Time and No Lotteries: A Simple
Axiomatization of Maxmin Expected Utility∗
Asen Kochov†
February 15, 2012
Abstract
The paper provides a simple axiomatization of the maxmin expected
utility model in a multi-period Savage-style framework of purely subjective uncertainty.
∗
I thank Paulo Barelli and Larry Epstein for helpful comments.
Department of Economics, University of Rochester, Rochester, NY 14627. E-mail:
asen.kochov@rochester.edu.
†
1
1
Introduction
The prevalent approach to modeling choice under uncertainty was introduced
by Savage [19]. In this approach uncertainty is represented by a space of
states of the world and actions take the form of mappings from states into
outcomes. This paper provides a simple axiomatization of maxmin expected
utility in a multi-period specification of the Savage framework. There are a
number of reasons why such an axiomatization is useful. Most importantly,
it gives a sharp understanding of the empirical content of the maxmin model.
The seminal paper on the subject, Gilboa and Schmeidler [12], assumes that
outcomes take the form of lotteries with exogenously specified probabilities.
This specification of the domain was introduced by Anscombe and Aumann
[3] and will be henceforth referred to as the AA framework. The AA framework is tractable and simplifies the analysis but is rarely used in applications.
Moreover, it is now clear that stark differences in predictions may arise if one
substitutes the AA for the Savage framework. The fact is evidenced by the
behavior of the multiplier model recently introduced in economics by Anderson et al. [2].
Most applications of the maxmin, or multiple-prior, model in finance and
macroeconomics involve intertemporal decisions. The key feature of the
model, which has been termed ‘preference for hedging’, interacts nontrivially
with other important aspects of intertemporal behavior such as stationarity.
This interaction is exploited in many applications. We refer the reader to
the recent survey by Epstein and Schneider [8]. Characterizing the maxmin
model in an intertemporal setting brings this interaction to the fore and
delineates it precisely.
Several other papers have recognized the first problem outlined above and
characterized the multiple-prior model in a general Savage framework without time or lotteries, e.g., [1, 4, 11]. These characterizations typically involve
many axioms or axioms of technical nature that are not directly verifiable.
The axiomatization provided here is arguably as simple as the original one by
Gilboa and Schmeidler [12]. In this sense, adopting an intertemporal framework turns out to be not only relevant to applied work but also amenable to
axiomatic analysis.
Even though the focus of this paper is on the multiple-prior model, the arguments can be easily adapted to characterize other models of ambiguity such
as the Choquet model of Schmeidler [20], and the multiplier model mentioned
2
above and recently axiomatized in an Anscombe-Aumann setting by Strzalecki [21]. The derivation of subjective expected utility is also technically
easier in the present multi-period setting and requires no restrictions on the
cardinality of the state space. This point has been made by Gorman [14] in a
little known paper. It is the subject of ongoing work to see if the setting can
be used to study the more general models of ambiguity aversion developed by
Maccheroni et al. [18] and Cerreia-Vioglio et al. [5]. It is nonetheless reasonable to conclude that the multi-period setting provides a viable alternative
to the Anscombe-Aumann setup. It is tractable and more relevant to applied
work.
2
Domain
Time is discrete and varies over an infinite horizon T := {0, 1, 2, ...}. The
information structure is described by a filtered space (Ω, {Ft }t ) where Ω is an
arbitrary set of states of the world and {Ft }t =: F is an increasing sequence
of algebras such that F0 = {Ω, ∅}. If Ω is finite, one may think of the
information structure as an event tree whose nodes correspond to time-event
pairs (t, ω) ∈ T × Ω.
In every period, outcomes lie in a connected, separable, compact space X.
An act is an X-valued, F-adapted processes. Each such process is the multiperiod counterpart of a Savage act; it has the form h = (ht ) where ht : Ω → X
is Ft -measurable for every t.
Let A0 be a subalgebra of ∪t Ft . An act h := (ht )t is A0 -adapted if ht is
A0 -measurable for every t. An act h is simple if h is A0 -adapted for some
finitely-generated algebra A0 ⊂ ∪t Ft . The domain of choice is the set H of
all simple acts. The domain excludes acts whose outcomes may depend on
the realization of tail events. Such events do not take place in finite time and
are therefore unobservable.1
An act is deterministic if its outcomes do not depend on the state of the
world. The space of deterministic acts may be identified with the space
X ∞ . Generic elements of X ∞ are denoted by bold-faced letters, e.g, x =
1
A tail event is an event in σ(∪t Ft ) \ ∪t Ft . A preference on the set of simple acts can
be extended to the set of all acts using the techniques developed in Epstein and Wang
[10].
3
(x0 , x1 , ...). Other important subsets of H include the subspaces X t+1 ×
H where t ∈ T . Generic elements of X t+1 × H will be denoted as tuples
(x0 , x1 , ..., xt , h).
3
Axioms
The primitive of the model is a preference relation over the space of simple
acts H.
Order The preference relation is complete and transitive.
If ∪t Ft is finitely generated, it is standard to require that preference be
continuous in the product topology. This is no longer the case if there are
infinitely many events. Consider a sequence of events An that increases
monotonically to Ω. Continuity in the product topology would require that
the certainty equivalent of a bet on An changes continuously as An % Ω. In
the context of ambiguity, a discontinuity may arise since Ω is unambiguous
whereas all of the events An may be ambiguous.2 The next axiom imposes
continuity on every subspace of acts that is adapted with respect to a finitelygenerated algebra.
Continuity For every finitely generated subalgebra A0 of ∪t Ft , the restriction of to the subspace of A0 -adapted acts is continuous in the respective
product topology.
For every act h and state ω, let h(ω) := (h0 (ω), h1 (ω), ...)) ∈ X ∞ denote
the deterministic act corresponding to the sample path of h in state ω. The
next axiom is a monotonicity requirement. It says that an act h is preferred
to h0 whenever h(ω) is preferred to h0 (ω) for every ω. The axiom is less
innocuous in the the present multi-period setting than it is in the atemporal
model of Gilboa and Schmeidler [12]. In particular, it presumes indifference
to the timing of uncertainty as made clear by the seminal work of Kreps and
Porteus [17].
2
See the related discussion in Epstein and Seo [9, p.347].
4
Monotonicity For all h, h0 ∈ H, h(ω) h0 (ω) for every ω ∈ Ω implies that
h h0 .
The next two assumptions are the core of the present axiomatization. The
first axiom captures a form of intertemporal hedging. It says that combining
some uncertain bet, ht , paying in period t, with another uncertain bet, ht+1 ,
paying in period t + 1, is at least as good as combining their respective
0
certainty equivalents zt , zt+1
. The intuition is that the two uncertain bets, ht
and ht+1 may hedge one another eliminating any ambiguity about the overall
outcome. In contrast, combining their certainty equivalents produces no such
complementarity.
Uncertainty Aversion For every t, h ∈ H, z, z 0 ∈ X and x ∈ X ∞ ,
(x−t , ht ) ∼ (x−t , z) and (x−(t+1) , ht+1 ) ∼ (x−(t+1) , z 0 )
⇒
(x−t,t+1 , ht , ht+1 ) (x−t,t+1 , z, z 0 )
Intertemporal hedging generates a complementarity between bets in different
time periods and may thus lead to nonstationary behavior. This interplay
between hedging and stationarity is delineated by the next axiom. First,
the axiom requires that ht and ht+1 do not hedge one another if one of
them is constant. In all these instances, the axiom requires that behavior be
stationary.
Certainty Independence For every t, h, h0 ∈ H and x0 , ..., xt ∈ X:
h h0 ⇔ (x0 , ..., xt , h) (x0 , ..., xt , h0 )
The next axiom is arguably restrictive but common in modeling intertemporal behavor. Since it deals with deterministic act only and the primary
focus is on ambiguity, the axiom is a natural starting point. The axiom rules
out complementarities among the utility levels of outcomes in different time
periods. Remarkably, Section 4.2 shows that the axiom is redundant except
in certain nongeneric cases.
Time Separability For every x, x0 , y, y 0 ∈ X and z, z̃ ∈ X ∞ ,
(x, y, z) (x0 , y 0 , z) ⇔ (x, y, z̃) (x0 , y 0 , z̃).
5
The final axiom is a weak nontriviality condition.
Nontriviality There exist x, y ∈ X and z ∈ X ∞ such that (x, z) (y, z).
4
Result and Discussion
Endow the space of finitely additive probability measures on ∪t Ft with the
weak∗ topology, i.e., the topology generated by all real-valued bounded measurable functions on Ω.
Theorem 1 The preference satisfies Order, Continuity, Monotonicity,
Certainty Independence, Uncertainty Aversion, Time Separability and Nontriviality if and only if there exist a discount factor β ∈ (0, 1), a nonconstant
continuous function u : X → R and a nonempty weak∗ -closed convex set P
of finitely additive probability measures on ∪t Ft such that:
Z X
Z X
0
t
h h ⇔ minp∈P
β u(ht ) dp ≥ minp∈P
β t u(h0t ) dp. (4.1)
Ω
t
Ω
t
Moreover, P and β are unique and u is unique up to positive affine transformations.
The simplicity of the axiomatization given in Theorem 1 depends critically
on the time-additivity of the ranking of deterministic acts. The framework,
however, leaves the door open for more general yet well-structured preferences
on X ∞ . For example, the stationary rankings studied in Koopmans [15]
have a number of separability properties that are jointly weaker but akin
to time additivity. It is an interesting question if those properties can be
exploited to generalize Theorem 1. Of course, a similar trade-off between
generality and simplicity arises in the context of the Gilboa and Schmeidler
[12] axiomatization in which the linearity of the ranking of objective lotteries
plays an analogous role.
Despite the obvious similarities, the proof of Theorem 1 is not a straightforward application of the arguments in [12]. A key property of the multipleprior functional is that it is positively homogenous.3 In an AA framework,
3
A functional I : V → R defined on a real vector space is positively homogenous if
I(αx) = αI(x) for every x ∈ V and every α > 0.
6
outcomes are objective lotteries and utility is linear in their probabilities.
By varying the latter, one obtains homogeneity with respect to any given
positive constant. In the present setting, the utility of deterministic acts is
additive across time periods and one may think of the discount factor β t as
the ‘probability’ of the period-t outcome. The problem is that β is fixed and
therefore homogeneity obtains in a limited sense only, that is, with respect
to powers of the fixed constant β. In the appendix, this property is called
β-homogeneity. The gist of the proof is to show that a β-homogenous, monotone, translation invariant and superadditive functional is in fact positively
homogenous.
4.1
Dynamic Extension
Even though the paper develops the static model only, the analysis can be
extended to a dynamic model of choice. In particular, it is possible to obtain
the prior-by-prior Bayes rule and the recursive utility formulation of Epstein
and Schneider [7] without assuming an AA framework. Because the necessary changes to their proof are minimal and can be easily deduced from
the proof of Theorem 1, it is enough to mention the primitives and additional axioms. As in [7], restrict the information structure so that every Ft
is finitely generated. To model dynamic choice, take as primitive a process
of conditional preferences {t,ω } and assume that each t,ω satisfies appropriate versions of the above axioms. The main additional axiom is Dynamic
Consistency which links together preferences at different nodes. The axiom
is standard and may be formulated in a manner identical to [7]. The same
applies to the other two axioms needed to derive the recursive formulation in
Epstein and Schneider [7], namely, Consequentialism and Conditional State
Independence.4
4.2
Essential Events
Say that an event A ∈ ∪t Ft is strongly essential if, for all {A, Ac }-adapted
acts h, h0 , h is strictly preferred to h0 whenever h(ω) h0 (ω) for all ω ∈ Ω
4
In [7], Consequentialism is called Conditional Preference whereas Conditional State
Independence is subsumed in an axiom called Risk Preference. The full force of the latter
is not needed here since it overlaps with Certainty Independence.
7
and the latter ranking is strict for some ω 0 . That strongly essential events
exist generically becomes evident if we consider the representation in (4.1).
In that case, an event A is strongly essential if and only if p(A) ∈ (0, 1) for all
p ∈ P . In dynamic settings and many applications, it is common to assume
that all events A ∈
/ {Ω, ∅} are strongly essential. The assumption is made to
avoid the problem of updating on null events. Remarkably, the next lemma
shows that Time Additivity is redundant whenever a strongly essential event
exists.
Lemma 2 If there exists a strongly essential event, Monotonicity, Continuity, Certainty Independence and Nontriviality imply Time Additivity.
It is important to note that strongly essential events play an indispensable
role in characterizations of the multiple-prior model which do not impose a
specific structure on the outcome space. See, for instance, [1, 4, 11]. In these
studies, the existence of a strongly essential event is used to ‘calibrate’ a
cardinal utility index over outcomes. In turn, cardinal utility is necessary to
derive a unique set of priors. Theorem 1 does not require the existence of a
strongly essential event. As in Gilboa and Schmeidler [12], a cardinal utility
index is derived from the richness of the outcome space, in conjunction with
an appropriate form of additivity.
5
5.1
Proofs
Theorem 1
Necessity of the axioms is straight-forward. Focus on sufficiency.
The first lemma is due to Koopmans [16]. It delivers an additively separable
utility function for the set of deterministic acts X ∞ .
Lemma 3 There exists a continuous function u : X → R and a discount
factor β ∈ (0, 1), such that the restriction of to X ∞ is represented by the
utility function:
X
U (x) := (1 − β)
β t u(xt )
(5.1)
t
Moreover, β is unique and u is unique up to positive affine transformations.
8
Proof. There are some minor differences in the the domain and axioms used
by Koopmans [16] to derive (5.1). First, he assumes that X is metrizable and
that is continuous in the uniform topology on X ∞ . Second, he assumes a
monotonicity axiom (P5) which is not made here. It is obvious that continuity
in the product topology, assumed here, implies uniform continuity. Moreover,
because X ∞ is compact in the product topology, there exist best and worst
elements in X ∞ . As Koopmans [16, p.84] notes, the existence of best and
worst elements implies his monotonicity postulate P5. Finally, Koopmans’
arguments are based on the results of [6, 13] for which metrizability is not
needed.
Because X is compact and connected, the range of u is a compact interval.
Rescaling appropriately, it is without loss of generality to set u(X) = [−1, 1].
Let x∗ ∈ X be an outcome such that u(x∗ ) = 0.
The product space X ∞ inherits the properties of X. In particular, X ∞ is
compact, separable and connected.
Let x and x be the best and worst elements in X ∞ , respectively. Fix an
arbitrary h ∈ H. It follows from Monotonicity that x h x. Because X ∞
is connected, a standard argument shows that there exists some xh ∈ X ∞
such that xh ∼ h. This allows us to extend the utility function U from X ∞
to the space of all acts H. For every h ∈ H, define Ũ (h) := U (xh ). The
transitivity of implies that Ũ : H → R is well-defined and represents the
preference relation .
For every act h ∈ H, define the utility act U ◦ h : Ω → R:
X
U ◦ h : ω 7→ (1 − β)
β t u(ht (ω))
(5.2)
t
Let U = {U ◦ h : h ∈ H} be the set of all utility acts. Define the functional
I : U → R:
I(U ◦ h) = Ũ (h)
Notice that, by Monotonicity, I is well-defined.
For any t, let Bt◦ be the set of simple Ft -measurable functions from Ω into
the closed interval [−β t , β t ].
Lemma 4 For every t, Bt◦ ⊂ U.
9
P
Proof. Let a = ki=1 αi 1Ai be the canonical representation of a ∈ Bt◦ and
note that, for every i = 1, .., k, | αβ ti | ≤ 1. Let xi be such that u(xi ) = αβ ti .
Define f (ω) = xi for every ω ∈ Ai , i ∈ {1, ..., k}. Define an act h by setting
hτ := x∗ for all τ < t and hτ = f for all τ ≥ t. It is easy to check that
U (h) = a.
Before stating the next lemma, notice that a ∈ Bt◦ implies that β k a ∈ Bt◦ for
every k, t.
Lemma 5 For every t, k ≥ 0, a ∈ Bt◦ , I(β k a) = β k I(a).
Proof. Fix some t and a ∈ Bt◦ . It suffices to show that the result holds for
k = 1. The rest of the proof follows by induction. By the previous lemma,
there exists an act h ∈ H such that U (h) = a. Let xh be a deterministic act
such that xh ∼ h. Consider the act (x∗ , h) and note that U (x∗ , h) = βa. By
Certainty Independence, (x∗ , h) ∼ (x∗ , xh ). Conclude that
I(βa) = I(U ◦ (x∗ , h)) = Ũ (x∗ , h) = U (x∗ , xh ) = βU (xh ) = β Ũ (h) =
= βI(U ◦ h) = βI(a)
Given any real number α, let α∗ : Ω → R be the constant function such that
α∗ (ω) = α for every ω.
Lemma 6 For every t, a, a + α∗ , α∗ ∈ Bt◦ , I(a + α∗ ) = I(a) + α.
Proof. By Lemma 5, it is without loss of generality to assume that α ∈
[β t − 1, 1 − β t ]. Let (x0 , ..., xt−1 ) be such that
(1 − β)
t−1
X
u(xτ ) = α
(5.3)
τ =0
and let h = (x∗ , .., x∗ , ht , ht+1 , ...) be such that a = U (h). If we define
h0 = (x0 , ..., xt−1 , ht , ht+1 , ...), it follows that U (h0 ) = a + α. Let xh be
such that
h = (x∗ , .., x∗ , ht , ht+1 , ...) ∼ (x∗ , .., x∗ , xh )
10
(5.4)
By Certainty Independence,
h0 = (x0 , ..., xt−1 , ht , ht+1 ) ∼ (x0 , ..., xt−1 , xh )
(5.5)
Conclude that
I(a + α∗ ) = U (x0 , ..., xt−1 , xh ) = α + U (x∗ , ..., x∗ , xh ) = α + Ũ (h) = α + I(a).
Lemma 7 For every t and a, b, a + b ∈ Bt◦ , I(a + b) ≥ I(a) + I(b).
Proof. By Lemma 5, it is without loss of generality to assume that a, b take
t+1
t+1
values in the interval [− β1−β , β1−β ]. If that is the case, there exist ht , ht+1
such that U (x∗−t , ht ) = a and U (x∗−(t+1) , ht+1 ) = b. Let za , zb ∈ X be such
that
(x∗−t , ht ) ∼ (x∗−t , za ) and (x∗−(t+1) , ht+1 ) ∼ (x∗−(t+1) , zb )
Then,
I(a + b) = U (x∗−t,(t+1) , ht , ht+1 ) ≥ U (x∗−t,(t+1) , za , zb )
= U (x∗−t , za ) + U (x∗−(t+1) , zb ) = I(a) + I(b),
where the inequality follows from Uncertainty Aversion.
Let B ◦ be the set of all simple, real-valued, ∪t Ft -measurable functions on Ω.
Lemma 8 U is an absorbing subset of B ◦ .
Proof. Fix some a ∈ B ◦ . Because a is simple, it must be Ft -measurable for
some t. Moreover, for k large enough, β k a(ω) ∈ [−β t , β t ] for every ω ∈ Ω.
Conclude that β k a ∈ U.
Fix any a ∈ B ◦ and let k, t be such that β k a ∈ Bt◦ . Define,
˜ := 1 I(β k a)
I(a)
βk
(5.6)
0
To see that I˜ is well-defined, let (k 0 , t0 ) be another pair such that β k a ∈ Bt◦0 .
0
If k 0 ≥ k, then β k a ∈ Bt◦ , If k > k 0 , then β k a ∈ Bt◦0 . It is therefore without
11
0
loss of generality to assume that t = t0 , k 0 > k and β k a, β k a ∈ Bt◦ . Lemma
5 then implies that:
0
0
0
I(β k a) = I(β k −k β k a) = β k −k I(β k a),
(5.7)
and hence that I˜ is well-defined.
The next lemma shows that the extension I˜ defined in (5.6) inherits the
properties of I established in Lemmas 5-7. Its proof is straightforward and
therefore omitted.
Lemma 9 The functional I˜ : B ◦ → R is β-homogenous, translation-invariant,
monotone and superadditive.
The next lemma is the main step of the argument.
Lemma 10 The functional I˜ is positively homogenous.
˜
Proof. By translation-invariance, it suffices to show that I(αa)
= 0 for
˜
all α > 0 and all a such that I(a) = 0. Suppose by way of contradiction
˜
that λ := I(αa)
6= 0. By translation-invariance, again, it is without loss
of generality to assume that λ < 0. Else, set b := αa − λ∗ and note that
˜ = 0 whereas I(
˜ 1 b) = − λ < 0. Moreover, by β-homogeneity, it is also
I(b)
α
α
without loss of generality to assume that α < 1.
˜ = 0 and λ := I(αa)
˜
So let α ∈ (0, 1) and let a be such that I(a)
< 0. Define
∗
˜
b := αa − λ and note that I(b) = 0. Let A be the positive ray through a:
A = {γa : γ ≥ 0}
(5.8)
Let V be the span of a, b and let B = {b0 ∈ V : b b0 }. Because a, and
hence b, is simple, the set B is open in V . Moreover, for every b0 ∈ B,
there exists a γ > 0, sufficiently small, such that b b0 + γ ∗ b0 . By
translation-invariance and monotonicity, conclude that:
˜ > I(b
˜ 0)
0 = I(b)
(5.9)
for all b0 ∈ B. By construction, αa belongs to A ∩ B. Since A is a ray and
B is convex and open, A ∩ B is an open line segment. Let (γ1 , γ2 ) be the
open interval such that γa ∈ A ∩ B for every γ ∈ (γ1 , γ2 ). If γ1 = 0, then
12
˜ k a) = β k I(a)
˜ =
β k a ∈ A ∩ B, for k sufficiently large. By β-homogeneity, I(β
˜
0 = I(b),
for every k, which contradicts (5.9). So suppose that γ1 > 0 and
fix any k such that γ1 > β k > 0. By superadditivity and β-homogeneity in
turn,
˜ k a + β t b) ≥ I(β
˜ k a) + I(β
˜ t b) = 0,
I(β
(5.10)
for every t. Applying superadditivity inductively, conclude that:
˜ k a + nβ t b) ≥ 0
I(β
(5.11)
A0 := {β k a + γb : γ ≥ 0}
(5.12)
for every n, t ∈ N. Let
be the ray through β k a parallel to b. By construction α > γ1 > β k which
k
implies that βα ∈ (0, 1). Recall that b = αa−λ∗ αa, since λ < 0. Conclude
that
βk
βk
b
(αa) + (1 − )b
α
α
(5.13)
and so A0 ∩ B is a nonempty open subset of A0 . It is not difficult to see that
{β k a + nβ t b : n ∈ N, t ∈ N}
(5.14)
is dense in the A0 . Conclude that b β k a + nβ t b for some n, t. But then
˜ > I(β
˜ k a + nβ t b)
0 = I(b)
(5.15)
contradicting (5.11).
5.2
Proof of Lemma 2
The result is an application of Gorman’s overlapping theorem [13]. Partition
the set of time periods into T 0 := {0, 1} and T 00 := {2, 3, ...}. Let A be a
strongly essential event and H(A) be the subset of {A, Ac }-adapted acts. To
recast our framework into Gorman’s setup, it is convenient to think of an act
h as a mapping from Ω × T into X.
13
Consider the following partition of Ω × T :
{A × T 0 , A × T 00 , Ac × T 0 , Ac × T 00 }
(5.16)
which we enumerate as {B1 , B2 , B3 , B4 }. Given this partition, we can identify
H(A) with the product space:
X 2 × X 2 × X ∞ × X ∞,
(5.17)
where X 2 is the restriction of H(A) to A × T 0 , etc. Accordingly, every act
h ∈ H(A) can be written as a vector h = (hB1 , hB2 , hB3 , hB4 ). Notice that
every element in the factorization (5.17) is a connected, separable space,
as needed to apply Gorman’s overlapping theorem. Because the event A is
strongly essential, it follows that
(hB1 , hB2 , hB3 , hB4 ) (ĥB1 , ĥB2 , hB3 , hB4 ) ⇔
(5.18)
(hB1 , hB2 , h0B3 , h0B4 )
(5.19)
(ĥB1 , ĥB2 , h0B3 , h0B4 )
for all h, h0 , ĥ ∈ H(A). In the language of Gorman, the set Y := B1 ∪ B2 ⊂
Ω × T is separable. It is not difficult to see that, by Stationarity, the set
Y 0 := B2 ∪ B4 is separable. If one more condition is satisfied, Gorman [13]
proved that the set difference Y \ Y 0 = B1 must be separable as well, which
is equivalent to Time Additivity.
The final condition is that for every h ∈ H(A), there exist h0 , h00 ∈ H(A)
such that
(hB1 , hB2 , hB3 , h0B4 ) (hB1 , hB2 , hB3 , h00B4 )
(5.20)
But this is easily seen to follow from Nontriviality, Stationarity and the fact
that A is strongly essential.
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