A Geometric Approach to Temptation and Self-Control

By making use of a geometry of preferences, Abe (2012) proves the Gul and Pesendorfer’s utility representation theorem about temptation without self-control. This companion paper provides a similar proof for the Gul and Pesendorfer's utility representation theorem about temptation and costly self-control. As a result, the both theorems are proved in the unified way.


Introduction
There is a large and growing literature on temptation and self-control in economics [1]  [2].Gul and Pesendorfer [3] propose basic models of choice under temptation and provide preference foundations for the models.We provide an alternative proof of the main theorem in [3], that is, the Gul and Pesendorfer's utility representation theorem about temptation and costly self-control.The proof makes use of a geometry of preferences and goes as follows.We first extract behaviors that display temptation and self-control.We then characterize the intuitive notions of temptation and self-control geometrically.Finally, we prove the utility representation theorem using the characterization.The proof highlights the reason why the self-control part can be written by sum of commitment utility and temptation utility.The proof also provides the refined testable implications of the Gul and Pesendorfer model.
This geometric approach is taken by the companion paper, [4], to prove the Gul and Pesendorfer's utility representation theorem about temptation without self-control.As a result, we prove the two representation theorems by an intuitive and unified approach.
This paper is organized as follows.Section 2 summarizes the Gul and Pesendorfer's utility representation theorem.In Section 3, we explore our notions of temptation and self-control and derive those cone representa-tions.Section 4 proves the Gul and Pesendorfer's representation theorem using the result of Section 3. In Section 5, we discuss relation between our approach and the Gul and Pesendorfer's approach.

The Gul and Pesendorfer Theorem
Let Z be a compact metric space of prizes.Let ∆ be the set of all Borel probability measures over Z and be endowed with the topology of weak convergence.Let  be the set of all compact (with respect to the topology of weak convergence) subsets of ∆ and be endowed with the topology induced by the Hausdorff metric.Let  be the set of continuous affine mappings from ∆ to real numbers; that is, f ∈  if and only if f is continuous on ∆ and satisfies ( ) ( ) ( ) ( ) ( ) and for all [ ] 0,1 α ∈ .Throughout this paper, we say that f is cardinally equivalent to a function g when f g for some positive α and real β .
We call the following model of utility function the for some , u v ∈  .Gul and Pesendorfer [3] provided preference foundations for this model.Let  be a binary relation over  .We say that  is are closed,  continuous if it is upper and lower semi-continuous.
We consider the following axioms.

Axiom 1 (Preference
 .Axiom 1 is a standard revealed preference axiom.Axioms 2 and 3 are variants of the von Neumann and Morgenstern axioms adapted to the preferences-over-menus setting.Axioms 4 is viewed as intuitive notion of costly self-control behaviors under temptation as we explain below. Imagine a situation in which an individual first chooses a menu and then selects an alternative from that menu.Suppose that the individual evaluates a menu by its best element.Such an individual's behavior is represented by a utility function U of the form ( ) ( ) for some u ∈  .Observe that an individual with this type of utility function follows a regularity called Strategic Rationality: A B  implies Ã A B  . 1 Clearly, any strategically rational decision maker does not exhibit a desire for commitment, where by `desire for commitment' we mean that an individual strictly prefers a subset of a menu to the menu itself.
Desire for commitment is an implication of temptation.An individual may strictly prefer menu A to menu A B  to avoid succumbing to temptation that is anticipated as follows: The individual anticipates that he/she will be tempted to select an alternative when facing menu A B  , and this alternative is undesired for him/her.Axiom 4 relaxes Strategic Rationality and allows a possibility that A A B B    .Suppose that B contains a tempting alternative.We can view A B B   as meaning that when facing menu A B  , the individual uses self-control and can resist the temptation.We then interpret A A B   as meaning that exercising self-control is costly.
Gul and Pesendorfer [3] showed the following representation theorem.Theorem 1.  satisfies Preference, Continuity, Independence, and Set Betweenness if and only if it has a Gul and Pesendorfer representation, that is, there exists a Gul and Pesendorfer model U such that A B  if and only if

Geometry of Temptation and Self-Control
This section explores some geometric properties of  that satisfies Set Betweenness (and von Neumann and Morgenstern type axioms).Specifically, as in [4], we extract behaviors that display temptation and self-control and geometrically characterize the behaviors.All lemmas in this section are proved almost in the same way as Abe [4] and hence omitted.Lemma 0. (Gul and Pesendorfer ([4], Lemma 1)). satisfies Preference, Continuity, and Independence if and only if there exists a continuous affine function : We define u by ( ) { } ( ) as in [3].Since u represents preferences that the individual would like to commit to, it is called a commitment utility.Any commitment utility defined in this manner is continuous and affine from Lemma 0.
Consider a nontrivial preference relation  , that is, there are . Set Betweenness induces the following four strict partial orders. 4A weak temptation relation T is defined by Two temptation relations display a desire for commitment in a binary menu.Suppose { } { } . We view x y  as meaning that the individual desires to commit to { } x because y is more tempting than x.Two resistance relations display self-control.We view { } { } , x y y  as meaning that the individual selects x when facing { } , x y .This means that when y tempts him/her, he/she uses self-control and resists the temptation.The next fact is worth pointing out, and we may use this fact repeatedly without warning below: When  satisfies Set Betweenness, { } { } The following properties of four relations are the fundamentals for our geometric approach.Lemma 1. Suppose that ≿ satisfies Preference, Continuity, Independence, and Set Betweenness.Then, the following hold. Four relations T, * T , R, and * R are Asymmetric and Transitive (that is, strict partial orders), and they satisfy Strong Independence. 5 The weak temptation relation T and the weak resistance relation R are Strong Archimedean. 6e now consider geometric representations of the four strict partial orders.Define four cones corresponding to the four relations as follows. 7 A weak temptation cone is defined by  A weak resistance cone is defined by  A strong resistance cone is defined by . Temptation cones are defined as the set of "tempting directions", and resistance cones are defined as the set of "resisting directions".Corresponding to Lemma 1, those cones possess the following properties. 2Gul and Pesendorfer [3] consider an extended preference relation over lotteries of menus that is defined in an obvious way and show that Axioms 1 to 3 naturally induce the same properties to the extended relation.They then obtain a function U as a von Neumann and Morgenstern preference-scaling function for expected utility representation of that relation and show by construction that U is indeed a continuous affine function. 3Alternatively, we can rely on [6] to prove Lemma 0. Kopylov [6] applies the mixture space theorem to  that is restricted on the set of all convex menus and directly obtains U as a von Neumann and Morgenstern expected utility of the restricted  .He then uses the property that every menu is indifferent to its convex hull, which is indeed implied from Axioms 1 to 3, and extends U naturally over  . 4The fact that these orders are strict partial orders is proved in Lemma 1 below.
Lemma 2. Suppose that  satisfies Preference, Continuity, Independence, and Set Betweenness.Then, the following hold. Four cones  , *  ,  , and *  are convex cones that represent their corresponding relations, respec- tively. 8 .  The weak temptation cone  and the weak resistance cone  are faceless. 9

A Geometric Proof for the Gul and Pesendorfer Theorem
In this section, we prove that any regular self-control preference relation admits a Gul and Pesendorfer representation.
If  satisfies Axioms 1, 2, 3, and 4 and there are ,  and *  are nonempty. 10e first obtain two functions that represent temptation and self-control.Lemma 3.There exist , v w∈  such that for any , x y  if and only if ( ) ( ) Proof.We can prove this lemma in much the same way as in Abe ([4], Section 4), and hence omit the detail of proof here.A sketch of proof is provided in Appendix.In there, the proof goes as follows.We openly separate  from *  and obtain v from their separating hyperplane.Similarly, we openly separate  from *  and obtain w from their separating hyperplane.∎ We call function v a temptation utility and w a self-control utility.Suppose that ( ) ( ) 11 Then, by Set Betweenness and Lemma 3, ( ) ( ) With this fact, we can show the following.Lemma 4. The self-control utility w must be written by w au bv c = + + for some constant , 0 a b > and c ∈  .
Proof.As stated above, when ( ) ( ) . Hence, we find that ( ) ( ) and ( ) ( ) Then, we can apply Harsanyi's [7] aggregation theorem and obtain some constant ˆ, , a b c ∈  such that ˆû aw bv c = + + .Furthermore, we show below that ˆ0 a > and ˆ0 b < .Because  is a regular self-control preference relation, we can take , , , Δ x y x y ′ ′ ′′ ′′ ∈ such that . From Lemma 3, we have ( ) ( ) , and ( ) ( ) ( ) ( ) ( ) ( ) , Lemma 3 and Set Betweeness ( ) , we obtain the desired result.∎ Lemma 4 means that the indifference curve of w lies between those of u and v when they pass a common point.From Lemma 4 together with Lemma 3, we further find the following fact that the self-control utility and the temptation utility exactly characterize temptation and costly self-control.The proof is immediate and thus omitted..We say that C represents R when ( ) x y ′ ′ in the domain of R imply x Ry ′ ′ . 9A face of a convex cone C is a nonempty convex subset F of C such that , s t C ∈ and ( ) A convex cone C is said to be faceless if C is the only face of C. 10 From Lemma 2, this is equivalent to the fact that there are , , , This is consistent with the concept of regularity proposed in Gul and Pesendorfer [3]. 11This commitment utility u is defined in Section 3.

Lemma 5. ( ) ( )
w x w y > and ( ) ( ) x y y   .We now characterize U using w and v.The next lemma essentially characterizes the functional form of U. Lemma 6.

{ } ( )
Proof.It immediately follows from Lemmas 1 and 5 that . 13t us now show that, for any ( ) ( ) ( ) . 14 Consider translations ( ) ( ) ( ) 15 Note then, under our supposition ( ) ( ) for all translations because v is a continuous affine function and hence satisfies Independence and Translation Invariance. 16On the other hand, and w is continuous.Hence, ( ) ( ) ( ) for any λ close to 1. Fix such a λ .We then have However then, since , where the last equality follows from Lemma 3.This is a contradiction.Therefore, Similarly, we can prove the converse implication.∎ This lemma says that the ranking of { } , x y and { } , x z is determined by the temptation ranking of y and z when both y and z are more tempting than x but the individual can resist the temptations. 17 .Take a z such that ( ) ( ) and ( ) ( ) . Recall from Lemma 4 that an appropriate scale-normalized commitment utility is the difference between the self-control utility and a scale-normalized temptation utility: au c w bv + = − .Therefore, we can calibrate utility value of { } , x y by the difference between the self-control utility of z and the normalized temptation utility of z.By the way of choosing z, we can hence calibrate utility 12 We can similarly show that . 13 To see it, note that . This is uniformly continuous and hence has a unique uniformly continuous extension over the closure of that domain (Kelly [8], Theorem 26, p. 195]), where . Moreover, since U  is affine, so is its extension. 14Assuming the existence of such x′ is without loss of generality.See Appendix for the detail. 15There is no loss of generality as for the footnote above.See Appendix for the detail. 16Function f on Δ satisfies Translation Invariance if ( ) ( ) for any translation, or equivalently for any signed measure t such that ( ) As in [9], any f ∈  satisfies Translation Invariance. 17Similarly, the ranking of { } , x y and { } , x z is determined by the self-control ranking of y and z when x is more tempting than both y and z but the individual can resist the temptation.Proof.Since Û is cardinally equivalent to U, it is clearly a representation of  .We now show that Û is a Gul and Pesendorfer model restricted on binary menus.Then, this lemma immediately follows from the extension result of Gul and Pesendorfer [3]. 18 . 19 Assume also that there is a z such that ( ) ( ) and ( ) ( ) . 20 Then, ( ) ( ) ( ) ( ) ( ) ( )

, , aU x y c aU x z c au c w bv w x y x bv c bv y
where the first equality follows from Lemma 6, the second from Lemma 3, and the third and the last from Lemma 4. This completes the proof.∎Remark.Until now, we have focused on regular self-control preferences.Let us comment about the other cases.As heretofore, suppose that  satisfies Axiom 1, 2, 3, and 4.
We first consider the case that { } { } .This case is the degenerate case of no self-control preferences.Suppose that there exist , , , We can observe such a situation in Figure 1 by rotating the indifference curve of  around x in anticlockwise direction and placing it over the indifference curve of v. But, Figure 1 indicates that it is incompatible with Continuity.Hence, in this case, there is no case other than the two extreme cases: . In both cases, Theorem 1 is trivial.Let us then consider the case of self-control preferences but not regular.There are three cases: (i , and (iii) * = ∅  and * = ∅  .We note that the first two cases are impossible.To see why intuitively, consider case (i).In this case, there is no , Δ x y ∈ such that . Self-control utility w must be car-18 Suppose that u and v are continuous affine functions on Δ.Let U be a continuous function that represents some  satisfying Set Bet- weenness and for all menus that have at most two elements.Then, that equation is valid for all menus. 19The other cases are straightforward. 20In general, for arbitrarily fixed , x y y   , there may be no such z.However, in that case, we can construct another triple , , Δ x y z ′′ ′′ ′′ ∈ having the requested property by mixing x and y with other lotteries.Hence, we can apply the proof presented here to the constructed , , x y z ′′ ′′ ′′ .We can then show that the shown result for , x y ′′ ′′ is maintained for the original , x y by the construction of , x y ′′ ′′ .See Appendix for the detail.
dinally equivalent to u.We can observe such a situation in Figure 1 by rotating the indifference curve of w around x in clockwise direction and placing it over the indifference curve of u.But, Figure 1 indicates that it is incompatible with Continuity.
Consider finally case (iii).In this case, { } { } x y y   .Hence,  restricted on singletons is equal to R and the inverse of T. Therefore, commitment utility, self-control utility, and (−1) × temptation utility are cardinally equivalent.In this degenerate case, we can easily prove Theorem 1 by constructing v directly. 21,22

Discussion
We provided an alternative proof of the Gul and Pesendorfer's utility representation theorem about temptation and self-control.In what follows, we clarify relations between our geometric approach and the Gul and Pesendorfer's original approach.
Gul and Pesendorfer [3] proved the theorem in a way different from ours.Their approach is constructive.
They directly define the temptation utility by ( ) tion utility v is viewed as measuring marginal utility for commitment.They showed under the conditions of Theorem 1 that v is indeed well-defined, continuous, and affine.This part serves as a building block to establish the desired representation. 23he constructive approach and the geometric approach taken here bring us additional but different benefits beyond just establishing the representation theorem. 24The former directly tells us how to calibrate temptation.On the other hand, the latter directly defines temptation and self-control in terms of preferences, so that it directly relates temptation and self-control utilities to particular intuitive properties of the underlying preferences.
The direct link between the two utilities and preferences promotes a better understanding of the Gul and Pesendorfer model.It highlights the reason why the self-control part can be written by sum of commitment utility and temptation utility.It is because we directly proved that all three dynamic considerations have linear structure and self-control considerations lie between commitment and temptation.
Moreover, the link provides the refined testable implications of the model.Our characterization of T, * T , R, and * R will be used to test the Gul and Pesendorfer model.First, it is helpful to design an experiment or a questionnaire.Since Independence and/or Set Betweenness are written in terms of choices over all menus, testing literally them entails a comprehensive examination of choices that uses not only small menus but large menus.The properties of T, * T , R, and * R provide simple testable implications of the model that are written by menus that include at most two elements.
Second, more importantly, because temptation utility v and self-control utility w are characterized by T, * T , R, and * R , the properties of those relations are testable predictions of a model with linear temptation utility and/or linear self-control utility.This means that if an individual's choices do not obey the prediction of the Gul for all Δ y ∈ . 22We note that our geometric approach does not work well in this degenerate case.Specifically, in the proof of Lemma 7, we cannot take a z by which we calibrate utility value of { } , x y . 23Kopylov [6] proved Theorem 1 for a more general choice object than the one considered here and applied it to characterize various models associated with temptation.In his proof, he also constructs the temptation utility directly in the same spirit with Gul and Pesendorfer [3]    .As Gul and Pesendorfer [3] did, he directly proved that ( ) U A can be written by the defined in the form of Theorem 1. 24 As Gul and Pesendorfer ([3], footnote 6) conjecture, there is another approach to prove Theorem 1 which is based on a representation theorem characterizing a general model called a finite additive expected utility representation.See Dekel, Lipman, and Rustichini [10] for the case of finite Z and Kopylov [11] for a more general choice object.and Pesendorfer model, then the properties of T, * T , R, and * R may be useful in exploring the nature of observed violations and in considering a minimally extended model that accommodates the violations. 25 typical element A of  is called a menu (of lotteries).
exactly one of either yTx or * xR y holds and (ii) exactly one of either xRy or * yT x holds.

5 A
binary relation R is said to be Asymmetric when xRy implies ( ) yRx ¬ , Transitive when xRy and yRz imply xRz , and satisfies Strong Independence when xRy if and only if

8
Consider a binary relation R on a domain.Let

1 .
on Δ as in Figure This observation leads us to the desired form of representation.Suppose { } { } { }

Figure 1 .
Figure 1.The Marschak-Machina triangle and Indifference curves of is a continuous affine function over Δ as uin the first part of the proof of Lemma 6.Easy (but tedious) calculation then shows that, on binary menus, U is the Gul and Pesendorfer model with u and v, where by are in a neighborhood of * x and x α ′ is an algebraically relative interior point.Moreover, those lotteries satisfy { } { } Note that there is a number Gul and Pesendorfer model.Definition 1.A utility function U on menus is said to be a Gul and Pesendorfer model if it is a function of the form: