_{1}

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.

There is a large and growing literature on temptation and self-control in economics [

This geometric approach is taken by the companion paper, [

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 representations. 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.

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

Let

We call the following model of utility function the 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:

for some

Gul and Pesendorfer [

upper semi-continuous if the sets

lower semi-continuous if the sets

continuous if it is upper and lower semi-continuous.

We consider the following axioms.

Axiom 1 (Preference).

Axiom 2 (Continuity).

Axiom 3 (Independence).

Axiom 4 (Set Betweenness).

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 ^{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

Axiom 4 relaxes Strategic Rationality and allows a possibility that

Gul and Pesendorfer [

Theorem 1.

This section explores some geometric properties of

Lemma 0. (Gul and Pesendorfer ( [^{2,3}

We define u by

Consider a nontrivial preference relation^{4}

A weak temptation relation T is defined by

A strong temptation relation

A weak resistance relation R is defined by

A strong resistance relation

Two temptation relations display a desire for commitment in a binary menu. Suppose

The next fact is worth pointing out, and we may use this fact repeatedly without warning below: When

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, ^{5}

The weak temptation relation T and the weak resistance relation R are Strong Archimedean.^{6}

We 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 strong 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.

Lemma 2. Suppose that

Then, the following hold.

Four cones^{8}

The weak temptation cone ^{9}

In this section, we prove that any regular self-control preference relation admits a Gul and Pesendorfer representation.

If ^{10}

We first obtain two functions that represent temptation and self-control.

Lemma 3. There exist

Proof. We can prove this lemma in much the same way as in Abe ( [

We call function v a temptation utility and w a self-control utility.Suppose that^{11} Then, by Set Betweenness and Lemma 3,

Lemma 4. The self-control utility w must be written by

Proof. As stated above, when

Because

Therefore, by the Harsanyi additive representation

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.

Lemma 5.

We now characterize U using w and v. The next lemma essentially characterizes the functional form of U.

Lemma 6. ^{12}

Proof. It immediately follows from Lemmas 1 and 5 that ^{13}

Let us now show that, for any^{14} Consider translations ^{15} Note then, under our supposition^{16} On the other hand,

This lemma says that the ranking of ^{17} Hence, we can plot indifference curves of

Suppose

ment utility is the difference between the self-control utility and a scale-normalized temptation utility:

value of

Formally, we prove the following.

Lemma 7. Define

Proof. Since ^{18} Assume that^{19} Assume also that there is a z such that ^{20} Then,

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

We first consider the case that

Let us then consider the case of self-control preferences but not regular. There are three cases: (i)

Consider finally case (iii). In this case, ^{21,22}

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 [^{23}

The constructive approach and the geometric approach taken here bring us additional but different benefits beyond just establishing the representation theorem.^{24} The 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,

Second, more importantly, because temptation utility v and self-control utility w are characterized by T, ^{25}

I would like to thank Fumio Dei, Hisao Hisamoto, Eiichi Miyagawa, and especially Hideo Suehiro, for their valuable comments and encouragement. I would also like to thank the anonymous reviewers for their many insightful comments and suggestions. Needless to say, the responsibility for any remaining errors rests with the author. This paper was supported by JSPS KAKENHI Grant Number 16K21038.

Koji Abe, (2016) A Geometric Approach to Temptation and Self-Control. Theoretical Economics Letters,06,539-548. doi: 10.4236/tel.2016.63060

Proof of Lemma 3 (Sketch). We first claim:

Claim 2.

Since

Claim 3. There are two linear functional

Note from Lemma 2 that

Define functions v and w on Δ by

Claim 4.

For

Similarly, using lower semi-continuity of U and the fact that

Supplement to the proof of Lemma 6. As we showed in Section 4,

Let us show that, for any

Step 1. We show that there exist

By regularity, there are

Step 2. Take

Suppose to the contrary that

Consider now translations

Fix such a

where the last equality follows from Lemma 3. This is a contradiction.

Step 3. For any

By Step 1, there are

Supplement to the proof of Lemma 7. We legitimate what we wrote in footnote 20. Observe first that there are

Then, by proof of Lemma 7, we have