Multi-utility representations of preferences over lotteries

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arXiv:2210.04739v2 [math.CA] 13 Jan 2024

EXPECTED MULTI-UTILITY REPRESENTATIONS OF

PREFERENCES OVER LOTTERIES

PAOLO LEONETTI

Abstract. Let %be a binary relation on the set of simple lotteries over a

countable outcome set Z. We provide necessary and suﬃcient conditions on

%to guarantee the existence of a set Uof von Neumann–Morgenstern utility

functions u:Z→Rsuch that

p%q⇐⇒ Ep[u]≥Eq[u]for all u∈U

for all simple lotteries p, q. In such case, the set Uis essentially unique. Then,

we show that the analogue characterization does not hold if Zis uncountable.

This provides an answer to an open question posed by Dubra, Maccheroni,

and Ok in [J. Econom. Theory 115 (2004), no. 1, 118–133]. Lastly, we show

that diﬀerent continuity requirements on %allow for certain restrictions on the

possible choices of the set Uof utility functions (e.g., all utility functions are

bounded), providing a wide family of expected multi-utility representations.

1. Introduction

Let %be a binary relation on a topological space X. The classical Debreu’s

utility representation theorem provides suﬃcient conditions on %and Xfor the

existence of a continuous utility function u:X→Rwhich represents %, so that

x%y⇐⇒ u(x)≥u(y),

for all x, y ∈X. In such case, %is necessarily complete and transitive.

However, since the seminal works of Aumann [5] and Bewley [7], it is known

that completeness may not be considered a basic trait of rationality of the decision

maker. A long line of research studies this topic, see e.g. [9,14,18,34,35] and

2020 Mathematics Subject Classiﬁcation. C79, D11.

Key words and phrases. Preferences over lotteries; closed convex cones; ﬁnitely-supported

probability measures; duality pairs; expected multi-utility representation.

2Paolo Leonetti

references therein. The main objective of this paper is to provide continuous multi-

utility representation theorems for preference relations over lotteries which may

fail to be complete. Thus, the basic question can be formulated as follows: Given

binary relation %on a suﬃciently well-behaved topological space X, we search

for necessary and suﬃcient conditions on %for which there exists a collection U

of continuous utility functions u:X→Rsuch that

x%y⇐⇒ ∀u∈U, u(x)≥u(y),

for all x, y ∈X. Note that, in such case, the binary relation is necessarily reﬂexive

and transitive, hence a preorder. In addition, we may interpret %as resulting from

the unanimity of a family of complete preference relations, each one admitting a

classical Debreu’s continuous utility representation.

In many instances, the underlying space Xhas also the structure of (convex

subset of a) vector space, so that it makes sense to require each continuous utility

function uto be, in addition, linear. To make an example, in the important work

of Dubra, Maccheroni and Ok [14], the authors provide necessary and suﬃcient

conditions on a binary relation %over the set Pof Borel probability measures

on a compact metric space to satisfy the representation

p%q⇐⇒ ∀u∈U, Ep[u]≥Eq[u],

for all p, q ∈P. Moreover, the set Uis essentially unique, in a precise sense. The

interpretation of such representation is that the decision maker can show a lack

of conﬁdence in the evaluation of lotteries because he is unsure about his future

tastes/risk attitudes, each represented by an utility function in U.

In the same work, the authors pose as open question in [14, Remark 1] whether

the analogue characterization holds replacing Pwith the set of simple lotteries

on a given outcome set Z. Hereafter, a simple lottery (or, simply, a lottery)

stands for a ﬁnitely-supported probability measure on Z.The main result of

this work is that the answer is aﬃrmative if and only if Zis ﬁnite or countably

inﬁnite. See Theorem 2.3 and Theorem 2.4 below for the positive and negative

case, respectively. Details follow in Section 2.

Our methods of proofs will be suﬃciently strong to provide a wide family of

expected multi-utility representations. Indeed, we show that diﬀerent continuity

Representations of preferences over lotteries 3

requirements on %allow for certain restrictions on the possible choices of the

utility functions (e.g., all utility functions are bounded, or countably supported).

Indeed, it is known that diﬀerent continuity properties in an inﬁnite dimensional

framework may correspond to diﬀerent behavioral implications, see e.g. [16].

Lastly, removing also the transitivity and continuity requirements, we provide a

representation for reﬂexive binary relations %which satisfy the independence ax-

iom over lotteries on arbitrary outcome sets Z, in the same spirit of the coalitional

representation given by Hara, Ok, and Riella in [24, Theorem 1].

1.1. Motivations. Before we provide the details of our contribution, we remark

that there are several economic reasons why one should be interested in the study

of incomplete preference relations. Indeed, many authors strongly suggest, from

both the normative and positive viewpoints, that preference relations should allow

for potential indecisiveness of the decision makers, see e.g. [5,7]. Accordingly,

within the revealed preference paradigm, it has been recently shown that incom-

pleteness of preferences may arise due to status quo bias and the endowment

eﬀect, see e.g. [15,33]. As an application, the resulting theory in [15] has been

able to cope with the classical preference reversal phenomenon.

In the case of our incomplete preferences, there are economic examples in which

a decision maker is in fact composed of several agents, each one with a given utility

function. For instance, in a social choice framework such as environmental policy,

healthcare, or public policy, one commonly uses the Pareto dominance to rank

alternatives (cf. First and Second Welfare Theorem), and such criterion is, of

course, incomplete. As another real-world example, when purchasing a car, one

needs to consider factors like price, fuel eﬃciency, safety, and style; here, the

decision maker has to compare preferences over these multiple criteria.

As anticipated before, we are going to study how to handle the problem of ac-

tually representing such preferences with expected multi-utility models as in [14].

It is worth to remark that the very same idea goes back at least to von Neumann

and Morgenstern, see [40, pp. 19–20], although without details. In this work, in

particular, we focus on preferences over simple lotteries (i.e., probability distri-

butions with ﬁnite support) on a given outcome set. There are many practical

4Paolo Leonetti

motivations which justify this choice (apart from answering an open question in

[14]), and we list below several ones.

First, simple lotteries provide a way to model and analyze situations involving

uncertainty and risk, cf. e.g. [8,13]. Accordingly, they can be considered as the

building blocks of utility theory, which is a cornerstone of microeconomics and

rational decision making. As remarked by Evren in [17], simple lotteries can also

be interpreted, in a game theory framework, as strategic decisions where players

and the nature are conﬁned to play simple strategies.

Second, in various ﬁelds such as ﬁnance, engineering, and operations research,

problems involve decision variables that are inherently ﬁnite. Preferences over

probability distributions on these variables are essential, e.g., for optimation prob-

lems. Simple lotteries are also employed in portfolio theory to assess the risk and

return proﬁles of diﬀerent investment portfolios. Diversiﬁcation strategies, aimed

at reducing risk, rely on preferences over (asset) lotteries. Similarly, in policy

analysis and resource allocation problems, decision makers often have to make

choices that aﬀect ﬁnite populations or groups. Representations of these pref-

erences help assess the impact of diﬀerent policies on such ﬁnite groups. As a

last example, in surveys or experiments, responses often take on discrete values,

and the resulting probability distributions are ﬁnitely-supported. In this sense,

studying preferences on the latter distributions allows decision makers to directly

deal with the empirical data.

Third, simple lotteries are often easier to work with and understand than con-

tinuous ones. This simplicity can make it more feasible for individuals to express

and communicate their preferences over such distributions. In addition, if pref-

erences are not fully known or cannot be easily elicited, the study of incomplete

preferences on probability measures oﬀers a practical approach. Indeed, decision

makers can express partial preferences over some aspects of a decision problem,

even when they are unsure about others.

Lastly, even if representations of such incomplete preferences may seem rather

technical, recent studies have shown that they are useful tools to understanding

certain classes of complete preferences. For example, the cautious expected util-

ity model by Cerreia-Vioglio, Dillenberger, and Ortoleva [10] can be derived by

applying the expected multi-utility representation of Dubra, Maccheroni, and Ok

Representations of preferences over lotteries 5

[14]. A similar remark holds for multi-prior expected multi-utility representations

due to Galaabaatar and Karni [20].

2. Preliminaries and Main results

Let ∆be the set of ﬁnitely-supported probability measures on a given nonempty

set Z, which is interpreted as the set of lotteries over the set of possible outcomes

Z, see e.g. [10,12,22,23,29]. A utility function over Zis a real-valued map

u:Z→R. Note that ∆is a convex subset of the vector space of ﬁnitely-

supported functions p:Z→R, hereafter denoted by RZ

0. Hence, a lottery is a

nonnegative function p∈RZ

0such that Pz∈Zp(z) = 1.

Deﬁnition 2.1. Let Zbe a nonempty set. A binary relation %on ∆satisﬁes the

independence axiom if

p%q⇐⇒ αp + (1 −α)r%αq + (1 −α)r(1)

for all lotteries p, q, r ∈∆and scalars α∈(0,1).

Hereafter, for each p∈RZ

0and utility function u:Z→R, deﬁne

Ep[u] := Xz∈Zu(z)p(z).

As anticipated in Section 1, our main objective is to characterize binary relations

%over lotteries through expected multi-utility representations in the sense of

Dubra, Maccheroni, and Ok [14], that is, to search for necessary and/or suﬃcient

conditions on %for which there exists a set Uof utility functions (possibly, of a

certain type) such that

p%q⇐⇒ ∀u∈U, Ep[u]≥Eq[u],(2)

for all lotteries p, q ∈∆.

To this aim, we endow the vector space RZ

0with the weak-topology

w:= σ(RZ

0,RZ),

so that a net (pi)i∈Iweak-converges to some p∈RZ

0if and only if

(Epi[u])i∈I→Ep[u]for all u∈RZ.

Note that (RZ

0, w)is a locally convex Hausdorﬀ topological vector space. Moreover,

we endow RZwith the weak⋆-topology σ(RZ,RZ

0)which is, equivalently, the

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摘要：

arXiv:2210.04739v2[math.CA]13Jan2024EXPECTEDMULTI-UTILITYREPRESENTATIONSOFPREFERENCESOVERLOTTERIESPAOLOLEONETTIAbstract.Let%beabinaryrelationonthesetofsimplelotteriesoveracountableoutcomesetZ.Weprovidenecessaryandsuﬃcientconditionson%toguaranteetheexistenceofasetUofvonNeumann–Morgensternutilityfunct...

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