A Geometric Approach to Temptation

Abstract

We provide a simple geometric proof of the Gul and Pesendorfer’s (Econometrica 69(6):1403–1435, 2001) utility representation theorem about choice under temptation without self-control. We extract two incomplete orders from preferences: temptation relation and resistance relation. We characterize those relations geometrically and obtain temptation utility using a separation method à la Aumann (Econometrica 30(3):445–462, 1962).

The original article first appeared in the Journal of Mathematical Economics 48:92–97, 2012. A newly written addendum has been added to this book chapter.

Appendices

Appendix: Omitted Proof

Proof (Proof of Lemma 4)

Take x, y ∈ Δ with xT ^∗ y arbitrarily. Let us first show that $z - y \in \mathrm{ span}(\mathcal{T}^{{\ast}})$ for any z ∈ Δ. Consider the linear subspace spanned by $S:=\{ x - y,z - y\}$ and denote it by span(S). If these underlying vectors are linearly dependent, then $z - y =\lambda (x - y) \in \mathrm{ span}(\mathcal{T}^{{\ast}})$ for some $\lambda \neq 0$. Assume that these vectors are linearly independent, that is, span(S) has dimension 2. Define the set $\mathcal{T}^{{\ast}}(y):=\{ z^{{\prime}}- y\vert z^{{\prime}}T^{{\ast}}y\}$ and let $\mathcal{T}^{{\ast}}_{s}(y)$ denote the intersection of $\mathcal{T}^{{\ast}}(y)$ and span(S). It is straightforward to verify that $\mathcal{T}^{{\ast}}_{s}(y)$ is a convex set. Moreover, this has dimension 2. Suppose to the contrary that $\mathcal{T}^{{\ast}}_{s}(y)$ has dimension 1. Then, for any α ∈ (0, 1),

$$\displaystyle{[\alpha x + (1-\alpha )z] - y =\alpha (x - y) + (1-\alpha )(z - y)\notin \mathcal{T}^{{\ast}}_{ s}(y).}$$

This means that $\mathit{yR}^{{\ast}}[\alpha x + (1-\alpha )z]$ for all α ∈ (α ^∗, 1), where α ^∗ ∈ (0, 1) is a number such that $\{y\} \succ \{\alpha ^{{\ast}}x + (1 -\alpha ^{{\ast}})z\}$. As we stated in the proof of Lemma 2, this contradicts Upper Semi-Continuity of $\succapprox$. Thus, $\mathcal{T}^{{\ast}}_{s}(y)$ has dimension 2. We can then take an algebraically relative interior point $z^{{\prime}}- y \in \mathcal{T}^{{\ast}}_{s}(y)$ (see p.9 in Holmes 1975). From its algebraic properties, there is a $\lambda> 0$ such that

$$\displaystyle{[z^{{\prime}}-\lambda (z - y)] - y = z^{{\prime}}- y -\lambda (z - y) \in \mathcal{T}^{{\ast}}_{ s}(y).}$$

This means that $[z^{{\prime}}-\lambda (z - y)]T^{{\ast}}y$ for some $\lambda> 0$. Hence,

$$\displaystyle{-\lambda (z - y) = [z^{{\prime}}-\lambda (z - y)] - y - (z^{{\prime}}- y) \in \mathrm{ span}(\mathcal{T}^{{\ast}}).}$$

As a result, $z - y \in \mathrm{ span}(\mathcal{T}^{{\ast}})$.

We now show that $\varDelta -\varDelta \subsetneq \mathrm{ span}(\mathcal{T}^{{\ast}})$. Take $z,z^{{\prime}}\in \varDelta$ arbitrarily. Then, as shown above, both z − y and $z^{{\prime}}- y$ are in $\mathrm{span}(\mathcal{T}^{{\ast}})$. Therefore, $z - z^{{\prime}} = (z - y) - (z^{{\prime}}- y) \in \mathrm{ span}(\mathcal{T}^{{\ast}})$. This completes the proof.

Addendum: On the Geometry of Temptation and Self-Control

This addendum has been newly written for this book chapter.

As stated in Remark 2, Abe (2011) demonstrates the usefulness of our geometric approach to menu preferences beyond the Strotz model. By taking the geometric approach, Abe (2011) provides an alternative proof of the costly self-control representation theorem in Gul and Pesendorfer (2001).^{13 ,  14} Here, we briefly supplement Remark 2 by providing an outline of the proof provided in Abe (2011).

We begin by summarizing Gul and Pesendorfer’s costly self-control representation theorem.

Definition 2

A utility function U on menus is said to be a Gul and Pesendorfer’s costly self-control model, hereafter referred to as a costly self-control model , if it is a function of the form:

$$\displaystyle{U(A) =\max _{x\in A}\big\{u(x) + v(x)\big\} -\max _{y\in A}\big\{v(y)\big\}}$$

for some $u,v \in \mathcal{C}$.

The costly self-control models are characterized by a behavioral regularity on menu preferences named Set Betweenness.

Axiom 4^′ (Set Betweenness )

$A \succapprox B$ implies $A \succapprox A \cup B \succapprox B$.

Set Betweenness relaxes No Self-Control and allows for the possibility that $A \succ A \cup B \succ B$. This possibility displays the notion of temptation and costly self-control. Suppose that B contains a tempting alternative. We can view $A \cup B \succ B$ as meaning that when facing menu $A \cup B$, the individual uses self-control and can resist the temptation. We then interpret $A \succ A \cup B$ as meaning that exercising self-control is costly. The main theorem in Gul and Pesendorfer (2001) is summarized as follows.¹⁵

The Costly Self-Control Representation Theorem. $\succapprox$ satisfies Preference, Continuity, Independence, and Set Betweenness if and only if it has a costly self-control representation , that is, there exists a costly self-control model U such that $A \succapprox B$ if and only if U(A) ≥ U(B).

We provide the proof outline on the sufficiency of the axioms in Abe (2011). In what follows, let U be a utility representation of $\succapprox$ derived as in Lemma 1 and u be a commitment utility derived by restricting U on singleton menus. Consider a nontrivial preference relation $\succapprox$, that is, there are x, y ∈ Δ such that $\{x\} \succ \{ y\}$. The proof outline is as follows. First, in a similar way as in Sect. 3, we can characterize the geometry of temptation under Set Betweenness. Specifically, Set Betweenness induces the following four strict partial orders satisfying Strong Independence.

• A weak temptation relation T is defined by yTx if $\{x\} \succ \{ x,y\}$.

• A strong temptation relation T ^∗ is defined by yT ^∗ x if $\{x\} \succ \{ x,y\} \sim \{ y\}$.

• A weak resistance relation R is defined by xRy if $\{x,y\} \succ \{ y\}$.

• A strong resistance relation R ^∗ is defined by xR ^∗ y if $\{x\} \sim \{ x,y\} \succ \{ y\}$.

Furthermore, the weak temptation relation T and the weak resistance relation R are both Strong Archimedean. Define four cones corresponding to the four relations as follows.

• A weak temptation cone is defined by $\mathcal{T} =\big\{\lambda (y - x)\ \vert \ \lambda> 0,\ \mathit{yTx}\big\}$.

• A strong temptation cone is defined by $\mathcal{T}^{{\ast}} =\big\{\lambda (y - x)\ \vert \ \lambda> 0,\ \mathit{yT}^{{\ast}}x\big\}$.

• A weak resistance cone is defined by $\mathcal{R} =\big\{\lambda (y - x)\ \vert \ \lambda> 0,\ \mathit{xRy}\big\}$.

• A strong resistance cone is defined by $\mathcal{R}^{{\ast}} =\big\{\lambda (y - x)\ \vert \ \lambda> 0,\ \mathit{xR}^{{\ast}}y\big\}$.

Then, the four cones $\mathcal{T}$, $\mathcal{T}^{{\ast}}$, $\mathcal{R}$, and $\mathcal{R}^{{\ast}}$ are convex cones that represent their corresponding relations, respectively. Furthermore, the weak temptation cone $\mathcal{T}$ and the weak resistance cone $\mathcal{R}$ are both faceless.

Second, we derive affine numerical representations of temptation and resistance (self-control) by the separation argument, and obtain $v,w \in \mathcal{C}$ such that for any x, y ∈ Δ with $\{x\} \succ \{ y\}$, (i) $\{x\} \succ \{ x,y\}$ if and only if v(y) > v(x) and (ii) $\{x,y\} \succ \{ y\}$ if and only if w(x) > w(y). We determine temptation utility v by openly separating $\mathcal{T}$ from $\mathcal{R}^{{\ast}}$ and self-control utility w by openly separating $\mathcal{R}$ from $\mathcal{T}^{{\ast}}$. Note that if u(x) > u(y), then, by construction, v(x) ≥ v(y) implies w(x) > w(y). With this fact, we can show that self-control utility w must be written by $w = au + bv + c$ for some constant a, b > 0 and $c \in \mathbb{R}$. This means that the indifference curve of w lies between those of u and v when they pass a common point, and this also implies w(x) > w(y) and v(y) > v(x) if and only if $\{x\} \succ \{ x,y\} \succ \{ y\}$.

Third, we can characterize U with w and v. The next lemma establishes this.

Lemma 7

U({x,⋅}) is cardinally equivalent to − v over set $\big\{y \in \varDelta \ \big\vert \ w(x) \geq \mathit{w}(y)\ \text{ and }\ \mathit{v}(y) \geq v(x)\big\}$.¹⁶

This lemma states 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 when the individual can resist the temptations.¹⁷ This observation leads us to the desired form of representation. Suppose $\{x\} \succ \{ x,y\} \succ \{ y\}$. Take a z such that w(x) = w(z) and v(y) = v(z). The facts derived in the second step imply $\{x\} \succ \{ x,z\} \sim \{ z\}$. Combining this with the above lemma, we then find $\{x,y\} \sim \{ x,z\} \sim \{ z\}$. Recall from the second step that an appropriate scale-normalized commitment utility is the difference between the self-control utility and the scale-normalized temptation utility: $\mathit{au} + c = w -\mathit{bv}$. Therefore, we can calibrate the utility value of {x, y} by the difference between the self-control utility of z and the normalized temptation utility of z. By way of choosing z, we can calibrate the utility value of {x, y} by the difference between the self-control utility of x and the normalized temptation utility of y, that is, w(x) −bv(y). This means that the utility value of {x, y} is measured by Gul and Pesendorfer’s costly self-control representation form $\hat{u}(x) +\hat{ v}(x) -\hat{ v}(y)$ if we define $\hat{u} = \mathit{au} + c$ and $\hat{v} = \mathit{bv}$. More formally, we can prove the next lemma in this line and hence complete the proof.

Lemma 8

Define $\hat{U}$ and $\hat{v}$ by $\hat{U}:= aU + c$ and $\hat{v}:= \mathit{bv}$ with a,b,c derived in the second step. Let $\hat{u}$ be the restriction of $\hat{U}$ on singleton menus. Then, $\hat{U}$ is a representation of $\succapprox$ and a costly self-control model.