ex post desires v. Thisdistortion takes the form

V (a) = (κ− λ)maxx∈a

[κ

κ−λu(x) + λ

κ−λv(x)−max

y∈a(v(y)− v(x))

](7)

if κ > λ, orV (a) = κmax

x∈a(u(x) + v(x)) (8)

if κ = λ. To interpret, compare (7) with (5) to see that interim desires Vin (7) modify the ex ante perspective U by replacing the ex ante normativeperspective u with

u∗ = κκ−λ

u+ λκ−λ

v.

The perspective underlying u∗ distorts u in the direction of the temptationutility v. It is as if the agent is tempted to believe that her ex ante normativeperspective u was too stoic, and thus to relax her normative standards andadopt u∗. The case (8) is a limiting case of (7) where the agent views u+ vas her normative perspective and turns a blind eye to any possibility oftemptation. It is as if by adjusting her normative perspective she believesthat she is resolving all internal conflict. Observe that in either case, since

17

κ ≥ λ > 0, it can never be that the distorted normative preference putstoo much weight on the temptation preference. This is strongly reminiscentof the fact that self-deception requires the agent to maintain a facade ofrationality. This constrains her from adopting rationalizations that revealtheir underlying motives too clearly.

Observe that interim choice maximizes

U(a) + V (a) = (1 + κ)U(a) + λmaxy∈a

v(y)

= (1 + κ− λ)maxx∈a

[1+κ

1+κ−λu(x) + λ

1+κ−λv(x)−max

y∈a(v(y)− v(x))

],

which has a similar interpretation to (7), that is, the agent’s interim be-havior correctly recognizes her ex post desires, but distorts her normativeperspective. The interim commitment ranking is represented by the func-tion 1+κ

1+κ−λu + λ

1+κ−λv that assigns the weight of at least 2

3to the ex ante

commitment utility u. This suggests that after engaging in rationalizationsthe agent’s perspective does not deviate considerably from the ex ante per-spective. This is reminiscient of the idea that rationalizations cannot be tootransparent – they should not suggest drastic deviations from the previousperspective – in order for self-deception to survive.

While our theorem confirms that the preference-over-menus frameworkcan be used to model a self-deceiver, a surprising consequence of using GP’smodel is that only a narrow class of rationalizations are permitted in therepresentation. We observe that while (7) and (8) accommodate a distortionin normative perspective and possible blindness to temptation, they cannotaccommodate a distortion in anticipated self-control ability. Specifically, aswe just noted, the agent’s anticipated ex post choice at the interim stagemaximizes the function 1+κ

1+κ−λu+ λ

1+κ−λv+v. But this is ordinally equivalent

to the function u + v, which appears in U . That is, the interim temptationperspective under both (7) and (8) leaves anticipated choice undistorted rela-tive to the ex ante anticipated choice. Evidently, of the three rationalizationsour axiom is consistent with, the third must always hold and the second mayhold simultaneously, but the first can never hold.13 Our intuitive discussionof the first revealed that MSD and RSD are consistent it. We conclude there-fore that such rationalizations are ruled out when these axioms are adoptedin conjunction with Independence and Set-Betweenness.

13These conclusions are not tied to the functional form. They can be expressed behav-iorally.

18

We conclude this subsection by considering whether the functional formis interpretable in terms of a usual GP-style overt temptation story. It ishard to interpret the blindness-to-temptation case (κ = λ) as one of overttemptation to become blind. The relaxing-standards case (κ > λ) may beread as an overt temptation to ‘live a little’, which goes beyond the urge toconsume tempting alternatives and includes an urge to view such consump-tion in a positive light. We see the latter urge as implausible. The psychologyliterature on motivated reasoning tells us that people try to maintain a com-mitment to rationality – they seek reasons for beliefs and actions. Thus,while it is certainly intuitive that normative standards may be temporarilyadjusted at times of clouded judgement, we find it implausible that a clear-headed agent would even take seriously an overt urge to change somethingas fundamental as normative standards, let alone struggle with such an urgeand incur psychological costs of resisting it.

4.3 Simple Motivated Reasoning

The distinction between simple motivated reasoning and self-deception isthat the latter is associated with internal tension while the former is not.We captured internal tension in the same way that GP capture self-controlin their model. The absence of internal tension can therefore be associatedwith the absence of self-control. In the language of our model, interim choicewould be optimal given beliefs. Thus the basic framework for studying simplemotivated reason would strengthen Set-Betweenness by maintaining (1) butstrengthening (2) to:14

A ≽ B ⇒ A ∼ A ∪B or A ∪B ∼ B.

Then U0 would take the Strotz form

U0(A) = maxargmaxa∈A V (a)

U(a).

This functional form suggests that interim choice maximizes V (though tiesare broken in favor of U). That is, the agent changes her ranking of menusto V , and proceeds to simply choose according to V in a manner completelydivorced from U . Indeed, choice is optimal given her beliefs and there is noreason to believe that she is experiencing any internal tension. Thus thisprovides a basic framework for studying simple motivated reasoning.

14This is ruled out by Continuity, and thus a weakening of Continuity is required. SeeGP.

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5 Multiple Self-Deceptions

Our self-deception model rules out the possibility of a virtuous distortion inher perceived ability to exert self-control. We present here an extension thataccommodates such virtuous distortion of self-perception without departingfrom the GP framework. Although the general model will not be compatiblewith an interpretation involving sophisticated desires with a single motive, ithelps identify some intuitive behaviors ruled out by the model in the previoussection and lends itself to discussion of welfare.

In the self-deception model, temptation by b is motivated by the choicethat the agents expects to make in b. The following axiom accommodatestemptations that are motivated also by the normative content in b.

Axiom 8 (Weak Motivated Self-Deception). For any a, b ∈ M1 such thata ∪ b ≻ b and a ≻ a, a ∪ b, there is z ∈ b such that z ≻ xfor all x ∈ a.

This condition (WMSD for short) relaxes MSD and allows the menu a∪bto tempt a when the anticipated ex post choices in both menus are the same,but the menu a∪b provides a more virtuous interim self-perception. Formally,the rankings a∪ b ≻ b and a ≻ a, a∪ b are allowed only if a∪ b hasan element z that is normatively better than any alternative in a. It is as ifthe agent has an exaggerated view of her propensity for self-control, that is,an excessively virtuous self-image.

A departure from a temptation-driven self-deception is evident here: in-terim temptation is no longer intimately connected with a desire to achieveex post satisfaction of desires. An excessively virtuous self-image may leadthe agent to temptation only by accident, and in some situations may evendefeat efforts at ex post desire satisfaction. It follows that the tendencytoward a virtuous self-perception satisfies a different desire – presumably adirect desire for a positive self-image – that is distinct from ex post tempta-tion. We thus interpret the axiom as adding a second kind of self-deception,one involved in positive illusions. However, due to the abstract nature of ourframework, we cannot strictly speaking justify this interpretation relative to,say, wishful thinking.

On the other hand, RSD remains intuitive even if the agent’s interimself-deception has virtuous rationalization and motivation.

20

Theorem 3. ≽ satisfies Axioms 1–6 and WMSD if and only if ≽ has autility representation (4)–(5) such that for all a ∈ M1,

V (a) = κU(a) + λmaxy∈a

v(y) + µmaxz∈a

u(z) (9)

where κ ≥ λ > 0, µ ≥ 0, and u, v ∈ U are independent.Moreover, ≽ has another representation (4), (5), (6) with parameters

κ′, λ′, µ′ ∈ R and functions u′, v′ ∈ U if and only if κ′ = κ, λ′ = λ, µ = µ′,u′ = αu+ βu, and v′ = αv + βv for some α > 0 and βu, βv ∈ R.

The difference between (9) and (6) is the additional term µmaxz∈a u(z)in the interim temptation utility V . Although the model offers up to threerationalizations for choosing a given menu (namely, distorted normative pref-erence, distorted temptation preference and perceived virtuosity), the ratio-nalizations may be inconsistent. For instance, if the menu contains irresistibletemptation, then one rationalization will recognize this and justify it accord-ing to a more relaxed normative perspective, while the other will refuse torecognize it.15 The rationalizations may also neutralize each other, such aswhen a is more virtuous than b and b contains greater temptation. These arereflections of the fact noted above that the model is not one of temptation-driven self-deception. Such self-deception would presumably involve a searchfor the strongest possible case for making a ‘bad’ decision, and such a casewould be as devoid of inconsistencies as possible. Nevertheless, as a modelof an agent who engages in rationalizations more broadly, it permits somesubstantive discussion of welfare, to which we now turn.

Formally, write a ≽0 b if there exists z ∈ a such that z ≽ x forall x ∈ b. Then the preference ≽ represented by (9) satisfies16

a ≻ a ∪ b and a ≽0 b ⇒ a, b ≽ a, a ∪ b.15Behaviorally, suppose x ≻ x, y ∼ y. Then x, y may be more tempting

than both x and y. That is, x, y ≻ x, y, x, y. Since one rationalizationfavors x and the other favors y, the fact that x, y is more tempting than eitherimplies the simultaneous use of both rationalizations.

16To show this claim, take any a, b ∈ M1 such that a ≻ a ∪ b and maxz∈a u(z) ≥maxz∈b u(z). Then V (a ∪ b) ≥ V (b) because U(a ∪ b) ≥ U(b) and maxy∈a∪b v(y) =maxy∈b v(y). Consider two cases.

(i) a ∼ a, a ∪ b. Then V (a) ≥ V (a ∪ b) ≥ V (b) and hence, a ∼ a, b.(ii) a ≻ a, a ∪ b. By WMSD, a ∪ b ∼ b, that is, U(a ∪ b) = U(b). Therefore,

V (a ∪ b) ≥ V (b) implies a, b ≽ a, a ∪ b.

21

This condition implies that if b is a ‘vice’ menu and a is a ‘virtuous’ menu,then the agent is generally better off if virtue and vice are kept separate(as in a, b) rather than combined (as in a, a ∪ b). The intuition is thatthe union a ∪ b lends itself to more excuses for the agent to lead herselfinto temptation. Indeed, in this case, given whatever rationalizations theagent may adopt to justify choosing b from a, b, virtuous self-perceptionis an additional rationalization that can be invoked to justify choosing a ∪b from a, a ∪ b. The behavioral implication suggests, for instance, thata procrastinor is better-off if a completely flexible option is not a feasiblechoice: if a is the option of completing the task sooner and b is the option ofcompleting it later, then having the opportunity to make this decision later(as in a, a ∪ b) makes her worse-off relative to a situation where she hasto decide today whether to complete the task sooner or later (as in a, b).For another example, view a menu as a physical location selling particularalternatives and a menu of menus as a town. Then agents in a town arebetter-off if virtue and vice are sold at distinct locations (as in a, b) relativeto when vice is always bundled with virtue (as in a, a∪ b). Zoning laws forcasinos may be welfare improving in this sense.

A common view is that optimal welfare policy for agents with self-controlproblems constitutes the provision of commitment opportunities. In theabove setting, this would correspond to providing the agent with a, a ∪ b,in which she may keep all her options by selecting a ∪ b or avoid tempta-tion by choosing the commitment option a. The above discussion suggeststhat when agents are subject to self-deception, then the simple provision ofcommitment opportunities may not always be optimal.

6 Comparative Self-Deception

To interpret the parameters κ, λ, and µ in terms of choice behavior, considera pair of preferences ≽ and ≽∗ over M0. Call this pair regular if both ≽ and≽∗ satisfy Axioms 1–5 and BSD, and the two rankings agree on the domainof singleton menus so that

a ≽ b ⇔ a ≽∗ b

for all a, b ∈ M1.By Theorem 3, any regular pair of preferences ≽ and ≽∗ can be rep-

resented by (4)-(6) with components (u, v, κ, λ, µ) and (u∗, v∗, κ∗, λ∗, µ∗) re-

22

spectively. Moreover, the functions U and U∗ represent the same preferenceon M1 and hence, by GP’s Theorem, one can take u = u∗ and v = v∗.

Say that ≽∗ is more self-deceptive than ≽ if for all menus a, b ∈ M1,

a ≻ a, b ⇒ a ≻∗ a, b, (10)

This definition requires that any self-deception that is tempting for ≽ shouldbe tempting for ≽∗ as well.

Theorem 4. Let ≽ and ≽∗ be a regular pair of preferences. Then ≽∗ is moreself-deceptive than ≽ if and only if the two preferences have representations(4)-(6) such that λ∗

κ∗ ≥ λκand µ∗

λ∗ = µλ.

This result suggests that the ratios λκand µ

κare both positively related to

the intensity of self-deception in our model. (Note that λ∗

κ∗ ≥ λκand µ∗

λ∗ = µλ

imply µ∗

κ∗ ≥ µκ.) If κ > λ, then the weight λ

κ−λthat is put on the function v

in (7) is a positive monotonic transformation of λκand hence, can serve as an

index of self-deception as well.Moreover, the equality µ∗

λ∗ = µλis necessary for an unambiguous compari-

son of self-deception revealed by the two rankings ≽ and ≽∗. This equalityrequires roughly that the proportion of the virtuous and motivated compo-nents of self-deception should be the same for ≽ and ≽∗. In particular, thismust be true when both ≽ and ≽∗ satisfy MSD and hence, µ = µ∗ = 0.

7 Concluding Remarks

This paper explores the behavioral foundations for self-deception. It seeksto behaviorally capture the phenomenon as described in Section 2, but alsoto distinguish it from overt temptation and motivated reasoning. Achievingthe distinction requires us to adopt a rich choice domain. For instance, whileself-deception may well take place in the ex post stage (choice from menus), itmay not be distinguishable from overt temptation: an agent may find reasonsto drink heavily while in a bar, but her behavior would be identical to thatof some overtly tempted agent. Thus we are led to focus on self-deceptionprior to the time of final consumption, that is, in the agent’s interim choice ofbars while in a neighborhood. In order to identify self-deception at this stageand to distinguish it from motivated reasoning, the choice data we requireis the agent’s choice of neighborhoods. Thus, avoiding neighborhoods with

23

particular bars – a preference for commitment – reveals the agent’s concernwith her capacity for self-deception. The distinction between self-deceptionand motivated reasoning (namely, the existence of internal tension) comesdown to whether the analog of self-control exists in the interim stage.

Rationalizations are germane to self-deception, but instead of taking theseunobservables as part of our primitives, we derive them from choice behavior.On an intuitive level, we expect that the overly virtuous self-image inherentin the rationalization “I have the will-power to stop myself..” is an importantstrategic tool for desires to induce the agent to make decisions that will leadher into temptation. Yet, while such rationalizations are consistent with keybehavioral properties of self-deception (the MSD and RSD axioms) on anintuitive level, an unexpected finding is that they are ruled out in a GP-stylemodel of self-deception (that is, when Independence and Set-Betweenness aresatisfied). A direction for future research is to provide a model of temptation-driven self-deception that can accommodate the strategic use of virtuousself-perceptions.

A APPENDIX: PROOFS

In the proofs, we use the following notation and terminology. For any func-tion u ∈ U and any menu a ∈ M1, write

u(a) = maxx∈a

u(x),

and letT (u) = αu+ β : α ≥ 0, β ∈ R

be the set of all non-negative transformations of the function u.For any S ∈ N, let S denote also the set 1, . . . , S. Kopylov [16, Lemma

A.1] shows that for any u1, . . . , uS ∈ U , there are elements x1, . . . , xS ∈ Xsuch that for all i, j ∈ S, and

ui /∈ T (uj) ⇔ ui(xi) > ui(xj). (11)

This equivalence implies that ui(xi) ≥ ui(xj) for all i, j ∈ S.Turn to Theorem 1. The necessity of Axioms 1–4 for representation (1) is

straightforward. Conversely, suppose that ≽ satisfies Axioms 1–4. Kopylov

24

[17, Theorem 1] shows that ≽ has a utility representation

U0(A) = maxa∈A,x∈a

[u(x)−max

y∈a(v(y)− v(x))−max

b∈A(V (b)− V (a))

]for some u, v ∈ U and V ∈ U1. Moreover, if ≽ satisfies Regularity, thenthe triple (u, v, V ) in this representation is unique up to a positive lineartransformation. The utility function U0 has the required form (4)-(5), where

U(a) = maxx∈a

[u(x)−maxy∈a

(v(y)− v(x))]

for all menus a ∈ M1. Let w = u+ v and W = U + V . Then

U0(A) = maxa∈A

W (a)−maxb∈A

V (b) (12)

U(a) = w(a)− v(a) (13)

for all A ∈ M0 and a ∈ M1.Turn to Theorems 2 and 3. Suppose that ≽ satisfies Axioms 1–6 and

WMSD. By Theorem 1, ≽ is represented by (12). By Regularity, there arex∗, y∗ ∈ X such that

x∗ ≻ x∗, y∗ ≻ y∗.

Then w(x∗) > w(y∗) and v(y∗) > v(x∗), and hence, w and v are not redun-dant. Without loss in generality, assume that

u(x∗) = v(x∗) = w(x∗) = V (x∗) = 0. (14)

The following two lemmas obtain the required form for V .

Lemma 5. There are κ, ρ, µ ∈ R such that for all a ∈ M1,

V (a) = κw(a) + ρv(a) + µu(a). (15)

Proof. We claim first that for all a, b ∈ M1,

w(a) = w(b), v(a) = v(b), u(a) = u(b) ⇒ V (a) = V (b). (16)

Show this claim by contradiction. Consider any a, b ∈ M1 such that w(a) =w(b), v(a) = v(b), u(a) = u(b), but V (b) > V (a). By (13), U(a) = U(b) andhence, W (b) > W (a). As W is continuous, then there is ε > 0 such that

W (εy∗+ (1− ε)b) > W (εx∗+ (1− ε)a).

25

Let a∗ = εx∗ + (1 − ε)a and b∗ = εy∗ + (1 − ε)b. As w(x∗) > w(y∗),v(y∗) > v(x∗), and u(x∗) > u(y∗), then by linearity, w(a∗) = w(a∗ ∪ b∗) >w(b∗), v(b∗) = v(a∗ ∪ b∗) > v(a∗), and u(a∗) = u(a∗ ∪ b∗) > u(b∗). By (13),

U(a∗) > U(a∗ ∪ b∗) > U(b∗).

As W (b∗) > W (a∗), then there are two possible cases.

• W (b∗) > W (a∗ ∪ b∗). Then V (b∗) > V (a∗ ∪ b∗). By (12),

U(b∗, a∗ ∪ b∗) = W (b∗)− V (b∗) = U(b∗) < U(a∗ ∪ b∗),

which contradicts RSD.

• W (a∗ ∪ b∗) > W (a∗). Then V (a∗ ∪ b∗) > V (a∗). By (12), a∗ ≻a∗, a∗ ∪ b∗, which contradicts WMSD because u(a∗) > u(b∗).

This contradiction shows (16).Take any four menus a1, a2, a3, a4 ∈ M1. Let a = ∪4

i=1ai. There is isuch that w(a) ≥ w(ai), v(a) ≥ v(ai), and u(a) ≥ u(ai). Let b = ∪j =iaj.Then w(a) = w(b), v(a) = v(b), and u(a) = u(b). By (16), V (a) = V (b).Kopylov [15, Theorem 2.1] implies that the ranking that V represents on M1

is represented also by

V ′(a) =S∑

i=1

γiui(a) (17)

such that S ≤ 3, γ1, . . . , γS ∈ −1, 1, and ui ∈ T (uj) for all i, j ∈ S suchthat i = j. As both V ′ and V are linear, then without loss in generality,V ′ = V .

We claim that for all i ∈ 1, . . . , S,

ui ∈ T (w) ∪ T (v) ∪ T (u). (18)

Wlog let i = 1, and suppose that u1 /∈ T (w)∪T (v)∪T (u). Take x1, . . . , xS+3

that satisfy (11), that is,

u1(x1) > u1(xj) for all j = 1

ui(xi) ≥ ui(xj) for all i > 1 and j = i

w(xS+1) ≥ w(xj) for all j = S + 1

v(xS+2) ≥ v(xj) for all j = S + 2

u(xS+3) ≥ u(xj) for all j = S + 3.

26

Let a = x1, . . . , xS+3 and b = x2, . . . , xS+3. Then u1(a) = u1(x1) >u1(b), but w(a) = w(b), v(a) = v(b), u(a) = u(b), and uj(a) = uj(b) for allj = 1. Thus,

V (b)− V (a) = V ′(b)− V ′(a) = γi(ui(xi)− ui(a)) = 0,

which contradicts (16).The claims (17) and (18) and the normalization (14) imply (15).

The previous lemma implies that

W (a) = U(a) + V (a) = (κ+ 1)w(a) + (ρ− 1)v(a) + µu(a)

= (κ+ 1)U(a) + (κ+ ρ)v(a) + µu(a)(19)

for all menus a ∈ M1.

Lemma 6. The functions u, v are independent. The parameters κ, ρ, µ areunique and satisfy ρ ≤ 0, κ+ ρ > 0, µ ≥ 0. If ≽ satisfies MSD, then µ = 0.

Proof. Let a = x∗ and b = y∗. By (13), a ≻ a ∪ b ≻ b. ByWMSD, V (a) ≥ V (a ∪ b). By (15),

V (a∪b)−V (a) = κ(w(x∗)−w(x∗))+ρ(v(y∗)−v(x∗))+µ(u(x∗)−u(x∗)) ≤ 0.

Thus, ρ ≤ 0. To prove the other claims of the lemma, consider two cases.

Case 1. w, v, u are redundant. Then u must be a positive linear trans-formation of w or v. If u = αv for some α > 0, then w = u + v and v areredundant. Thus, u = αw for some α > 0. Then v = (α− 1)w. As w and vare not redundant, then α ∈ (0, 1). For all a ∈ M1,

V (a) = κw(a) + ρv(a) + µu(a) = (κ′ + ρ)U(a) + ρv(a),

W (a) = (κ′ + 1)U(a) + (κ′ + ρ)v(a),

where κ′ = κ+ µα. Suppose that κ′ + ρ < 0. Take α, β ∈ (0, 12) such that

1 <α

β

w(x∗)− w(y∗)

v(y∗)− v(x∗)< 1 +

∣∣∣∣κ′ + ρ

κ′ + 1

∣∣∣∣ . (20)

Let a = x∗, y∗ and b = αy∗ + (1− α)x∗, βx∗ + (1− β)y∗. Then

w(a ∪ b)− w(b) = w(x∗)− w(αy∗ + (1− α)x∗) = α(w(x∗)− w(y∗)) > 0

v(a ∪ b)− v(b) = v(y∗)− v(βx∗ + (1− β)y∗) = β(v(y∗)− v(x∗)) > 0.

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By (20) and (19),

U(a ∪ b)− U(b) = α(w(x∗)− w(y∗))− β(v(y∗)− v(x∗)) > 0

|(κ′ + 1)(U(a ∪ b)− U(b))| < |κ′ + ρ| β(v(y∗)− v(x∗))

W (a ∪ b)−W (b) = (κ′ + 1)(U(a ∪ b)− U(b)) + (κ′ + ρ)β(v(y∗)− v(x∗)) < 0.

Therefore, a∪ b ≻ b, but b, a∪ b ∼ b, which contradicts RSD. Thus,κ′ + ρ ≥ 0. Thus, for all x ∈ X,

V (x) = (κ′ + ρ)U(x) + ρα−1α

u(x) = γU(x),

where γ = (κ′ + ρ) + ρα−1α

is positive. By (12) and (13), for all x, y ∈ X,

x ≽ y ⇒ x ∼ x, y ≽ y,

which violates Regularity.

Case 2. w, v, u are not redundant. Then u and v are independent, andthere are x, y, z ∈ X such that

w(x) > w(y) ∨ w(z)

v(y) > v(x) ∨ v(z)

u(z) > u(x) ∨ u(y).

Suppose that µ < 0. Take α ∈ (0, 1) such that

α(κ+ 1)(w(x)− w(y)) + µ(u(z)− u(x)) < 0.

Let a = x, y, z and b = y, αy + (1− α)x. Then

w(a ∪ b)− w(b) = w(x)− w(αy + (1− α)x) = α(w(x)− w(y)) > 0

v(a ∪ b)− v(b) = v(y)− v(y) = 0

u(a ∪ b)− u(b) = u(z)− u(αy + (1− α)x) ≥ u(z)− u(x) > 0.

By (13), U(a ∪ b)− U(b) = w(a ∪ b)− w(b) > 0. By (19),

W (a ∪ b)−W (b) = (κ+ 1)(w(a ∪ b)− w(b)) + µ(u(a ∪ b)− u(b)) ≤α(κ+ 1)(w(x)− w(y)) + µ(u(z)− u(x)) < 0.

Therefore, a ∪ b ≻ b and b, a ∪ b ∼ b. These rankings violate RSD.Thus, µ ≥ 0.

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Suppose that κ+ ρ < 0. Take α, β ∈ (0, 12) such that

1 <α

β

w(x)− w(y)

v(y)− v(x)< 1 +

∣∣∣∣κ+ ρ

κ+ 1

∣∣∣∣ . (21)

Let a = x, y, z and b = αy + (1− α)x, βx+ (1− β)y, z. Then

w(a ∪ b)− w(b) = w(x)− w(αy + (1− α)x) = α(w(x)− w(y)) > 0

v(a ∪ b)− v(b) = v(y)− v(βx+ (1− β)y) = β(v(y)− v(x)) > 0

u(a ∪ b)− u(b) = u(z)− u(z) = 0.

By (21) and (19),

U(a ∪ b)− U(b) = α(w(x)− w(y))− β(v(y)− v(x)) > 0

|(κ+ 1)(U(a ∪ b)− U(b))| < |κ+ ρ| β(v(y)− v(x))

W (a ∪ b)−W (b) = (κ+ 1)(U(a ∪ b)− U(b)) + (κ+ ρ)β(v(y)− v(x)) < 0.

Therefore, a∪ b ≻ b, but b, a∪ b ∼ b, which contradicts RSD. Thus,κ+ ρ ≥ 0.

Suppose that κ+ ρ = 0. Then for all x′ ∈ X,

V (x′) = κU(x′) + µu(x′) = (κ+ µ)U(x′),

where κ + µ = −ρ + µ ≥ 0. This equality contradicts Regularity (see theproof of Case 1.) Thus, κ+ ρ > 0.

Suppose that ≽ satisfies MSD. Let a = x, y and b = y, z. Thena ∪ b ≻ b. By MSD,

V (a)− V (a ∪ b) = µ(u(x)− u(z)) ≥ 0.

Thus, µ = 0.Finally, note that

κ =V (x, y, z − V (αz + (1− α)x, y, z)

α(w(x)− w(z))

ρ =V (x, y, z − V (x, αx+ (1− α)y, z)

α(v(y)− v(x))

µ =V (x, y, z − V (x, y, αx+ (1− α)z)

α(u(z)− u(x))

for all sufficiently small α. These equations show that all of these parametersare unique.

29

Lemmas 5 and 6 imply that V has the required form

V (a) = κU(a) + λv(a) + µu(a) = κw(a) + (λ− κ)v(a) + µu(a), (22)

where κ ≥ λ = κ+ ρ > 0, µ ≥ 0, and u, v ∈ U are independent.Conversely, suppose that ≽ has representation (12), (13), and (22). Take

any a, b ∈ M1 such that a ∪ b ≻ b. By (22),

V (a ∪ b)− V (b) = κ(U(a ∪ b)− U(b)) + λ(v(a ∪ b)− v(b))+

µ(u(a ∪ b)− u(b)) ≥ 0

because U(a ∪ b) > U(b), v(a ∪ b) ≥ v(b), and u(a ∪ b) ≥ u(b). By (12),

a ∪ b ∼ b, a ∪ b ≻ b,

and ≽ satisfies RSD. (If b ≽ a∪ b, then b, a∪ b ≽ a∪ b follows fromSet-Betweenness.)

Moreover,

V (a)− V (a ∪ b) = κ(w(a)− w(a ∪ b)) + (λ− κ)(v(a)− v(a ∪ b))+

µ(u(a)− u(a ∪ b)) ≥ µ(u(a)− u(a ∪ b))

because w(a) = w(a ∪ b), v(a ∪ b) ≥ v(a), and λ − κ ≤ 0. Therefore, theranking a ≻ a, a∪b implies that µ > 0 and u(b) > u(a). Thus ≽ satisfiesWMSD, and if µ = 0, then ≽ satisfies MSD.

As u and v are independent, then w and v are not redundant. Takex, y ∈ X such that w(x) > w(y) and v(y) > v(x). By (13),

x ≻ x, y ≻ y.

Let v′ = (κ + µ)u + λv and w′ = u + v′. As u and v are independent andλ > 0, then the functions w′ and v′ are not redundant. Take x′, y′ ∈ X suchthat w′(x′) > w′(y′) and v′(y′) > v′(x′). By (12),

x′ ≻ x′, y′ ≻ y′.

Thus, ≽ satisfies Regularity.

Turn to Theorem 4. Suppose that ≽ and ≽∗ have representations (4)-(6)with components (u, v, κ, λ, µ) and (u, v, κ∗, λ∗, µ∗) such that κ ≥ λ > 0,κ∗ ≥ λ∗ > 0, and µ, µ∗ ≥ 0.

30

Suppose that λ∗

κ∗ ≥ λκand µ∗

λ∗ = µλ. Then U = U∗ and for all a, b ∈ M1,

a ≻ a, b ⇒ U(a) > U(b) and V (b) > V (a) ⇒[U(b)− U(a)] + λ

κ[v(b)− v(a) + µ

λu(b)− µ

λu(a)] > 0 > U(b)− U(a) ⇒

[U∗(b)−U∗(a)] + λ∗

κ∗ [v(b)− v(a) + µ∗

λ∗u(b)− µ∗

λ∗u(a)] > 0 > U∗(b)−U∗(a) ⇒U∗(a) > U∗(b) and V ∗(b) > V ∗(a) ⇒ a ≻∗ a, b.

Thus, ≽∗ is more self-deceptive than ≽.Conversely, suppose that ≽∗ is more self-deceptive than ≽. As u and v

are independent, then the functions u, v, and w = u+ v are not redundant.By (11), there are x, y, z ∈ X such that

w(x) > w(y) ∨ w(z)

v(y) > v(x) ∨ v(z)

u(z) > u(x) ∨ u(y).

As w, v, u ∈ U , then for any α, γ > 0, there exist x′, y′, z′ ∈ X such that

w(x) > w(x′) > w(y) ∨ w(y′) ∨ w(z) ∨ w(z′)

v(y) > v(y′) > v(x) ∨ v(x′) ∨ v(z) ∨ v(z′)

u(z) > u(z′) > u(x) ∨ u(x′) ∨ u(y) ∨ u(y′)

w(x)− w(x′)

v(y)− v(y′)= α

v(y)− v(y′)

u(z)− u(z′)= γ.

(23)

Show the inequalities λ∗

κ∗ ≥ λκand µ∗

λ∗ = µλby contradiction. Consider three

cases.

Case 1. µ∗

λ∗ > µλ. Take γ such that µ∗

λ∗ > γ > µλand α such that 1 >

α > 1 − γλ−µγκ

. Take x′, y′, z′ ∈ X that satisfy (23). Let a = x′, y′, z and

b = x, y, z′. Then

U(a)− U(b) = (w(x′)− v(y′))− (w(x)− v(y)) = (1− α)(v(y)− v(y′)) > 0

V (b)− V (a) =(−κ(1− α) + λ− µ

γ

)(v(y)− v(y′)) > 0

V ∗(b)− V ∗(a) =(−κ∗(1− α) + λ∗ − µ∗

γ∗

)(v(y)− v(y′)) < 0

31

because−κ(1− α) + λ− µ

γ> 0 > −κ∗(1− α) + λ∗ − µ∗

γ.

Thus, a ≻ a, b, but a ∼∗ a, b, which contradicts the assumptionthat ≽∗ is more self-deceptive than ≽.

Case 2. µ∗

λ∗ < µλ. Take γ such that µ∗

λ∗ < γ < µλand α such that 1 <

α < 1 + µ−γλγκ

. Take x′, y′, z′ ∈ X that satisfy (23). Let a = x, y, z′ and

b = x′, y′, z. Then

U(a)− U(b) = (w(x)− v(y))− (w(x′)− v(y′)) = (α− 1)(v(y)− v(y′)) > 0

V (b)− V (a) =(−κ(α− 1)− λ+ µ

γ

)(v(y)− v(y′)) > 0

V ∗(b)− V ∗(a) =(−κ∗(α− 1)− λ∗ + µ∗

γ∗

)(v(y)− v(y′)) < 0

because−κ(α− 1)− λ+ µ

γ> 0 > −κ∗(α− 1)− λ∗ + µ∗

γ.

Thus, a ≻ a, b, but a ∼∗ a, b, which contradicts the assumptionthat ≽∗ is more self-deceptive than ≽.

Case 3. µ∗

λ∗ = µλand λ∗

κ∗ < λκ. Take α such that 1− λ∗

κ∗ > α > 1− λκ. Take

x′, y′ ∈ X that satisfy (23). (z′ is not required here.) Let a = x′, y′, z andb = x, y, z. Then

U(a)− U(b) = (w(x′)− v(y′))− (w(x)− v(y)) = (1− α)(v(y)− v(y′)) > 0

V (b)− V (a) = (−κ(1− α) + λ) (v(y)− v(y′)) > 0

V ∗(b)− V ∗(a) = (−κ∗(1− α) + λ∗) (v(y)− v(y′)) < 0

because−κ(1− α) + λ > 0 > −κ∗(1− α) + λ∗.

Thus, a ≻ a, b, but a ∼∗ a, b, which contradicts the assumptionthat ≽∗ is more self-deceptive than ≽.

Thus, µ∗

λ∗ = µλand λ∗

κ∗ ≥ λκ.

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