Risk Neutrality Incompleteness and Degenerate Indifference



Risk Neutrality, Incompleteness and

Degenerate Indifference

Georgios Gerasimou

School of Economics and Finance

Online Discussion Paper Series

issn 2055-303X


info: econ@st-andrews.ac.uk

School of Economics and Finance Discussion Paper No. 1609

23 Jun 2016

JEL Classification: D01, D03, D11

Keywords: Bewley preferences; incomplete preferences; risk neutrality;

degenerate indifference

Risk Neutrality, Incompleteness and Degenerate


Georgios Gerasimou ∗†

June 23, 2016


We show that strictly monotonic and risk-neutral Bewley preferences on the set of

purely uncertain monetary acts over a finite state space are associated with a degenerate

indifference relation. This conclusion is valid irrespective of whether strict or weak

preferences are taken as primitive. In the latter case, it follows from this result that such

a decision maker is indifferent between distinct acts only if her preferences are complete.

Keywords: Bewley preferences; incomplete preferences; risk neutrality; degenerate indifference.

JEL Classification: D01, D03, D11

∗ School of Economics & Finance, University of St Andrews. Email: gg26@st-andrews.ac.uk

† I thank Igor Rivin for a useful suggestion.

1 Introduction

In Bewley’s (2002) model of preferences over monetary acts the primitive is an incomplete strict

preference relation that is represented by a set of priors over the states of the world together

with the decision rule whereby one act is preferred to another if and only if its expected value is

strictly higher under every prior in that set. Rigotti and Shannon (2005), Kajii and Ui (2009)

and Ma (2015) are papers in general equilibrium theory under Knightian uncertainty where agents

are assumed to have Bewley preferences that may be of this type, 1 while Lopomo, Rigotti, and

Shannon (2011) and Chiesa, Micali, and Zhu (2015) are papers in contract theory and mechanism

design, respectively, where risk-neutral Bewley preferences are assumed directly.

Following Ghirardato, Maccheroni, Marinacci, and Siniscalchi (2003) and Gilboa, Maccheroni,

Marinacci, and Schmeidler (2010), an alternative approach to modelling preferences over general

Savage (1954) or Anscombe-Aumann (1963) acts that are defined by Bewley’s unanimity rule is

to take a possibly incomplete weak preference relation as the primitive, and let it be determined

by weak rather than strict dominance with respect to every prior. Each of these approaches is

associated with certain advantages and disadvantages, briefly reviewed below.

A thing in common for these two approaches that is important for the purposes of this note

is that the agent is assumed to be indifferent between two acts if and only if their expected

value/expected utility is the same under every prior. We show, in particular, that when risk-neutral

Bewley preferences of either type are defined on the linear space of purely uncertain monetary acts

over a finite state space, and also satisfy strict monotonicity (as in Bewley’s (2002) Theorem 1, for

instance), then the indifference relation associated with them is degenerate: there exist no distinct

acts between which the decision maker is indifferent. An important implication of this result in

the case where weak preferences are primitive is that strict monotonicity is incompatible with

indifference non-triviality unless preferences are complete.

2 Degenerate Indifference: A Sufficient Condition

The choice domain is the set of all acts f : S → R, where S = {1, . . . , n} is the finite state space.

This domain coincides with R n and will be denoted by X. We will write 0 for the zero vector in

X. Moreover, for i ≤ n, we let e i stand for the element of X whose i-th coordinate is 1 and all

other coordinates are zero. For x, y ∈ X, we write x ≥ y whenever x i ≥ y i for all i ≤ n.

The set of all probability measures on S is denoted by ∆(S). As in Rigotti and Shannon

(2005), for P ⊆ ∆(S) we define the interior of P relative to the set {x ∈ R n : ∑ n

i=1 x i = 1}

by rint(P ) = {p ∈ ∆(S) : p i > 0 for all i ≤ n} if P = ∆(S), and rint(P ) = {p ∈ ∆(S) :

B ɛ (p) ∩ rint(∆(S)) ⊂ P for some ɛ > 0} if P ⊂ ∆(S), where B ɛ (p) is the Euclidean open ball

around p. When we make a reference to a set P ⊆ ∆(S) we will use the terms interior and relative

interior interchangeably.

The following result is key in what follows.

Proposition 1.

Let ∼ be an equivalence relation on X and P ⊆ ∆(S) a compact, convex set with nonempty interior

such that

x ∼ y ⇐⇒ px = py for all p ∈ P (1)

1 Kajii and Ui (2009), in particular, showed that when all agents’ Bewley preferences are risk-neutral, an allocation is interim efficient

if and only if there exists a common element in the agents’ sets of priors.



x ∼ y ⇐⇒ x = y (2)

The proof appears in the Appendix. In the case where n = 2 this result can be easily illustrated

with an example: Suppose x ∼ y for x, y ∈ R 2 . It follows from (1) and the nonempty-interior

assumption that there exist distinct probability measures (p, 1 − p) and (q, 1 − q) in P such that

px 1 + (1 − p)x 2 = py 1 + (1 − p)y 2

qx 1 + (1 − q)x 2 = qy 1 + (1 − q)y 2

Moreover, since P has a nonempty interior, (p, 1 − p) and (q, 1 − q) can be chosen to be strictly

positive. In this case, dividing the first equation by p and the second by q, and then subtracting

the former from the latter yields x 2 = y 2 and x 1 = y 1 . The proof of the general case generalizes this

argument mainly by showing that there exists an n-tuple of strictly positive probability measures

that span X ≡ R n .

3 Strict Bewley Preferences

As in Bewley (2002), we first explore the implications of Proposition 1 in the case where the primitive

is a strict preference relation ≻ on X. The following four axioms are standard in this context

and were also employed in the statement of Bewley’s Theorem 1, which is of direct relevance here.

Strict Partial Order

≻ is asymmetric and transitive.


For all x, y, z ∈ X and all α ∈ (0, 1), x ≻ y implies αx + (1 − α)z ≻ αy + (1 − α)z.

Strict Monotonicity

For all x, y ∈ X, x ≥ y and x ≠ y implies x ≻ y.

Upper Hemicontinuity

For all x ∈ X, the set U ≻ (x) := {y ∈ X : y ≻ x} is open.

We restate Bewley’s Theorem 1 below in a way that reflects the indifference-degeneracy implications

of Proposition 1 and also allows for one additional clarification to be made.

Corollary 2.

A binary relation ≻ on X is a Strict Partial Order that satisfies Independence, Strict Monotonicity

and Upper Hemicontinuity if and only if there exists a compact, convex set P ⊆ rint(∆(S)) such

that, for all x, y ∈ X,

x ≻ y ⇐⇒ px > py for all p ∈ P (3)

x = y ⇐⇒ px = py for all p ∈ P (4)

Although Bewley (2002) did not assume an indifference relation as part of the model’s primitives,

he noted that once a set of priors P that represents ≻ in the sense of (3) is known to exist, an


indifference relation ∼ on X could be defined by x ∼ y ⇔ px = py for all p ∈ P . Driven by

Proposition 1, part (4) of this result clarifies that, under the conditions assumed in Bewley’s

(2002) Theorem 1, this indifference relation is actually degenerate. Moreover, the statement of

Corollary 2 also clarifies that the relevant conditions in Bewley’s (2002) Theorem 1 are not merely

sufficient but also necessary for ≻ to be represented as in (3).

We note, finally, that when a strict relation ≻ is assumed to be the primitive in a model of

incomplete preferences, Galaabaatar and Karni (2013) suggested to define an indifference relation

∼ from ≻ by letting x ∼ y if and only if U ≻ (x) = U ≻ (y). 2 Since this ∼ is an equivalence relation,

it follows from Proposition 1 that this too is degenerate in the present context whenever the set

P that represents ≻ has a nonempty interior.

4 Weak Bewley Preferences

A conceptual difficulty with the strict-preference version of Bewley’s unanimity model is that it

does not specify a relevant “mild preference” label for the cases where all priors weakly favour an act

x over another act y and some but not all strictly favour x. An approach that addresses this issue

was followed by Ghirardato, Maccheroni, Marinacci, and Siniscalchi (2003), Gilboa, Maccheroni,

Marinacci, and Schmeidler (2010) and Ok, Ortoleva, and Riella (2012), who took as the primitive

a (possibly incomplete) weak preference relation . Unlike Bewley’s (2002) framework, under this

approach the relations ≻ and ∼ that define strict preference and indifference are derived from

as its asymmetric and symmetric parts in the usual way. While this is an advantage of the weakover

the strict-preference approach, a well-known difficulty that arises here is that even though

the relation is continuous, its strict part ≻ cannot be unless is complete (see below).

To study the analogue of Corollary 2 in a weak-preference framework we state a few axioms

that are particular to such a relation.


is reflexive and transitive.


There exist x, y ∈ X such that x ̸ y and y ̸ x.

Restricted Hemicontinuity

The set U (0) := {y ∈ X : y 0} is closed.

The former two axioms ensure that is a transitive and incomplete weak preference relation.

It will be clear in the statement of the result below why requiring incompleteness explicitly is

important for our purposes. Although the third axiom is weaker, in principle, than the standard

axiom of closed-graph continuity, it turns out that in the present context the two are equivalent. 3

As far as the Independence axiom for is concerned, it coincides with the one that was stated in

the previous section provided that ≻ is replaced by .

2 Bewley (2002) called two acts x and y equivalent when, in addition, to the above condition of Galaabaatar and Karni (2013) that

requires equality between the strict upper contour sets of x and y, their strict lower contour sets are also equal.

3 The reader is referred to Theorem 1 in Gerasimou (2015) and p. 156 in Ghirardato, Maccheroni, and Marinacci (2004) for more

details on this.


Corollary 3.

A binary relation on a X is a Preorder that satisfies Incompleteness, Independence, Strict

Monotonicity and Restricted Hemicontinuity if and only if there exists a compact, convex set P ⊆

∆(S) with nonempty interior such that, for all x, y ∈ X

x y ⇐⇒ px ≥ py for all p ∈ P (5)

x = y ⇐⇒ px = py for all p ∈ P (6)


(x y ⇔ x ≥ y) ⇐⇒ P = ∆(S) (7)

It is an implication of Proposition A.1 in Ghirardato, Maccheroni, and Marinacci (2004) that

the stated axioms are sufficient for the existence of a set P ⊆ ∆(S) that represents as in (5).

It is easy to verify that Strict Monotonicity implies a nonempty interior for P . In light of this

fact, the indifference-degeneracy implication (6) again follows from Proposition 1. Given (6), it is

easily seen that (7) is true. This equivalence shows that, in the present setting, Bewley preferences

coincide with the usual partial ordering if and only if P contains all possible priors, hence that the

model becomes uninformative in this special case.

Importantly, Corollary 3 can also be stated as a preference-completion theorem. To this end,

we write explicitly as an axiom the requirement that there exist two distinct acts between which

the decision maker is indifferent.

Nontrivial Indifference

There exist distinct x, y ∈ X such that x ∼ y.

Corollary 4.

If is a Preorder on X that satisfies Independence, Strict Monotonicity, Nontrivial Indifference

and Restricted Hemicontinuity, then is complete.

This result parallels the well-known and general preference-completion theorem that is due to

Schmeidler (1971). The latter provides natural topological conditions on the preference domain

(i.e. connectedness) as well on the relation and its asymmetric part ≻ (i.e. hemicontinuity)

that are sufficient for to be complete. Although the domain in Corollary 4 is a special case in

the class of domains that Schmeidler’s theorem allows for, we note that, unlike the latter result,

almost every condition in the statement of Corollary 4 is a preference postulate with a behavioral



Proof of Proposition 1.

We will first show that there is a set {q 1 , . . . , q n } of n linearly independent elements in rint(P ).

Let p ∈ rint(P ). For ɛ > 0, let B ɛ (p) := {v ∈ R n ++ : d(v, p) < ɛ}, where d denotes Euclidean

distance. For all i ≤ n, define q i ∈ R n ++ by

q i := p + ɛ 2 ei .


Observe that the set {q 1 , . . . , q n } ⊂ R n ++ consists of n linearly independent vectors. Hence, it is a

basis of R n . Now define q i by

q i :=

q i

||q i || 1

where ||q i || 1 := ∑ n

j=1 qj i . Observe also that, for all i, j ≤ n, ||q i|| 1 = ||q j || 1 = 1 + ɛ 2 . Define

β := 1

1 + ɛ .


Notice that since {q 1 , . . . , q n } is a basis of R n and q i ≡ βq i for all i ≤ n, the set {q 1 , . . . , q n } is also

a basis of R n . Moreover, by construction, q i ∈ B ɛ (p). Since p ∈ rint(P ), choosing ɛ > 0 sufficiently

small ensures that q i ∈ rint(P ) for all i ≤ n.

Now consider x, y ∈ X and suppose x ∼ y. Let z := x − y. To establish (2) it suffices to show

that if pz = 0 for all p ∈ P , then z = 0. To this end, suppose pz = 0 for all p ∈ P . It holds that

( n∑


q i z = 0, for all i = 1, . . . , n (8)


λ i q i ≠ 0, for all (λ 1 , . . . , λ n ) ∈ R n \ {0} (9)


λ i q i


z = 0, for all (λ 1 , . . . , λ n ) ∈ R n \ {0} (10)

where (8) is true by assumption, (9) follows from the fact that {q 1 , . . . , q n } is a basis of R n , and

(10) is implied by (8). In particular, it follows from (9) and (10) that z = 0, which is equivalent

to x = y.


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