A Non-Bayesian Theory of State-Dependent Utility · The binary relation ' on Adepicts the decision maker’s preferences over acts. The symmetric and asymmetric parts of ', and ¡,

A Non-Bayesian Theory of State-Dependent Utility∗

Brian Hill

GREGHEC, HEC Paris & CNRS†

December 7, 2017

Abstract

Many decision situations involve two or more of the following divergences from

subjective expected utility: imprecision of beliefs (or ambiguity), imprecision of tastes

(or multi-utility), and state dependence of utility. Examples include multi-attribute de-

cisions under uncertainty, such as some climate decisions, where trade-offs across

attributes may be state dependent. This paper proposes and characterises a model of

uncertainty averse preferences that can simultaneously incorporate all three phenom-

ena. The representation supports a principled separation of (imprecise) beliefs and

(potentially state-dependent, imprecise) tastes, and we pinpoint the axiom that ensures

such a separation. Moreover, the representation supports comparative statics of both

beliefs and tastes, and is modular: it easily delivers special cases involving various

combinations of the phenomena, as well as state-dependent multi-utility generalisa-

tions of popular ambiguity models.

Keywords: State-dependent utility, uncertainty aversion, multiple priors, ambiguity, im-

precise tastes, multi-utility.

JEL classification: D81

∗The author gratefully acknowledges support from the ANR project DUSUCA (ANR-14-CE29-0003-01).†1 rue de la Libération, 78351 Jouy-en-Josas, France. E-mail: [email protected].

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Brian Hill Non-Bayesian Theory of State-Dependent Utility

1 Introduction

Consider a decision maker who has been diagnosed with a serious yet rare degenerative

disease, and is faced with the choice between not doing anything or paying to receive a

new, expensive but not very well-understood treatment. If he does nothing, there is a 60%

chance of an acute form of the disease, involving a significant reduction in life expectancy,

and a 40% chance of mild form, with moderate reduction in life expectancy. The treat-

ment’s chances of success are between 25% and 75%; a successful treatment fully cures

the disease (so life expectancy is normal); failure results in a short and painful life. This

decision has a marked resemblance with Ellsberg’s (1961) examples, insofar as the prob-

abilities of various outcomes are known for one of the options, and they are not known

precisely for the other. Borrowing a term from statistics, we will say that there may be

imprecision in beliefs.1 It is well-known that in such cases the decision maker may exhibit

non-neutral attitudes to uncertainty, such as uncertainty aversion. Moreover, given the lack

of familiarity with some of the potential consequences, this decision may also involve what

could be called imprecision in tastes: the decision maker may have trouble trading off the

reductions of life expectancy associated with inaction against the reduction in wealth as-

sociated with treatment. Finally, as is well-known (Drèze, 1987; Arrow, 1974; Cook and

Graham, 1977; Karni, 1983b), such health-related decisions often involve state-dependent

utility: the decision maker’s utility for the money saved by not undergoing the treatment

may depend on his state of health.

Examples such as this are relevant beyond health economics. Indeed, interpreting the

decision maker as the human race and the disease as man-made climate change arguably

yields a simplified version of some choices facing our society today. Certainly, climate

change decisions involve imprecision of beliefs (lack of fully specified probabilities on

the basis of the scientific evidence), imprecision of tastes (difficulty in comparing future

possible outcomes for mankind) and state dependence of utility (the value of the resources

saved or spent today may depend on the climate).

From a decision-theoretic perspective, these cases involve several distinct violations of

the standard axioms of subjective expected utility (Savage, 1954; Anscombe and Aumann,

1The term ‘imprecise probabilities’ is widely used in statistics and philosophy; see for example Walley

(1991); Bradley (2014) and the references therein.

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1963).2 To date, these violations have only been studied in isolation: the literature on

ambiguity, for example, almost universally assumes precise tastes and state-independent

utility, and that on state-dependent utility generally works in the context of precise beliefs

and tastes. So a model of uncertainty averse decision making is needed that can handle all

of these violations, whilst ideally preserving the separation of beliefs and tastes present in

the standard subjective expected utility model. The current paper addresses this need.

We propose and characterise a state-dependent non-expected utility representation of

which the following is a suggestive yet paradigmatic special case:

(1) minpPC

ÿ

sPS

ppsq minuPυpsq

upfpsqq

where C is a (closed convex) set of probability measures over states and υ is a function

assigning to each state s a (closed convex) set of affine utility functions over consequences.

This representation can accommodate all the aforementioned phenomena. Indeed, it dis-

plays: (i) imprecise beliefs, in the use of multiple priors C; (ii) imprecise tastes, in the

multiple utility functions υpsq; and (iii) state dependence of utility, in the possible depen-

dence of the set of utility functions used to evaluate consequences on the state.

On the axiomatic front, given the absence of expected utility both over states and con-

sequences, no form of the Independence axiom appears among our conditions. In its stead,

there is an uncertainty aversion condition that retains the same spirit as, though strengthens

the Uncertainty Aversion axiom due to Schmeidler (1989): it basically says that hedging

(via mixing) can only improve the relative evaluation of acts. Similarly, given the state de-

pendence of utility, there is no Monotonicity or State-Independence axiom. In their stead is

State Consistency, which basically says that one’s preferences regarding the consequence

obtained in a given state is independent of what one would get in the other states. This is the

minimal axiom ensuring the coherence of the standard notion of preferences conditional on

a state. We show that it is a necessary condition for a separation of beliefs and (potentially

state-dependent) tastes of the sort present in (1): no such separation is possible in its ab-

sence. This suggests thinking of State Consistency as a basic axiom for state-dependent

utility in the context of imprecise beliefs or tastes. The need for a specific condition is in2More precisely, in the Anscombe-Aumann framework and in the context of complete preferences: vi-

olations of the Independence axiom for imprecision of beliefs, of the restriction of Independence to risky

prospects for imprecision of tastes, and of the Monotonicity axiom for state dependence of utility.

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contrast with the expected utility context adopted for most work on state-dependent utility,

and seems to have gone previously unnoticed in the literature.

Whilst our benchmark model is general, insofar as it can accommodate all of the afore-

mentioned violations of subjective expected utility, it provides a springboard for the study

of models incorporating some but not all of the violations. As an illustration, we show how

to easily obtain special cases in which different factors are ‘shut down’. We obtain axioma-

tisations of state-dependent (precise) utility and state-independent multi-utility uncertainty

averse models, as well as of state-dependent multi-utility model with probabilistic beliefs.

Moreover, the treatment of uncertainty in the benchmark model is also extremely general,

covering virtually all existing uncertainty averse models in the ambiguity literature. As

such, our approach straightforwardly yields state-dependent multi-utility generalisations of

standard ambiguity models, such as the generalisation of the Gilboa and Schmeidler (1989)

multiple prior model in (1) above. We also explore an alternative treatment of imprecision

in tastes to that in (1), and provide results underlining the particularity of this representa-

tion’s capacity to separate beliefs and tastes whilst accommodating imprecision in both.

To corroborate the interpretations of the different elements of the model, we provide

comparative statics exercises separating their impacts on choice. Finally, an area where

the phenomena considered may be particularly relevant are multi-attribute decisions under

uncertainty. Beyond issues of imprecision, there are important cases where trade-offs may

be state-dependent: how one weighs a monetary payoff against a delay in its reception

could depend on one’s state of health, and such dependence could have an impact on, say,

long-term investment decisions or the choice of life insurance. We illustrate some of the

tools provided by our approach on a simple multi-attribute decision under uncertainty.

The paper is organised as follows. The framework is set out in Section 2. The bench-

mark model is stated and characterised in Section 3. Section 4 maps out the space of special

cases, and in doing so clarifies the relationship to the existing literature on the various vio-

lations. Section 5 provides a comparative statics analysis of the model. Section 6 focusses

on the State Consistency axiom, characterising a representation resulting from dropping it

from our benchmark result, whereas Section 7 examines the consequences of adopting an

alternative decision rule for multi-utility. Section 8 considers an illustrative application,

loosely related to that with which the paper began, and Section 9 discusses extensions and

remaining related literature. Proofs are contained in the Appendix.

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2 Preliminaries

Let S be a finite set of states; ∆ is the set of probability measures over S. Let X be the set

of consequences. An act is a function from states to consequences; let A “ XS be the set

of acts. For state s and acts f and g, the act fsg is defined as follows: fsgpsq “ fpsq and

fsgptq “ gptq for all t ‰ s. With slight abuse of notation, a constant act taking consequence

x for every state will be denoted x and the set of constant acts will be denoted X .

We assume that X is a compact polyhedral convex subset of a finite-dimensional vector

space.3 For example, the Anscombe-Aumann (1963) framework is a special case of that

used here, taking X as the set of lotteries over a finite set of prizes. However, X could

also be a set of commodity bundles, or multi-attribute outcomes (with a finite number

of continuous-valued attributes taking values in bounded intervals), as in multi-attribute

decision making. Given the structure of X , A is also a vector space with the following

mixture relation, defined pointwise: for f, h P A and α P r0, 1s, the mixture αf `p1´αqh

is defined by pαf ` p1 ´ αqhqpsq “ αfpsq ` p1 ´ αqhpsq. We write fαh as short for

αf ` p1´ αqh.

The binary relation ľ on A depicts the decision maker’s preferences over acts. The

symmetric and asymmetric parts of ľ, „ and ą, are defined in the standard way. A state

s is said to be null if fsh „ h for all f, h P A; otherwise it is non-null. A functional

V : AÑ < represents ľ if, for all f, g P A, f ľ g if and only if V pfq ě V pgq.

A utility function is an affine function u : X Ñ <. We endow U, the set of utility

functions, with the Euclidean topology. A set of utility functions U Ď U is non-trivial if it

contains a non-constant function. For any c, c1 P X , U Ď U is c, c1-precise if there exists

u P U with updq ď u1pdq for d “ tc, c1u and all u1 P U . We define the following order on

sets of utility functions: U1 ď U2 if and only if, for every x P X and u P U2, there exists

u1 P U1 with u1pxq ď upxq. Affine transformations of sets of utility functions are defined

as follows: for any U Ď U, κ P <ą0 and λ P <, κU ` λ “ tκu` λ| u P Uu.A function υ : S Ñ 2UzH is non-trivial, closed and convex if υpsq is non-trivial, closed

and convex for every s P S. Moreover, for any h, h1 P A, it is h, h1-precise if υpsq is

hpsq, h1psq-precise for every s P S, and it is h, h1-constant if, for every α P r0, 1s and

s, t P S, minuPυpsq uphpsqαh1psqq “ minuPυptq uphpsqαh

1psqq. Containment and unions are

3A polyhedral convex set is a set of points satisfying a finite collection of linear inequalities (Rockafellar,

1970).

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defined pointwise: for every pair of υ1, υ2 : S Ñ 2UzH, υ1 Ď υ2 if and only if υ1psq Ď

υ2psq for all s P S, and υ1 Y υ2 : S Ñ 2UzH is defined by υ1 Y υ2psq “ υ1psq Y υ2psq for

all s P S. For any such υ, we let υpXq “Ť

sPStminuPυpsq upxq | x P Xu.

Finally, we introduce the following properties of functions α : ∆ Ñ 2<ą0ˆ<. α is

non-trivial if there exists p P ∆ such that αppq ‰ H. It is null-consistent if, for every p, q

such that αppq, αpqq ‰ H and every s P S, ppsq “ 0 if and only if qpsq “ 0. It is convex if,

whenever pa, bq P αppq and pa1, b1q P αpp1q, then for all λ P r0, 1s, pλa`p1´λqa1, λb`p1´

λqb1q P α´

λaλa`p1´λqa1

p` p1´λqa1

λa`p1´λqa1p1¯

. It is upper hemicontinuous if, whenever pan, bnq P

αppnq with pn Ñ p and pan, bnq Ñ pa, bq P <ą0 ˆ <, then pa, bq P αppq. It is grounded

if there exists p P ∆ such that p1, 0q P αppq. α is calibrated with respect to a function

υ : S Ñ 2UzH if, for all p P ∆, pa, bq P αppq and x P υpXq, ax ` b ě x (where it is

evident from the context, mention of υ may be omitted).4 The union of these functions

is defined pointwise. An order over them is defined analogously to that for sets of utility

functions: α1 ď α2 if and only if, for every c P <S and pa, bq P α2ppq for p P ∆, there

exists pa1, b1q P α1pp1q for p1 P ∆ such that a1ř

sPS p1psq.cpsq ` b1 ď a

ř

sPS ppsq.cpsq ` b.

3 Benchmark model

3.1 Axioms

Throughout the paper, we assume the following standard Basic Axioms on preferences.

Axiom A1 (Weak Order). ľ is complete and transitive.

Axiom A2 (Non-degeneracy). There exists f, g P A such that f ą g.

Axiom A3 (Mixture Continuity). For all f, g, h P A, the sets tα P r0, 1s| fαh ĺ gu and

tα P r0, 1s| fαh ľ gu are closed in r0, 1s.

Each of the following axioms has been used in standard treatments to impose state

independence of utility.

Axiom A4 (Monotonicity). For all f, g P A, if fpsq ľ gpsq for all s P S, then f ľ g.4The definition of calibration with respect to a single set of utilities or a state-dependent utility follows

from this definition, considering a single set of utilities and a state-dependent utility to be special cases of

functions υ : S Ñ 2U that are constant and singleton-valued respectively.

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Axiom A5 (State Independence). For all h, h1 P A, x, y P X and non-null s, t P S,

xsh ľ ysh if and only if xth1 ľ yth1.

In the presence of the standard Independence axiom5 (and Weak Order), these two ax-

ioms are equivalent; as we shall see in Section 4, this is no longer be the case in the context

of imprecise beliefs and tastes. Since the aim is to go beyond state-independent utility,

neither of these axioms will be imposed in our main model. Rather, we shall argue that the

following is the fundamental axiom permitting a state-dependent utility representation.

Axiom A6 (State Consistency). For every h, h1 P A, x, y P X and s P S, if xsh ľ ysh,

then xsh1 ľ ysh1.

This is evidently a severe weakening of State Independence, concerning only cases

involving a single state. As such, it says nothing about the relationship between preferences

conditional on different states, and so does not imply any form of state independence of

utility.

On the other hand, State Consistency does ensure that the standard notion of preference

conditional on a non-null state s can be (coherently) defined, as follows: for every x, y P X

and s P S, x ľs y if and only if there exists h P A with xsh ľ ysh. The Basic Axioms

imply that ľs is a continuous weak order. It follows in our setup that, for every non-null

s P S, there exists a minimal and maximal element of ľs; for each such s, let cs and csrespectively be such elements. Without loss of generality, we can pick cs and cs such that

cs ăs pcsqβpcsq ăs cs for all β P p0, 1q. (If this is not the case, take the maximum and

minimum pcsqβpcsq for which it is.) Pick any non-null s1 P S, and define the acts h, h

by: hpsq “ cs (respectively hpsq “ cs) for every non-null s P S; and hptq “ cs1 (resp.

hptq “ cs1) otherwise. By State Consistency and the Basic Axioms, h, h are best and worst

acts in ľ: h ľ f ľ h for all f P A.

In the absence of Monotonicity and State Independence, constant acts—acts taking the

same consequence in every state—cease to have any special status. This poses a significant

challenge, given their central role both as concerns state independence of utility and uncer-

tainty aversion. For the former, they are the acts that have the same utility in all states, and

hence are key to ‘tying’ together the utilities assigned to consequences in different states.

For the latter, they constitute the ‘safe options’ that play a role in the axiomatic charac-

terisation of many popular ambiguity models (Schmeidler, 1989; Gilboa and Schmeidler,5Independence states that, for all f, g, h P A and α P p0, 1q, f ľ g if and only if fαh ľ gαh.

7


1989; Ghirardato et al., 2004; Maccheroni et al., 2006; Chateauneuf and Faro, 2009), as

well as in the definition of notions such as relative ambiguity aversion (Ghirardato and

Marinacci, 2002). In the face of this challenge, we mobilise an insight from the literature

on state-dependent utility (Drèze, 1987; Karni, 1993b,a), namely to use essentially con-

stant acts: acts that, though they are not constant—they yield different consequences in

different states—yield constant utility—the utility of the consequences in different states

are the same. As discussed below, such acts may be useful in application because they

can often play the role of constant acts, hence permitting access to well-known notions and

techniques developed for state-independent utility. In our model, the set of mixtures of h, h

(ie. thα h : α P r0, 1su) will be a set of essentially constant acts, and this comes out in the

formulation of the uncertainty aversion axiom.

Recall firstly the classic uncertainty aversion axiom from Schmeidler (1989).

Axiom A7 (Uncertainty aversion). For all f, g P A and α P r0, 1s, if f „ g then fαg ľ g.

We shall use the following strengthening of this axiom.

Axiom A8 (Strong Uncertainty aversion). For all f, g P A and α, β, β1 P r0, 1s, if f ľ hβ h

and g ľ hβ1 h then fαg ľ phβ hqαphβ1 hq.

To understand this condition, consider first the case where consequences are lotter-

ies over monetary outcomes, preferences over lotteries are represented by a single state-

independent utility function, and h, h are constant acts. The axiom then implies6 that if the

decision maker values the act f at over $x, and values the act g at over $y, then he prefers

the 50-50 mixture of the two acts, f 12g, to the 50-50 lottery over the two monetary pay-

ments $x and $y. This is just a strong version of the preference-for-hedging motive behind

the classical Uncertainty Aversion axiom: mixing over acts may bring a hedging advantage

with respect to (the mixing of) sure monetary payments.

Strong Uncertainty Aversion is the formulation of precisely this condition in the ab-

sence of any independence or monotonicity axioms, and hence where nothing can be as-

sumed about the precision, form and state independence of utility. In this context, mixtures

between best and worst acts (eg. hβ h) are a proxy for ‘sure’ acts—monetary payments

in the previous example. The axiom says that this hedging motive holds on this ‘scale’ of

essentially constant acts.7

6The implication is immediate noting that, in this case, Independence holds over constant acts.7See also Section 9 for further discussion of this use of best and worst acts.

8


3.2 Representation Theorem

The previously discussed axioms yield the following representation.

Theorem 1. Let ľ be a preference relation on A. The following are equivalent:

(i) ĺ satisfies the Basic Axioms, State Consistency and Strong Uncertainty Aversion

(ii) there exists a non-trivial, closed, convex, h, h-constant, h, h-precise function υ :

S Ñ 2UzH and a non-trivial, null-consistent, grounded, calibrated, upper hemicon-

tinuous, convex function α : ∆ Ñ 2<ą0ˆ< such that ľ is represented by:8

(2) V pfq “ minpP∆, pa,bqPαppq

˜

aÿ

sPS

ppsq minuPυpsq

upfpsqq ` b

¸

The axioms thus yield a general state-dependent utility representation, incorporating

imprecision of both beliefs and tastes. On the one hand, tastes for consequences are rep-

resented by a function υ assigning a set of utility functions to each state. To the extent

that sets of utilities are involved, this captures imprecision of tastes; to the extent that the

set may depend on the state, state-dependence is also accomodated. The non-triviality,

closure and convexity of υ are familiar in the literature. The other properties translate the

(double) role of h, h-mixtures as essentially constant acts: h, h-constancy says that they

receive the same evaluation in all states, and h, h-precision ensures that they can be seen as

‘sure options’, insofar as hedging among them provides no particular advantage.9 Since the

set of utilities depends on the state, we call the non-trivial, closed, convex, h, h-constant,

h, h-precise function υ a state-dependent multi-utility.

On the other hand, the function α provides a weighing of probability measures over

states, in a way analogous to several models in the ambiguity literature. As shall be fur-

ther discussed in Section 4.3, this part of the representation is very general, containing

virtually every uncertainty averse model in the ambiguity literature as a special case. The

groundedness and calibration of α are normalisation conditions, and null-consistency is

a non-nullness condition: it guarantees that if a state is non-null according to one prob-

ability measure assigned to some pair by α, then it is non-null according to all of them.

8We adopt the convention that, when αppq is empty, p is not involved in the minimisation.9h, h-precision implies that the restriction of V to the set of h, h-mixtures is affine.

9


The other properties correspond to familiar non-triviality, continuity and convexity con-

ditions. By analogy with similar functions in the literature, we shall call the non-trivial,

null-consistent, grounded, calibrated, upper hemicontinuous, convex α an ambiguity index.

3.3 Uniqueness

To discuss the uniqueness of the representation, we first introduce some terminology.

A state-dependent multi-utility and ambiguity index pair pυ, αq representing ľ according

to (2) is said to be tight if there exists no state-dependent multi-utility and ambiguity index

pυ1, α1q representing ľ according to (2) such that υ1psq Ď υpsq and α1ppq Ď αppq for all

s P S and p P ∆, with strict containment for some s or p. A tight pair is as small as a

representation can be, in the sense that there are no extraneous members of the relevant

sets.

Proposition 1. Let ľ satisfy the Basic Axioms, State Consistency and Strong Uncertainty

Aversion. Then there exists a tight pair pυ, αq representing ľ according to (2). Moreover,

for any other tight pair of state-dependent multi-utility and ambiguity index pυ1, α1q repre-

senting ľ according to (2), there exists κ P <ą0 and λ P < such that υ1psq “ κυpsq ` λ

for all non-null s P S, and α1ppq “ tpa, κb` λp1´ aqq | @pa, bq P αppqu for all p P ∆.

A central challenge in the state-dependent utility literature (in the context of expected

utility) is to provide a suitably unique representation, separating in particular the (state-

dependent) utility part from the beliefs. This result shows that our representation has the

desired uniqueness: the state-dependent multi-utility is unique up to a positive affine trans-

formation (of all utilities at all states), and the ambiguity index is unique up to a corre-

sponding linear re-normalisation. Not only is the uniqueness of the utility comparable to

the standard case of precise state-independent utility, but the same goes for the ambiguity

index, whose renormalisation is reminiscent of similar re-normalisations in recent ambigu-

ity models, such as Maccheroni et al. (2006); Cerreia-Vioglio et al. (2011).

Remark 1. Proposition 1 does not require that the state-dependent multi-utilities υ and

υ1 are precise and constant with respect to the same best and worst acts: it continues to

hold if they are h, h-precise and -constant and h1, h 1-precise and -constant respectively, for

different h, h and h1, h 1. See the proof (Appendix A) for further details.

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4 Special cases

The three phenomena—state-dependence of utility, imprecision of tastes and imprecision

of beliefs—can be straightforwardly separated in representation (2); indeed, any combi-

nation of them can be ‘shut down’, yielding potentially useful special cases. We consider

each one in turn.

4.1 State-independent multi-utility with imprecise beliefs

Firstly, one obtains an uncertainty averse representation involving imprecision of beliefs

and tastes but state-independent (multi) utility by adding the standard Monotonicity axiom.

Proposition 2. Let ľ be a preference relation on A. The following are equivalent:

(i) ľ satisfies the Basic Axioms, State Consistency, Strong Uncertainty Aversion, and

Monotonicity

(ii) there exists a tight pair10 consisting of a non-trivial, h, h-precise, closed, convex set

of utility functions U Ď U and an ambiguity index α : ∆ Ñ 2<ą0ˆ< such that ľ is

represented by:


˜

aÿ

sPS

ppsqminuPU

upfpsqq ` b

¸

Moreover, for any such pair pU 1, α1q representing ľ according to (3), there exists κ P <ą0

and λ P < such that U 1 “ κU `λ and α1ppq “ tpa, κb` λp1´ aqq | @pa, bq P αppqu for all

p P ∆.

In this representation the set of utilities U is state independent. So state-dependence

of utility can be ‘shut down’ by adding one of the standard state-independence axioms,

Monotonicity. One might wonder about the other well-known state-independence axiom,

State Independence (Section 3.1). Whilst in the context of expected utility preferences,

they are equivalent, this is not the case here: the following Proposition reveals the rela-

tionship between them, and shows that representation (3) can (equivalently) be obtained by

strengthening State Consistency to State Independence.10Tightness is defined as above, considering a set of utility functions as a constant state-dependent multi-

utility.

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Proposition 3. In the presence of the Basic Axioms and Strong Uncertainty Aversion,

Monotonicity and State Consistency are equivalent to State Independence.

These results can be thought of as characterising a general uncertainty averse represen-

tation under state independent utility, involving imprecision of both tastes and beliefs.

4.2 State-dependent precise utility with imprecise beliefs

A representation involving a single state-dependent utility (rather than a multi-utility)

and uncertainty averse preferences can be obtained by adding the following restriction of

the standard Independence axiom to preferences conditional on states.

Axiom A9 (State-wise Independence). For every x, y, z P X , h P A, α P p0, 1q and s P S,

xsh ľ ysh if and only if pxαzqsh ľ pyαzqsh.


(i) ľ satisfies the Basic Axioms, State Consistency, Strong Uncertainty Aversion and

State-wise Independence

(ii) there exists a tight pair consisting of a non-trivial, h, h-constant function u : S ˆ

X Ñ < that is affine in its second coordinate and an ambiguity index α : ∆ Ñ

2<ą0ˆ< such that ľ is represented by:11


˜

aÿ

sPS

ppsqups, fpsqq ` b

¸

Moreover, for any such pair pu1, α1q representing ľ according to (4), there exists κ P

<ą0 and λ P < such that u1 “ κu ` λ and α1ppq “ tpa, κb` λp1´ aqq | @pa, bq P αppqu

for all p P ∆.

The function u in this representation is a standard (precise) state-dependent utility func-

tion, and its uniqueness is comparable to that obtained in the state-dependent utility liter-

11Tightness and h, h-constancy are as defined above, considering a state-dependent utility function as

a singleton-valued state-dependent multi-utility; in particular, h, h-constancy means that ups, hα hq “

upt, hα hq for all s, t P S.

12


ature (Karni, 1993b,a, 2011a).12 So the imprecision of tastes can be ‘shut down’ by the

addition of an appropriate, but weak, independence axiom, yielding a familiar notion of

state-dependent utility. Note that, unlike most of the work on state-dependent utilities, it is

not required that the preferences are expected utility; indeed, this is the first precise state-

dependent utility uncertainty averse representation of which we are aware in the literature.

4.3 α and Ambiguity

In its treatment of uncertainty, representation (2) generalises many of the main theories

of uncertainty averse preferences proposed in the literature. One way to see this is to

compare such ambiguity models, which generally assume precision and state-independence

of utility, with the special case of (2) with precise state-independent utility. A corollary of

the previous two results is that adding Monotonicity and State-wise Independence to the

axioms in Theorem 1 yields this case: that is, the representation of preferences by


˜

aÿ

sPS

ppsqupfpsqq ` b

¸

where u is a precise state-independent utility, and α is as in Theorem 1.

It is clear from the representations that:

• preferences represented by (5) are variational (Maccheroni et al., 2006) whenever

a “ 1 for all p P ∆ and pa, bq P αppq;

• preferences represented by (5) are confidence preferences (Chateauneuf and Faro,

2009) whenever minuPυpsq uphpsqq “ 0 and b “ 0 for all p P ∆ and pa, bq P αppq;13

• preferences represented by (5) are maxmin EU preferences (Gilboa and Schmeidler,

1989) when there exists a closed convex C Ď ∆ with αppq “ tp1, 0qu for all p P Cand αppq “ H for all p R C.

12Other approaches in that literature, such as Karni et al. (1983); Karni and Schmeidler (2016, 1993); Karni

and Mongin (2000), yield a weaker uniqueness: up to cardinal unit comparable transformation (Karni et al.,

1983) rather than positive affine transformation. See also Section 9.13If one removes the condition that minuPυpsq uphpsqq “ 0, one obtains the more general class of homo-

thetic preferences, as Cerreia-Vioglio et al. (2011) call them. They do not involve Chateauneuf and Faro’s

assumption of a minimal element of the preference order.

13


• preferences represented by (5) are EU preferences when there exists a p P ∆ with

αppq “ tp1, 0qu and αpqq “ H for all q ‰ p.

Thus, the α in (5) clearly generalises similar ‘ambiguity indices’ and ‘confidence func-

tions’ in the variational and confidence models respectively, as well as the set of priors in

the maxmin EU model. The most general uncertainty averse model in the literature to date

is the ‘uncertainty averse preferences’ characterised by Cerreia-Vioglio et al. (2011), which

are represented by the following functional form:

(6) V pfq “ infpP∆

G

˜

ÿ

sPS

ppsqupfpsqq, p

¸

Representation (5) is clearly the special case where Gpt, pq “ minpa,bqPαppq at ` b. By

duality, it is essentially the subclass of Cerreia-Vioglio et al’s uncertainty averse preferences

where G is concave in the first coordinate. Indeed, as is clear from the proof of Theorem

1, (5) is the class of uncertainty averse preferences that are represented by a concave func-

tional on A. Since the vast majority of existing uncertainty averse models involve concave

functionals—including (beyond those discussed above) uncertainty averse Choquet prefer-

ences (Schmeidler, 1989) and uncertainty averse smooth ambiguity preferences (Klibanoff

et al., 2005)—they are special cases of representation (5). This suggests that these exist-

ing ambiguity models can be extended to incorporate state dependence and imprecision of

tastes using techniques similar to those adopted above. As an illustration, we present an

axiomatisation for the maxmin EU specification of our general representation (2).14

Firstly, consider the following axiom.

Axiom A10 (EC-Independence). For all f, g P A and α P r0, 1s, β P p0, 1q, f ľ g if and

only if fβphα hq ľ gβphα hq

This is the standard C-independence axiom introduced by Gilboa and Schmeidler (1989),

but with constant acts replaced by mixtures of best and worst acts, h, h. As noted in Section

3.1, in the absence of state independence of utility, constant acts may not be valued con-

stantly across states by the decision maker, and so the standard C-independence axiom loses

its original sense. By contrast, mixtures of h, h are essentially constant: they do have the

14Similar exercises can be carried out in analogous ways for the other models mentioned.

14


same utility in all states. So, in using these in the place of constant acts, EC-Independence

(for Essentially Constant-Independence) retains the essence of the original axiom.

This state-dependence-corrected version of the standard C-independence condition is

all that is required to get a maxmin EU version of (2).

Theorem 2. Let ľ be a preference relation on A. The following are equivalent:


EC-Independence

(ii) ľ satisfies the Basic Axioms, State Consistency, Uncertainty Aversion and EC-Independence

(iii) there exists a pair of a tight state-dependent multi-utility υ : S Ñ 2UzH and a

null-consistent, closed, convex set of priors C Ď ∆ such that ľ is represented by:15

(1) V pfq “ minpPC

ÿ

sPS

ppsq minuPυpsq

upfpsqq

Moreover, C is unique, and υ is unique up to positive affine transformation.

This representation can be thought of as a natural extension of the maxmin EU model

to incorporate state-dependence of utility and imprecision of tastes. Moreover, as the result

makes clear, Strong Uncertainty Aversion can be weakened to Gilboa and Schmeidler’s

Uncertainty Aversion in the presence of EC-Independence. So this result can be thought of

as a natural generalisation of theirs, making only minimal axiomatic changes.

For the sake of completeness, we remark that the following version of the Independence

axiom ‘shuts down’ the imprecision in beliefs, yielding a state-dependent multi-utility rep-

resentation with a single-prior belief.

Axiom A11 (Restricted Independence). For all f, g, h P th1

β h1 : β P r0, 1suS and α P

p0, 1q, f ľ g if and only if fαh ľ gαh.


15The definition of null-consistency applies immediately to sets of priors, which, as noted above, can be

written as special cases of ambiguity indices. The definition of tightness of a state-dependent multi-utility is

analogous to that in Section 3.2: no strict subset can represent the same preferences.

15



Restricted Independence

(ii) there exists a pair of a tight state-dependent multi-utility υ : S Ñ 2UzH and a

probability measure p P ∆ such that ľ is represented by:

(7) V pfq “ÿ

sPS

ppsq minuPυpsq

upfpsqq

Moreover, p is unique, and υ is unique up to positive affine transformation.

Remark 2. Propositions 2 and 5 combined provide a characterisation of the special case of

a single prior and state-independent multi-utility representation. The closest representation

of which we are aware in the literature, (Riella, 2015, Thms 5 & 6), concerns precisely

this case. Unlike the representations involved here, his uses the certainty equivalents of

consequences rather than their utility values, in the style of Cerreia-Vioglio et al. (2015);

see Section 7 and Remark 3 for further discussion.

5 Comparative Statics

We now explore the comparative statics of our general representation (2) (corollaries for

the special cases in Section 4 are immediate). A popular notion of relative uncertainty

aversion is formulated in terms of the propensity of decision makers to prefer an act over

a constant act. Such comparisons rely on the constant act having a fixed value across

states for the decision maker, and so lose their intuition in the absence of state-independent

utility. Fortunately, in the context of (2), there is a suitable proxy for constant acts, namely

mixtures of h, h. Substituting them into the standard definition of comparative uncertainty

aversion yields the following notion.

Definition 1. The (decision maker with) preference ľ1 is more imprecision averse than

(that with) preference ľ2 if and only if, for each f P A and α P r0, 1s,

(8) f ľ1 h 1α h

1 implies f ľ2 h 2α h

2

16


Throughout this section, we use h1, h1 for the best and worst acts determining the es-

sentially constant acts for decision maker 1, and similarly for decision maker 2. So this

definition does not assume that the decision makers are using the same best and worst acts,

h, h, and hence it does not assume that the decision makers share the same essentially

constant acts. For a given imprecision aversion comparison, this makes it difficult to disen-

tangle aspects pertaining to the decision makers’ tastes from those concerning their beliefs:

what is a comparison of essentially constant acts for one decision maker—and hence one

involving only tastes—may not be for the other. Indeed, to conduct comparative statics in

the context of state-dependent utility, it is not uncommon to invoke some assumption of

comparability between the decision makers’ preferences (Karni, 1979, 1983a,b; Drèze and

Rustichini, 2004). For our main result, we only require the mild assumption that the less

imprecision-averse decision maker’s best act h2

is considered a maximal act by the more

imprecision-averse decision maker.16

Proposition 6. Let ľ1 and ľ2 be represented according to (2) by tight pairs of state-

dependent multi-utilities and ambiguity indices pυ1, α1q and pυ2, α2q, and suppose that

they are normalised so that υ1pXq “ υ2pXq. Suppose that h2

is a maximal element of ľ1.

Then the following are equivalent.

(i) ľ1 is more imprecision averse than ľ2

(ii) pυ1 Y υ2, α1 Y α2q represents ľ1 according to (2)

(iii) α1 ď α2 and υ1psq ď υ2psq for all ľ1-non-null states s P S.

Furthermore, pυ1, α1q is the unique tight subset of pυ1 Y υ2, α1 Y α2q representing ľ1.

Imprecision aversion corresponds to separate comparison of the belief and taste ele-

ments. The utilities involved in the representation of less imprecision-averse preferences

are ‘redundant’ with respect to the more imprecision-averse preferences: adding them does

not change the fact of representing preferences ľ1. The same holds for the weights as-

signed to probabilities. This essentially means that both the state-dependent utilities and

the ambiguity indices can be ordered: less imprecision-averse preferences have both higher

state-dependent utilities and ambiguity indices than more imprecision-averse ones.16See Proposition 10 in Appendix A.3 for a characterisation of imprecision aversion in the absence of this

assumption.

17


These effects on tastes and beliefs can be characterised separately by using more refined

notions of comparative imprecision aversion.

Definition 2. Preference ľ1 is more imprecision averse on consequences than ľ2 if and

only if, for every ľ1-non-null s P S, h P A, x P X and α P r0, 1s,

(9) xsh ľ1ph 1

α h1qsh implies xsh ľ2

ph 2α h

2qsh

Moreover, preference ľ1 is more imprecision averse on states than ľ2 if and only if,

for every α P r0, 1s, every f P th1

β h1 : β P r0, 1suS , and for f P th

2

β h2 : β P r0, 1suS

with fpsq “ h2

β h2psq if and only if fpsq “ h

1

β h1psq for all s P S,

(10) f ľ1 h 1α h

1 implies f ľ2 h 2α h

2

These notions are as to be expected. Imprecision aversion on consequences compares

consequences with the ‘essentially constant’ mixtures of best and worst acts on each (non-

null) state, but eschews comparisons of acts differing on several states. Imprecision aver-

sion on states considers only acts f yielding ‘essentially constant’ consequences, and asks

that if f is preferred to an essentially constant act by decision maker 1, then 2 retains the

same preference. A complication with the latter condition is that the decision makers may

have different essentially constant acts: because of this, it uses the ‘equivalent’ act to f but

formulated in terms of decision maker 2’s essentially constant acts (f ).

Proposition 7. Let ľ1 and ľ2 be represented according to (2) by pairs of state-dependent

multi-utilities and ambiguity indices pυ1, α1q and pυ2, α2q, and suppose that they are nor-

malised so that υ1pXq “ υ2pXq. Then:

(i) ľ1 is more imprecision averse on consequences than ľ2 if and only if υ1psq ď υ2psq

for all ľ1-non-null s P S.

(ii) ľ1 is more imprecision averse on states than ľ2 if and only if α1 ď α2.

These results are consistent with known characterisations of uncertainty attitudes. As

remarked above, a standard notion of relative uncertainty aversion (Ghirardato and Mari-

nacci, 2002) is as Definition 1 with mixtures hα h replaced by constant acts c P X . In the

18


special case of precise state-independent utility (representation (5)), the two definitions are

equivalent. So it is a straightforward corollary of Proposition 6 that in this case, ľ1 is more

uncertainty averse than ľ2 if and only if α1 ď α2 and the decision makers share the same

(normalised) utilities. This corroborates the interpretation of α as an ambiguity index.

Similarly, applying Proposition 6 in the case of precise probabilistic beliefs (represen-

tation (7)), or alternatively the first clause of Proposition 7, reveals that, once beliefs are

put aside, comparisons of imprecision aversion translate to orderings of the υ, corroborat-

ing its interpretation as a state-dependent multi-utility. This echoes comparative statics of

multi-utility models under complete preferences in the case of (monetary) risk, where the

standard notion of risk aversion has been found to be connected to—though not entirely

equivalent to—similar orderings in terms of sets of utilities (see Cerreia-Vioglio (2009);

Cerreia-Vioglio et al. (2015) and Section 7).

6 State Consistency as state dependence

We now examine in more detail the issue of state dependence of utility in the presence

of imprecise beliefs and tastes. It was suggested that State Consistency can be thought of

as the axiom for state dependence; to evaluate this claim, we now characterise the conse-

quences of dropping it from the general state-dependent utility representation (2).

To this end, let E be the set of real-valued functions on S ˆ X that are affine in the

second coordinate, endowed with the Euclidean topology.17 Elements of E are called eval-

uations. U P E is constant ifř

sPS Ups, fpsqq “ř

sPS Ups, gpsqq for all f, g P A. A set

E Ď E is non-trivial if it contains a non-constant evaluation. h, h1-precision is defined in

an analogous way as for state-dependent multi-utilities.18 Finally, as before, let h, h, the

elements with respect to which the Strong Uncertainty Aversion axiom is formulated, be

maximal and minimal acts of ľ such that h ą hα h ą h for every α P p0, 1q.

The following result characterises the consequences of dropping State Consistency.

Theorem 3. Let ľ be a preference relation on A. The following are equivalent:17E “ tU P <SˆX : Ups, λx ` p1 ´ λqyq “ λUps, xq ` p1 ´ λqUps, yq @s P S, x, y P X, λ P r0, 1su.

Since X is finite-dimensional and the elements of E are affine, E is isomorphic to a finite-dimensional vector

space; the topology on E is that generated by the Euclidean topology under this isomorphism.18For any h, h1 P A, a set E Ď E is h, h1-precise if there exists U P E with

ř

sPS Ups, gpsqq ďř

sPS U1ps, gpsqq for g “ th, h1u and all U 1 P E .

19


(i) ľ satisfies the Basic Axioms and Strong Uncertainty Aversion,

(ii) there exists a non-trivial, h, h-precise, closed, convex set of evaluations E Ď <SˆX

such that ľ is represented by:

(11) V pfq “ minUPE

ÿ

sPS

Ups, fpsqq

In the absence of State Consistency, preferences are represented by a set E of evalu-

ations. Evaluations appear in the literature on state-dependent utility, where they can be

considered as a stepping stone to providing a full state-dependent utility representation

(Hammond, 1998; Hill, 2010). This literature generally works under expected utility, and

so is concerned with the special case of (11) with singleton E . The challenge is to pro-

vide a principled way of factorising that evaluation into a suitably unique probability and

(potentially state-dependent) utility.

In the case of imprecise beliefs and tastes, the challenge is more severe: as Theorem 3

shows, one needs to factorise a set of evaluations into a belief and a taste factor. Comparing

representations (11) and (2) shows how this may be done. For a state-dependent multi-

utility υ and ambiguity index α, define the following set of evaluations, which we call

the product of υ and α: υ b α “ tU P E : Dp P ∆, pa, bq P αppq, us P υpsq @s P

S s.t. @s P S, x P X, Ups, xq “ auspxqppsq ` bu. This is the set of evaluations obtained

by combining υ and α as specified in representation (2). Theorem 1 shows that, in the

presence of the other axioms, State Consistency guarantees that the set of evaluations E in

representation (11) can effectively be factorised into such a product: there exists υ, α with

υ b α representing the same preferences. In its absence, no such separation exists. The

following example illustrates such a case.

Example 1. Suppose that there are two states, s and t, two prizes c and d, and consequences

are lotteries over prizes. Evaluations are fully specified by real-valued functions on ts, tuˆ

tc, du. Let U1, U2 be evaluations with U1ps, cq “ U1pt, cq “ 0, U1ps, dq “14, U1pt, dq “

34

and U2ps, cq “14, U2pt, cq “

34, U2ps, dq “ U2pt, dq “ 0 and let E be the convex closure

of tU1, U2u.19 Consider ľ represented by E according to (11). By simple calculation, it is

clear that csd ą d but c ă dsc, contradicting State Consistency. It is also clear that there19It is straightforward to check that this set is pc 1

2d, c 1

2dq, pc, cq-precise.

20


is no factorisation υ b α that could represent these preferences: for any state-dependent

multi-utility υ, if minuPυpsq upcq ă minuPυpsq updq then it can accomodate the preference

csd ă d but not c ą dsc, and similarly for the reverse order.

So, in the presence of imprecision, State Consistency is required to separate out a belief

element from a (state-dependent) taste; it may thus be thought of as the axiom guaranteeing

a meaningful notion of (state-dependent) utility. There is an interesting contrast with the

well-studied case of precise beliefs and tastes. In that case, there are many ways of fac-

torising a single evaluation into a probability measure and state-dependent utility function:

the problem is the multiplicity of separations, not the lack of them. In the imprecise case,

by contrast, there is also an existence problem: no separation of the sort involved in (2)

may exist. Theorems 1 and 3 tell us that State Consistency is the condition guaranteeing

existence. Note that it holds automatically in the the expected utility context (it is implied

by the Sure Thing Principle or the standard Independence Axiom).20

Note finally that there is little reason to think that this general moral is specific to

uncertainty averse preferences: as is clear in the proof of Theorem 1, a ‘factorisation’

into a state-side part and consequence-side part is established without using uncertainty

aversion, so similar points should apply to more general non-EU preferences.

7 Multi-utility, Risk and Cautious Expected Utility

Our modelling of imprecise tastes and their role in decision—using a set of affine utility

functions and evaluating a consequence by the minimum utility value over them—follows

representations proposed in the literature on decision under risk. Maccheroni (2002) stud-

ies a representation with this functional form (though under a normalisation condition on

utility functions that is not imposed here); Cerreia-Vioglio (2009) provides a more general

representation of convex preferences, of which the representation used here is a special

case.21 Recently, Cerreia-Vioglio et al. (2015) have proposed, for the case of risk involving

monetary prizes, a different multi-utility decision rule, which looks at the infimum of the

20See Section 9 for further discussion of the relationship with the state-dependent utility literature.21The relationship between the functional form used on consequences in (2) and that proposed by Cerreia-

Vioglio (2009) is analogous to the relationship between the treatment of uncertainty in (2) and uncertainty

averse preferences (Cerreia-Vioglio et al., 2011), discussed in Section 4.3.

21


certainty equivalents of a lottery rather than of its utility values. It is natural to ask whether

their approach can be combined with the tools developed above to provide an alternative

state-dependent utility representation with imprecise beliefs and tastes.

To treat this question, let us first recall their framework, where the objects of choice

are lotteries—Borel probability measures on a compact interval I “ rm,M s of monetary

prizes, where m ă M . Let ∆pIq be the set of such measures, with the topology of weak

convergence; generic members will be denoted with l, l1, l2. A preference relation ľ over

∆pIq is Cautious Expected Utility if it is represented by a continuous functional:

(12) V plq “ minvPW

v´1pElpvqq

whereW is a set of continuous strictly increasing (utility) functions v : I Ñ < and Elpvqis the expectation of v with respect to l.

To extend the Cerreia-Vioglio et al. (2015) approach to the case of uncertainty, take a

framework precisely as set out in Section 2 but with X “ ∆pIq. So preferences are over

acts defined with this consequence space: A “ ∆pIqS . (The topology on A is the product

topology.) For every z P I , δz will denote the degenerate lottery giving all mass to z;

moreover, we also use δz to denote the constant act yielding this lottery in every state. Let

U csi be the set of continuous strictly increasing functions from I to < that are normalised

so that vpmq “ 0 and vpMq “ 1.

The central axiom in Cerreia-Vioglio et al. (2015) is Negative Certainty Independence.

Though they state it on lotteries, it can naturally be formulated on acts as follows.

Axiom A12 (Negative Certainty Independence). For every f, f 1 P A, α P r0, 1s and z P I ,

f ľ δz implies that fαf 1 ľ pδzqαf1.

A major motivation behind this axiom is its capacity to capture the certainty effect

(Cerreia-Vioglio et al., 2015), and this presumably applies as much in the case of uncer-

tainty as in the case of risk. It is comparable to, though different from the Strong Uncer-

tainty Aversion axiom used in previous sections. Both axioms imply Uncertainty Aversion.

Moreover, they share a common intuition: namely, that there is a special category of acts—

essentially constant acts in previous sections, constant acts yielding sure monetary amounts

here—such that mixing other acts can only procure extra advantage compared to these ones.

Although it is natural to use constant sure amounts in the current, monetary context, it is

22


possible to formulate a version of this axiom in terms of the essentially constant acts used

in previous sections, with similar results.

Given the contrast and similarities between Negative Certainty Independence and

Strong Uncertainty Aversion, a natural strategy for developing a Cautious Expected state-

dependent Utility model would be to replace the latter axiom by the former in our bench-

mark result (Theorem 1).22 For this, two other conditions from the Cerreia-Vioglio et al.

(2015) characterisation will be needed. The first is a stronger continuity requirement to that

used previously, but one that is more appropriate for the current consequence space.

Axiom A13 (Topological continuity). For every f 1 P A, the sets tf P A| f ĺ f 1u and

tf P A| f ľ f 1u are closed.

The second is the assumption—which is standard in monetary contexts—that more

money is better than less in every non-null state.

Axiom A14 (Weak Monetary Monotonicity). For all non-null s P S, h P A, z, w P I ,

z ě w if and only if pδzqsh ľ pδwqsh.

Finally, for any function v : S Ñ U csi and prior p P ∆, we define vp : I Ñ < by

vppzq “ř

sPS ppsqvpsqpzq. This function gives the expected utility value of each constant

sure amount, so, for any f P A, v´1p

`ř

sPS ppsqEfpsqpvpsqq˘

is the certainty equivalent of f

under v and p. Note that vp is strictly increasing, so the inverse is a well-defined function.

We have the following characterisation.

Theorem 4. Let ľ be a preference relation onA “ ∆pIqS with at least two non-null states.

The following are equivalent:

(i) ľ satisfies Weak Order, Topological Continuity, Weak Monetary Monotonicity, Neg-

ative Certainty Independence and State Consistency

(ii) there exists a function v : S Ñ U csi and a set of probability measures C Ď ∆ such

that ľ is represented by:

22Theorem 1 does not strictly speaking apply in the framework considered here, notably because the con-

sequence space is infinite-dimensional. Nevertheless, we conjecture that the basic form of the result extends

to such cases. In particular, a version of the theorem would seem to hold whenever consequences are compact

subsets of Banach spaces, by the Brondsted & Rockafellar Theorem (1965).

23


(13) V pfq “ minpPC

v´1p

˜

ÿ

sPS

ppsqEfpsqpvpsqq

¸

with V continuous.

This representation takes the minimum over certainty equivalents for each utility-

probability pair, in a similar manner to (12). The function v assigns (strictly increasing)

utility values to every prize in every state, so it is basically a utility function. More-

over, since the elements of U csi are strictly increasing in every state, v is ordinally state

independent—more money has higher utility than less in every state. However, it is not

necessarily cardinally state independent—the utility of a particular monetary amount may

vary across states (and this will be the case when the vpsq differ). So the representation

permits (cardinal) state dependence of utility. In the monetary context considered here,

ordinal state independence is standardly assumed.

However, somewhat surprisingly, representation (13) involves a single state-dependent

utility function—and hence imposes precision of tastes. To the extent that, as argued pre-

viously, State Consistency is necessary for the separation of beliefs and tastes, this could

be read as a negative result, suggesting that the Cautious Expected Utility representation

applied in the context of uncertainty cannot support separation of beliefs and tastes with im-

precision in both. Note that this conclusion is independent of the issue of state dependence,

as shown by the following immediate corollary.23

Corollary 1. Let ľ be a preference relation on A “ ∆pIqS with at least two non-null

states. The following are equivalent:

(i) ľ satisfies Weak Order, Continuity, Weak Monetary Monotonicity, Negative Cer-

tainty Independence and Monotonicity

(ii) ľ is a continuous maxmin EU preference (à la Gilboa and Schmeidler 1989)

So it would seem that any representation in the context of decision under uncertainty

that satisfies the Negative Certainty Independence axiom characteristic of Cautious Ex-

pected Utility and standard Monotonicity coincides with the maxmin EU representation,23This corollary is immediate from (13) once one notes that, in this context, Monotonicity implies State

Consistency; see Proposition 11 in the Appendix.

24


and hence does not involve any imprecision in tastes. This is in contrast with the Strong

Uncertainty Aversion axiom, and the treatment of taste imprecision employed in previous

sections.

Remark 3. Riella (2015, Theorem 6) provides a characterisation of the following Cautious

Expected Utility representation in the context of uncertainty, involving a single prior p P ∆

and a set of state-independent utility functionsW Ď U csi:

(14) V pfq “ minvPW

v´1p

˜

ÿ

sPS

ppsqEfpsqpvq

¸

The monotonicity axiom he uses—One Side Mixture Monotonicity—is weaker than stan-

dard Monotonicity (though it implies it in the presence of the full Independence axiom). In-

deed, the results above entail that this representation (for non-singletonW) violates Mono-

tonicity, as well as State Consistency, and thus does not admit a coherent definition of pref-

erences conditional on a state. Relatedly, note that the set of evaluations involved in (14)

is not necessarily of the product form identified in Section 6. This leads us to conjecture

that the conflict between State Consistency and Negative Certainty Independence is related

to the incompatibility between two different notions of state independence of utility under

uncertainty aversion in the presence of imprecise beliefs and tastes (see Hill, 2017). In this

paper, we work with one of the notions—that characterised by Monotonicity—where the

Cautious Expected Utility representation may fit better with the other notion. Investigation

of this possibility is left as a topic for future research.

8 An application

Consider the choice of policy for adapting to the effects of climate change. Such choices

will have consequences for the future average per capita consumption and population size;

for simplicity, let us focus solely on these two aspects. A policy maker must make her

choice in the face of uncertainty about the future state of the climate.

For simplicity, suppose that there are two states s1, s2 (mild and severe climate change),

and that consequences are consumption-population pairs pc, pq P r0, Csˆr0, P s taking val-

ues in closed bounded intervals of possible consumption and population values. Available

25


policies are characterised by an adaptation expenditure e P r0, Es, which yields the follow-

ing consumption-population profile across states: xpeq “ ppc1 ´ e, p1 ´ σeq, pc2 ` δe, p2 `

τeqq, where σ, δ, τ ą 0. In other words, expenditure raises the consumption and population

in the severe climate state, at the expense of consumption and population in the mild climate

state; as such, it can be thought of as a form of insurance against severe climate change.

The values are such that c1 ą c2, p1 ą p2 (the do-nothing policy yields higher consumption

and population values in the state with mild climate change), and c1 ´ e, c2 ` δe P r0, Cs

and p1 ` σe, p2 ` τe P r0, P s for all e P r0, Es.

Clearly, this is a multi-attribute decision under uncertainty (Keeney and Raiffa, 1976).

Moreover, it involves many of the issues motivating the model proposed here. Firstly, as is

well-known, climate decisions are plagued by uncertainty, of the sort that is not amenable to

expression by precise probabilities (Stainforth et al., 2007; Heal and Millner, 2014; Millner

et al., 2013). Secondly, there is no reason to assume the consumption-population trade-offs

in each state to be linear, as would be the case with a (single) affine utility.24 Finally, how

consumption and population are traded off could be climate dependent—for instance, in

climates with less habitable land, overcrowding may lead to different evaluations of future

population sizes. This is basically a form of state-dependence of utility. Our model applies

naturally to such cases, which in turn illustrate its relevance for multi-attribute decision

making.

Consider two policy makers i “ t1, 2u, each of which has

pp0, 0q, p0, 0qq, ppC,P q, pC 1, P 1qq-precise preferences ľi represented by (1), with set

of priors Ci and state-dependent multi-utility υi. Allowing C ‰ C 1 and P ‰ P 1 leaves

open the possibility that the ideal consumption and population size are state dependent:

they might depend on the state of the climate. Suppose, to make the problem non-trivial,

that both policy makers agree that expenditure involves a sacrifice in state s1: for every

e ě e1, they prefer pc1 ´ e1, p1 ´ σe1q to pc1 ´ e, p1 ´ σeq conditional on s1.

The following fact sheds some light on the impact of belief imprecision.

Proposition 8. Let 1 and 2 have identical state-dependent multi-utilities and let 1 be more

imprecision averse than 2 in the sense of Definition 1. Then:

(a) if there exists an essentially constant act x such that xpe˚2q ľ2s1

x and xpe˚2q ĺ2s2

x,

24By contrast, a form concavity or quasi-concavity, of the sort implied by representation (2), is a more

innocent and fairly standard assumption.

26


then e˚1 ě e˚2;

(b) if there exists an essentially constant act x such that xpe˚2q ĺ2s1

x and xpe˚2q ľ2s2

x,

then e˚1 ď e˚2 .

where e˚i denotes i’s optimal adaptation expenditure.

This result says that imprecision aversion leads to more investment in the direction of

the ‘worse-off’ state. Despite the state dependence of utility, it is possible to get a grasp on

which state is ‘worse-off’ using essentially constant acts. For instance, the condition in (a)

says that there is an essentially constant act that is worse than the consumption-population

profile resulting from expenditure e˚2 in state s1 but better than it in s2: this means that the

outcome in s2 is worse than that in s1. Similarly, the condition in (b) says that s1 is the

worse-off state. So an increase in imprecision aversion leads to an expenditure that is more

compensatory towards the worse-off state: that is, to an increase in expenditure to further

insure s2 when it is worse-off (in case (a)) and a decrease otherwise (in case (b)).

Turning to the state-dependent utility dimension of the model, let us say that a policy

choice problem is symmetric if δ “ 1, τ “ σ, and c1 ´ E2“ c2 ` E

2, p1 ´ σE

2“ p2 ` σE

2

(so expenditure E2

yields the same consumption-population profiles in each state). A policy

maker i has symmetric beliefs when minpPCi pps1q “ minpPCi pps2q.

Proposition 9. Suppose that the choice problem is symmetric and consider a policy maker

with symmetric beliefs. If the policy maker has a state-independent multi-utility (ie.

υips1q “ υips2q “ U), then her optimal expenditure yields a consumption-population

profile that is constant across states. By contrast, this need not be the case for a policy

maker with state-dependent multi-utility.

This fact highlights one of the consequences of the state dependence of utility, which

can be captured by the model: even in symmetric situations, it can drive asymmetries across

states in the outcomes resulting from optimal choices.

Naturally, this is but a simple example, but the preceding observations illustrate the

potential relevance of two of the main phenomena incorporated in our model—belief im-

precision and state-dependence of utility.

Remark 4. The paper’s opening example can be formalised in a similar way to that con-

sidered here, taking the consequences to be wealth-life expectancy pairs and taking four

27


states reflecting the form of the disease and the success of the treatment. Representation

(1) can account for belief imprecision using the appropriate set of priors, the imprecision

in wealth–life expectancy trade-offs with its use of multi-utilities, and the dependence of

these trade-offs on the state (of health) via the state dependence of the multi-utility.

9 Discussion

Much of the analysis performed here can be extended in several directions. For instance, as

noted in the state-dependent utility literature, it may be that the consequence space depends

on the state: in the case of life insurance, for example, some consequences attainable in

a state of life are unattainable in a state of death (Hammond, 1998; Hill, 2009a). The

tools developed here can be extended to such cases: indeed, it is still possible with state-

dependent consequence spaces to define best and worst acts state-wise in the presence of

State Consistency, which allows application of the machinery of essentially constant acts.

More technically, although the essentially constant acts feature in the Strong Uncer-

tainty Aversion axiom, Hill (2017) provides a slightly more general version of our Theorem

3, which uses a weaker, though less intuitive axiom in place of Strong Uncertainty Aver-

sion. Substituting this axiom for Strong Uncertainty Aversion, and making other relevant

technical changes yields a version of Theorem 1 that makes no reference to the essentially

constant acts in the axioms. Moreover, that paper relates representation (11) to the in-

complete ‘unambiguous’ preference relation generated in the manner of Ghirardato et al.

(2004), and those results carry over here. It also contains a discussion of the related liter-

ature on incomplete preferences, to which we refer the reader. Hill (2017) can be thought

of as a sister paper to the current one: it studies state independence of utility in the context

of belief and taste imprecision, distinguishing two different notions of state independence.

The approach in this paper extends upon one of those notions, namely that corresponding to

the Monotonicity axiom; indeed, the simplicity of our axiomatic treatment testifies to this

notion’s relevance for the development of non-Bayesian models of state-dependent utility.

As noted previously, we adopt one technique for separating out state-dependent utili-

ties from beliefs in this paper, namely that based on essentially constant acts (also called

‘constant valuation acts’; Karni (1993a,b); Hill (2009b)). In fact, our notion of essentially

constant act is technically weaker than that used in the literature: taking, for concrete-

28


ness, the Anscombe-Aumann framework used by the last two cited papers, they require

that any essentially constant act that yields a prize (degenerate lottery) as consequence in

one state does so in all states, whereas no such condition is imposed here. This difference

is behind the weaker state-dependence axioms required here. Other notable approaches

to state dependence in the literature employ a richer framework, for instance by involving

other preference relations—such as preferences over state-prize lotteries (Karni et al., 1983;

Karni and Schmeidler, 2016), updated preferences (Karni, 2007) or preferences in differ-

ent situations (Hill, 2009a)—or by modifying the setup (Karni, 2011b, 2006). Of course,

each approach has its strengths and weaknesses, and the one we adopt is no exception.

This paper is a first attempt at extending this literature to imprecise beliefs and tastes under

uncertainty aversion; the hope is that it will serve as a springboard for developing similar

extensions for other state-dependent utility approaches.

10 Conclusion

In many important real-life decisions, decision makers have difficulty forming precise prob-

abilities, but also furnishing precise utilities. In some such situations, there may be a natural

dependence of tastes on states. Notable situations of this sort include some multi-attribute

decisions under uncertainty, when the trade-offs between various attributes may depend on

the state. This paper proposes a theory of uncertainty averse decision making that incorpo-

rates all three of these phenomena.

The benchmark model is general but modular, in the sense that it is straightforward,

both axiomatically and in the representation, to ‘shut down’ any combination of the phe-

nomena. As such, our model yields state-independent utility models with imprecise beliefs

and tastes, state-dependent precise utility uncertainty averse models, and state-dependent

multi-utility generalisations of many ambiguity models in the literature as special cases.

Accompanying analyses reveal the role of State Consistency—the axiom that guaran-

tees a coherent notion of preferences conditional on a state—as a necessary condition for

separating beliefs and (potentially state-dependent) utilities, as well as the particularity of

our representation in being able to support this separation along with imprecision of beliefs

and tastes. Comparative statics analyses shed light on the roles of the various elements of

the model, and corroborate their proposed interpretations.

29


Appendix A Proofs

Throughout the Appendices, ď on <n is the standard order, given by a ď b iff ai ď bi for

all 1 ď i ď n, for all a, b P <n. ¨ is the standard scalar product of vectors. Finally, let

ZR “ te P <X |ř

xPX epxq “ 0u.

A.1 Proofs of results in Section 3

Proof of Theorem 1. Consider firstly the (i) implies (ii) direction.

By assumption, hpsq ąs hpsqα hpsq for all α P p0, 1q, whence it follows from A1 and

A6 that h ą hα h for all α P p0, 1q. We now show the following stochastic dominance

property for h, h: for every α, β P r0, 1s, α ě β iff hα h ľ hβ h. If α ě β, then hα h “

h α´β1´βphβ hq. Since hβ h ľ hβ h and h ľ hβ h, it follows from A8 that hα h ľ hβ h, as

required. For the other direction, suppose for reductio that there exist α, β P r0, 1s with

α ă β, and hα h ľ hβ h. It follows from the previous argument that hα h „ hβ h. Without

loss of generality, we can assume that rα, βs is a maximal interval with this property, in

the following sense: for every α1 ă α and every β1 ą β, hα1 h ă hα h „ hβ h ă hβ1 h.

Since hα1 h ă h for all α1 P p0, 1q, β ă 1. Take any γ P`

0, β´α1´α

˘

(this set is non-

empty since β ą α). So hγphα hq “ hδ h for some α ă δ ă β. By the previous result,

hδ h „ hα h „ hβ h. However, since hβ h ĺ hα h, it follows from A8 that hγ`βp1´γq h “

hγphβ hq ĺ hγphα hq “ hδ h „ hβ h, whence, by the previous result hγ`βp1´γq h „ hβ h,

contradicting the maximality of rα, βs. So hα h ă hβ h whenever α ă β, as required.

It follows from this observation, the fact that h ľ f ľ h and A3 that, for each f P A,

there exists a unique αf P r0, 1s with f „ hαf h. Define V : AÑ r0, 1s by V pfq “ αf for

each f P A. By definition and A1, V represents ľ. By A2, V is non-constant. It follows

from A8 that, for all f, g P A, α P r0, 1s, V pfαgq ě αV pfq ` p1 ´ αqV pgq. So V is

concave. It follows that tf P A : V pfq ě ru is convex for all r; by the arguments in Dubra

et al. (2004, Proposition 1), this set is closed, and so V is upper semi-continuous. Since X ,

and hence A is a polyhedral convex set, it follows by Rockafellar (1970, Theorem 10.2)

that V is continuous.

For each non-null s P S, define ąs on X as follows: for every c, d P X and s P S,

c ľs d if and only if there exists h P A with csh ľ dsh. By A6 and A1, ľs is a weak order.

Moreover it is clearly mixture continuous, by A3.

We now show that for every α, β P r0, 1s, α ě β iff hpsqα hpsq ľs hpsqβ hpsq. If α ě β,

30


then hα h “ h α´β1´βphβ hq. Since hβ h ľ hβ h and hsphβ hq ľ hβ h, it follows from A8 that

phα hqsphβ hq ľ hβ h, so, by A6, hpsqα hpsq ľs hpsqβ hpsq as required. For the other

direction, suppose for reductio that there exist α, β P r0, 1s with α ă β, and hpsqα hpsq ľs

hpsqβ hpsq. It follows from the previous argument that hpsqα hpsq „s hpsqβ hpsq. Without

loss of generality, we can assume that rα, βs is a maximal interval with this property, in

the same sense as above. Since hpsqα1 hpsq ăs hpsq for all α1 P p0, 1q, β ă 1. Take any

γ P`

0, β´α1´α

˘

(this set is non-empty since β ą α). So hpsqγphpsqα hpsqq “ hpsqδ hpsq

for some α ă δ ă β. By the previous result, hpsqδ hpsq „s hpsqα hpsq „s hpsqβ hpsq.

However, since, by A6, hβ h ĺ phα hqsphβ hq, it follows from A8 that hγ`βp1´γq h “

hγphβ hq ĺ hγpphα hqsphβ hqq “ phδ hqsphγ`βp1´γq hq „ phβ hqsphγ`βp1´γq hq, whence,

by the previous result hγ`βp1´γq h „s hβ h, contradicting the maximality of rα, βs. So

hpsqα hpsq ăs hpsqβ hpsq whenever α ă β, as required.

So, by A6, for every x P X , there is a unique αxs such that x „s hpsqαxs hpsq. Define

the function Vs : X Ñ < by Vspxq “ αxs . By A1 and A6, Vs represents ľs. Moreover,

for any x, y P X , by A6, xsphαxs hqsc „ hαxs h and ysphαys hqsc „ hαys h, whence it follows

from A8 that pxγyqsphγαxs`p1´γqαys hqsc ľ hγαxs`p1´γqαys h for all γ P r0, 1s, so Vspxγyq ě

γVspxq`p1´γqVspyq and Vs is concave. As for V , Vs is continuous. We have the following

consequence.

Lemma A.1. For each non-null s P S, there exists a non-trivial hpsq, hpsq-precise closed

convex set of utility functions Us such that Vspxq “ minuPUs u ¨ x for all x P X . Moreover

there exists such a set which is tight, in the sense that every other set of utility functions

with these properties is a superset.

Proof. Since X is a convex subset of a finite-dimensional space with non-empty interior

(if not, convert to the smallest linear subspace containing it), and Vs is concave and finite,

for each x P ripXq,25 there exists a utility function supporting Vs at x, ie. u P U such that

u ¨ x ě Vspyq for all y P X and u ¨ x “ Vspxq. Since Vs is not a constant function, not all

of these are constant. Let Us be convex closure of the set of all utility functions supporting

Vs at some x P ripXq. By construction, Vspxq “ minuPUs u ¨ x for all x P ripXq; by the

continuity of V , this holds for all x P X . By construction Us is closed and convex. Since,

by construction Vs is linear on thα h | α P r0, 1su, there exists u P Us supporting Vs at every

point in thα h | α P r0, 1su; so Us is h, h-precise. Since there is a dense subset of ripAq25For a set X , ripXq is the relative interior of X .

31


on which V is differentiable – and hence at which it has unique supergradients – and the

supergradients at all other points are contained in the convex closure of the supergradients

at points where V is differentiable (Rockafellar, 1970, Theorems 25.5 and 25.6), Us is tight.

Take any non-null s P S, and for all null s1 P S, define Vs1 “ Vs and Us1 “ Us. By

construction, Vsphpsqα hpsqq “ α “ Vs1phps1qα hps

1qq for every α P r0, 1s and s, s1 P S, so

υ : S Ñ 2UzH defined by υpsq “ Us for all s P S, is h, h-constant. Moreover, by Lemma

A.1, it is non-trivial, closed, convex and h, h-precise.

Let B “ r0, 1sS . For every f P A, define V pfq P B by V pfqpsq “ Vspfpsqq for all

s P S. By A1 and A6, for every f, g P A, if fpsq „s gpsq for every s P S, then f „ g,

so the functional I : B Ñ < defined by Ipaq “ V pfq for any f such that V pfq “ a is

well-defined. By definition V pfq “ IpV pfqq.

Lemma A.2. I is concave, continuous, monotonic, and normalised (ie. for all z P r0, 1s,

Ipz˚q “ z, where z˚ is the constant function in B taking value z). Moreover, for every

z, w P r0, 1s with z ‰ w, a P B and s P S, Ipzsaq “ Ipwsaq if and only if s P S is null.26

Proof. Normalisation follows from the definition of I , Vs and A6. Monotonicity of I is

immediate from A6. Continuity follows from the continuity of V and the fact that V

is a quotient map. As concerns concavity, for every a P B, let ha P A be such that

hapsq “ hapsq h for all s P S. By construction, V phaq “ a. Note moreover that, for every

a, b P B and γ P r0, 1s, V phaγhbq “ γa` p1´ γqb. By concavity of V , Ipγa` p1´ γqbq “

V phaγhbq ě γV phaq ` p1´ γqV phbq “ γIpaq ` p1´ γqIpbq, so I is concave. As concerns

the final property, if s P S is null, then V pxsfq “ V pysfq for all x, y P X , f P A by

definition, and the corresponding property for I follows immediately. If s P S is non-null,

then, since, as shown above hpsqα hpsq s hpsqβ hpsq for α ‰ β, it follows from A6 that

V pphpsqα hpsqqqsfq ‰ V pphpsqβ hpsqqsfq for all f P A; the corresponding property for I

follows immediately.

Since B is closed, bounded and convex and I is concave, for each a P ripBq, there

exists an affine functional φ : <S Ñ < supporting I at a: i.e. such that φpbq ě Ipbq for all

b P B and φpaq “ Ipaq. Each such φ can be written as φpbq “ η ¨ b ` µ for some η P <S ,

26zsa is defined in an analogous way to xsf P A.

32


µ P <. We first show that ηs ě 0 for every s P S. Take any a P ripBq such that φ supports

I at a, and consider as P B defined by aspsq “ apsq ` ε, asps1q “ apsq for s1 ‰ s, where

ε ą 0 such that apsq ` ε P r0, 1s. By the monotonicity of I , Ipasq ě Ipaq; since φ supports

I at a, we have that φpasq “ η ¨ as ` µ ě Ipasq ě Ipaq “ η ¨ a ` µ, so ηs.ε ě 0. Since

this holds for every s P S, we have that ηs ě 0 for every s P S. We now show that ηs “ 0

if and only if s is null. Take any a P ripBq such that φ supports I at a, and consider as as

defined previously and a´s P B defined by a´spsq “ apsq ´ δ, a´sps1q “ apsq for s1 ‰ s,

where δ ą 0 such that apsq ´ δ P r0, 1s. If ηs “ 0, then φpasq “ φpaq “ Ipaq, whence it

follows, since φpasq ě Ipasq, that Ipasq “ Ipaq, so s is null, by the final clause in Lemma

A.2. Conversely, by the final clause in Lemma A.2, if s is null, then Ipa´sq “ Ipaq; since

φpa´sq “ η ¨ a´s ` µ “ φpaq ´ ηs.ε, it follows from the fact that φ supports I at a that

ηs ď 0, and thus, in the light of the previous result, that ηs “ 0, as required.

Let Φ “ cl pconvtpη, µq | Da P ripBq s.t. φpbq “ η ¨ b` µ supports I at auq. By the

continuity of the superdifferential mapping and of I , Ipaq “ minpη,µqPΦ η ¨ a ` µ for all

a P B. Since I is differentiable on a dense subset of B and supergradients at differentiable

points determine the supergradients elsewhere (Rockafellar, 1970, Theorem 25.6), this does

not hold for any proper closed convex subset of Φ. Moreover, since I is normalised, for all

µ1 ă 1, p0, µ1q R Φ. (It is clear from the construction that p0, µ1q R Φ for all µ1 ą 1.) Hence,

for every pη, µq P Φztp0, 1qu, there is a unique p P ∆ Ď <S and a P <ą0 such that η “ a.p.

Let P “ tpp, a, bq P ∆ˆ <ą0 ˆ < | pa.p, bq P Φztp0, 1quu.

By the previous observations, for all a P B, Ipaq ď minpp,a,bqPPpa.pq ¨ a ` b with

equality whenever a R I´1p1q. Since I is normalised, it follows that ax ` b ě x for all

pp, a, bq P P and x P r0, 1s. We now show that there exists p P ∆ such that pp, 1, 0q P P .

To establish this, note firstly that for every x ą 0, there exists no pp, a, bq P P such that

a ą 1 and a.x ` b “ x: if there were such a, b, then for any y P r0, 1s with y ă x,

a.y ` b “ x ´ a.px ´ yq ă y, contradicting the fact that P represents I and that I is

normalised. Similarly, for every x ă 1, there exists no pp, a, bq P P such that a ă 1 and

a.x ` b “ x. Since I is normalised, it follows from the previous observation about the

representation of I by P that there exists p P ∆ with pp, 1, 0q P P . Since, for any p P ∆

and a P B, p ä ď 1, it follows from the fact that P represents I on pI´1p1qqc that, for every

pp, 1, 0q P P and a P I´1p1q, p ä “ 1. Hence Ipaq “ minpp,a,bqPPpa.pq ä` b for all a P B.

Define α : ∆ Ñ <ą0 ˆ < by αppq “ tpa, bq P <ą0 ˆ < | pp, a, bq P Pu. By the

33


definition of P and the properties of Φ (in particular non-emptiness, closure, convexity,

and the fact about null states), α is non-trivial, null-consistent, upper hemicontinuous and

convex. By the last two properties of P mentioned above, α is calibrated (with respect

to υ) and grounded. So α is a non-trivial, null-consistent, grounded, calibrated, upper

hemicontinuous, convex function representing ľ along with υ according to (2), as required.

Since no proper closed convex subset of Φ represents I in the specified way, there is no

upper hemicontinuous, convex α1 : ∆ Ñ <ą0 ˆ < with α1ppq Ď αppq for all p P ∆ where

the inclusion is proper for at least one p P ∆ that represents ĺ along with υ according to

(2). It follows from this and the tightness of υ that the pair pυ, αq is tight.

Now consider the (ii) to (i) implication. It is standard for the Basic Axioms. State-

Consistency follows immediately from the form of the representation, and the fact that α

is null-consistent. Since υ is h, h-constant and h, h-precise, for each β P r0, 1s, V phβ hq “

βV phq`p1´βqV phq; and hence, by the representation, f ľ hβ h iff ař

sPS ppsquspfpsqq`

b ě βV phq ` p1 ´ βqV phq for all p P ∆, pa, bq P αppq P U and us P υpsq. Strong

Uncertainty Aversion follows immediately from the form of the representation.

Proof of Proposition 1. The existence clause for tight representations was established in

the proof of Theorem 1. Now consider the uniqueness clause, and suppose that pυ, αq,

and pυ1, α1q are tight pairs of state-dependent multi-utilities and ambiguity indices rep-

resenting ľ. Let Vυ,α and Vυ1,α1 be the functionals defined from them according to (2).

By Lemma A.3 below, we can assume without loss of generality that υ and υ1 are cal-

ibrated and constant with respect to the same h, h. Let κ P <ą0 and λ P < be such

that κVυ1,α1phq ` λ “ Vυ,αphq and κVυ1,α1phq ` λ “ Vυ,αphq (it is straightforward to

show that such κ and λ exist). Define V 2 “ κVυ1,α1 ` λ, υ2 “ κυ1 ` λ and α2 by

α2ppq “ tpa, κb` λp1´ aqq | @pa, bq P α1ppqu for all p P ∆; it is clear that V 2, υ2 and α2

are related according to (2) and that V 2 represents ľ. υ2 is h, h-precise and h, h-constant

since pυ1, α1q is, and because υ is h, h-precise and h, h-constant, V 2phα hq “ V phα hq for

all α P r0, 1s. It follows by the fact that for each f P A there exists an unique α P r0, 1s

with f „ hα h and the fact that they both represent ľ that V 2 “ V . Since υ and υ2 are h, h-

constant, it follows that, for all s, t P S, and β P r0, 1s, Vsphpsqβ hpsqq “ Vtphpsqβ hpsqq “

V phβ hq “ V 2phβ hq “ V 2t phpsqβ hpsqq “ V 2s phpsqβ hpsqq, where Vspxq “ minuPυpsq u ¨ x

and similarly for V 2s . Since, for each non-null s P S and x P X , there exists a unique

34


β with x „s hpsqβ hpsq, it follows that V 2s “ Vs for each non-null s P S. Since υpsq is

tight, by the reasoning in the proof of Lemma A.1, it is the convex closure of the set of

supergradients of Vs; however, since υ2psq is tight, the same holds for it, and so the two

sets are identical. So υpsq “ υ2psq “ κυ1psq ` λ for all non-null s P S as required.

Let K “ V pXq “ V 2pXq; since υ and υ2 are h, h-constant, K “ tminuPυpsq u ¨ x | x P

Xu “ tminuPυ2psq u ¨ x | x P Xu for all non-null s P S. Let I be the functional on KS

defined from V as in the proof of Theorem 1, and similarly for I2 and V 2. Since V “ V 2,

I “ I2. By the construction in the proof of Theorem 1, since α is tight, it is generated by

the set of supergradients of I; however, since α2 is tight, the same holds for it, so α “ α2.

So α “ tpa, κb` λp1´ aqq | @pa, bq P α1ppqu for all p P ∆, as required.

Lemma A.3. Let pυ, αq and pυ1, α1q be pairs of non-trivial closed convex functions and

ambiguity indices representing ľ according to (2) where υ is h, h-constant h, h-precise

and υ1 is h1, h 1-constant h

1, h 1-precise. Then hα h „ h

1

α h1 for all α P r0, 1s. It follows

in particular that υ and υ1 are both h, h-constant, h, h-precise and h1, h 1-constant, h

1, h 1-

precise.

Proof. By Theorem 1 and its proof, since υ is h, h-constant and h, h-precise, the Strong

Uncertainty Aversion holds when formulated with h, h, and similarly for υ1 and h1, h 1.

Since pυ1, α1q represents ľ, we have that h1

and h 1 are maximal and minimal elements of

ľ, respectively, and similarly for h and h. So h „ h1

and h „ h 1. It follows that, for

any α P r0, 1s, by Strong Uncertainty Aversion formulated with h, h, that h1

α h1

ľ hα h,

and by Strong Uncertainty Aversion formulated with h1, h 1, that hα h ľ h

1

α h1, whence

hα h „ h1

α h1, as required. The remaining clauses in the lemma follow immediately.

A.2 Proofs of Results in Section 4

Proof of Proposition 2. The necessity of the axioms is straightforward to check; we con-

sider sufficiency. Let ąs be defined as in the proof of Theorem 1. We first show that

ľs“ľs1 for all non-null s, s1 P S. Take any such s, s1 and suppose, for x, y P X , x ąs y.

By A4, it follows from the fact that hps1q and hps1q are ľs1-maximal and ľs1-minimal

elements in X that they are ľ-maximal and ľ-minimal elements of X , and hence of

ľs-maximal and ľs-minimal elements, respectively. By A3, it follows that there exist

α, β P p0, 1q, α ą β, such that x ąs hps1qα hps

1q, hps1qβ hps1q ąs y. By A4, it follows that

35


x ą hps1qα hps1q, hps1qβ hps

1q ą y, and hence that x ľs1 hps1qα hps

1q, hps1qβ hps1q ľs1 y. It

follows from the stochastic dominance property for ľs established in the proof of Theorem

1 that x ľs1 hps1qα hps

1q ąs1 hps1qβ hps

1q ľs1 y, as required.

By the reasoning in the proof of Theorem 1, there exists a non-constant, continuous,

concave functional V : A Ñ <, linear on thβ h | β P r0, 1su, representing ľ, and contin-

uous concave functionals Vs, linear on thpsqβ hpsq | β P r0, 1su, representing ľs for each

non-null s P S. Take any such Vs. Since ľs“ľs1 for all non-null s, s1 P S, Vs repre-

sents ľs1 . Moreover, for similar reasons, it represents the restriction of ľ to X . Note that

since Vs is linear on thpsqβ hpsq | β P r0, 1su, with hpsq and hpsq maximal and minimal

elements of ľs1 respectively, it is a minimal concave representation of ľs1 , in the sense

of Debreu (1976) (see also Kannai (1977)): every other concave representation V 1 of ľs1

with V 1pXq “ V pXq is such that V 1pxq ě V pxq for all x P X . Since the same holds

for Vs1 , and since minimal representations are unique (Kannai, 1977, pp11-13), it follows

that Vs “ Vs1 , and more generally that Vs “ Vt for every non-null s, t P S. In particular,

it follows that, for every non-null s, s1 P S, hpsqβ hpsq „s hps1qβ hps1q for all β P r0, 1s.

Since υ in representation (1) can be taken to be h, h-constant and tight (by Proposition 1),

it follows that υpsq “ υptq for all non-null s, t P S. Setting U “ υpsq for any non-null

s P S yields the desired representation. The uniqueness clause follows immediately from

Proposition 1 and the fact just mentioned that h, h-constant υ are constant-valued.

Proof of Proposition 3. State Independence clearly implies State Consistency (taking t “

s). Moreover, in the presence of Weak Order, State Independence implies that, if x ľ y,

then xsf ľ ysf for every non-null s P S and f P A: if this were not the case, then

xsf ă ysf for some and hence every non-null s P S (by State Independence), whence

x ă xs1y ă xts1,s2uy ă ¨ ¨ ¨ ă y (with indifferences for null states), contradicting the

fact that x ľ y. So, for f, g P A satisfying the conditions in the Monotonicity axiom,

f ľ gs1f ľ gts1,s2uf ľ ¨ ¨ ¨ ľ g, where each step is an application of the previous fact;

hence State Independence implies Monotonicity in the presence of Weak Order.

The other direction was established by the reasoning at the beginning of the proof of

Proposition 2.

Proof of Proposition 4. The necessity of the axioms is straightforward. As concerns suffi-

ciency, by State-wise Independence, ľs satisfies the standard Independence axiom for each

36


state s P S. It follows by standard arguments that the Vs defined in the proof of Theorem

1 is affine for every s P S, so υpsq, being tight, is a singleton for all non-null s P S. It

can thus be assumed without loss of generality that υpsq is a singleton for all s P S (for

instance, by setting υ on null states as in the proof of Theorem 1). Setting ups, xq “ upxq

for the unique u P υpsq for all x P X and s P S yields representation (4). The uniqueness

clause follows immediately from Proposition 1.

Proof of Theorem 2. We first consider the (i) to (iii) implication. By Theorem 1, ľ is

represented according to (2) by a tight pair pυ, αq. Let V and I be as in the proof of

Theorem 1, so Ipcq “ minpP∆,pa,bqPαppq a.c ¨ p` b for all c P V pAqS . By standard arguments

(see for instance Gilboa and Schmeidler (1989)), it follows from EC-Independence that I

is positive homogeneous (Ipλcq “ λIpcq for all c P V pAqS , λ P <ą0) and constant additive

(Ipc ` r˚q “ Ipcq ` r, where r˚ is the constant function taking value r P < everywhere).

Consider any p, a, b with pa, bq P αppq such that ap ` b is the unique support to I at some

c P ripV pAqSq; such points exist by the arguments in the proof of Theorem 1. So ap¨c`b “

Ipcq. Now consider c` ε˚ for some ε with |ε| sufficiently small, so that c` ε˚ P V pAqS . By

the representation Ipc` ε˚q ď ap ¨ pc` ε˚q ` b “ ap ¨ c` b` a.ε “ Ipcq ` a.ε. But by the

constant additivity of I , Ipc` ε˚q “ Ipcq` ε. It follows, taking ε ą 0, that a ě 1; however,

taking the case of ε ă 0 implies that a ď 1. So a “ 1. Similarly, considering λc for λ ą 0,

λ ‰ 1 such that λc P V pAqS , we have Ipλcq ď ap ¨ pλcq ` b “ λpap ¨ c` bq ` p1´ λqb “

λIpcq ` p1 ´ λqb. Positive homogeneity implies that Ipλcq “ λIpcq. Again, taking δ ă 1

implies that b ě 0 whereas the case with δ ą 1 implies that b ď 0; so b “ 0. It thus

follows that at all points where I has a unique support, a “ 1 and b “ 0 for the supporting

ap ` b. It follows from the arguments in the proof of Theorem 1, and in particular the fact

that α is determined as the closure of such points, that a “ 1 and b “ 0 for every pa, bq and

p P ∆ such that pa, bq P αppq. So α reduces to a null-consistent closed convex subset of ∆,

yielding representation (1). Uniqueness follows from Proposition 1.

The (iii) to (i) implication is straightforward to show, as is the (i) to (ii) implication.

To establish the (ii) to (i) implication, it suffices to show that Uncertainty Aversion and

EC-Independence imply Strong Uncertainty Aversion, in the presence of the other axioms.

By the usual arguments, for every α, β P r0, 1s, hα h ľ hβ h if and only if α ě β. Consider

any f, g P A; by A3 and the fact that h and h are maximal and minimal elements of ľ,

37


there exists unique β, β1 P r0, 1s such that f „ hβ h and g „ hβ1 h. To establish Strong

Uncertainty Aversion, it suffices to show that, for any α P r0, 1s, fαg ľ phβ hqαphβ1 hq: the

condition for γ, γ1 with f ľ hγ h and g ľ hγ1 h follows from the fact, noted above, that this

implies that β ě γ and β1 ě γ1. If β “ β1, then Uncertainty Aversion directly implies that

fαg ľ hβ h “ phβ hqαphβ1 hq, as required. Now suppose not, and assume without loss of

generality that β ą β1. Let γ “ β1

β. By EC-Independence, it follows from the previous in-

differences that fγ h „ phβ hqγ h “ hβ1 h. So, by Uncertainty Aversion, pfγ hq αα`p1´αqγ

g ľ

hβ1 h. Since pfγ hq αα`p1´αqγ

g “ pfαgq γα`p1´αqγ

h and hβ1 h “ pphβ hqαphβ1 hqq γα`p1´αqγ

h, it

follows from EC-Independence that fαg ľ phβ hqαphβ1 hq as required.

Proof of Proposition 5. The (ii) to (i) direction is straightforward. The (i) to (ii) is a direct

extension of the proof of Theorem 2, noting that Restricted Independence implies that I

is affine, and hence, by standard arguments, is generated by a (single) probability measure

p.


The following result characterises imprecision aversion in the absence of the assump-

tion about the less imprecision-averse decision maker’s best act.

Proposition 10. Let ľ1 and ľ2 be represented according to (2) by pairs of state-dependent

multi-utilities and ambiguity indices pυ1, α1q and pυ2, α2q, and suppose that they are nor-

malised so that υ1pXq “ υ2pXq. The following are equivalent.

i. ľ1 is more imprecision averse than ľ2

ii. pυ1 b α1q Y pυ2 b α2q represents ĺ1 according to (11), where b is as defined in

Section 6.

Proof of Proposition 10. Define the representing functionals V 1 and V 2 from pυ1, α1q and

pυ2, α2q according to (1); since, as shown in the proof of Theorem 1, υ1pXq “ V 1pAq,it follows from the normalisation assumption that V 1pAq “ V 2pAq. We first show that

ľ1 is more imprecision averse than ľ2 iff V 1pfq ď V 2pfq for all f P A. By the rep-

resentation and the h i, h i-precision of the pυi, αiq, for every f P A, f „1 h 1V 1pfq h

1

and f „2 h 2V 2pfq h

2. If ľ1 is more imprecision averse, then f „1 h 1V 1pfq h

1 implies

38


that f ľ2 h 2V 1pfq h

2, from which it follows, by the stochastic dominance property for

h 2, h 2 (see proof of Theorem 1), that V 2pfq ě V 1pfq. Conversely, if V 2pfq ě V 1pfq,

then by the same stochastic dominance property, f ľ1 h 1α h

1 iff V 1pfq ě α, and this

implies that V 2pfq ě α and hence f ľ2 h 2α h

2, establishing the claim. We have

thus established that (i) holds iff V 1pfq ď V 2pfq for all f P A, which is the case iff

mintV 1pfq, V 2pfqu “ V 1pfq for all f P A. However, since, as noted in Section 6,

V 1pfq “ minUPυ1bα1

ř

sPS Ups, fpsqq for all f P A and similarly for V 2, it follows that

mintV 1pfq, V 2pfqu “ minUPpυ1bα1qYpυ2bα2q

ř

sPS Ups, fpsqq for all f P A. Hence (i)

holds if and only if pυ1 b α1q Y pυ2 b α2q represents ľ1, as required.

Proof of Proposition 6. It is straightforward to show that (ii) implies (i) (using, in particu-

lar, the reasoning in the proof of Proposition 10). It is also clear that (iii) implies (ii), once

one notes that the values of υ1Y υ2 are immaterial on ľ1-null states. We now show that (i)

implies (iii); henceforth, assume (i) to hold.

Define the representing functionals V 1 and V 2 from pυ1, α1q and pυ2, α2q according to

(1), and let V 1s pxq “ minuPυ1psq upxq for all x P X and s P S, and similarly for V 2

s . We

first show that h1

α h1„1 h

2

α h2 for all α P r0, 1s. Recall firstly that h

2ľ1 f for all f P A (it

is a maximal element of ľ1), and note that h2 is a minimal element of ľ1: for if not, there

would exist α ą 0 such that h2ľ1 h

1

α h1, whence h2

ľ2 h2

α h2 by the fact that ľ1 is more

imprecision averse than ľ2, contradicting the stochastic dominance property of ľ2 with

respect to h2

α h2 (see proof of Theorem 1). By the stochastic dominance property of ľ1

with respect to h1

α h1 (see proof of Theorem 1), for every α P r0, 1s, there exists a unique

βα P r0, 1s with h2

α h2„1 h

1

βα h1; let µpαq “ βα for all α P r0, 1s. Since V1 is concave and

continuous, and V1ph1

β h1q “ β for all β P r0, 1s, µ is a concave continuous function. Since

h2

and h2 are maximal and minimal elements of ľ1 respectively, µp0q “ 0 and µp1q “ 1.

Finally, if µpαq ą α, then, by the stochastic dominance property of ľ2 with respect to

h2

α h2, h

2

µpαq h2

ą2 h2

α h2, whence it follows, by the fact that ľ1is more imprecision averse

than ľ2, that h1

µpαq h1

ą1 h2

α h2, contradicting the definition of µ. So µpαq ď α for all

α P r0, 1s. So µ is the identity function, as required.

It follows that ľ1 satisfies the Strong Uncertainty Aversion axiom with respect to

th2

α h2u: for all f, g P A and α, β, β1 P r0, 1s, if f ľ1 h

2

β h2 and g ľ1 h

2

β1 h2 then

fαg ľ1 ph2

β h2qαph

2

β1 h2q.

39


Observe furthermore that for every s P S, if s is ľ1-non-null, then it is ľ2-non-null.

For if s is ľ1-non-null, then h2

s h2

ľ1 h1

α h1 for some α ą 0, so by the fact that ľ1 is more

imprecision averse, h2

s h2

ľ2 h2

α h2

ą h2, and hence s is ľ2-non-null. Now we have the

following claim.

Claim 1. For every ľ1-non-null s P S, h1

α h1„1s h

2

α h2.

Proof. Fix a ľ1-non-null s P S, and let ľ1s be as defined in the proof of Theorem 1. Since

υ1 is h1, h1-constant and ľ1

s is represented by V 1s , ľ1

s satisfies the stochastic dominance

property with respect to h1

α h1: for every α, β P r0, 1s, α ě β iff h

1

α h1

ľ1s h

1

β h1. Since

h2

is a maximal element of ľ1, it follows that h2psq is a maximal element of ľ1

s; similarly

h2psq is a minimal element of ľ1

s. We now show that the stochastic dominance property

holds for ľ1s with respect to th

2

α h2u. Let α ą β; by the stochastic dominance property

for ľ2s (see proof of Theorem 1), h

2

α h2

ą2s h

2

β h2, and so h

2

α h2

ą2 ph2

β h2qsph

2

α h2q. It

follows from the fact that ľ1 is more imprecision averse and the observation above that

h2

α h2„1 h

1

α h1

ą1 ph2

β h2qsph

2

α h2q. Hence h

2

α h2

ą1s h

2

β h2. For the other direction,

suppose that h2

β h2

ľ1s h

2

α h2. So ph

2

β h2qsph

2

α h2q ľ1 h

2

α h2„1 h

1

α h1, whence, by the fact

that ľ1 is more imprecision averse, ph2

β h2qsph

2

α h2q ľ2 h

2

α h2 and hence h

2

β h2

ľ2s h

2

α h2.

It follows from the stochastic dominance property for ľ2s that β ě α. So α ą β iff

h2

α h2

ą1s h

2

β h2 as required.

It follows from this property, and the fact that ľ1 satisfies the Strong Uncertainty Aver-

sion axiom with respect to th2

α h2u (as well as State Consistency) that, by the reasoning

in the proof of Theorem 1, there is a continuous concave functional Vs : X Ñ r0, 1s rep-

resenting ľ1s which is linear on th

2

α h2u. Since this function is linear on a set on which

it takes values ranging over its whole co-domain, it is minimal in the sense of Debreu

(1976); Kannai (1977). However, the same holds for V 1s , which is (by the proof of Theo-

rem 1) a continuous concave functional repxresenting ľ1s which is linear on th

1

α h1u. Since

minimal concave representations are unique up to positive affine transformation (Kannai,

1977), and VspXq “ V 1s pXq, we have that Vs “ V 1

s . It follows in particular that V 1s is

linear on both th1

α h1u and th

2

α h2u, taking the value 1 and α “ 1 and 0 at α “ 0 in both

cases, so V 1s ph

1

α h1q “ V 1

s ph2

α h2q for all α P r0, 1s. Since V 1

s represents ľ1s, it follows that

h1

α h1„1s h

2

α h2 for all α P r0, 1s.

40


We now show that ľ1 is more imprecision averse on consequences than ľ2, in the

sense of Definition 2. For every x P X and α P r0, 1s, by Claim 1 (and Weak Order

applied repeatedly), xsph2

α h2q „1 xsph

1

α h1q. So the fact that ľ1 is more imprecision averse

implies that, for any ľ1-non-null s P S, whenever xsh ľ1 ph1

α h1qsh, then xsph

2

α h2q „1

xsph1

α h1q ľ1 h

1

α h1 (by State Consistency), and hence xsph

2

α h2q ľ2 h

2

α h2 (by Imprecision

Aversion), and so xsh ľ2 ph2

α h2qsh. So ľ1 is more imprecision averse on consequences

than ľ2; by Proposition 7, υ1psq ď υ2psq for all ľ1-non-null states, as required.

Finally, we show that ľ1 is more imprecision averse for states than ľ2. Consider any

f P th1

β h1 : β P r0, 1suS . By Claim 1, f „1 f , defined as in Definition 2. So the fact that

ľ1 is more imprecision averse implies that, whenever f „1 f ľ1 h1

α h1, then f ľ2 h

2

α h2.

Hence ľ1 is more imprecision averse for states than ľ2. By Proposition 7, α1 ď α2 as

required.

The uniqueness clause follows from Proposition 1 and the fact that the representations

by pυ1, α1q and pυ1 Y υ2, α1 Y α2q give the same range of values.

Proof of Proposition 7. Part i. For each s P S, define the functionals V 1s : X Ñ < and

V 2s : X Ñ < by V 1

s pxq “ minuPυ1psq upxq and similarly for V 2s and υ2. By the assumption

and Proposition 1, we can assume without loss of generality that V 1s ph1q “ V 2

s ph2q “ 1

and V 1s ph1q “ V 2

s ph2q “ 0 for all s P S. We first show that ľ1 is more imprecision averse

on consequences than ľ2 iff V 1s pxq ď V 2

s pxq for all x P X and ľ1-non-null s P S. By

the representation, the previous normalisation and the h i, h i-precision and -constancy of

the υi, for every x P A, x „1s h 1

V 1s pxq

h 1 and x „2s h 2

V 2s pxq

h 2. If ľ1 is more imprecision

averse on consequences, then x „1s h

1α h

1 implies that x ľ2s h

2α h

2, from which it follows,

by the stochastic dominance property for ľ2s with respect to h 2, h 2 (see proof of Theorem

1), that V 2s pxq ě V 1

s pxq. Conversely, if V 2s pxq ě V 1

s pxq for all x P X , then by the

same stochastic dominance property, x ľ1s h 1

α h1 iff V 1

s pxq ě α, and this implies that

V 2s pxq ě α and hence x ľ2

s h 2α h

2; since this holds for every ľ1-non-null s P S, it

establishes the claim. We have thus established that ľ1 is more imprecision averse on

consequences iff V 1s pxq ď V 2

s pxq for all x P X and ľ1-non-null s P S. This is the case iff

mintV 1s pxq, V

2s pxqu “ V 1

s pxq for all x P X and ľ1-non-null s P S. By the definition of V is ,

we have that mintV 1pfq, V 2pfqu “ minuPυ1psqYυ2psqř

sPS upxq for all x P X , so ľ1 is more

imprecision averse on consequences iff minuPυ1psqř

sPS upxq “ minuPυ1psqYυ2psqř

sPS upxq

41


for all x P X and ľ1-non-null s P S, and this is the case if and only if υ1psq ď υ2psq for

all ľ1-non-null s P S, as required.

Part ii. By Proposition 1 and the normalisation of the υi, we can assume without loss

of generality that υipXq “ r0, 1s for i P t1, 2u. Define V i : A Ñ r0, 1sS by V ipfqpsq “

minuPυipsq upfpsqq for all f P A and i P t1, 2u. Note that, by the hi, h i-precision and the

fact that the υi are hi, h i-constant, V 1pfqpsq “ α whenever fpsq “ h

1

α h1, and similarly for

V 2. Now define I1 : r0, 1sS Ñ < by I1pcq “ minpP∆, pa,bqPα1ppq pař

sPS ppsqcpsq ` bq, and

similarly for I2. By definition, I i ˝ V i represents ľi.

We now show that ľ1 is more imprecision averse on states than ľ2 iff I1pcq ď I2pcq

for all c P r0, 1sS . Take any f P th1

β h1 : β P r0, 1suS with f as in Definition 2. For each

s P S, fpsq “ h1

β h1 and f “ h

2

β h2 for some β P r0, 1s, so by the previously observed

property of V i, V 1pfqpsq “ β “ V 2pfqpsq. So V 1pfq “ V 2pfq. Moreover, by the rep-

resentation and the h i, h i-precision of the pυi, αiq, for each g P A, g „1 hI1pV 1pgqq h, and

similarly for ľ2. It follows that, for every f P th1

β h1 : β P r0, 1suS , f „1 hI1pV 1pfqq h and

f „2 hI2pV 1pfqq h. Condition (10) for a given f P th1

β h1 : β P r0, 1suS implies, given the

stochastic dominance property for ľ2 with respect to h2, h2, that I1pV 1pfqq ď I2pV 1pfqq.

Conversely, if I1pV 1pfqq ď I2pV 1pfqq, then, by the same stochastic dominance prop-

erty, f ľ1 h1

β h1 iff I1pV 1pfqq ě β, and this implies that I2pV 1pfqq ě β and hence

f ľ2 h1

β h2; so Condition (10) holds for this f . So ľ1 is more imprecision averse on

states than ľ2 iff I1pV 1pfqq ď I2pV 1pfqq for all f P th1

β h1 : β P r0, 1suS . However,

since, for each c P r0, 1sS , there exists f P th1

β h1 : β P r0, 1suS with V 1pfq “ c—

namely fpsq “ h1

cpsq h1 for all s P S—this holds if and only if I1pcq ď I2pcq for all

c P r0, 1sS . This is the case iff mintI1pcq, I2pcqu “ I1pcq for all c P r0, 1sS . By the defini-

tion of I i, we have that mintI1pcq, I2pcqu “ minpP∆, pa,bqPα1ppqYα2ppq pař

sPS ppsqcpsq ` bq.

So ľ1 is more imprecision averse on states iff minpP∆, pa,bqPα1ppq pař

sPS ppsqcpsq ` bq “

minpP∆, pa,bqPα1ppqYα2ppq pař

sPS ppsqcpsq ` bq, and this is the case if and only if α1 ď α2,

as required.


Proof of Theorem 3. The necessity of the axioms is shown similarly to the proof of Theo-

rem 1. We now consider the sufficiency of the axioms. By the argument at the beginning of

42


the proof of Theorem 1, for every α, β P r0, 1s, α ě β iff hα h ľ hβ h. So for each f P A,

there exists a unique αf P r0, 1s such that f „ hαf h. Define V : A Ñ < by V pfq “ αf

for all f P A. As argued in the proof of Theorem 1, V represents ľ, and is non-constant,

continuous and concave.

We say that U,U 1 P E are cardinally equivalent if there exists b P <S withř

sPS bpsq “

0 such that U 1 “ U ` b. Note that cardinally equivalent evaluations represent the same

preferences over acts (according to the version of (11) with singleton E). Define the relation

« on E by U « U 1 iff U and U 1 are cardinally equivalent. This is clearly an equivalence

relation, and consider any mapping σ : E{ «Ñ E such that σprU s«q P U . Let CE “σpE{ «q; we call CE the set of canonical evaluations. By construction, for each evaluation,

there exists exactly one cardinally equivalent canonical one.

Since X is finite-dimensional, it is a subset of <n for some n; take any basis teiuiďnwith ei P X for all i ď n. (n can be chosen so that such a basis exists.) Hence A “ XS

is isomorphic to a compact convex subset of <|S|ˆn, via the mapping taking f P A to

f P <|S|ˆn, where fps, iq “ fpsqi for all s P S, i ď n, where fpsq “řni“1 fpsq

iei. Via this

isomorphism, V generates a concave continuous functional on a compact convex subset of

<|S|ˆn. Moreover, there is an obvious value-preserving one-to-one mapping from affine

functions on <|S|ˆn and canonical evaluations, taking the affine function characterised by

hpzq “ ν ¨ z ` µ (where ν P Re|S|ˆn, µ P <) to the canonical evaluation σprUpν,µqsq where

Upν,µq is defined as follows: for all s P S x P X , Upν,µqps, xq “řni“1 x

iνps, eiq `µ|S|

where x “řni“1 x

iei. Given these correspondences, we will implicitly identify A with

– and work in – the appropriate subset of <|S|ˆn: for example, we shall say that V is

differentiable when its image in <|S|ˆn is, and that a (canonical) evaluation supports V at a

point when the corresponding affine function supports the image of V at the image of the

point.

Since A is a compact convex subset of a finite-dimensional space, and V is concave

and finite, for each f P ripAq, there exists a canonical evaluation supporting V at f , ie.

U P CE such that U ¨ g ě V pgq for all g P A and U ¨ f “ V pfq. Since V is non-constant,

there is at least one point at which the supporting U is too. Let U be convex closure of the

set of all canonical evaluations supporting V at some f P ripAq. By construction, V pfq “

minUPU U ¨ f for all f P ripAq; by the continuity of V , this establishes representation (11).

By construction U is closed and convex. Since V is linear on thα h | α P r0, 1su, there

43


exists U P U supporting V at every point in thα h | α P r0, 1su; so U is h, h-precise. We

note in passing that, since there is a dense subset of ripAq on which V is differentiable –

and hence at which it has unique supergradients – and the supergradients at all other points

are contained in the convex closure of the supergradients at points where V is differentiable

(Rockafellar, 1970, Theorems 25.5 and 25.6), E is tight, in a sense analogous to that defined

for state-dependent multi-utilities.27


Proof of Theorem 4. The (ii) to (i) implication is straightforward for the first four axioms,

relying on the arguments in Cerreia-Vioglio et al. (2015) for Negative Certainty Inde-

pendence in particular. As concerns State Consistency, since vpsq is strictly increasing

for all s P S, if lsh ľ l1sh for some h P A, non-null s P S and l, l1 P ∆pIq, then

Elpvpsqq ě El1pvpsqq, so lsh1 ľ l1sh1 for any other h1 P A, by the representation.

As for the (i) to (ii) direction, the centrepiece is the following Lemma.

Lemma A.4. For every non-null state s P S, there exists a continuous strictly increasing

utility function vs P U csi such that ľs is represented by Elpvsq.

Proof. Fix a non-null state s P S. Note first of all that ľs satisfies Negative Certainty

Independence: for every l, l1 P ∆pIq, α P r0, 1s and z P I , l ľs δz iff lsδz ľ δz (by

A6), which implies, by A12, that plsδzqαl1 ľ pδzqαl1 which holds iff lαl1 ľs pδzqαl

1, by

A6 again. Since ľs evidently also satisfies Weak Order, Weak Monetary Monotonicity and

Topological Continuity, Cerreia-Vioglio et al. (2015, Theorem 1) implies that ľs can be

represented according to a continuous functional Vs defined as in (12), with a set Us Ď U csi.By the Von-Neumann Morgenstern theorem (Grandmont, 1972), for ľs to be repre-

sented by a (single) strictly increasing continuous utility function vs : I Ñ <, given that

ľs satisfies Topological Continuity and Weak Monetary Monotonicity, it suffices that it

satisfy an Independence axiom. The representation thus follows immediately from the fol-

lowing Lemma.

Lemma A.5. For any l, l1, l2 P ∆pIq, and α P r0, 1s, l „s l1 iff lαl2 „s lαl2.27That is: a h, h-precise, closed, convex set E Ď E representing ľ according to (11) is tight if there exists

no closed, convex, proper subset E 1 P E representing the same preferences.

44


Proof. We say that a lottery l P ∆pIq is simple if it has finite support. Let ∆pIqs Ď ∆pIq be

the set of simple lotteries. We first establish the independence property for mixtures with

degenerate lotteries.

Claim 2. For any l, l1 P ∆pIq, α P r0, 1s, and any z P I , l „s l1 iff lαδz „s l1αδz.

Proof. Take any l, l1 P ∆pIq. We first establish the claim for z P I such that lsδm ĺ δz ĺ

lsδM . For each such z, by A13 there exists h P A such that lsh „ δz. (For instance, it

is possible to take h “ δmβδM for appropriate β.) If l „s l1, then it follows from A6

that l1sh „ δz. But then, by A12 applied twice, it follows that δz „ lsh ľ plshqαδz ľ δz

and similarly for l1sh, so plshqαδz „ δz „ pl1shqαδz, whence lαδz „s l1αδz, as required.

For the other direction, suppose for reductio that lαδz „s l1αδz but l ąs l1, and take h

as above, ie. such that lsh „ δz. Note that it follows directly from the fact that ľs is

represented according to (12) that it respects first-order stochastic dominance: if l2 strictly

first-order stochastically dominates l3, then l2 ąs l3. So, if lαδz „s l1αδz „s δM , then

z “ M and l „s l1 „s δM contradicting the assumption. We henceforth assume that

lαδz „s l1αδz ăs δM . By A13 and A14, there exists l2 P ∆pIq that strictly first-order

stochastically dominates l1 and such that l2 „s l. So, by A6, l2sh „ δz, and hence by

a similar argument to above, pl2shqαδz „ δz „ plshqαδz. But, since l2 strictly first-order

stochastically dominates l1 and α ą 0, this contradicts l1αδz „s lαδz. So l „s l1, as required.

Let Z0 “ tz P I | lsδm ĺ δz ĺ lsδMu, take a decreasing sequence of εi P p0, 1q,

εi Ñ 0, and define the sequence Zi iteratively by: Zi`1 “ tz P I | plεiδminZiqsδm ĺ

δz ĺ plεiδmaxZiqsδM & δminZi ĺ pδzqsδM & δmaxZi ľ pδzqsδmu. Arguing iteratively,

the claim can be established for each Zi in the sequence, as follows. Suppose that it

holds for some Zi and consider z P Zi`1zZi. Suppose that z ă minZi, the case of

z ą maxZi being treated similarly. Since minZi P Zi (by A13 it is closed), l „s l1

iff lεiδminZi „s l1εiδminZi . But since lεiδminZi ľs δminZi by definition, plεiδminZiqsδM ľs

δminZi . So, applying the result in the previous paragraph to lεiδminZi and l1εiδminZi , we have

that lεiδminZi „s l1εiδminZi iff plεiδminZiqαδw „s pl

1εiδminZiqαδw for every α P r0, 1s and

w P I with plεiδminZiqsδm ĺ δw ĺ plεiδminZiqsδM . So, for every α P r0, 1s, l „s l1 iff

lεiδminZi „s l1εiδminZi iff plεiδminZiq α

εip1´αq`αδz „s pl

1εiδminZiq α

εip1´αq`αδz iff lαδz „s l1αδz,

where the last equivalence is due to another application of the result in the previous para-

graph, the fact that plαδzqsδM ľ pδzqsδM ľ δminZi (by the definition of Zi`1), and the fact

that plεiδminZiq αεip1´αq`α

δz “ plαδzq εiεip1´αq`α

δminZi .

45


Let Z “Ť8

i“1 Zi. We now show that pm,Mq Ď Z. Suppose not, and without loss

of generality, suppose that m ă inf Z (the case of M ą sup Z is treated similarly). By

definition, applying the iterative procedure to Z yields itself, so either plεδinf Zqsδm ľ δinf Z

for every ε ą 0 or δinf Z ą pδinf ZqsδM ; since these both contradict A14, it follows that

min Z “ m. So pm,Mq Ď Z. It follows that, for every z P pm,Mq, and every α P r0, 1s,

l „s l1 iff lαδz „s lαδz.

Finally, we establish that this holds for m and M . Consider the former, the claim for

the latter being established similarly. Take any decreasing sequence zi P I , zi Ñ m. If

l „s l1, then by the fact that the claim holds for all z P pm,Mq, lαδzi „s l

1αδzi for all i.

Since ľ respects first-order stochastic dominance, it follows that lαδzi ľs l1αδm for all i

and hence, by A13, lαδm ľs l1αδm. The same argument establishes the opposite preference,

so lαδm „s l1αδm. Finally, suppose that l s l1 and consider without loss of generality the

case where l ąs l1, so, by A13, there exists w with l ąs δw ąs l

1. By the established

property, lαδzi s lαδzi for all i. Moreover, using the established property and the fact

that ľs respects first order stochastic dominance, we have, for w1 with l1 „s δw1 , that

lαδzi ąs pδwqαδzi ąs pδw1qαδzi „s l1αδzi for all i. If follows from A13 that lαδm ľs

pδwqαδm ąs pδw1qαδm „s l1αδm, so lαδm s l1αδm, as required.

Since each simple lottery is a convex combination of degenerate lotteries, the following

claim follows immediately from the previous one.

Claim 3. For any l, l1 P ∆pIq, α P r0, 1s, and any l2 P ∆pIqs, l „s l1 iff lαl2 „s lαl2.

To complete the proof, it needs to be shown that this independence property extends to

mixtures by any element of ∆pIq. Consider l, l1 P ∆pIq, α P p0, 1q and l P ∆pIqz∆pIqs.

Since ∆pIqs is dense in ∆pIq (Aliprantis and Border, 2007, Thm 15.10), there exists a

sequence li P ∆pIqs with li Ñ l. Suppose that l „s l1, and, for each i, let zi be such that

lαli „s δzi and let z be such that lαl „s δz. By A13, lαli Ñ lαl; moreover, since ľ is

represented according to (12) with a continuous functional Vs, and, by the representation,

Vsplαliq “ zi for all i, and Vsplαlq “ z, it follows that zi Ñ z. By Claim 3, l1αli „s δzifor each i, so Vspl1αliq “ zi; since, by A13, l1αli Ñ l1αl, it follows that Vspl1αlq “ z, so

l1αl „s δz „s lαl, as required. The other direction can be established by an argument

similar to that used in the last paragraph of the proof of Claim 2.

46


We can suppose without loss of generality that the vs are normalised so that vspδmq “ 0

and vspδMq “ 1. Pick a non-null s1 P S, and define v : S Ñ U csi by vpsq “ vs when s is

non-null; and vpsq “ vs1 otherwise. By A1 and A13, there exists a continuous functional

V : A Ñ < representing ľs. Since V is strictly increasing in δz, by A14, it can be chosen

such that V pδzq “ z for all z P rm,M s. For every act f P A, let vpfq P r0, 1sS be defined

by vpfqpsq “ Efpsqpvpsqq. By A1 and A6, I : r0, 1sS Ñ < defined by Ipaq “ V pfq such

that vpfq “ a is well-defined. Moreover, V “ I ˝ v. ľ generates a well-defined preference

relation on r0, 1sS , which we also denote ľ: a ľ b iff f ľ g for some f, g P A with

vpfq “ a and vpgq “ b. ľ satisfies versions of the axioms satisfied by the preference over

A (in particular, mixtures in A translate to mixtures in r0, 1sS), and is represented by I .

The rest of the proof largely repeats techniques from Ghirardato et al. (2004); Gilboa

et al. (2010). We define the relationship ľ˚ on r0, 1sS by a ľ˚ b iff αa ` p1 ´ αqc ľ

αb ` p1 ´ αqc for all α P r0, 1s and c P r0, 1sS . By well-known arguments (Ghirardato

et al., 2004; Gilboa et al., 2010), there exists a closed convex set of probability measures

C Ď ∆ such that a ľ˚ b iff:

ÿ

sPS

ppsqapsq ěÿ

sPS

ppsqbpsq @p P C

For each z P rm,M s, let δ˚z “ vpδzq. Note that, by construction, Ipδ˚z q “ z. Note

that δ˚z need not be a constant function; however, by A13, for each a P r0, 1sS , there exists

z P rm,M s such that a „ δ˚z . By the definition of ľ˚, ifř

sPS ppsqapsq ěř

sPS ppsqδ˚z psq

for all p P C, then a ľ δ˚z . On the other hand, since ľ satisfies Negative Certainty In-

dependence, ifř

sPS ppsqapsq ăř

sPS ppsqδ˚z psq for some p P C, then it is not the case

that αa ` p1 ´ αqc ľ αδ˚z ` p1 ´ αqc for all α P r0, 1s and c P r0, 1sS , so, by Neg-

ative Certainty Independence, a ń δ˚z . Noting thatř

sPS ppsqapsq ěř

sPS ppsqδ˚z psq

for all p P C if and only if v´1p p

ř

sPS ppsqapsqq ě z for all p P C, it follows that

a „ δ˚z iff minpPC v´1p p

ř

sPS ppsqapsqq “ z. Since Ipδ˚z q “ z, it follows that Ipaq “

minpPC v´1p p

ř

sPS ppsqapsqq, establishing the representation.

Proposition 11. When X “ ∆pIq and under Weak Order, Topological Continuity and

Weak Monetary Monotonicity, Monotonicity implies State Consistency.

47


Proof. Suppose lsh ľ l1sh for some l, l1 P ∆pIq, non-null s P S and h P A. By Topological

Continuity, there exist z, w P I such that l „ δz and l1 „ δw. By Monotonicity, lsh „

pδzqsh and l1sh „ pδwqsh. By Weak Monetary Monotonicity, since lsh ľ l1sh, z ě w. By

Monotonicity again, for any h1 P A, lsh1 „ pδzqsh1 and l1sh

1 „ pδwqsh1. It follows from

Weak Monetary Monotonicity (and Weak Order) that lsh1 „ pδzqsh1 ľ pδwqsh1 „ l1sh

1, as

required.


Proof of Proposition 8. By Proposition 6 and the assumption of identical multi-utilities

(which implies that they have the same best acts), α1 ď α2, whence it follows, by the

special form of α in the case of maxmin EU preferences (Section 4.3), that C2 Ď C1. Since

e˚2 is 2’s optimal expenditure, it maximises V , where

(15) V “ minpPC2

pps1qV1peq ` p1´ pps1qqV2peq

with V1peq “ minuPυ21 upc1 ´ e, p1 ´ σeq and V2peq “ minuPυ22 upc

2 ` δe, p2 ` τeq. Hence,

taking the first order conditions, we have that, if V1 and V2 are differentiable at e˚2 :

(16) q˚ps1qV1

1pe˚2q ` p1´ q

˚ps1qqV

12pe

˚2q “ 0

where q˚ P C2 supports V at e˚2 . It follows that q˚ is such that q˚ps1q “ maxpPC2 pps1q if

V1pe˚2q ă V2pe

˚2q, q

˚ps1q “ minpPC2 pps1q if V1pe˚2q ą V2pe

˚2q and q˚ can take any value in

C2 otherwise. Note moreover that, by assumption, V 11peq ď 0 for all e P r0, Es, V 12pe˚2q ě 0.

1’s optimisation problem is to maximise V , defined as in (15) with C2 replaced by C1. If

V1 and V2 are differentiable at e˚2 , the first order conditions are as in (16), with a supporting

probability measure from C1. Since C1 Ě C2, we have the following consequences for the

various cases:

• if V1pe˚2q ă V2pe

˚2q, then arg minpPC1 pps1qV1pe

˚2q ` p1 ´ pps1qqV2pe

˚2q “ q with

qps1q ě q˚ps1q. Since V 11pe˚2q ď 0 and V 12pe

˚2q ě 0,

qps1qV1

1pe˚2q ` p1´ qps1qqV

12pe

˚2q ď 0

48


and by the concavity of this functional,

e˚1 ď e˚2

• if V1pe˚2q ą V2pe

˚2q, then arg minpPC1 pps1qV1pe

˚2q ` p1 ´ pps1qqV2pe

˚2q “ q with

qps1q ď q˚ps1q. Since V 11pe˚2q ď 0 and V 12pe

˚2q ě 0,

qps1qV1

1pe˚2q ` p1´ qps1qqV

12pe

˚2q ě 0

and by the concavity of this functional,

e˚1 ě e˚2

• if V1pe˚2q “ V2pe

˚2q, then the first order condition for 1 follows immediately from (16)

and e˚1 “ e˚2 .

This establishes the result if V1 and V2 are differentiable at e˚2 . If not, then, by standard

results in convex analysis (Rockafellar, 1970, Section 27), it follows from the optimality of

e˚2 that 0 P BV pe˚2q, so the equivalent of (16) holds with derivatives of V1 and V2 replaced

by supergradients. Consider any sequence of ei Ñ e˚2 such that, for all i, V1 and V2 are

differentiable at ei and q˚i supports V at ei. Suppose that V1pe˚2q ă V2pe

˚2q and consider a

sequence such that V1peiq ă V2peiq for all i (such a sequence exists by continuity). By the

reasoning above, qips1qV1

1peiq ` p1 ´ qips1qqV1

2peiq ď q˚i ps1qV1

1peiq ` p1 ´ q˚i ps1qqV1

2peiq

with qi “ arg minpPC1 pps1qV1peiq`p1´pps1qqV2peiq, for every i and every such sequence.

It follows from Rockafellar (1970, Theorem 25.6) that either 0 P BV pe˚2q or d ă 0 for all

d P BV pe˚2q. It follows from the concavity of V that e˚1 ď e˚2 . The other cases are treated

similarly.

Proof of Proposition 9. The optimisation problem is to maximise

(17) V peq “ minpPC

pps1qV1peq ` p1´ pps1qqV2peq

where V1peq “ minuPU upc1é, p1´σeq and V2peq “ minuPU upc

2è, p2`σeq. Note that,

by the symmetry of choice problem, for any u P U and for p 12P ∆ where p 1

2ps1q “ 0.5:

49


p 12ps1q

Bupc1 ´ e, p1 ´ σeq

BepE

2q ` p1´ p 1

2ps1qq

Bupc2 ` e, p2 ` σeq

BepE

2q “ 0

Since p 12P C by symmetric beliefs, it follows from standard results (Rockafellar, 1970)

and the concavity of V that E2

is an optimum. So 1’s optimal expenditure is E2

, resulting in

a consumption-population profile which is constant across states.

As for the final claim, consider U1 : r0, Csˆ r0, P s Ñ < defined as follows: U1pc, pq “`

cC

˘α ` p

P

˘1´α, for α P p0, 1q. This function is linear on mixtures of p0, 0q, pC,P q. More-

over, it is differentiable, continuous and concave. Similarly, define U2 : r0, Csˆr0, P s Ñ <by: U2pc, pq “

`

cC

˘δ ` p

P

˘1´δ, for δ P p0, 1q; it too is concave, differentiable and linear on

the same mixtures.

It follows (see the proof of Theorem 1) that there exists

pp0, 0q, p0, 0qq, ppC,P q, pC,P qq-precise closed convex sets of utility functions U ,U 1 with

U1pc, pq “ minuPU upc, pq and U2pc, pq “ minuPU 1 upc, pq for all pc, pq P r0, Cs ˆ r0, P s.

Let υ be the state-dependent multi-utility with υps1q “ U and υps2q “ U 1, and let C be a

symmetric set of priors with mintpps1q | p P Cu “ mintpps2q | p P Cu “ q. We consider a

decision maker represented according to (1) with C and υ.

Since U1 and U2 are differentiable, there exists a single u1 P υ1ps1q and a single

u2 P υ2ps1q supporting the representing functional at e “ E

2; moreover, for every p P C

supporting the functional at this point:

pps1qdu1pc

1 ´ e, p1 ´ σeq

depE

2q ` p1´ pps1qq

du2pc2 ` e, p2 ` σeq

depE

2q

“pps1qdU1pc

1 ´ e, p1 ´ σeq

depE

2q ` p1´ pps1qq

dU2pc2 ` e, p2 ` σeq

depE

2q

“pps1q

¨

˝´α

C

˜

c1 ´ E2

C

¸α´1 ˜

p1 ´ σE2

P

¸1´α

´p1´ αqσ

P

˜

c1 ´ E2

C

¸α˜

p1 ´ σE2

P

¸´α˛

‚

`p1´ pps1qq

¨

˝

δ

C

˜

c2 ` E2

C

¸δ´1 ˜

p2 ` σE2

P

¸1´δ

`p1´ δqσ

P

˜

c2 ` E2

C

¸α˜

p2 ` σE2

P

¸´δ˛

‚

“pps1qXα

ˆ

´α

CX´1

´p1´ αqσ

P

˙

` p1´ pps1qqXδ

ˆ

δ

CX´1

`p1´ δqσ

P

˙

50


where X “

´

c2È2

C

¯´

p2`σE2

P

¯´1

“

´

c1É2

C

¯´

p1´σE2

P

¯´1

.

It follows that, for every triple α, δ and q such that

Xδ

ˆ

δ

CX´1

`p1´ δqσ

P

˙

R

„

q

1´ qXα

ˆ

α

CX´1

`p1´ αqσ

P

˙

,1´ q

qXα

ˆ

α

CX´1

`p1´ αqσ

P

˙

E2

is not an optimum. Whenever X ‰ 1 or 1XC

‰ σP

such triples exist: for instance, q “ 12

and any δ ‰ α satisfy this inequality. Hence, except for these special cases, there are

policy makers (characterised, under this parametrisation, by α, δ, and q) for whom E2

is

not an optimum; since no other values of e yield constant consumption-population profiles,

this establishes the result.

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