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
http://ideas.repec.org/s/san/wpecon.html
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
Indifference
Georgios Gerasimou ∗†
June 23, 2016
Abstract
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.
1

Then,
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.
Independence
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
2

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.
Preorder
is reflexive and transitive.
Incompleteness
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.
3

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)
Moreover,
(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
content.
Appendix
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 .
4

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 .
2
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∑
i=1
q i z = 0, for all i = 1, . . . , n (8)
n∑
λ i q i ≠ 0, for all (λ 1 , . . . , λ n ) ∈ R n \ {0} (9)
i=1
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.
References
Anscombe, F. J., and R. J. Aumann (1963): “A Definition of Subjective Probability,” Annals
of Mathematical Statistics, 14, 477–482.
Bewley, T. F. (2002): “Knightian Decision Theory. Part I,” Decisions in Economics and Finance,
25, 79–110.
Chiesa, A., S. Micali, and Z. A. Zhu (2015): “Knightian Analysis of the Vickrey Mechanism,”
Econometrica, 83, 1727–1754.
Galaabaatar, T., and E. Karni (2013): “Subjective Expected Utility with Incomplete Preferences,”
Econometrica, 81, 255–284.
Gerasimou, G. (2015): “(Hemi)Continuity of Additive Preference Preorders,” Journal of Mathematical
Economics, 58, 79–81.
Ghirardato, P., F. Maccheroni, and M. Marinacci (2004): “Differentiating Ambiguity
and Ambiguity Attitude,” Journal of Economic Theory, 118, 133–173.
Ghirardato, P., F. Maccheroni, M. Marinacci, and M. Siniscalchi (2003): “A Subjective
Spin on Roulette Wheels,” Econometrica, 71, 1897–1908.
5

Powered by TCPDF (www.tcpdf.org)
Gilboa, I., F. Maccheroni, M. Marinacci, and D. Schmeidler (2010): “Objective and
Subjective Rationality in a Multiple Prior Model,” Econometrica, 78, 755–770.
Kajii, A., and T. Ui (2009): “Interim Efficient Allocations Under Uncertainty,” Journal of
Economic Theory, 144, 337–353.
Lopomo, G., L. Rigotti, and C. Shannon (2011): “Knightian Uncertainty and Moral Hazard,”
Journal of Economic Theory, 146, 1148–1172.
Ma, W. (2015): “The Existence and Efficiency of General Equilibrium with Incomplete Markets
under Knightian Uncertainty,” Economics Letters, 134, 78–81.
Ok, E., P. Ortoleva, and G. Riella (2012): “Incomplete Preferences under Uncertainty:
Indecisiveness in Beliefs Versus Tastes,” Econometrica, 80, 1791–1808.
Rigotti, L., and C. Shannon (2005): “Uncertainty and Risk in Financial Markets,” Econometrica,
73, 203–243.
Savage, L. J. (1954): The Foundations of Statistics. New York: Wiley.
Schmeidler, D. (1971): “A Condition for the Completeness of Partial Preference Relations,”
Econometrica, 39, 403–404.
6