Menu Choice, Environmental Cues andTemptation: A “Dual Self” Approach

to Self-control∗

Kalyan Chatterjee† R. Vijay Krishna‡

July, 2006

Abstract

We consider the following two-period problem of self-control. In the first pe-riod, an individual has to decide on the set of feasible choices from which she willselect one in the second period. In the second period, the individual might choosean alternative that she would find inferior in the first period. This eventuality neednot occur with certainty but might be triggered by the nature of the set chosen inthe first period. We propose a model for this problem and axioms for first-periodpreferences, in which the second period choice could be interpreted as being madeby an “alter ego” who appears with some probability. We provide a discussion ofthe behavioural implications of our model as compared with existing theories.

1 Introduction

Casual observation and introspection (especially about unhealthy but tasty food items),as well as stories like that of Ulysses and the Sirens, suggest that otherwise rational indi-

∗We wish to thank Sophie Bade, Siddhartha Bandyopadhyay, Susanna Esteban, Kfir Eliaz, Drew Fu-denberg, Ed Green, Faruk Gul, John Hardman Moore, Jawwad Noor, József Sákovics, Rani Spieglerand Jonathan Thomas for helpful comments and Avantika Chowdhury for research assistance. We areespecially grateful to Bart Lipman for insightful and thought-provoking discussions.

†Department of Economics, Pennsylvania State University. Email: kchatterjee@psu.edu, URL:http://www.econ.psu.edu/people/biographies/chatterjee_bio.htm

‡Department of Economics, University of North Carolina, Chapel Hill. Email: rvk@email.unc.edu.

1 Introduction 2

viduals act to constrain their future choices. At first sight, this seems inconsistent withstandard notions of rational choice, where increasing the set of available choices cannotmake anyone strictly worse off. Strotz (1955), who discusses intertemporal choice, callsthis constraining of one’s future choices the “strategy of pre-commitment” and suggeststhat it results from the agent today being a “different person with a different discountfunction from the agent of the past” (p 168). Schelling (1978, 1984) gives several examplesof this kind of behaviour and also speaks of “multiple selves,” though he writes also thathe is somewhat hesitant to use this terminology outside the circle of professional econo-mists. Hammond (1976), in a paper on coherent dynamic choice, discusses an exampleof addiction as a reflection of changing tastes.

It is important here to distinguish between changing tastes and choices being madethat maximise a different utility function. An agent whose tastes might change wouldnever (strictly) prefer a smaller set of alternatives to a larger set of alternatives. Anexample of this is a person who knows in the morning that he might be in the moodfor either seafood or vegetarian food for dinner (depending on his changed taste in theevening), even though he would go for seafood if he were forced to choose a menu itemin the morning. It is clear that he would prefer making a reservation at a restaurantthat serves both rather than one that serves just seafood. If, instead, the agent perceivesthat future choices might be made that maximise a utility function different from his, hemight prefer to limit his options. For instance, if someone had to choose between goingto a party with tea and coffee being served, versus another one in which tea, coffee andcocaine would be served (to modify Amartya Sen’s example), the fact that this personmight choose cocaine in the evening would possibly lead to him pre-committing in themorning to going to the party where cocaine would not be available. We are interested inthe second notion mentioned above, namely that of choices in the future possibly beingmade to maximise a different utility function.

One way to conceptualise both types of problem is to consider, following Kreps(1979), an implicit two-stage choice problem. In the morning, an individual chooses amenu of objects. In the evening she chooses an item from the previously chosen menu.As mentioned above, we are interested in the following behaviour, which the individualconsiders a possibility. The menu chosen might trigger temptation in the next period.Temptation is thought of as the (utility maximising) choice of a virtual alternate self or al-

1 Introduction 3

ter ego. The probability of the alternate self taking over is menu-dependent.1 Whicheverself is in charge of making a choice from the menu in the second period makes its ownmost preferred choice. Each self does not particularly care about the utility of the otherself, so this is not an interdependent utilities model, but does care about the choice made(although we shall assume that the alter ego breaks ties in favour of the decision maker).Let x be a typical menu and β a typical member of the menu. This gives the decision-maker a utility function over menus of the following form:

U(x) = (1− ρx)maxβ∈x

u(β)+ ρx maxβ∈Bv(x)

u(β)

where U is the individual’s utility from a menu, u her utility from individual items, vis the alter ego’s utility from individual items and Bv(x) is the set of v-maximisers inx. We shall take ρx to be the probability that the individual gets tempted (when facedwith the choices in the menu x), which results in the alter ego making a choice. Theaforementioned asymmetry between the selves requires that the alter ego, if he has tomake a choice, will choose, among his most preferred alternatives, that which is mostpreferred by the decision-maker. Since we are characterising the decision-maker’s utility,how she breaks ties does not really matter to us.

Since the loss of self-control is systematic or regular, it makes sense to think of the“virtual” alter ego as having a von Neumann-Morgenstern utility function v(·) as well.However, it is important to remember that the alter ego is just a convenient way ofdescribing the actual tempted behaviour of our decision maker. Thus v(·) cannot bearbitrary; it must be consistent with the the description of what alternative elements ofa menu tempt the decision-maker and which ones do not. (We note the von Neumann-Morgenstern utility has, similarly, to be consistent with a decision-maker’s choices underrisk.)

Note the four important features of this representation: First, choices of single itemsare evaluated using the utility function u(·). Second, the utility of a menu U(x) liesbetween the decision maker’s utility of her most preferred alternative in x and her utilityof the tempted choice (as if made by the alter ego) and is therefore a menu-dependentconvex combination of the two. Third, the tempting choices are systematically tempting,which is why we hesitate to call them mistakes. What tempts the decision maker must

1We can think of this as a generalisation of Strotz, in that the alternate self appears here with someprobability instead of with certainty, as in Strotz.

1 Introduction 4

be shown to be the result of consistent, utility maximising choice by the alter ego; thatis, we have to demonstrate that the tempting choices maximise some utility function.Finally, our formulation enables us to use the metaphor of the alter ego while still makingunambiguous welfare statements. Thus, the link between choice and welfare remainsunbroken and we do not have to impose our subjective judgement on the weights givento the decision maker and the alter ego in some social welfare function.

We shall say that a utility function over menus that takes such a form admits a dualself representation. We provide axioms for first period preferences over menus so that thedecision-maker’s utility from a menu is given by the equation above. Thus, a decision-maker who satisfies our axioms behaves as if there is a probability of her getting tempted,when a choice has to be made from a menu, which is represented as the choice beingmade by an alter ego. It should be emphasised that the alter ego (and his utility functionv) is subjective, as is the probability, ρx, of getting tempted. The only observables arefirst period choices over menus and v and ρx must be inferred from these choices. Theinference works as follows.

Consider two prizes {a} and {b}. If the decision-maker’s preferences are given by{b} � {a,b} å {a}, then it must be the case that (i) v(a) > v(b), ie the alter ego prefersb to a and (ii) ρ{a,b} > 0, ie the decision-maker subjectively assesses that there is a positiveprobability that the alter ego will make the choice from the menu {a,b}. We may saythat a is a revealed temptation for b. If either of the requirements above does not hold,then temptation would not be an issue.

We will have more to say later on about the nature of the dependence on x of ρx. Atthis stage, we can mention that ρx could depend solely on the best and the maximallytempting elements in a menu (ie the u- and v-maximisers respectively; this is analysed inTheorem 3.7); however, it could also, more generally, depend on other elements in themenu (for example, the presence of sorbet in the menu makes it easier to be tempted byfull-fledged ice cream). This is our Dual Self Theorem, Theorem 3.4. Our axioms willalso imply certain restrictions on ρx resulting from the linearity of the utility function.These restrictions are not assumptions, they are implications. We consider ρx to be po-tentially dependent on the menu x. The case where ρx is constant (independent of x) isan important special case, which we consider in detail in §3.1. It is worthwhile to notethat the well-known representation of Gul and Pesendorfer (2001) of temptation prefer-ences implies our representation with ρx dependent on the menu. The case where ρ > 0

2 The Model 5

is constant and not equal to 1 is not capable of being captured by the Gul-Pesendorferpreferences (as discussed in §4.1).

We emphasise that this paper seeks to characterise behaviour that exhibits temporaryloss of self-control by determining a set of axioms on choice of menus that yield thisrepresentation. Note that since we are interested in choice under temptation, the variouscomponents of the representation above cannot be unrestricted. If we restrict ourselvesto menus consisting of a single alternative each (an alternative is an “objective lottery”like the ones in the classic von Neumann-Morgenstern formulation of expected utility),there is no scope for temptation and the utility of a singleton “menu” α is just the vonNeumann-Morgenstern utility u(α).2

The remainder of the paper is structured as follows. In §2 we introduce our model, in§3 the axioms and our representation theorem and sketch the proof of the representationtheorem in §3.2. In §4 we first compare our axioms with those of Gul and Pesendorfer(2001) (henceforth GP) in §4.1 and with those of Dekel, Lipman and Rustichini (2005)in §4.2 (henceforth DLR05) and then review other related literature. §5 concludes andproofs are in the Appendix.

2 The Model

We have in mind a decision-maker who faces a two-period decision problem. In thefirst period, the agent chooses the set of alternatives from which a consumption choicewill be made in the second period. Nevertheless, as in Kreps (1979), Dekel, Lipman andRustichini (2001) (henceforth DLR) and GP, we shall only look at first period choices.Let us now describe the ingredients more formally. (The basic objects of analysis areexactly the same as in GP.)

The set of all prizes is Z where (Z,d) is a compact metric space. The space of prob-ability measures on Z is denoted by ∆ (with generic elements being denoted by α,β, . . .)and is endowed with the topology of weak convergence. This topology is metrisable andwe let dp be a metric which generates this topology. The objects of analysis are subsetsof ∆. Let A be the set of all closed subsets of ∆ (with generic elements, called menus,

2This implies that the axioms in our paper should (and do) reduce to those of von Neumann andMorgenstern for singleton menus. These latter axioms are not uncontroversial; nevertheless since ourfocus is on issues relating to self-control, we think they are acceptable assumption in our framework.

3 Axioms and Representations 6

denoted by x,y, . . .) endowed with the Hausdorff metric

dh(x,y) := max{

maxx

minydp(α,β),max

yminxdp(α,β)

}.

Convex combinations of elements x,y ∈ A are defined as follows. We let λx + (1 −λ)y :=

{γ = λα+ (1− λ)β : α ∈ x,β ∈ y

}where λ ∈ [0,1]. (This is the so-called

Minkowski sum of sets.) We are interested in binary relations å which are subsets ofA × A . (In the sequel, read A → B as A implies B unless the arrow is a limit. In ei-ther case, it should be clear from the context what the intended arrow denotes and noconfusion should arise.)

Before we impose axioms on å, it may be worthwhile to dwell on the implicationsof the model. The use of subsets of lotteries over Z as the domain for preferences insteadof subsets of Z itself was first initiated by DLR in this context and is reminiscent ofthe approach pioneered by Anscombe and Aumann (1963). From a normative point ofview, this approach should not be troublesome as long as our decision-makers are able toconceive of the lotteries they consume and agree with the axioms we impose on them.But from a revealed preference perspective, are decision-makers faced with menus oflotteries? As noted by Kreps (1988, pp. 101) (in the context of the Anscombe-Aumanntheory), if decision-makers are not faced with choices of lotteries, our assumption thatthey are can be quite burdensome, especially from a descriptive point of view.

Nevertheless, it could be argued that such menus of lotteries are, in fact, objects ofchoice. A patient who chooses to go to a hospital is, arguably, choosing a menu oflotteries with the level of pain being an uncontrolled random event. Similarly, a seafoodfancier who goes to a restaurant not knowing the quality of the shrimp he is about to get,is doing the same. It is also possible that the menu of lotteries could arise from a non-degenerate mixed strategy played by an opponent, for instance in determining the set ofobjects available for sale by, say, a car dealer. There is, of course, the analytical benefit ofour approach, which is the use of the additional structure a linear space provides.

3 Axioms and Representations

We impose the following axioms on our preferences.

Axiom 1 (Preferences) å is a complete and transitive binary relation.

3 Axioms and Representations 7

Axiom 2 (Continuity) The sets {y : y å x} and {y : x å y} are closed.

Axiom 3 (Independence) x � y and λ ∈ (0,1] implies λx + (1− λ)z � λy + (1− λ)z.

The first axiom is standard. Axiom 2 is a continuity requirement in the Hausdorfftopology. The motivation for Independence is the familiar one and some normative ar-guments in its favour are given in DLR and GP. It basically says that our decision-makerdoes not distinguish between simple and compound lotteries and all that matters to herare the prizes. (Nevertheless, as noted by Fudenberg and Levine (2005), this may not bean innocuous assumption.) Let us define

B(x) :=⋂β∈x

{α ∈ x : {α} å {β}

}and W(x) :=

⋂β∈x

{α ∈ x : {β} å {α}

}to be the sets of best and worst elements respectively in x. In light of Continuity and thecompactness of x, it follows that both B(x) and W(x) are well defined. Our next axiomcaptures the essence of temptation.

Axiom 4 (Temptation) For all x, B(x) å x å W(x).

Temptation says that insofar as the presence of alternatives different from the best alter-native in the menu affects the decision-maker, it does not make the decision-maker worseoff than her worst choice in the menu. It also says that the cost of temptation (i.e. thecost of not being able to choose the best alternative) is bounded. In particular, it rulesout situations like Sen’s rational donkey, which starves because it is unable to make achoice between two equally acceptable alternatives. In other words, it is never the casethat “analysis is paralysis.”

A decision-maker who faces no temptation would simply pick the best lottery inany menu. Our decision-makers however do not always do so. Consider three prizes,broccoli (b), rich chocolate cake (c) and deep-fried Mars bars (m). Let us suppose thedecision-maker has the following preferences over the prizes in the morning: {b} � {c} �{m}. A “standard” decision-maker would always pick her most preferred alternative inany menu she encounters in the afternoon. But we are interested in decision-makerswho are not immune to temptations. Suppose the decision-maker has preferences overthe following menus: {b} � {b, c} å {c}. We can then conclude that the presence ofc in the menu, which makes the decision-maker worse off, is the source of temptation.

3 Axioms and Representations 8

Formally, let {β} be any lottery. Say that {α} tempts {β} if {β} is superior to {α} and theaddition of {α} to {β} makes the agent strictly worse off, i.e. {β} � {α,β} å {α}.

Our next axiom says that a decision maker finds lotteries over tempting alternativesto be tempting. To see why this might be the case, think of a decision maker for whom{m} and {c} tempt {b}. Let us also suppose that before being offered a 1/2– 1/2 lotteryover the two tempting alternatives, she is allowed a whiff of the two temptations. Tosay that the decision maker also finds this lottery tempting says that she does not usefortune as a cover for her frailty. We now formally state the axiom which says that theset of lotteries that tempts any {β} is convex.

Axiom 5 (Regularity) {α1} and {α2} tempt {β} implies {λα1 + (1 − λ)α2} tempts {β}for all λ ∈ [0,1].

Our next axiom is an excision axiom in that it allows us to excise elements from amenu without affecting the value of the menu to the decision-maker. Consider onceagain the three prizes, broccoli (b), rich chocolate cake (c) and deep-fried Mars bars(m). As before, our decision-maker has the following preferences over the prizes in themorning: {b} � {c} � {m} and both m and c tempt b, i.e. {b} � {b, c}, {b,m}. Thus,the presence of c and m make the decision-maker strictly worse off. Now, suppose thatadding m to the menu {c} does not affect the decision-maker, i.e. {c} ∼ {c,m}. Thisimplies that the “real” temptation comes from the rich chocolate cake and the additionof m to the menu {b, c} should leave the agent indifferent, (i.e. {b, c} ∼ {b, c,m}). Ournext axiom formalises this idea.

Axiom 6 (AoM: Additivity of Menus) For x,y finite

{β} ∼ {β} ∪y implies {β} ∪ x ∪y ∼ {β} ∪ x.

To recap, the axiom says that if the presence of y does not affect the decision makerwhen she is faced with {β}, adding a set x to {β} should not change this irrelevance.3

This axiom has the flavour of additivity, namely adding the same set to both sides of anexpression does not change the relation between the left and right hand sides. Notice

3This is similar to the axiom used by Kreps (1979) who requires that x ∼ x ∪ x′ implies x ∪ x′′ ∼x ∪ x′ ∪ x′′. Notice that we only require this axiom to be true when x is a singleton.

3 Axioms and Representations 9

that we only require the axiom to hold for finite menus. Requiring it hold for arbitrarymenus will neither change any of the theorems below nor simplify their proofs.

We now define the linear functionals relevant to a dual self representation. As isstandard, we shall say that U : A → R is linear if U(λx+(1−λ)y) = λU(x)+(1−λ)U(y)for all x,y ∈ A and λ ∈ (0,1) and that it represents å if it is the case that U(x) á U(y)if and only if x å y. The functions u,v : ∆ → R are linear if similar conditions hold.Let Bv(x) = arg maxβ∈x v(β) be the set of v-maximisers in x (with a similar definitionfor Bu). Let β∗x ∈ Bu(x) and let β̂x ∈ Bu (Bv(x)).

For any menu x, we shall say that the decision-maker, when confronted with a choicefrom the menu x, gets “tempted” with probability ρx. If ρ is to be consistent withlinearity of U , then it must be the case that for all λ ∈ (0,1),

ρλx+(1−λ)y =λρxδx + (1− λ)ρyδyλδx + (1− λ)δy

(♣)

where δx := u(β∗x) − u(β̂x) and δy := u(β∗y) − u(β̂y). (The expression follows fromthe linearity of u and v and the observation that if β∗x (resp. β̂x) maximises u (resp. v)over x and if β∗y (resp. β̂y ) maximises u (resp. v) over y, then λβ∗x + (1 − λ)β∗y (resp.λβ̂x + (1 − λ)β̂y ) maximises u (resp. v) over λx + (1 − λ)y. Call a menu y inessentialto x ∪y if there exists β ∈ x such that U({β}) = U({β} ∪y) and U(x) = U

(x ∪y

). Let

B ⊂A be any collection of menus. This leads us to the following definition.

Definition 3.1. A linear functional U : B → R admits a dual self representation (u,v, ρ)if there exist continuous linear functionals u,v : ∆ → R unique up to affine transformationswith u := U|∆ and ρ : B → [0,1] such that

U(x) = (1− ρx)maxβ∈x

u(β)+ ρx maxβ∈Bv(x)

u(β).

Moreover, (i) ρx = ρx+c for all signed measures c such that c(∆) = 0 and x + c ∈ B andx such that Bu(x) ∩ Bv(x) = ∅, (ii) ρ satisfies equation (♣) and (iii) ρx∪y = ρx > 0 for allfinite x,y ∈B such that y is inessential to x ∪y and Bu(x ∪y)∩ Bv(x ∪y) = ∅.

The condition Bu(x)∩Bv(x) = ∅ says that the decision-maker and her alter ego don’tshare a common best element in the menu x, thereby making temptation meaningfulfor such a menu. The third condition in the definition above says that if there are someelements in the menu x whose removal doesn’t affect its valuation, then their removalalso doesn’t affect the probability of getting tempted. The second condition requires that

3 Axioms and Representations 10

ρ be consistent with the linearity of U . The first condition says that for all menus wheretemptation is meaningful, ρ is translation invariant. This is due to the more general factthat Independence and Continuity ensures that the preferences themselves are translationinvariant. This is made precise below.

Definition 3.2. A binary relation å is translation invariant if x å y implies x+c å y+cfor all signed measures c such that c(∆) = 0 and x + c,y + c ∈A .

Lemma 3.3 (Translation Invariance). Let å satisfy Axioms 1, 2 and 3. Then å is translationinvariant.

Proof. See appendix.

We can now state our main representation theorem.

Theorem 3.4 (Dual Selves). A binary relation å satisfies Axioms 1–6 if and only if thereexists a continuous linear functional U : A → R, unique up to affine transformation, thatrepresents å and U admits a dual self representation.

Proof. See appendix.

The dual self theorem below says that when faced with choices of menus, the decision-maker who satisfies Axioms 1–6 behaves as if he has an alter ego who has a utilityfunction over lotteries given by v. Moreover, in the event that the alter ego makesthe choice, he chooses the lottery in his most-preferred set (in x) which maximises thedecision-maker’s utility. Also, the decision-maker behaves as if he will be tempted (i.e.the probability that the choice will be made by the alter ego) with a probability of ρxwhen faced with the menu x.

One way to imagine the representation is that the decision-maker expects an internalbattle for self-control with her alter ego and ρx represents the probability that she losesthis battle. It is entirely conceivable that ρx depends on more than the u and v max-imisers in x. For instance, consider an decision-maker whose choices for dessert includefruit (f ), sorbet (s) and chocolate cake (c). Suppose the decision-maker’s preferencesare u(f) > u(s) > u(c) and suppose the alter ego is such that v(c) > v(s) > v(f).Then, the decision-maker can have U({f , c}) > U({f , s, c}), ie the presence of moretemptations increases the probability that the decision maker will lose the battle for selfcontrol.

3.1 Other Representations 11

It should be emphasised that ρx and v are subjective and hence unobservable. Theyhave to be inferred from first period choices. The only observables here are first periodbehaviour. Furthermore, the decision-maker behaves as if second period choice from aconvex menu is from an extreme face of the menu. (That the decision-maker is indifferentbetween any menu and its convex hull is proved in Lemma A.1 in the Appendix.)

We should point out that in the dual self representation above, ρ cannot be constantas this would violate the continuity of å. To see this, suppose ρ is constant and U(·)is continuous. Consider an α and β such that u(β) > u(α) and v(α) = v(β) so thatU({β}) = U({β,α}). Let (αk) be a sequence such that (i) limkαk = α, (ii) u(αk) = u(α)for all k and (iii) v(αk) > v(αk+1) > v(α) for all k. Then, U({β}) > U({β,αk}) forall k but, as noted above, U({β}) = U({β,α}) which contradicts the continuity of U(·).Moreover, the following is also true.

Proposition 3.5. Let ρ satisfy equation (♣) and be continuous. Then ρ is constant.

Proof. Suppose ρ is continuous and suppose it is not constant. Then, there exist menusx,y such that ρx ≠ ρy . Then, for any singleton {β}, ρλ{β}+(1−λ)x = ρx and ρλ{β}+(1−λ)y =ρy . (This follows from equation (♣) above.) But for any ε > 0, there exists λ largeenough such that

dh(λ{β} + (1− λ)x, λ{β} + (1− λ)y

)< ε

but ρx and ρy remain just as far apart, contradicting the continuity of ρ.

Thus, no matter what additional axioms we choose, we cannot have continuous ρexcept if ρ is constant. Furthermore, if ρ is constant, then it must be the case that å isdiscontinuous. We examine the case of constant ρ in the next section.

3.1 Other Representations

The dual self representation allows many possibilities. Indeed, it encompasses the GPrepresentation (see §4.1 below) and allows ρ to depend on arbitrarily many items in amenu. We can, nevertheless, say that there can be no continuous selection ρ. This wasdemonstrated in Proposition 3.5. Moreover, the dual self theorem rules out situationswhere the decision maker has no self control, i.e. decision makers who have a dual selfrepresentation with ρx = 1 for all x. This is because such a decision maker’s preferences

3.1 Other Representations 12

are typically only upper-semicontinuous (and not continuous). In this section, we intro-duce another excision axiom which will give us this possibility and also the case whereρx ∈ [0,1] is a constant for all x. Let us first weaken Continuity appropriately.

Axiom 2a: (Upper Semicontinuity) The sets {y ∈A : y å x} are closed.

Axiom 2b: (Lower von Neumann-Morgenstern Continuity) x � y � z implies λx+(1− λ)z � y for some λ ∈ (0,1).

Axiom 2c: (Lower Singleton Continuity) The sets {α : {β} å {α}} are closed.

Axioms 2a–c are identical to Axioms 2a–c in §3 of GP. They weaken Continuity justenough to enable us to have a linear utility representation that admits a dual self represen-tation. Notice also that Axioms 2a–c are strictly weaker than Axiom 2. It is worthwhileto record what we lose by not requiring the preference relation to satisfy Continuity. LetA0 ⊂ A be the collection of all sets x that are either finite or the convex hulls of finitesets. Thus, x ∈A0 if and only if x is finite or x is the convex hull of a finite set. We thenhave the following representation.

Theorem 3.6. A binary relation å satisfies Axioms 1, 2a–c, 3–6 if and only if there exists anupper semicontinuous linear functional U : A0 → R, unique up to affine transformation, thatrepresents å and U admits a dual self representation.

Proof. The proof follows from the proof of Theorem 3.4.

Observe that the representation above allow permits ρ to be constant. We nowproceed towards the axiom which isolates this property. We have already seen one kindof excision in AoM. Another kind of excision is the notion that the only items in a menuthat matter to a decision-maker are the alternative he would have chosen were he nottempted and the item in the menu that causes him maximal temptation. For instance,suppose the decision-maker’s preferences are as follows: {b} � {b, c} � {c} � {c,m} �{m}. Thus, although c tempts b, c itself is tempted by m. Then, whenever both arepresent, we will require that the decision-maker is unaffected by the presence of c. Inother words, {b, c,m} ∼ {b,m}. We shall formalise this below. Let us say that β ∈ x istempted if there exists y ⊂ x such that {β} � {β} ∪y.

3.2 Proof-sketch of Theorem 3.4 13

Axiom 7 (SoM: Separability of Menus) If x is finite and β ∉ B(x ∪ {β}) is tempted,

x ∪ {β} ∼ x.

SoM says that the only alternative that matters in a menu (other than the decision-maker’sbest alternative in the menu) is the object that is maximally tempting. We want toexpress the idea that if the agent succumbs to temptation, she will fall all the way andchoose the most tempting alternative (from the perspective of the alter ego). This is astrong assumption, but it has the advantage of providing a lot of structure to the dualself representation. As with AoM, we only require SoM to hold for finite menus. Heretoo, requiring it for arbitrary menus neither changes Theorem 3.7 below nor simplifiesits proof.

Theorem 3.7. A binary relation å satisfies Axioms 1, 2a–c, 3–7 if and only if there exists anupper semicontinuous linear functional U : A → R, unique up to affine transformation, thatrepresents å and admits a dual self representation wherein

U(x) = (1− ρ)maxβ∈x

u(β)+ ρ maxβ∈Bv(x)

u(β).

Proof. See Appendix B.

We note again that SoM is a strong axiom and some readers may not find it particu-larly compelling. Nevertheless, from a consequentialist perspective, this axiom gives usthe utility of a menu as characterised by (a fixed convex combination of) the best optionand the most tempting option. For the readers who find behaviour of this type intu-itively appealing as a description, SoM is necessary and sufficient to generate it. We nowturn to a brief sketch of the proof of Theorem 3.4.

3.2 Proof-sketch of Theorem 3.4

The “only if” part of the proof is straightforward and is omitted. Here, we only sketchthe “if” part. The proof proceeds through a number of simple of arguments which wedescribe below.

1. Representing å. An application of the mixture space theorem (lemma A.2) showsthat Preferences, Continuity and Independence guarantee the existence of a contin-uous linear functional U unique up to affine transformation which represents å.Also U restricted to singletons is continuous.

4 Related Literature 14

2. The alter ego’s preferences. For lotteries α,β (with finite support) such that {β} �{β,α} å {α}, we stipulate that this must be because the alter ego strictly prefers αto β. From Regularity we see that for each β ∈ ∆, the set β+ := {α : {β} � {β,α} å {α}}is convex. Repeated application of AoM tells us that β− := {α : {β} ∼ {β,α} � {α}}is also convex. Thus, β+ and β− are disjoint, convex sets and there exists a linearfunctional vbeta that separates them on finite dimensional faces of ∆. Further-more, α ∈ β+ if and only if vβ(α) > vβ(β), i.e. adding a lottery to {β} makesthe decision-maker worse off if and only if the “alter ego” strictly prefers the newlottery to β. However, this linear functional depends on β and some face of ∆. Weshow next that “the same” linear functional performs the separation action for alllotteries in the domain.

3. Translation Invariance. We say that U is translation invariant if U(x) á U(y) ifand only if U(x + c) á U(y + c) for all signed measures c such that c(∆) = 0

and x + c,y + c ∈ A . That U is translation invariant follows from Continuityand Independence. We use this property to show that there is an essentially uniquelinear functional which performs the separation described in the previous step foreach lottery β. Thus, there exists a continuous linear functional which representsthe alter ego.

4. Finite menus. For any finite menu x, let β∗x ∈ x be such that u(β∗x) = maxβ∈x u(x)and let β̂x ∈ x be such that u(β̂x) = maxBv(x)u(β). Temptation and repeatedapplication of AoM implies that u(β∗x) á U(x) á u(β̂x).

5. Arbitrary menus. Using Continuity, we show that for all x, it is the case thatu(β∗x) á U(x) á u(β̂x) where β∗x and β̂x are defined as above.

6. Representation. A simple application of the intermediate value theorem gives usthe desired representation and ρx for each x.

4 Related Literature

4.1 The Requirement of Set Betweenness

Gul and Pesendorfer (2001) consider an agent who faces the same two-period problemas above. They introduce a condition on preferences called Set Betweenness (SB). This

4.1 The Requirement of Set Betweenness 15

requirement says that for all menus x,y ∈A ,

x å y -→ x å x ∪y å y. (SB)

Their representation theorem (as it pertains to us) says that if å satisfies axioms 1, 2a–c,3 and Set Betweenness, then the utility of a menu is given by a function UGP defined eitheras

UGP(x) = maxβ∈x

{u(β)+ v(β)} −maxβ∈x

v(β)

orUGP(x) = max

β∈Bv(x)u(β)

where UGP : A → R is linear and upper-semicontinuous, u,v : ∆ → R are continuous,linear functionals unique up to the same affine transformation and u = UGP |∆. GP referto u as the commitment utility and to v as the temptation utility. The first representationobtains if å is continuous and the second obtains if å is upper semicontinuous but notcontinuous. The second kind of representation described above is to account for the pos-sibility of “overwhelming temptation” where the agent always succumbs to temptation.To better understand the representation, let us assume, for the moment, that UGP(x) iscontinuous. This immediately rules out the overwhelming temptation representation.Let us write

c(β,x) := maxβ′∈x

v(β′)− v(β)

and interpret c(β,x) to be the cost imposed by the temptation whenever the most tempt-ing item is not chosen. We can now rewrite

UGP(x) = maxβ∈x

{u(β)− c(β,x)}

which says that the utility to the decision maker of a menu is determined (additively) bythe utility of the best choice in the menu and the cost it imposes through its selection.The interpretation is that in the second period, the agent makes the choice which max-imises the utility function u + v, which represents a compromise between the agent’scommitment utility and his temptation utility.

In spite of the very different motivation underlying GP’s model, it is surprising tonote that an agent satisfying axioms 1, 2a–c, 3 and Set Betweenness also satisfies our ax-ioms. (Whether he satisfies SoM or not depends on whether or not the preferences arecontinuous.) To see this, suppose UGP(·) is continuous so that for any menu x,

maxβ∈x

u(x) á UGP(x) á maxβ∈Bv(x)

u(β)

4.1 The Requirement of Set Betweenness 16

so that there exists some ρx such that

UGP(x) = (1− ρx)maxβ∈x

u(β)+ ρx maxβ∈Bv(x)

u(β).

Moreover, ρ also satisfies the other requirements of Theorem 3.4 so that UGP(x) admits adual self representation. The case where UGP(·) is only upper semicontinuous, ie where

UGP(x) = maxβ∈Bv(x)

u(β)

corresponds to the case where ρx = 1 for all x, i.e. the decision-maker gets tempted withprobability 1. Here too, UGP(·) admits a dual self representation.

Thus, GP’s representation implies our representation. Put another way, a decision-maker who satisfies Axioms 1–3 and Set Betweenness also behaves as if he has an alterego who makes a choice with probability ρx when the menu chosen is x. Thus anydifferences between the two models can only be detected by looking at second periodchoice. Nevertheless, there exist preferences which have dual self representations but donot satisfy GP’s axioms. This is demonstrated in Example 4.1 below. Before proceedingto the example, we need a little more simplifying algebra.

An obvious consequence of GP’s representation theorem is that any menu x is equiv-alent to a certain two-element subset. For two-element sets {α,β} where α tempts β, itis straightforward to note that

UGP({α,β}) = maxβ′∈x

{u(β′)+ v(β′)} − v(α).

Since the preferences also satisfy our axioms, we find that

ρ{α,β} = min{v(α)− v(β)u(β)−u(α), 1

}, (♥)

so thatUGP({α,β}) = (1− ρx)u(β)+ ρxu(α).

Thus, for general x,

UGP(x) = (1− ρx)maxβ∈x

u(β)+ ρx maxβ∈Bv(x)

u(β)

where ρx = ρ{α,β} with β ∈ Bu+v(x) and α ∈ Bv(x). We are now ready to present theaforementioned example (from Dekel, Lipman and Rustichini (2005)).

4.1 The Requirement of Set Betweenness 17

Example 4.1. Consider a weak-willed dieter who faces choices over broccoli (b)multipletemptations in the form of rich chocolate ice cream (c) and low-fat yogurt (y). A naturalset of rankings over menu is:

{b,y} � {y} and {b, c,y} � {b, c}.

Dekel, Lipman and Rustichini (2005) show that there is no GP representation that isconsistent with the above ordering. In other words, any extension of the ordering aboveto the space of all menus would violate one of GP’s axioms.

Nevertheless, there is a dual self representation that is consistent with the aboveordering. To see this most transparently, let u and v be as following:

b c y

u 6 0 4

v 0 8 6

so that utility over menus is given by

U(x) = (1− ρ̃x)maxβ∈x

u(β)+ ρ̃x maxβ∈Bv(x)

v(β)

where ρ̃x = 1/2ρ{α,β} and ρ{α,β} is given by equation (♥). We can interpret the utilityfunction U as representing an agent who is uncertain about the probability with whichhe will be tempted, wherein with probability 1/2 he has ρx = 0 for all x and with equalprobability has ρx = ρ{α,β} for all x where α ∈ Bu+v(x) and β ∈ Bv(x). ♦

We reiterate that for an agent who satisfies Set Betweenness, the implied second-periodbehaviour is vastly different under GP’s interpretation as compared to ours. Under GP’sinterpretation (in the no overwhelming temptation case), second period choice is madeaccording to the utility function u + v and the agent knows with certainty in the firstperiod that this will happen. Under our interpretation, the second period choice is madewith some (menu dependent) probability according to one utility function and withcomplementary probability according to another and the agent does not know in thefirst period which event will occur. (Needless to say, this does not apply in case the agenthas the overwhelming temptation representation in which case she knows with certaintythat the choice will be made by her alter-ego.)

4.2 Temptation and Multiple States 18

4.2 Temptation and Multiple States

In another recent paper, Dekel, Lipman and Rustichini (2005) consider generalisations ofGP preferences. They want to look at the class of preferences where the decision-makeris certain about the preferences of his untempted self. The only uncertainty is aboutwhat form the temptation takes. A further generalisation that they consider is in theform of the temptation cost. This is described below.

To model the uncertainty in temptation, DLR05 look at different linear functionsvj : Z → R which give rise to different cost functions

ci(β,x) :=∑j∈Ji

maxβ′∈x

vj(β′)−∑j∈Ji

vj(β).

The key to uncertainty about temptation then, is that the decision-maker is uncertainabout which cost function he will be facing. Thus, the generalisation of GP whichobtains, namely the temptation representation, can be written as

U(x) :=∑qi max

β∈x{u(β)− ci(β,x)}

where qi > 0 and∑qi = 1 which means that qi can be interpreted as the probability that

the decision-maker is faced with the ith temptation in the form of the cost function ci.(The temptation representation is a special case of a finite state additive EU representation.For details, the reader is referred to Dekel, Lipman and Rustichini (2005). It suffices hereto point out that a consequence of the finite state additive EU representation is that thereexists an N > 0 such that for all menus x, there exists a submenu x′ ⊂ x such that x′ ∼ xand |x′| à N.) The function u is such that u(β) := U({β}) and is called the commitmentutility. (All the functions u and vj are expected utility functions.)

To characterise such a preference, DLR05 introduce two more axioms. The first saysthat if the decision-maker could commit himself to a certain item in a menu, he would.This is made precise in the following axiom.

Axiom (DFC: Desire for Commitment) There exists α ∈ x such that {α} å x.

DFC strongly resembles (the first half of) Temptation. Indeed, there exists preferencesthat admit temptation representations that also satisfy Temptation but do not admit adual self representation. This is demonstrated in example 4.3 below. The second axiomintroduced in DLR05 is as follows.

4.2 Temptation and Multiple States 19

Axiom (Domination) 4 If there exists α ∈ B(x ∪ {β}) such that {α} ∼ x and β ∉

B(x ∪ {β}), then there exists ε > 0 such that for all β̂ ∈ Nε(β) and all x′,

x′ ∪ x å x′ ∪ x ∪ {β̂}.

The main representation theorem in DLR05 is then the following:

Theorem 4.2. A preference relation å has a temptation representation if and only if it has afinite state additive EU representation and satisfies DFC and Dominance.

It is straightforward to show that our representation implies Domination. Therefore,it would seem that any continuous preference (in the Hausdorff topology) which has adual self representation strictly belongs to the class of preferences identified in DLR05.But this is not the case, primarily because we do not have a finiteness axiom. But let usfirst look at a a couple of examples where the decision-maker satisfies the DLR05 axiomsbut does not have a dual self representation.

Example 4.3. Let Z be a finite set and let ∆ be the space of lotteries over Z. Define utilityfunction u, v1 and v2 so that

u v1 v2α 0 2 2

β 0 1 6

Now let c(γ,x) :=∑j[maxγ′∈x vj(γ′)

]−∑j vj(γ). Then U(x) := maxβ∈x [u(γ)− c(γ,x)]

is a temptation representation.

Let x := {α,β}. Then c(α,x) = v1(α) + v2(β) − v1(α) − v2(α) = 4 and c(β,x) =v1(α) − v1(β) = 1. This means that U(x) = maxγ∈x

{u(γ)− c(γ,x)

}= −1. Thus,

{α} ∼ {β} � x which is not possible in a dual self representation. (Note that thisexample does not violate Temptation because B(x) = x = W(x).) ♦

Notice that if in the temptation representation, each cost function depends on onlyone temptation i.e. each Ji is a singleton, such an example could not arise. In a dualself representation, the decision maker does not care about the utility level of the alterego. She only cares about the choice made by the alter ego insofar as it affects her ownutility level. If we were to interpret the different vj’s in a temptation representation as

4This axiom is from an earlier version of their paper. In the present version, it is called AIC: Approxi-mate Improvements are Chosen.

4.2 Temptation and Multiple States 20

belonging to different selves who are the cause of the temptation, then we could say that atemptation representation is an interdependent utilities model, where the decision maker(or the “commitment self” in the terminology of DLR05) cares about the utility levelsof the different selves. We shall now see another example of a temptation representationwhich does not admit a dual self representation. This time our axiom Regularity will beviolated.

v1

α1

v2

α2

uβ

½α1 +½α2

Example 4.4. Suppose the decision-maker has utility function u and is faced with twotemptations denoted by expected utility functions v1 and v2. Also, suppose that vj isnot a convex combination of vi and u for i ≠ j. Then, such a configuration might lookas in the figure above. But such a configuration would violate our axiom Regularity.The violation is because the lottery α1 and α2 which both tempt β, but there also existlotteries over α1 and α2 which do not tempt β. ♦

It should be remarked that the construction above holds in any temptation repre-sentation where there is uncertainty about the form the temptation will take. We shallnow show that there exist preferences that have dual self representations but do not havetemptation representations. Recall that GP preferences of the following form

Ui(x) = maxβ∈x

{u(β)+ kv(β)} −maxβ∈x

kv(β)

for k ∈ [0,∞), have a dual self representation with

ρx(k) = min{kv(α)− v(β)u(β)−u(α), 1

}where β ∈ Bu+kv(x) and α ∈ Bkv(x). Now, let (ki) be a sequence such that ki ∈ [0,∞)and the following holds: for x :=

⋃i{βi} ∪ α, βi ∈ Bu+kiv(x), βj ∈ Bu+kiv(x) implies

ki = kj and α ∈ Bv(x).

4.3 Other Papers 21

Let (λi) be another sequence such that λi ∈ [0,1] for each i and∑i λi = 1. Now, it

easy to see thatU(x) = (1− ρx)max

β∈xu(β)+ ρx max

β∈Bv(x)u(β)

where ρx =∑i λiρx(ki), is part of a dual self representation with ρx satisfying all the

necessary conditions for this to be so. By choosing the ki’s and λi’s appropriately, itis therefore possibly to construct utility functions that admit dual self representationsbut not temptation representations. In other words, these utility functions are such thatthe utility of a menu can depend on an arbitrarily large (countable) sub-menu. This isbecause each ρ(ki) depends on {βi, α} and we can make the set of βi’s arbitrarily large.Indeed, this is the spirit of Example 4.1. We can further generalise this construction(as suggested by Bart Lipman). Let µ be a Borel probability measure on [0,1) and letρx =

∫ρx(k) dµ(k) so that we have a dual self representation but not a temptation repre-

sentation.

4.3 Other Papers

Several recent papers have focused on the problems raised by Schelling. The paper closestin spirit to ours is the innovative paper by Bernheim and Rangel (2004), who specificallydeal with addiction and are clear that, in their view, the individual who takes drugs ismaking a mistake caused by overestimating the amount of pleasure consumption wouldinvolve relative to the long-term costs of such consumption. The selves are not treatedsymmetrically; drug consumption is anomalous and abstaining from it rational. Theirmodel also explicitly takes into account the effect of environmental cues in triggering thechange of the controlling self, from cold to hot. Here the cold self is supposed to be thepreference that usually represents the agent, while the hot self is the one who makes theanomalous choices. Fudenberg and Levine (2005) adopt an explicitly dual self model forthese dynamic choice problems and focus on the game between the selves rather than onthe axiomatisation of a virtual dual self model, as we do here. Eliaz and Spiegler (2004)study contracting issues with several preference representations, including dual selves.5

The spirit of our approach (ie considering the planning stage of a choice problem)is introduced in Kreps (1979) who characterises preferences that value flexibility whilelooking at a finite set of prizes. Dekel, Lipman and Rustichini (2001) extend Kreps’model by looking at lotteries over a finite set of prizes and menus that are sets of lotteries.

5In this context, also see Esteban and Miyagawa (2005a,b) and Esteban, Miyagawa and Shum (2003).

4.4 Exogenous States of the World 22

Gul and Pesendorfer (2001) extend this environment to a compact metric space of prizesto provide a characterisation of temptation. They also emphasise the importance of notbreaking the link between choice and welfare. Samuelson and Swinkels (2006) explorethe evolutionary foundations of temptation. They develop a model where endowinghumans with utilities of menus that depend on unchosen alternatives is an optimal choicefor nature from an evolutionary perspective.

4.4 Exogenous States of the World

Our representation admits a straightforward extension to finite exogenous states. Thiswould be the formal equivalent of the model studied by Bernheim and Rangel (2004)limited to two periods. Formally, let S be a finite set of states with the probabilitythat state s ∈ S occurs being given by πs. The state is realised after the decision-makerchooses the menu. We take this to be some set of exogenous circumstances that affect theagent only inasmuch as they affect the likelihood of her getting tempted. Note that theagent’s utility function does not change across states nor does her alter ego’s. The onlything that changes is the probability of getting tempted. In particular, we are looking fora utility function (over menus) that looks like the following:

U(x) =∑s∈Sπs

((1− ρsx)max

β∈xu(β)+ ρsx max

β∈Bv(x)u(β)

).

In particular, note that the probability of getting tempted can depend on both the menuand the state.

Example 4.5. Let S := {0,1, . . . , n} and let ρxi be the probability of getting tempted inthe state of the world i. Then one specification could be the following:

ρix < ρi+1x

for all x and ρ0x = 0 for all x. If in addition we assumed ρix = ρiy for all x,y ∈ A for alli ∈ S, we would get the Bernheim-Rangel model. ♦

5 Conclusion

In this paper, we consider a decision-maker faced who has to decide on the set of feasiblechoices from which an actual choice will be made at a later point in time. We rule

5 Conclusion 23

out the case where the decision-maker may prefer larger sets of feasible choices due to apreference for flexibility.

Our main contribution is to provide axioms on first period preferences that enableus to interpret this problem as a decision-maker who behaves as if he has an alter ego(with preferences different from her own), who makes the actual choice from the menuwith some probability. Doing so enables us to address problems where decision-makersdemonstrate apparent dynamic inconsistency (i.e. make ex-post choices that are inferiorfrom an ex-ante perspective) and make unambiguous welfare statements in these situa-tions. We also relate our model to the papers of Gul and Pesendorfer (2001) and Dekel,Lipman and Rustichini (2005).

Possible extensions include more explicit study of dynamic applications. In our ear-lier working paper, Chatterjee and Krishna (2005), we considered an example of a bar-gaining problem as an application of our framework. We plan to develop this in a laterpaper.

A Proof of Theorem 3.4 24

A Proof of Theorem 3.4

The “only if” part of the proof is straightforward. Here we shall demonstrate the “if”part of the proof. We first begin with a crucial lemma which can also be found in DLR.The proof is presented here for expositional ease. Recall that for any set x, its convexhull is denoted by conv (x).

Lemma A.1. Let å satisfy Independence. Then for all finite x, x ∼ conv (x). Furthermore,if å satisfies Continuity, then x ∼ conv (x) for all x ∈A .

Proof. We shall proceed through a series of simple arguments.Step 0. Let us denote the standard (n − 1)-simplex by ∆n−1 := {p ∈ Rn :

∑pi = 1}

and its vertices by Vert (∆n−1). It is straightforward to verify that for all λ ∈ (0,1/n),λVert (∆n−1) + (1 − λ)∆n−1 = ∆n−1. Now, let x be the vertices of the (n − 1)-simplexconv (x). In other words, |x| = n, dim(conv (x)) = n − 1 and conv (x) is linearlyisomorphic to ∆n−1. Thus, for all λ ∈ (0,1/n), λx + (1− λ) conv (x) = conv (x).

Step 1. Let x now be any finite set. We know that conv (x) has a simplicial de-composition, K such that Vert (K) = x and each simplex has the same dimension asconv (x). Now apply the result of Step 0 to each simplex to conclude that for anyλ ∈

(0, 1

dim(conv (x))+1

), λx + (1− λ) conv (x) = conv (x).

Step 2. For any finite x, suppose conv (x) � x. Then, for any λ ∈(0, 1

dim(conv (x))+1

),

conv (x) = λ conv (x) + (1 − λ) conv (x) � λx + (1 − λ) conv (x) = conv (x) which isa contradiction (where the relation in the middle follows from Independence). Alterna-tively, suppose x � conv (x). Then conv (x) = λx+ (1−λ) conv (x) � λ conv (x)+ (1−λ) conv (x) = conv (x) which is again a contradiction. Thus, x ∼ conv (x).

Step 3. Let x be an arbitrary closed set. Then (because x is compact), there exists asequence (xk) of finite sets such that xk ⊂ x and xk → x (in the Hausdorff metric). Thus,conv (xk)→ conv (x). From the continuity of å, we see that x ∼ conv (x).

We shall now show that there exists a continuous linear functional that representspreferences. (This is Proposition 2 in DLR.) Recall that A is the space of all closedsubsets of ∆.

Lemma A.2. If å satisfies Axioms 1, 2 and 3, then there exists a continuous linear functionalU : A → R that represents å. Furthermore, U is unique up to affine transformations.

A Proof of Theorem 3.4 25

Proof. Let X ⊂ A be the space of all closed convex subsets of ∆. Notice that X endowedwith the Minkowski sum is a mixture space. It only remains to verify the mixturespace axioms (see Kreps, 1988, page 52). By assumption, Independence holds. Continuityensures that vN-M continuity is also satisfied. Thus, by the mixture space theorem, thereexists a linear functional V : X → R so that for all x,y ∈ X, V(x) á V(y) if and only ifx å y.

We now extend V to all menus. Let us define U : A → R as follows: for all x ∈A , letU(x) := V (conv (x)). It is easily seen that U represents å. All that remains to be shownis that U is linear.

From lemma A.1, it follows that λx + (1 − λ)y ∼ conv(λx + (1− λ)y

). Also

x ∼ conv (x) and y ∼ conv (y). From Independence it follows that λx + (1 − λ)y ∼λ conv (x) + (1 − λ)y and λ conv (x) + (1 − λ)y ∼ λ conv (x) + (1 − λ) conv (y), i.e.λx + (1− λ)y ∼ λ conv (x)+ (1− λ) conv (y). Therefore,

U(λx + (1− λ)y) = U(λ conv (x)+ (1− λ) conv (y)

)= V(λ conv (x)+ (1− λ) conv (y))

= λV(conv (x))+ (1− λ)V(conv (y))

= λU(x)+ (1− λ)U(y)

which is the desired result.

Note the heavy use of Independence in the proof. We are identifying the mixturewhich gives x with probability µ and y with probability 1 − µ with the convex combi-nation of x and y, µx+ (1−µ)y, an identification which lies at the heart of the mixturespace theorem. Let us define u(α) := U({α}) and interpret it to be the decision-maker’sutility from a lottery (in the untempted state). It is clear that u is a continuous, lin-ear function. Another important property of preferences that we shall make us of istranslation invariance. This is made precise below.

Definition A.3. A binary relationå is translation invariant if x å y implies x+c å y+cfor all signed measures c such that c(∆) = 0 and x+ c,y + c ∈A . Such a c will be called anadmissible translate.

Lemma A.4. Let å satisfy Axioms 1 – 3. Then, å is translation invariant.

Proof. As before, let X ⊂A be the space of closed convex subsets of ∆. Notice again thatX endowed with the Minkowski sum is a mixture space.

A Proof of Theorem 3.4 26

Let x,y ∈ X be such that x å y and c an admissible translate. Simple geometryshows that for all λ ∈ (0,1),

λy + (1− λ)(x + c) = λ{λy + (1− λ)(y + c)} + (1− λ){λx + (1− λ)(x + c)}.

Then, since ∼ is reflexive,

λy + (1− λ)(x + c) ∼ λ{λy + (1− λ)(y + c)} + (1− λ){λx + (1− λ)(x + c)}.

From Independence we get

λx + (1− λ)(x + c) å λy + (1− λ)(x + c).

Combining the relations above

λx + (1− λ)(x + c) å λ{λy + (1− λ)(y + c)} + (1− λ){λx + (1− λ)(x + c)}.

From Independence we see that

λx + (1− λ)(x + c) å λy + (1− λ)(y + c).

But from lemma A.2 above, there exists a linear functional V that represents å (on X).Thus,

V(λx + (1− λ)(x + c)) á V(λy + (1− λ)(y + c)).

Using the linearity of V and rearranging terms gives us for each λ ∈ (0,1),

λ[V(x)− V(y)]+ (1− λ)[V(x + c)− V(y + c)] á 0. («)

Now suppose by way of contradiction, y + c � x + c, i.e. V(y + c) > V(x + c). Thiswould mean that there exists some λ ∈ (0,1) which does not satisfy («) yielding thedesired contradiction.

Since the space of all measures on (Z,d), denoted by M(Z), is a locally convex topo-logical vector space (endowed with the weak* topology), it is the case that for all x ∈A ,and for all admissible translates c, conv (x + c) = conv (x) + c. (This follows from thefact that in a locally convex topological vector space, the convex hull of a compact set isclosed. See Aliprantis and Border, 1999, lemma 5.57, pp. 193.) From the reflexivity of ∼,conv (x + c) ∼ conv (x)+ c.

Suppose x,y ∈ A and x å y. Then, conv (x) å conv (y) and for an admissibletranslate c, conv (x) + c å conv (y) + c. Then x + c ∼ conv (x + c) ∼ conv (x) + c åconv (y)+ c ∼ conv (y + c) ∼ y + c.

A.1 The Alter ego’s Preferences 27

Lemma A.5. Suppose Axioms 1–6 hold. Then there exists a continuous, linear functionalv : ∆ → R such that (i) {β} � {α,β} å {α} if and only if v(β) < v(α) and u(β) > u(α),and (ii) for all x, there exists β̂x ∈ x such that U(x) á u(β̂x) and u(β̂x) = max

β∈Bv(x)u(β).

Proof. See §A.1 below.

Now, there exists β∗ ∈ x so that u(β∗x) á U(x) from where we can determine ρxusing the Intermediate Value Theorem, which completes the proof. The translationinvariance of ρ follows from the translation invariance of U (see lemma A.4 above) andthe other properties of ρ are also easily obtained.

We should note that the proof above requires the following to hold.

Lemma A.6. Let x ∈A such that {β} ∼ {β′} for all β,β′ ∈ x. Then, x ∼ {β}.

Proof. There exist β◦, β◦ ∈ ∆ such that u(β◦) á u(β) á u(β◦) and at least one of theinequalities is strict since u(β◦) > u(β◦). Let x be such that {β} ∼ {β′} for all β,β′ ∈ xand let β̄ ∈ Bv(x). Suppose u(β̄) > u(β◦). Let (xn) be a sequence such that xn → x,|xn| = n and β̄ ∈ xn for all n so that we can write xn := {β̄, β1, . . . , βn−1}. Now definexλn := {β̄, λβ◦+ (1−λ)β1, . . . , λβ◦+ (1−λ)βn−1}. Then, for λ small enough, AoM impliesxλn ∼ {β̄} so that Continuity implies that limλ→0 xλn = xn ∼ {β̄}. Once again Continuitygives us {β̄} ∼ x since xn → x.

A.1 The Alter ego’s Preferences

In this section, we shall construct the alter ego’s preferences via some revealed preferencearguments thereby providing a proof of lemma A.5.

Let us define β+ := {α : {β} � {β,α} å {α}}. From Regularity, it follows that β+ isconvex. Let us also define β− := {α : {β,α} ∼ {β} � {α}}. The lemma below shows thatβ− is also convex.

Lemma A.7. Suppose å satisfies Axioms 1, 3 and 6. Then, β− is convex.

Proof. Let α1, α2 ∈ β−. By Independence and {β} ∼ {β,α2},

{β} ∼ λ{β} + (1− λ){β,α2}.

Independence and {β} ∼ {β,α1} also implies

λ{β} + (1− λ){β,α2} ∼ λ{β,α1} + (1− λ){β,α2}.

A.1 The Alter ego’s Preferences 28

Transitivity of å implies

{β} ∼ λ{β,α1} + (1− λ){β,α2}.

But note that

λ{β,α1} + (1− λ){β,α2} = {β,λα1 + (1− λ)β, λβ+ (1− λ)α2, λα1 + (1− λ)α2}.

Applying AoM twice, we find {β} ∼ {β,λα1+(1−λ)α2}. Since {β} � {α1}, Independencegives us {β} � (1−λ){β}+λ{α1}. Also, {β} � {α2} and Independence implies (1−λ){β}+λ{α1} � λ{α1} + (1− λ){α2}. By the transitivity of �, {β} � {λα1 + (1− λ)α2}. Thus,β− is convex.

Let us recall some definitions of objects in linear spaces. An affine subspace (or linearvariety) of a vector space is a translation of a subspace. A hyperplane is a maximal properaffine subspace. If H is a hyperplane in the vector space M(Z), then there is a linearfunctional f on M(Z) and a constant c such H = {x : f(x) = c}. Moreover, H is closedif and only if f is continuous (Luenberger, 1969, pp. 129, 130). For notational ease, weshall write H as [f = c]. Similarly, (two of) the negative and positive half spaces arerepresented as [f à c] and [f > c] respectively. For any subset S ⊂ V, let aff (S) denotethe affine subspace generated by S, i.e. the smallest affine subspace that contains S. (Inwhat follows, if β∗+ = ∅, let v = u = U|∆. Henceforth, we shall assume β∗+ is not empty.)We shall also denote the zero vector by θ. We shall first demonstrate that the alter ego’spreferences can be represented on any finite dimensional face of ∆. For this, we needsome definitions. Let x ∈ A so that Fx := aff (x) ∩ ∆ is a face of ∆. If x is essentiallyfinite, then Fx is a finite dimensional face.

Lemma A.8. For any x ∈ A0, let β∗ ∈ ri Fx. Then there exists vx : ∆ → R which iscontinuous and linear so that β∗− ⊂ [vx à vx(β∗)] and β∗+ ⊂ [vx > vx(β∗)].

Proof. Since β∗− and β∗+ are disjoint convex subsets of a finite dimensional Euclidean space(which we can take to be aff (Fx) − β∗), there exists a continuous linear functional thatseparates them. We denote this functional by vx.

We have thus far established that for some β∗ ∈ x, there exists a continuous linearfunctional vx that represents the alter ego’s preferences at that point. We will now show

A.1 The Alter ego’s Preferences 29

that there is a single continuous, linear functional which represents the alter ego’s pref-erences over all of x. (We shall use Translation Invariance towards this end.) Note thatfor β ∈ ri Fx, there exists ε > 0 such that Nε(β)∩ ∆ ⊂ ri Fx. Also recall a fact about theHausdorff metric, dh. For all λ ∈ [0,1], dh ({β}, {β,λα+ (1− λ)β}) = λdh ({α}, {β}) .

Lemma A.9. For all β ∈ Fx, [vx = vx(β)] separates β− and β+.

Proof. Suppose not. Then there exists β ∈ Fx such that either(i) ∃ α ∈ β− such that vx(α) á vx(β), or(ii) ∃ α ∈ β+ such that vx(β) á vx(α).

Let us consider the first possibility.Let c = β∗ − β. Since β∗ ∈ ri Fx, there exists ε > 0 such that Nε(β∗) ∩ ∆ ⊂ ri Fx.

From Translation Invariance, we can assume α ∈ Nε(β)∩ Fx. Thus, α+ c ∈ Nε(β∗)∩ Fx.Since vx is continuous and linear, vx(α + c) > vx(β + c) = vx(β∗). This implies that{β+ c} � {β+ c,α+ c} which, by Translation Invariance,6 is equivalent to {β} � {β,α}which is a contradiction of the hypothesis that α ∈ β−.

The second possibility is taken care of with a similar argument, thus establishing thedesired result.

We now define the binary relation R ⊂ ∆ × ∆ as follows: α R β if (i) ∀ (αn) suchthat αn → α and for all n, αn tempts β (ie {β} � {αn, β}) or (ii) ∃ (βn) such that βn → β

and for all n, {α} ∼ {α,βn} (ie β does not tempt α). Notice that for any x ∈ A0, R|Fx isrepresented by vx. We shall show that R has an expected utility representation. We firstclaim that R is complete and transitive.

Claim A.10. R is complete and transitive.

Proof. For any α,β ∈ ∆, let A0 3 x 3 α,β so that dim(Fx) á 2. It is easy to see nowthat either α R β or β R α so that R is complete. To see that R is transitive, suppose notso that there exists α,β, γ ∈ ∆ such that α R β and β R γ but ¬(α R γ). Consider againx ∈ A0 such that α,β, γ ∈ x and dim(Fx) á 2. Since R|Fx is represented by vx, this isimpossible.

We shall now show that R is continuous.

Claim A.11. R is continuous.6Notice that we only require Translation Invariance to hold for two-element subsets.

A.1 The Alter ego’s Preferences 30

Proof. For any β ∈ ∆, define β− :={α ∈ ∆ : u(α) > u(β) and {α} � {α,β}

}and

β+ :={α ∈ ∆ : u(α) á u(β) and {α} ∼ {α,β}

}. Thus, the R-lower contour set at β is

β− ∪ cl (β−) and similarly for the upper contour set. We shall show that R is continuousby demonstrating that β− is closed and β+ is open. A symmetric argument for the otherregions will complete the proof.

We first recall that U : A → R is upper semicontinuous. To show that β− is closed,consider a convergent sequence (αn) in β− and let αn → α. Then, by definition of β−and from the upper semicontinuity of U , U({α,η}) = lim supnU({αn, β}) = U({β}) sothat α ∈ β− which proves that β− is closed.

To show that β+ is open, fix α ∈ β+ and let ε = U({β}) − U({α,β}) > 0. Then,by the upper semicontinuity of U , there exists δ > 0 such that for all α′ ∈ Nδ(α),U({α′, β}) < U({α,β})+ ε = U({β}). Hence, β+ is open in ∆.

Recall that R, by definition, satisfies Translation Invariance. Chatterjee and Krishna(2006) show that a preference relation R that is continuous and is translation invarianthas an expected utility representation.7 Let v : ∆→ R represent R.

Lemma A.12. For all finite x

maxβ∈x

u(β) á U(x) á maxβ∈Bv(x)

u(β).

Proof. Let β∗ ∈ x such that u(β∗) = maxβ∈x u(β) and let β̂ ∈ x such that u(β̂) =maxβ∈Bv(x)

u(β). Let x′ := {α ∈ x : u(α) á u(β̂)} and y := {α ∈ x : u(α) < u(β̂)}.

Then, x = x′ ∪ y and by Temptation, β∗ å x′ å β̂. Let y := {α1, α2, . . . , αn}. It thenfollows that for each αi ∈ y, {β} ∼ {β,αi}. By AoM, it follows that {β} ∼ {β,α1, α2}.Repeatedly applying AoM implies {β} ∼ {β} ∪ y. Once again applying AoM implies{β} ∪ x′ ∪y ∼ {β} ∪ x′ = x′. Thus, x ∼ x′ and {β} å x å {β̂}.

Lemma A.13. For any x,

maxβ∈x

u(β) á U(x) á maxβ∈Bv(x)

u(β).

7It follows, therefore, that for a continuous preference relation, Independence is equivalent to Transla-tion Invariance. This is easy to see once one notices that Translation Invariance implies Betweenness in thesense of Dekel (1986) from which it is easy to show that Independence follows.

B Proof of Theorem 3.7 31

Proof. Let β∗ ∈ x such that u(β∗) = maxβ∈x u(β) and let β̂ ∈ x such that u(β̂) =maxβ∈Bv(x)u(β). Let (xk) be a sequence of finite sets where |xk| = k, xk ⊂ x, xk → x andβ∗, β̂ ∈ xk for each k.

By Temptation, u(β∗) á U(xk) for each k. Hence, by Continuity, u(β∗) á U(x).Also, for each k, U(xk) á u(β̂). Once again, Continuity implies that U(x) á u(β̂). Thisgives us the desired result.

B Proof of Theorem 3.7

We shall prove the theorem for finite menus. Towards this end, we show that thereexists a linear functional that represents preferences over essentially finite menus (whichare defined below). Also, for finite menus x, x ∼ conv (x) and Translation Invarianceholds. SoM is then used to derive the representation for finite menus. A straightforwardcontinuity argument then extends the result to arbitrary menus.

Let us call a set x ⊂ ∆ essentially finite if x is finite or if x is the convex hull of afinite set. Let us denote by A0, the space of all essentially finite menus, i.e. the space ofall essentially finite subsets of ∆. Also define X0 ⊂ X as the space of all closed, essentiallyfinite, convex subsets of ∆. Before we begin, recall that lemma A.1 shows that for eachfinite x, x ∼ conv (x).

Lemma B.1. Let å satisfy Axioms 1, 2a–c and 3. Then there exists an upper-semicontinuouslinear functional U : A0 → R such that for all x,y ∈A0, x å y if and only if U(x) á U(y).Also, U is unique up to affine transformation.

Proof. The proof is similar to that of lemma A.2. We shall only provide a sketch here.Notice that X0 is a mixture space. Since Independence and von Neumann-Morgensterncontinuity also hold, there exists a V : X0 → R, unique up to affine transformation, so thatfor all x,y ∈ X0, x å y if and only if V(x) á V(y). Now define U(x) := V(conv (x))for all x ∈ A0. Linearity of U is demonstrated as in lemma A.2. Upper-semicontinuityfollows from Axiom 2a.

Another property that we shall establish for essentially finite menus is translationinvariance.

Lemma B.2. Let å satisfy Axioms 1, 2a–c and 3. Then for all x,y ∈ A0, x å y impliesx + c å y + c where c is an admissible translate.

B Proof of Theorem 3.7 32

Proof. The proof is similar to the proof of lemma A.4. Notice that the proof of lemmaA.4 only relied on the existence of linear functional that represented preferences. Fromlemma B.1, such a functional exists which gives us the desired result. (Indeed, all that isto be changed in the proof of lemma A.4 is to read X0 for X and A0 for A .)

We shall now construct the alter ego’s preferences. The construction in §A.1 goesthrough without change. Recall that in the construction of v, we only used the UpperSemicontinuity of preferences and Translation Invariance for two-element subsets. Re-peated application of AoM as in §A.1 gives us the following lemma.

Lemma B.3. Suppose Axioms 1, 2a–c, 3–6 hold. Then there exists a continuous, linearfunctional v : ∆ → R such that (i) {β} � {α,β} å {α} if and only if u(β) > u(α) andv(β) < v(α), and (ii) for all x ∈ A0, there exists β̂x ∈ x such that U(x) á u(β̂x) andu(β̂x) = max

β∈Bv(x)u(β).

It follows from Temptation that for all x ∈ A0, maxβ∈x u(β) á U(x) á u(β̂x). Thedual self representation follows immediately giving us ρx for each x. The properties ofρ which are required for it to be part of a dual self representation are easily verified.

We now prove a simple lemma which shows that we can restrict attention to essen-tially finite menus that lie entirely in the relative interior of ∆. (In what follows, we shalldenote the ε-neighbourhood (in ∆) of a point β ∈ ∆ by Nε(β) and the diameter of a setx by diam (x). We shall also repeatedly use the fact that ρ must be consistent with thelinearity of U , i.e. (♣) holds.) For any y ∈ A , notice that Fy := aff (y)∩∆ is a face of ∆(where aff (y) is the affine hull of y in the vector space M(Z)).

The lemma shows the for any essentially finite set y, there exists another menu x thatlies in the relative interior of some finitely generated face of ∆. Also, the diameter of xcan be made arbitrarily small such that ρy = ρx. This result is useful because restrictingattention to such an x enables us to consider arbitrary perturbations of x that lie inaff (x).

Lemma B.4. For all y ∈A0 and for all ε̄ > 0, there exists x ∈A0 so that x ⊂ ri Fy , ρy = ρxand diam (x) < ε̄.

Proof. Let y ∈ A0 and let β̂ be an extreme point of conv y. Also, let ε̄ > 0. Then,for all λ ∈ (0,1), ρλ{β̂}+(1−λ)y = ρy . Moreover, for all ε > 0, there exists λε ∈ (0,1)

such that λε{β̂} + (1− λ)y ⊂ Nε(β̂)∩ Fy . Let us now take ε ∈ (0, ε̄/2) so that for some

B Proof of Theorem 3.7 33

β∗ ∈ ri Fy , Nε(β∗) ⊂ ri Fy . Let c := β∗ − β̂ be a signed measure so that c(∆) = 0. Bythe translation invariance property of ρ, it follows that for x := λε{β̂} + (1 − λε)y + c,ρx = ρλε{β̂}+(1−λ)y = ρy . (The case where u(β̄) = u(β◦) is dealt with similarly by notingthat in such a case, u(β◦) > u(β̄).)

Lemma B.5. Let å have a dual self representation and satisfy SoM. Then, for all finite x, forany β ∈ Bu(x) and for any α ∈ Bu (Bv(x)), x ∼ {β,α}.

Proof. Let x̂ := {β1, . . . , βm} ∪ x′ ∪ {α1, . . . , αn} ∪ y where βi ∈ Bu(x̂) for i = 1, . . . ,m,αj ∈ Bu(Bv(x̂)) for j = 1, . . . , n, u(β1) > u(γ) á u(α1) for all γ ∈ x′ and u(α1) > u(γ′)for all γ′ ∈ y. (Note that by definition, v(α1) > v(βi) and v(α1) á v(γ′) for all γ′ ∈ y.)

Recall that β ∈ x is untempted in x if there exists α′ ∈ x

u

v

β1β2

α1

γ1

γ2

γ′1 γ′2

β1

2

such that {β} � {α′} and for all α ∈ x, {β} æ {β,α}. Sinceα1 is untempted in x̂ (which means, among other things, that{α1} ∼ {α1} ∪ {α2, . . . , αm} ∪ y), by AoM, x̂ ∼ {β1, . . . , βn} ∪x′ ∪ {α1}. Also, by AoM, x̂ ∼ {β1, . . . , βm} ∪ x′ ∪ {α1}. BySoM, x̂ ∼ {β1, . . . , βm} ∪ {α1}. Let x := {β1, . . . , βm} ∪ {α1}.From lemma B.4, we can assume, without loss of generality,that x ⊂ ri Fx.

Let (βki ) be a sequence in ri Fx such that {βki } � {βk+1i } �{βi} and βki ∈ conv

({β1i , α1}

)for each k and limk→∞ βki = βi.

Since x ⊂ ri Fx, it is clear that such a sequence always exists.

Let xki = {βki , α1}. By SoM, it follows that {β1, . . . , βki , . . . , βm}∪{α1} ∼ xki . Furthermore, xki = λkx1i + (1 − λk){α1} for someλk ∈ (0,1) and λk > λk+1. Therefore, ρxki = ρx1i . This impliesthat xki � xk+1i . By Upper-Semicontinuity, it now follows that

x = limk{β1, . . . , βki , . . . , βm} ∪ {α1} ∼ lim

kxki = xi

which gives us the desired result.

Lemma B.6. Let å have a dual self representation and satisfy SoM. Then for all β,α,α′

such that {β} � {α} ∼ {α′} and α,α′ tempt β, it is the case that {β,α} ∼ {β,α′}. Hence,ρ{β,α} = ρ{β,α′}.

Proof. By lemma B.4, we can assume that β ∈ ri F{β,α,α′}, so there exists ε > 0 such thatNε(β) ⊂ ri F{β,α,α′}. We can also assume that α,α′ ∈ Nε/4(β).

B Proof of Theorem 3.7 34

Let v(β) < v(α) < v(α′) and let c := α′ − α, c′ := α − β. By hypothesis, α temptsβ, so that α + c = α′ tempts β + c = β′. By Translation Invariance, {β,α} ∼ {β′, α′}.Also, {β} ∼ {β′}. To see this, suppose the contrary, ie suppose {β} � {β′}. By definition,β′ = β+c = β+(α′−α). By Translation Invariance, {β}+c′ � {β′}+c′, i.e. {β}+(α−β) �{β}+(α′−α)+(α−β)which is equivalent to {α} � {α′}which contradicts the hypothesis.

Hence β,β′, α′ and α form the vertices of a parallelogram. By Lemma B.5, {β,α′} ∼{β,β′, α′} ∼ {β′, α′}. This proves that {β,α} ∼ {β,α′}. Since {α} ∼ {α′}, it follows fromthe representation that ρ{β,α} = ρ{β,α′}.

Proof of Theorem 3.7 for finite x. From lemma B.5, it follows that for any x, there existelements β,α ∈ x such that {β,α} ∼ x. Therefore, we can restrict attention to twoelement subsets. Let x = {β,α} and y = {β′, α′} where x,y ∈ ri Fx∪y , ε = diam (x) á

diam (y) > 0 and Nε(β)∩∆ ⊂ ri Fx∪y .

Let c := β− β′. Then, y + c ⊂ Nε(β) and ρy ∼ ρy+c. If u(α) > u(α′ + c), then thereexists λ ∈ (0,1) so that u(λβ + (1 − λ)(α′ + c)) = u(α). Appealing to lemma B.6 nowgives us the desired result. (The case where u(α) à u(α′ + c) is dealt with in a similarfashion.)

We can now prove Theorem 3.7 for arbitrary menus.

Proof of Theorem 3.7. Let x ∈ A , β ∈ Bu(x) and α ∈ Bu(Bv(x)). If β ∈ Bu(Bv(x)), thenby Temptation, we are done. Let us assume this isn’t the case.

Consider a sequence (xk) such that for each k, xk ∈ A0, xk ⊂ x, |xk| < |xk+1| andlimk xk = x. Define αk := λkβ + (1 − λk)α for λk ∈ (0,1). We will also require that foreach k, β ∈ xk and αk ∈ Bu(Bv(xk)). Then, xk ∼ {β,αk} and U(xk) = ρU({β}) + (1 −ρ)U({αk}). Now, limk{β,αk} = {β,α}. Also, for each k, {β,αk} � {β,αk+1}, i.e. xk �xk+1. From Upper Semicontinuity, it follows that U(x) = limkU(xk) = limkU({β,αk}) =U({β,α}).

REFERENCES 35

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