unique induced preference representations

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Journal of Mathematical Economics 44 (2008) 951–963

Unique induced preference representations

Mihai ManeaDepartment of Economics, Harvard University, Cambridge, MA 02138, United States

Received 25 May 2007; received in revised form 30 October 2007; accepted 31 October 2007Available online 19 November 2007

Abstract

Machina [Machina, M.J., 1984. Temporal risk and the nature of induced preferences. Journal of Economic Theory 33, 199–231]considers an individual who has to choose from a set of alternative temporal uncertain prospects, and must take an action beforethe uncertainty is resolved, seeking to maximize the expected value of an (action determined) von Neumann-Morgenstern utilityindex. It is natural to ask if the set of underlying von Neumann-Morgenstern utility indices can be uniquely recovered solely on thebasis of the thus induced (ordinal) preferences over temporal prospects. Machina’s conclusion is that “ordinal preferences alone willnot suffice.” However, we show that it is possible to recover the action–utility set inducing the preferences uniquely if we restrictattention to action–utility sets for which no two actions induce the same preference relation on the space of temporal prospects, noaction is redundant, and no action leads to a risk free outcome.© 2007 Elsevier B.V. All rights reserved.

JEL Classification: D81; D90; E21; G11

Keywords: Induced preferences; Simple action–utility sets; Consumption–investment optimal decisions; Intertemporal income substitution; Portfoliochoice

1. Action–utility induced preferences

In many dynamic choice settings some decisions must be made before uncertainty is completely resolved. Choicesregarding saving and consumption, portfolio choice, and insurance, for example, often involve a time lapse betweenthe moment when the relevant decisions must be made and the moment when all the uncertainty is resolved. As Kreps(1988) notes, the objects of choice in dynamic settings with temporal resolution of uncertainty are usually more complexthan the lotteries of the von Neumann-Morgenstern theory; “the objects of choice in [the von Neumann-Morgenstern]theory, marginal distributions, aren’t rich enough to capture all that is essential”1 to the individual’s preferences ina dynamic decision problem with temporal resolution of uncertainty.2 Kreps observes that one can build a modelwith temporal resolution of uncertainty in which lotteries can be regarded as the sole objects of choice only if all theuncertainty resolves prior to the time any decision has to be made, or after the time all the decisions are due.

Exploiting such a reduction in the set of objects of choice, Machina (1984) considers an individual who faces a set(Fi)i∈P ⊂ D[0,M] of alternative temporal uncertain prospects (lotteries) with prizes in the interval [0,M] (D[0,M]

E-mail address: [email protected] Kreps (1988, p. 173).2 Kreps and Porteus (1978) allow for uncertainty dated by the time of its resolution. In their framework, as a multiple-stage compound lottery

resolves, the individual makes choices conditioning on the outcomes of the previous-stage lotteries.

0304-4068/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.jmateco.2007.10.005

mailto:[email protected]

dx.doi.org/10.1016/j.jmateco.2007.10.005

952 M. Manea / Journal of Mathematical Economics 44 (2008) 951–963

denotes the set of cumulative distributions on [0,M]), and must make another auxiliary choice α out of a compactset A before the uncertainty is resolved, seeking to maximize the expected value of a continuous and increasing vonNeumann-Morgenstern utility index Φα(·). We assume that Φα(x) varies continuously in α, uniformly in x. We callthe pair (A, (Φα(·))α∈A) an action–utility set. Therefore, the individual’s preferences over temporal prospects arerepresented by the function

Y (F ) = supα∈A

∫Φα(x) dF (x). (1.1)

We call the functional Y (·) action–utility induced preference functional over temporal distributions, and the ranking itgenerates, action–utility induced preferences.

Many dynamic choice applications can be accommodated by this model. Kreps and Porteus (1979) discuss appli-cations of induced preferences to consumption, savings and portfolio choices, money versus bundles consumptiondecisions, and preferences over different activities. We formalize and discuss some applications in Section 4.

Induced preferences over temporal prospects are complete, transitive, and continuous. Kreps and Porteus (1979)make the key observation that any action–utility induced preference functional needs to be convex, which gives someintuition for their result that induced preferences admit an expected utility representation only under very stringentconditions. Mathematically, Kreps and Porteus’ result is driven by the observation, made by Mossin (1969) and Spenceand Zeckhauser (1972), and reinforced by the criticism of the separability across mutually exclusive outcomes argumentfor dynamic choice settings,3 that induced preferences typically violate the independence axiom.

We are interested in uniquely identifying the set of underlying von Neumann-Morgenstern utility indices solely fromobservation of the individual’s ordinal induced preferences over temporal prospects. For an arbitrary continuous convexfunctional, Machina (1984) offers a method of recovering a set of von Neumann-Morgenstern utility indices generatingthat functional as an induced preference functional. Machina argues that different continuous convex functionals thatrepresent the same preference may lead to sets of underlying von Neumann-Morgenstern utility indices that are notaffinely isomorphic. Hence, the underlying von Neumann-Morgenstern utility indices cannot be uniquely recovered ifwe set no restrictions on the action–utility sets. We identify a key restriction on the action–utility sets that, togetherwith two other innocuous assumptions, allows us to uniquely determine the action–utility set underlying any ordinalinduced preference up to a positive affine transformation. Specifically, we assume that no two actions induce the samepreference relation on the space of temporal prospects. The next section presents Machina’s conclusions in order tobetter qualify our question, and provide some intuition for our assumptions.

2. Induced preference representations

One immediate question concerning induced preferences is: what functionals are induced preference functionalsfor some set of actions A and some set {Φ(·, α)}α∈A of underlying von Neumann-Morgenstern utility indices?

2.1. Machina’s representation theorem

Machina (1984) proves that continuity and convexity are necessary and sufficient for the existence of an inducedpreference representation. Besides a characterization, Machina’s theorem also offers a way of recovering a set ofunderlying von Neumann-Morgenstern utility indices, Z∗

Y , from an arbitrary continuous convex functional, that servesto generate that functional as an induced preference functional. Denote by C the set of all continuous functions from[0,M] to R endowed with the topology of uniform convergence.

Theorem. [Machina (1984), Theorem 2] If Y : D[0,M] → R is continuous and convex, then

Y (F ) = maxΦ(·) ∈Z∗

Y

∫Φ(x) dF (x), (2.1)

3 See Kreps (1988, pp. 44–45).

M. Manea / Journal of Mathematical Economics 44 (2008) 951–963 953

where

Z∗Y := {Φ∈ C|

∫Φ(x) dF (x) ≤ Y (F ),∀F ∈D[0,M]}. (2.2)

The purpose of the next subsection is to investigate how the setZ = {Φα(·)|α∈A}, inducing the preference functionalY, may differ from the set Z∗

Y recovered by Machina’s result.

2.2. Recoverability of the underlying utility indices with known Y

We call a subset B of C comprehensive if for each Φ in B, all Φ∗ in C that are uniformly dominated by Φ are also inB, that is, Φ∈B and Φ∗(x) ≤ Φ(x),∀x∈ [0,M] imply that Φ∗ ∈B. The convex comprehensive closure of a set B in Cis the smallest (with respect to inclusion) convex, closed and comprehensive subset of C that contains B. A subset of Cis non-redundant if all of its points are extreme points of its convex comprehensive closure.

Theorem. [Machina (1984), Theorem 3] If

Y (F ) = maxΦ(·) ∈Z

∫Φ(x) dF (x),

where Z = {Φα(·)|α∈A}, then the set Z∗Y (as defined by (2.2)) is the convex comprehensive closure of Z.

2.3. One step towards uniqueness: the Y → ZY correspondence

It is easy to show that (2.2) defines a bijective correspondence between the continuous convex functionals and theconvex closed comprehensive sets. Also, the correspondence associating to each convex closed comprehensive set itssubset of extreme points is a bijection from the convex closed comprehensive subsets of C to the non-redundant subsetsof C. Let

Y → ZY (2.3)

be the correspondence obtained by composing these two bijections. Therefore, for any action–utility induced preferencefunctional Y, Machina’s theorems implicitly identify the unique non-redundant set of underlying utility indices thatgenerate it.

Corollary. Any continuous convex functional Y (·) can be uniquely generated by a non-redundant set ZY ⊂ C ofunderlying von Neumann-Morgenstern utility indices.

2.4. Non-unique representations

It is natural to ask if the set of underlying von Neumann-Morgenstern utility indices can be recovered solely onthe basis of the individual’s ordinal induced preferences over temporal prospects. Machina points out that for anycontinuous convex functional Y (·) and any δ : R→ R such that δ ◦ Y is continuous and convex (this can happen, e.g.,for δ increasing, continuous and convex) the sets

Z∗Y =

{Φ∈ C|

∫Φ(x) dF (x) ≤ Y (F ),∀F ∈D[0,M]

}(2.4)

Z∗δ◦Y =

{Φ∈ C|

∫Φ(x) dF (x) ≤ (δ ◦ Y )(F ),∀F ∈D[0,M]

}(2.5)

are not typically affinely isomorphic. Indeed, they can differ very significantly because the restrictions on δ(·) arevery light. Conceivably, the non-redundant utility sets ZY and Zδ◦Y representing Y and δ ◦ Y , respectively, could evenhave different cardinalities. Therefore, Machina’s discussion implies an impossibility to even pin down uniquely thecardinality of the action space. Machina (1984) concludes that “ordinal preferences alone will not suffice” to recoverthe underlying von Neumann-Morgenstern utility indices uniquely. However, we are going to show that it is possible torecover the action–utility set inducing the preferences uniquely if we restrict attention to action–utility sets for whichno two actions induce the same preference relation on the space of temporal prospects, no action is redundant, and noaction leads to a risk free outcome.


3. The unique induced preference representation result

For simplicity, we prove our main result for the finite equivalent of Machina’s framework. Specifically, we modelinduced preferences over temporal prospects with a finite set of prizes assuming that only finitely many actions areavailable.

3.1. The finite action–utility framework

Let X be a finite set of prizes, |X| ≥ 2, and denote by L the set of all lotteries on X,

L ={p : X → [0, 1]|

∑x∈X

p(x) = 1

}. (3.1)

We consider an individual who has to take actions out of a non-empty finite set A. When facing a lottery p∈L, theindividual takes an action α∈A (before the uncertainty represented by p resolves; to emphasize the temporal nature ofthe resolution of uncertainty we shall, on some occasions, call the elements ofL temporal lotteries) seeking to maximizethe expectation of a von Neumann-Morgenstern utility indexΦα : X → R. Hence, the individual’s preference relationover lotteries can be represented by the utility function Y(A,(Φα)α∈A) defined by

Y(A,(Φα)α∈A)(p) = maxα∈A

φα(p), where φα(p) :=∑x∈X

p(x)Φα(x). (3.2)

We call the pair (A, (Φα)α∈A) action–utility set, the function Y(A,(Φα)α∈A), induced preference utility function overlotteries, and the ranking �(A,(Φα)α∈A) it generates, action–utility induced preference. As stated in the previous section,we are concerned with recovering the underlying von Neumann-Morgenstern utility indices (Φα)α∈A solely from theindividual’s ordinal induced preference over lotteries.

3.2. Non-unique representations without any restrictions: an example

Let n = |X|. We regard L as a subset of Rn, given its obvious embedding in the hyperplane H ={(x1, x2, . . . , xn)|∑n

i=1xi = 1} ⊂ Rn. As already discussed in the previous section, the utility function Y(A,(Φα)α∈A)is continuous and convex on L. When there is no risk of confusion, we omit the index on Y(A,(Φα)α∈A) and redefinethe correspondence Y → ZY (2.3) for our finite framework (Machina’s recoverability result and its corollary carrythrough).

As argued in Section 2, for any δ : R→ Rsuch that δ ◦ Y is continuous and convex on L, ZY and Zδ◦Y are theunique non-redundant action–utility sets generating Y and δ ◦ Y . The next example illustrates Machina’s observationthat the sets ZY and Zδ◦Y may be significantly different.

3.2.1. An exampleLet X = {1, 2, 3} and define the preference �ex on the corresponding lottery space L (3.1) by

p�exq ⇔ p(1) ≥ q(1).

One utility function that represents �ex is Y, defined by Y (p) = p(1). Obviously,ZY = {Φα}, whereΦα(1) = 1 andΦα(2) = Φα(3) = 0.

If we define δ by

δ(x) =

⎧⎪⎨⎪⎩x, if x ≤ 1

2

3x− 1, if x >1

2

,


we get that

(δ ◦ Y )(x) =

⎧⎪⎨⎪⎩p(1), ifp(1) ≤ 1

2

3p(1) − 1, ifp(1) >1

2andZδ◦Y = {Ψβ1 , Ψβ2},withΨβ1 = Φα andΨβ2 defined byΨβ2 (1) = 2, Ψβ2 (2) = Ψβ2 (3) = −1, or, in concise notation,Ψβ2 = 3Φα − 1.

Both action–utility sets ({α}, {Φα}) and ({β1, β2}, {Ψβ1 , Ψβ2}) induce the preference �ex. Nevertheless, they repre-sent different decision frameworks and suggest dissimilar action choice behaviors. In the first framework, the individualhas only one possible action, α, which he uses for each temporal lottery, so there is no optimization problem for himto solve, while in the second framework, the individual has two possible actions, β1 and β2, and optimally chooses totake the former when facing lotteries that offer the first prize with probability at most one half, and the latter for thecomplementary set of lotteries.

3.3. The assumptions that deliver a unique representation

3.3.1. The main assumptionWe identify the key restriction on the allowable action–utility sets that enables us to uniquely determine, up to a

positive affine transformation, the action–utility set underlying any ordinal induced preference.

Non-dissociativity: No two actions in A induce the same preference relation on L, i.e. (by the von Neumann-Morgenstern expected utility representation theorem), there do not exist two actions α �= α′ such thatΦα is a positiveaffine transformation of Φα′ .

The non-dissociativity assumption is based on the observation that for each lottery p for which the optimal action,α, is unique, the p-indifference hyperplane of φα needs to be tangent at p to the p-indifference surface of the ordinalinduced preference. Assume that we can represent an induced preference relation � by two continuous and convexutility indices, Y and δ ◦ Y . Then, for any lottery p at which the optimal actions in ZY and Zδ◦Y correspond to utilityindices φ and ψ, respectively,4 the p-indifference hyperplanes of φ and ψ coincide as they both need to be tangent at pto the p-indifference surface of �. Hence, Φ and Ψ are affine transformations of each other.5 Therefore, for any otherlottery q at which the optimal action inZY is the one corresponding toΦ, the utility index corresponding to the optimalaction in Zδ◦Y needs to be an affine transformation of Φ.

We denote by LΦ the set of lotteries q at which the optimal action in ZY is the one corresponding to Φ. For each qin LΦ, we denote by Ψq the utility index corresponding to the optimal action in Zδ◦Y at q. For all q∈LΦ, we have thatY (q) = φ(q) and (δ ◦ Y )(q) = ψq(q), so ψq(q) = δ(φ(q)). Hence, in order for Ψq to be the same affine transformationof Φ for all the lotteries q in a subset lΦ of LΦ, we need δ to be linear on φ(lΦ). Therefore, the richness of the set{Ψq|q∈LΦ} of affine transformations of Φ that Zδ◦Y needs to include depends on the “linear properties” of δ.

In the example of the previous subsection, we haveLΦα = L, thereforeΨβ1 andΨβ2 need to be affine transformationsof Φα. Indeed, we found that Ψβ1 = Φα and Ψβ2 = 3Φα − 1. The function δ is linear on [0, 1/2] and [1/2, 1], andconsequently, when the utility-action set is ({β1, β2}, {Ψβ1 , Ψβ2}), it is optimal to use the action β1 for the lotteries inY−1([0, 1/2]), and the action β2 for the lotteries in Y−1([1/2, 1]). The utility indices Ψβ1 and Ψβ2 recovered by thecorrespondence (2.3) applied to δ ◦ Y are different affine transformations ofΦα because the “linearity” of δ changes at1/2.

Imposing non-dissociativity on Zδ◦Y guarantees that for any lottery q∈LΦ, the optimal action in Zδ◦Y needs tobe the one corresponding to Ψ , i.e., Ψq = Ψ,∀q∈LΦ.6 Therefore, for all lotteries q∈LΦ, we have Y (q) = φ(q) and(δ ◦ Y )(q) = ψ(q), so, becauseΦ and Ψ are affine transformations of each other, δ needs to be linear on the set Y (LΦ).

4 We assume for the sake of this argument that the optimal actions in both ZY and Zδ◦Y are unique at the lotteries considered.5 We denote von Neumann-Morgenstern utility indices with upper-case Greek letters and the associate expected utility functions with the

corresponding lower-case letters.6 This holds true as long as we can prove that Ψq is a positive affine transformation of Φ for all q in LΦ.


Repeating this analysis for each lottery p, and relating indifferent pairs of lotteries, we can hope to show that thelinearity regions of δ thus obtained overlap. Consequently δ is linear on Y (L), hence δ ◦ Y is an affine transformationof Y, so ZY is an affine transformation of Zδ◦Y .

3.3.2. Two other assumptionsObviously, there is no way of identifying actions that are never used, so one natural condition to impose, if we

desire a uniqueness of representation result, is that for each action α there be a lottery p such that α is the uniqueoptimal action at p. However, because we wish for assumptions that make direct statements about the primitives ofour framework – action–utility sets and their ordinal induced preferences – we make a slightly stronger assumptionregarding action non-redundancy.

Non-redundancy: No action α′ ∈A is inessential to the representation, i.e., for any α′ ∈A the induced preferencerelations �(A,(Φα)α∈A) and �(A\{α′},(Φα)α∈A\{α′}) are not identical.

It is not difficult to prove that if non-dissociativity holds, then non-redundancy is equivalent to requiring that forany action α′ there exists a lottery p such that7

φα′ (p) = Y(A,(Φα)α∈A)(p) > Y(A\{α′},(Φα)α∈A\{α′})(p). (3.3)

Hence, if we are willing to regard the individual’s utility Y(A,(Φα)α∈A) as comparable across different action–utility sets(A, (Φα)α∈A) (that can be compared with respect to set inclusion), then, granted non-dissociativity, non-redundancyis equivalent to the fact that the individual would be made strictly worse off at least in one instance (for one lottery) ifhis action set were restrained.

In order to avoid thick indifference surfaces for the induced preferences, we also make the assumption that no actionhas a constant underlying utility function.

Non-constancy: No action α in A has the corresponding von Neumann-Morgenstern utility index Φα constant on X.

The non-constancy property is included only to simplify the exposition. All the statements we make can be alteredto apply to action–utility sets that do not have this property.

We say that (A, (Φα)α∈A) is a simple action–utility set if it satisfies non-dissociativity, non-redundancy and non-constancy, and that a preference relation over temporal lotteries is a simple action–utility induced preference if it canbe induced by a simple action–utility set.

3.4. The main result

If we restrict attention to action–utility sets that satisfy non-dissociativity, non-redundancy and non-constancy, theunderlying von Neumann-Morgenstern utility indices are uniquely determined by the ordinal induced preference upto an affine transformation.

Theorem. If (A, (Φα)α∈A) and (B, (Ψβ)β∈B) are two simple action–utility sets that induce the same preferencerelation � over L then (A, (Φα)α∈A) is a positive affine transformation of (B, (Ψβ)β∈B), i.e., there exist a bijectionf : A → B and real numbers a(> 0), b such that

Φα(x) = aΨf (α)(x) + b,∀x∈X∀α∈A. (3.4)

7 One implication of this equivalence, the one we use in the proof of our theorem, is trivial: if Y(A,(Φα)α∈A)(p) = Y(A\{α′},(Φα)α∈A\{α′ })(p),∀p∈L,then the induced preference relations �(A,(Φα)α∈A) and �(A\{α′},(Φα)α∈A\{α′ }) are identical. The reverse implication can be established using thearguments of the first step in the proof of the theorem.


Proof. Consider two simple action–utility sets (A, (Φα)α∈A) and (B, (Ψβ)β∈B) that induce the same preferencerelation � over L. For all lotteries p, we define

φα(p) =∑x∈X

p(x)Φα(x) (3.5)

ψβ(p) =∑x∈X

p(x)Ψβ(x) (3.6)

YA(p) = maxα∈A

φα(p) (3.7)

YB(p) = maxβ∈B

ψβ(p) (3.8)

α(p) = arg maxα∈Aφα(p) (3.9)

β(p) = arg maxβ∈Bψβ(p). (3.10)

Let n = |X|. L’s obvious embedding in the hyperplane H = {(x1, x2, . . . , xn)|∑ni=1xi = 1} ⊂ Rn enables us to

alternatively regard L as an (n− 1)-dimensional subset of a real vector space, as a metric space (with the metricinduced by the Euclidean metric of Rn) and as a measurable space (with the measure inherited from the measureinduced on H by the Lebesgue measure on Rn−1 through the obvious isometry between H and Rn−1). We exploit allthese structural properties of the lottery space at different points in the proof.

The proof of the theorem proceeds in a few steps.

3.4.1. The intuition for the first stepAs the example (Section 3.2.1) and the discussion motivating the non-dissociativity assumption suggest (Section

3.3.1), for each lottery at which the optimal action inA is unique, non-dissociativity allows us to identify the underlyingutility function and the corresponding action uniquely for both (A, (Φα)α∈A) and (B, (Ψβ)β∈B).

Step 1. For any action α∈A there exist a unique triplet (β, x, y) ∈B × (0,∞) × R such that

φα(q) = xψβ(q) + y,∀q∈L. (3.11)

Moreover, for the unique action β∈B thus associated with α,

α(q) = {α} ⇒ β∈ β(q),∀q∈L. (3.12)

Fix α∈A. From the discussion of the non-redundancy property it follows that there exists a lottery p such thatφα(p) > φα′ (p) for all actions α′ ∈A − {α}. Because the set A is finite and the functions (φα)α∈A are continuous onL, there exists an ε > 0 such that φα(p′) > φα′ (p′) for all p′ with ‖p′ − p‖ ≤ ε(α �= α′). Therefore,

YA(p′) = φα(p′) if ‖p′ − p‖ ≤ ε. (3.13)

Consider the equatorial (n− 2)-dimensional ball

O = {p′′ ∈L|‖p′′ − p‖ ≤ ε&φα(p′′) = φα(p)} (3.14)

situated at the intersection of the (n− 1)-dimensional ball {p′′ ∈L|‖p′′ − p‖ ≤ ε} with the (n− 2)-dimensionalhyperplane {p′′ ∈L|φα(p′′) = φα(p)}.

Note that

YA(p′′) = YA(p),∀p′′ ∈O. (3.15)

Because we assumed that (A, (Φα)α∈A) and (B, (Ψβ)β∈B) induce the same preference relation � over L, it followsthat

YB(p′′) = YB(p),∀p′′ ∈O, (3.16)


therefore,

O =⋃β∈B

{p′′ ∈O|φβ(p′′) = YB(p)}. (3.17)

Let μ be the measure induced on O by its embedding in Rn−2 with the Lebesgue measure. Since μ(O) is positive,(3.17) implies that there exists β′ ∈B such that the set {p′′ ∈O|φβ′ (p′′) = YB(p)} has positive μ-measure. All subsetsof O of dimension smaller than n− 2 have zero μ-measure, hence {p′′ ∈O|φβ′ (p′′) = YB(p)} needs to have dimensionn− 2.

It is easy to see that if {p′′ ∈O|φβ′ (p′′) = YB(p)} has dimension n− 2 then

{p′′ ∈L|φβ′ (p′′) = YB(p)} = {p′′ ∈L|φα(p′′) = YA(p)}, (3.18)

which implies that there exist reals x′(�= 0), y′ such that

φα(q) = x′ψβ′ (q) + y′,∀q∈L. (3.19)

To find a representation for φα as in (3.19) with x′ > 0, non-constancy allows us to find a sequence of lotter-ies (pi)i≥1 converging to p with ‖pi − p‖ < ε and φα(pi) < φα(pi+1) for all i ≥ 1. By ε’s definition, YA(pi) =φα(pi) < φα(pi+1) = YA(pi+1) for all i ≥ 1. Since YA represents the preference relation � on L it should be thecase that pi �� pi+1 for each i ≥ 1. Because YB also represents � on L, it follows that YB(pi) < YB(pi+1) for everyi ≥ 1.

Applying the argument we made for p to each pi (i ≥ 1) we can find a sequence of triplets (βi, xi, yi)i≥1 satisfying

YB(pi) = ψβi (pi) andφα(q) = xiψβi (q) + yi, ∀q∈L. (3.20)

One β∈B will appear in the sequence (βi)i≥1 infinitely many times. We can therefore pick 1 ≤ i < j such thatβi = βj = β and xixj > 0. The non-dissociativity of (B, (Ψβ)β∈B) and (3.20) imply that

(βi, xi, yi) = (βj, xj, yj) ≡ (β, x, y). (3.21)

We have that ψβ(pi) = YB(pi) < YB(pj) = φβ(pj), which, together with the fact that φα(pi) < φα(pj), (3.20) and(3.21), implies that x > 0. Thus (β, x, y) yields the desired representation. Using (pi)i≥1’s convergence to p we caneasily establish that β∈ β(p).

The uniqueness of such a representation follows from the non-dissociativity of (B, (Ψβ)β∈B) and the proof of thestep is over.

We define f : A → B by f (α) = β, and set aα = x and bα = y. Obviously, (A, (Φα)α∈A)’s non-dissociativityimplies that f is injective. Repeating the argument above with the roles of (A, (Φα)α∈A) and (B, (Ψβ)β∈B) interchangedit follows that f is a bijective function.

3.4.2. The intuition for the other stepsSince f (α) ∈ β(p) whenever α(p) = {α}, YB needs to be a linear transformation of YA, YB = aαYA + bα, on the

set Jα = {YA(p)|α(p) = {α}}. The intuition for the rest of the proof is that any pair of actions (α, α′) that are usedon the same indifference surface of the induced preference � (for lotteries at which they are the respective uniqueoptimal actions) causes an overlap of Jα and Jα′ , hence YB needs to be the same linear transformation of YA onthe union of Jα and Jα′ , i.e., (aα, bα) = (aα′ , bα′ ). The non-redundancy assumption and YA’s continuity guaran-tee the existence of a connected graph of such (α, α′) pairs, so we can string together the equalities (aα, bα) =(aα′ , bα′ ) to prove that all the pairs (aα, bα) are identical, thus (A, (Φα)α∈A) is a positive affine transformation of(B, (Ψβ)β∈B).

We definemA = minp∈L

YA(p) andMA = maxp∈L

YA(p) (YA achieves its extremes since it is a continuous function) and,

for any α∈A, let

Iα = {YA(p)|α(p) = {α}} ∩ (mA,MA). (3.22)

The non-redundancy and non-constancy of (A, (Φα)α∈A) imply that Iα is non-empty for all α∈A. If we rewrite

Iα = {φα(p)|φα(p) > φα′ (p), ∀α′ ∈A \ {α}} ∩ (mA,MA) (3.23)


it is easy to prove, using (A, (Φα)α∈A)’s non-constancy, that Iα is an open set (in the standard topologyon R).

Define the relation ∼ on A by α ∼ α′ ⇔ (aα, bα) = (aα′ , bα′ ). Obviously, ∼ is an equivalence relation.

Step 2. If Iα ∩ Iα′ �= ∅ then α ∼ α′.

Fix α and α′ in A such that Iα and Iα′ are not disjoint. Iα ∩ Iα′ has cardinality ℵ0 since it is open and non-empty, sowe can find two different reals, u and v, belonging to Iα ∩ Iα′ . Therefore, there exist lotteries p, q, p′ and q′ such that

YA(p) = YA(p′) = u and YA(q) = YA(q′) = v, and (3.24)

α(p) = α(q) = {α} and α(p′) = α(q′) = {α′}. (3.25)

We have

YB(p) = YB(p′) and YB(q) = YB(q′) (3.26)

as (A, (Φα)α∈A) and (B, (Ψβ)β∈B) induce the same preference relation over L.Note that

YB(p) = ψf (α)(p) (3.27)

= aαφα(p) + bα (3.28)

= aαu+ bα (3.29)

because f (α) ∈ β(p) by the previous step. We can get analogous expressions for YB(p′), YB(q), YB(q′), so the followingequalities hold

YB(p) = aαu+ bα (3.30)

YB(p′) = aα′u+ bα′ (3.31)

YB(q) = aαv+ bα (3.32)

YB(q′) = aα′v+ bα′ . (3.33)

Subtracting the sum of (3.31) and (3.32) form the sum of (3.30) and (3.33) and using (3.26) we obtain that

(aα − aα′ )(u− v) = 0, or aα = aα′ (u �= v). (3.34)

Then (3.26), (3.30) and (3.31) imply that bα = bα′ , proving that α ∼ α′.

Step 3. The set {Iα|α∈A} is an open cover of (mA,MA).

Assuming the contrary, it follows that there exists yA ∈ (mA,MA) such that yA /∈ ⋃α∈AIα, or α(p) is not a singleton

for any lottery p∈YA−1(yA).From YA’s continuity it follows that there exist lotteries q and r and a positive real ε with the property that

YA(r) < yA < YA(q′), ∀q′ ∈L, ‖q′ − q‖ ≤ ε. (3.35)

Also from YA’s continuity we obtain that for all lotteries q′ with ‖q′ − q‖ ≤ ε there exists a tq′ ∈ (0, 1) such that

YA(tq′q′ + (1 − tq′ )r) = yA. (3.36)

A measure theoretical argument similar to the one used in the proof of the first step shows that there exists A′ ⊂ A,|A′| ≥ 2, such that

Q = {q′|q′ ∈L, ‖q′ − q‖ ≤ ε, α(tq′q′ + (1 − tq′ )r) = A′} (3.37)

has dimension n− 1.If α and α′ are two different elements of A′ then

φα(tq′q′ + (1 − tq′ )r) = yA = φα′ (tq′q′ + (1 − tq′ )r) (3.38)


or equivalently,

yA = tq′φα(q′) + (1 − tq′ )φα(r) (3.39)

yA = tq′φα′ (q′) + (1 − tq′ )φα′ (r) (3.40)

for all q′ in Q. Solving for tq′ we obtain that

yA − φα(r)

φα(q′) − φα(r)= tq′ = yA − φα′ (r)

φα′ (q′) − φα′ (r), ∀q′ ∈Q, (3.41)

which can be rewritten

φα(q′) = yA − φα(r)

yA − φα′ (r)φα′ (q′) − yA − φα(r)

yA − φα′ (r)φα′ (r) + φα(r), ∀q′ ∈Q. (3.42)

(Note that yA > YA(r) ≥ max(φα(r), φα′ (r)).)φα’s linearity onL and Q’s full dimensionality with respect toL imply that (3.42) holds for all q′ ∈L, a contradiction

with (A, (Φα)α∈A)’s non-dissociativity.

Step 4. Any two actions in A are equivalent under ∼.

Assuming the contrary, we can find a partition A′ ⋃A′′ of A such that no action in A′ is equivalent to an action inA′′. Hence, by the second step,⎛

⎝ ⋃α′ ∈A′

Iα′

⎞⎠ ⋂ ⎛

⎝ ⋃α′′ ∈A′′

Iα′′

⎞⎠ = ∅. (3.43)

However, from the third step it follows that⎛⎝ ⋃α′ ∈A′

Iα′

⎞⎠ ⋃ ⎛

⎝ ⋃α′′ ∈A′′

Iα′′

⎞⎠ (3.44)

forms a partition of (mA,MA) into two non-empty open sets, a contradiction with the fact that (mA,MA) is a connectedset, which finishes the proof of the step and ends the proof of the theorem. �

4. Applications of the uniqueness result

Many dynamic choice applications can be accommodated by our framework. Consumption–investment decisionsand rate of return preferences are only two examples (Kreps and Porteus, 1979; Machina, 1984). The uniqueness ofthe representation has one very powerful implication in all applications: we can elicit the individual’s preferencesover risky prospects described by action–lottery pairs from his primitive ordinal induced preference over temporallotteries. Indeed, if we restrict attention to simple action–utility induced preferences, the individual’s ordinal inducedpreference determines the action–utility set uniquely up to an affine transformation, and the action–utility set revealsthe individual’s ranking of action–lottery pairs.8

Therefore, under our assumptions, exclusive knowledge of the individual’s ordinal induced preference over temporallotteries should enable us to find out how the individual ranks any two action–lottery pairs, hence, for any α, α′ ∈Aand p, q∈L, it reveals which of the two scenarios

• take action α, face lottery p• take action α′, face lottery q

8 We have in mind applications for which the action space A is self-evident and only the underlying utility indices are not known.


the individual prefers. This information reveals what action the individual would optimally take when facing anytemporal lottery, and what changes in the individual’s behavior one should expect if the set of actions he is allowed totake is restricted.

Moreover, unique determination of the underlying utility indices suggests that we can decompose the risk attitude ofan individual whose preferences over temporal lotteries are action–utility induced, via Machina’s analysis of risk undertemporal resolution of uncertainty into the various risk attitudes he displays when he compares lotteries based on thepremise of pre-commitment to a specific action. The set of induced preference functionals is a subset of the non-linearpreference functionals introduced by Machina (1982) in order to provide an “expected utility” analysis without theindependence axiom. Most of the risk attitude analysis existent for the expected utility theory can be extended to thenon-linear preference functionals by sequencing together local expected utility arguments.

For example, a non-linear preference functional satisfies first (second) order stochastic dominance if and only ifthe local utility function is increasing (concave). Similarly, the necessary and sufficient conditions of expected utilitytheory for comparative risk aversion and absolute or relative risk aversion have direct analogues for characterizing riskunder non-linear preference functionals. Machina (1984) implies that the induced preference functional is a non-linearpreference functional with local utility functions given by the corresponding maximizing utility indices. Therefore,we can make inferences about the risk-taking behavior of an individual with action–utility induced preferences byapplying the expected utility risk theory to each of the von Neumann-Morgenstern utility indices Φα(·) separately.

4.1. Consumption–investment optimal decisions

We live in a world in which most of the transfers and prices belong to the set

M = {1 cent, 2 cents, 3 cents, . . . , 1020 cents}. (4.1)

We look at an individual with (known) initial endowment e∈M facing uncertainty about his future income, x∈M,described by the temporal lottery p over M. The individual can invest part of his initial endowment in a portfolio of kavailable assets with random return vector ρ, with ρ independent of his future income. If we denote by α the vector ofamounts invested in each asset,9α∈Mk, then the individual’s utility index for future income is given by

Φα(x) = Eρ[U(e− 1K · α, x+ ρ · α)], (4.2)

where U(·, ·) is the von Neumann-Morgenstern utility for two-period consumption (Eρ is the expectation over therandom vector ρ).10

When facing a temporal lottery over future income, p, the individual chooses his portfolio α in order to maximizeφα(p) (φα(·) is the expected utility function corresponding to the von Neumann-Morgenstern utility index Φα(·)).Let A be a minimal subset of Mk with respect to inclusion such that the action–utility induced preference relations�(A,(Φα)α∈A) and �(Mk,(Φα)

α∈Mk ) are identical. The individual’s preferences for temporal lotteries over future incomeare induced by the action–utility set (A, (Φα)α∈A). This action–utility set satisfies non-redundancy (by A’s definition),and it does not seem unreasonable to think that it also satisfies the other two assumptions that deliver the uniquenessof the representation result. Non-dissociativity follows from the assumption that different investment decisions inducedistinct risk attitudes towards future income, and non-constancy follows from the assumption that higher income isstrictly preferred to lower income.

The uniqueness result implies that the individual’s induced preference for lotteries over future income uniquelydetermines, up to an affine transformation, the generating simple action–utility set (A, (Φα)α∈A). Therefore, theindividual’s ordinal induced preference reveals information about all his underlying von Neumann-Morgenstern utilityindices. This information facilitates many comparative statics results. First, we can find out how much the individualwould optimally invest in each asset when facing a fixed future income lottery p. Second, we can find out how theindividual ranks scenarios of the type: buy portfolio α and face uncertainty p about future income. Third, we can findout how the individual would change his investment behavior if we place restrictions on the amounts he can invest

9 We shall slightly abuse language, sometimes calling α a portfolio. In those instances, we have in mind the portfolio that is made up by vector αworth of the corresponding assets.10 The assumption that future income and asset return are independent is essential to obtaining this representation.


in different assets, for example, if a quota is imposed on how much he is allowed to invest in a specific asset, or if aspecific asset becomes unavailable.

4.2. Consumption–savings and intertemporal income substitution

An even richer set of comparative statics results can be obtained if the only asset that the individual can invest in isa safe asset, i.e., in the application of the previous subsection, ρ (and hence α) is one-dimensional and deterministic.In that case, our result implies that we can uniquely recover the underlying von Neumann-Morgenstern utility indices,which are given by

Φα(x) = U(e− α, x+ ρα). (4.3)

Therefore, the individual’s ordinal induced preference over future income lotteries is sufficient to uniquely traceout his multiperiod von Neumann-Morgenstern utility function U(·, ·) (over the range in which his first periodconsumption is optimal for some future income lottery p). From information on U(·, ·) we can determine the indi-vidual’s attitude towards intertemporal income substitution, find out how his saving behavior changes under variousincome and interest tax schemes, and study the effect of many other macroeconomic policies on consumption–savingschoice.

4.3. Portfolio choice and intertemporal income substitution

In a model of portfolio choice the primitive uncertainty needs to be represented by the future value of the portfolio,and an action needs to represent the amount invested in a portfolio. However, we note that the linear structure of thelottery space L introduced in the previous section cannot be used directly to model this portfolio set. For in this case,given two portfolios p, q∈L, the linear combination 1/2p+ 1/2q does not stand for a portfolio that can be replicatedby buying 1/2 of portfolio p and 1/2 of portfolio q, but for the portfolio generated by flipping a fair coin and deliveringportfolio p if the outcome is heads and portfolio q if the outcome is tails.

We say that a portfolio q that can be generated by a randomization device that delivers one of the portfoliosp1, p2, . . . , pm formed with the assets available on the market is an idiosyncratic portfolio or, more precisely, a convexprobability combination ofp1, p2, . . . , pm. Some idiosyncratic portfolios may be non-existent on the market. However,if p1, p2, . . . , pm are portfolios formed with the assets available on the market that have the same cost c, arbitrageimplies (assuming ability to buy and sell any such portfolio for c) that any idiosyncratic portfolio that is delivered bya convex probability combination of them should cost c as well, if the market offered it. It should also be noted thatthe portfolio formed by combining 1/2 of portfolio p and 1/2 of portfolio q will typically not be a convex probabilitycombination of p and q.

We are led to consider the set N of all portfolios that cost $1, and let the set of prizes X be a maximal subset (withrespect to inclusion) of N such that no element of X is a convex probability combination of other elements of X. Weassume that X consists of finitely many $1 portfolios. Let L be the set of convex probability combinations of portfoliosin X. Thus an element of L will be an idiosyncratic portfolio not necessarily offered by the market, but which shouldcost $1, if it were offered. We can think of L as the completion of the market with randomization devices. It is implicitlyassumed that the uncertainty concerning the random generation of an idiosyncratic portfolio is resolved immediatelyafter the investment decision is made.

If we denote by α the amount invested in an idiosyncratic portfolio (α belongs to M, as defined by (4.1)), then theindividual’s utility index for future portfolio payoff x is given by11

Φα(x) = U(e− α, f + αx), (4.4)

11 Note the notation difference from Section 4.1(necessary for consistency with the notation established in Section 3): here x stands for the realizedportfolio payoff, and α for the amount invested in a portfolio that belongs to the set of rates of return L, while in Section 4.1, x stands for the realizedincome, and α for the amount invested in the fixed portfolio ρ. The set of temporal prospects for which induced preferences are observed is the setof rates of return in this subsection, and the set of income lotteries in Section 4.1.


where e and f are the (known) initial endowment and future income, respectively, and U(·, ·) is the von Neumann-Morgenstern utility for two-period consumption.12 If we let A be a minimal subset (with respect to inclusion) of Msuch that (A, (Φα)α∈A) and (M, (Φα)α∈M) induce the same preference relation over idiosyncratic portfolios, we canargue, as in the consumption–investment application, that the action–utility set (A, (Φα)α∈A) should be simple.

The uniqueness of representation result will have implications similar to the ones we have established for theapplications considered in the previous subsections. The individual’s ordinal induced preference over idiosyncraticportfolios is sufficient to uniquely trace out his multiperiod von Neumann-Morgenstern utility functionU(·, ·) (over therange in which the individual’s first period consumption is optimal for some idiosyncratic portfolio p). The individual’sordinal induced preference over idiosyncratic portfolios – which we could elicit if the market is complete with respectto randomization devices, or there is some other mechanism that can exact the individual to choose between twoidiosyncratic portfolios – uniquely determines the individual’s optimal investment in each portfolio and the individual’sranking of any two scenarios: invest α in portfolio p and invest α′ in portfolio p′; in addition, it uniquely determineshis investment behavior adjustment if a quota is imposed on how much he is allowed to invest in a specific asset,or if a specific asset becomes unavailable. Exclusive knowledge of the individual’s ordinal induced preference overidiosyncratic portfolios also uniquely determines the individual’s attitude towards income substitution over time, hisconsumption–investment response to various income and capital tax schemes, and to many other changes in themacroeconomic environment driven by the government or by the free market.

Acknowledgements

This paper represents the main contents of my senior thesis at Princeton University. I thank my adviser, Faruk Gul,for valuable guidance, and Dilip Abreu, Drew Fudenberg, and Wolfgang Pesendorfer for helpful discussions.

References

Kreps, D.M., 1988. Notes on the Theory of Choice. Westview Press Inc., Boulder, Colorado, pp. 43–45, 171–175.Kreps, D.M., Porteus, E.L., 1978. Temporal resolution of uncertainty and dynamic choice theory. Econometrica 46, 185–200.Kreps, D.M., Porteus, E.L., 1979. Temporal von Neumann-Morgenstern and induced preferences. Journal of Economic Theory 20, 81–109.Machina, M.J., 1982. “Expected utility” analysis without the independence axiom. Econometrica 50, 277–324.Machina, M.J., 1984. Temporal risk and the nature of induced preferences. Journal of Economic Theory 33, 199–231.Mossin, J., 1969. A note on uncertainty and preference in a temporal context. American Economic Review 59, 172–173.Spence, M., Zeckhauser, R., 1972. The effect of the timing of consumption and the resolution of lotteries on the choice of lotteries. Econometrica

40, 401–403.

12 To obtain these underlying utility indices we use the fact that any idiosyncratic portfolio that belongs to L should cost $1, if it were offered bythe market.

unique induced preference representations

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