This note provides a behavioral characterization of mutually absolutely continuous multiple priors.Mutual absolute continuity, Multiple priors

Larry G. Epstein

Massimo Marinacci

English

Research Papers in Economics

Mutual Absolute Continuity of Multiple 
Priors 
Larry G. Epstein
Massimo Marinacci
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No.19, September 2006
www.carloalberto.orgMutual Absolute Continuity of Multiple Priors1
Larry G. Epstein Massimo Marinacci
August 2006
1Epstein is at Department of Economics, University of Rochester, USA,
lepn@troi.cc.rochester.edu; Marinacci is at Dipartimento di Statistica e Matematica Appli-
cata and Collegio Carlo Alberto, Università di Torino, Italy, massimo.marinacci@unito.it.
Epstein gratefully acknowledges the ﬁnancial support of the NSF (award SES-0611456) and
Marinacci that of the Ministero dell’Istruzione, dell’Università e della Ricerca.
c ° 2006 by Larry Epstein and Massimo Marinacci. Any opinions expressed here are those of
the authors and not those of the Fondazione Collegio Carlo Alberto.Abstract
This note provides a behavioral characterization of mutually absolutely continuous
multiple priors.
JEL classiﬁcation: D81
Keywords: Mutual absolute continuity, Multiple priors1 Introduction
Beginning with Epstein and Schneider [2], in recent years there has been a growing in-
terest in dynamic versions of the multiple-priors (MP) model of Gilboa and Schmeidler
[4], motivated by the importance of dynamic considerations in many economic appli-
cations. In these dynamic versions of the MP model a very convenient property that is
often assumed is mutual absolute continuity of the priors, that is, their mutual agree-
ment on which events are null. This agreement makes possible dynamically consistent
Bayesian updating of the priors, without having to deal with updating on events that
only some priors regard as null, something that greatly simpliﬁes the analysis of these
dynamic MP models.1
Here we provide a behavioral characterization of this agreement property of the
priors. Interestingly, the relevant behavioral condition turns out to be the translation
into the Savage framework of a condition introduced by Kreps [5] in his seminal paper
on menu choices.
2S e t t i n g
We use a standard Savage-style setting. Throughout, Σ is an event algebra in a state
space S. Subsets of S are understood to be in Σ even where not stated explicitly.
We denote by ∆ the set of all priors, that is, the set of all ﬁnitely additive probability
measures P on Σ. Two such measures P1 and P2 are mutually absolutely continuous
when, for all E ∈ Σ, P1 (E)=0if and only if P2 (E)=0 .
We denote by X the space of consequences.A nact is a map f : S → X and it is
simple when it is ﬁnite valued; L0 denotes the set of all simple Σ-measurable acts.
The decision maker has a preference relation % on L0, which in turn induces a
preference over X, obtained in the standard way by identifying consequences with
constant acts.
Say that an event E is null (resp. universal) if xEy ∼ y (resp. xEy ∼ x)f o r
all x,y ∈ X such that x Â y.H e r e xEy is the act that pays x if E obtains, and y
otherwise.
A binary relation % on L0 is a multiple-priors (MP) preference relation if there
exists a utility index u : X → R, and a non-empty weak∗-compact set C ⊆ ∆ such
that % is represented by the preference functional V : L0 → R deﬁned by
V (f)=m i n
P∈C
R
u(f(s))dP (s), ∀f ∈ L0.( 1 )
1Similar considerations apply to the closely related topic of dynamic coherent risk measures (see
[1]).
1Subjective expected utility (SEU) preferences are the special case of singleton sets of
priors C.
For MP preferences, an event E is null iﬀ P (E)=0for some P ∈ C, while E is
universal iﬀ P (E)=1for all P ∈ C. Hence, the complement of a universal event is
null, but the converse is in general false.
3 The Axiom and Results
Given any two consequences x and y,l e t
x ∨ y =
(
x if x % y,
y else;
and given any two acts f and g,d e ﬁne the act f ∨ g by
(f ∨ g)(s)=f (s) ∨ g(s), ∀s ∈ S.
The next axiom translates the Kreps axiom (see [5, Eq. (1.5)]) into our setting.
Axiom 1 (General Kreps (GK)) For all f,f0,g∈ L0,
f ∼ f ∨ f
0 =⇒ f ∨ g ∼ (f ∨ g) ∨ f
0.
In every state, the act f ∨f0 gives the better of the two outcomes associated with f
and f0.T h u ss a yt h a tf ∨f0 weakly improves f in ‘the direction’ f0.G Kr e q u i r e st h a t
if an improvement of f in direction f0 has no value, then the same must be true for an
improvement in direction f0 of any act (here f ∨g)t h a ti m p r o v e sf. The scope of this
seemingly innocuous axiom will be fully understood in light of the results to follow.
First observe that SEU preferences always satisfy the GK axiom: f ∼ f ∨ f0 =⇒
Z
u(f)dP =
Z
u(f ∨ f
0) dP =
R
(u(f) ∨ u(f
0)) dP =⇒
u(f
0 (s)) ≤ u(f (s)) P-a.e. =⇒
u(f
0 (s)) ∨ u(g(s)) ≤ u(f (s)) ∨ u(g(s)) P-a.e. =⇒
f ∨ g ∼ (f ∨ g) ∨ f
0.
In general MP preferences do not satisfy GK, as the next example shows.
2Example 1 Let S = {1,2} and X ⊆ R. Consider acts f =( x1,x 2), f0 =( x0
1,x 0
2) and
g =( y1,y 2),w h e r e
x
0
1 <y 2 <x 1 <x 2 <x
0
2 <y 1.
We have f ∨ f0 =( x1,x 0
2), f ∨ g =( y1,x 2),a n d(f ∨ g) ∨ f0 =( y1,x 0
2).
Consider the case of complete ignorance and risk neutrality: u(x)=x and C =
{(p,1 − p):p ∈ [0,1]}. Then, contrary to GK,
V (f)=V (f ∨ f
0)=x1 and V (f ∨ g)=x2 <x
0
2 = V (f ∨ g ∨ f
0). N
The following result characterizes MP preferences that satisfy GK, and it shows
that the latter is the sought-after behavioral characterization of the mutual absolute
continuity of priors.
Theorem 1 For any MP preference %, the following properties are equivalent:
(i) % satisﬁes GK;
(ii) all measures in C are mutually absolutely continuous;
(iii) any event E is null iﬀ its complement Ec is universal.
Proof. The equivalence of (ii) and (iii) is easily established. We prove that (i) and (ii)
are equivalent.
(ii) implies (i): Assume that all measures in C are mutually absolutely continuous.
Suppose that f ∼ f ∨ f0. Then there exists P∗ ∈ C such that
min
P∈C
R
u(f ∨ f
0) dP =m i n
P∈C
R
(u(f) ∨ u(f
0)) dP =
R
(u(f) ∨ u(f
0)) dP
∗
≥
R
u(f)dP
∗ ≥ min
P∈C
R
u(f) dP =m i n
P∈C
R
u(f ∨ f
0) dP.
Hence,
R
u(f)dP
∗ =
R
(u(f) ∨ u(f
0)) dP
∗ =⇒
u(f
0 (s)) ≤ u(f (s)) P
∗-a.e. =⇒
u(f
0 (s)) ∨ u(g(s)) ≤ u(f (s)) ∨ u(g(s)) P
∗-a.e. =⇒
u(f
0 (s)) ∨ u(g(s)) ≤ u(f (s)) ∨ u(g(s)) P
∗-a.e. ∀P ∈ C =⇒
u(f
0 (s)) ∨ u(g(s)) ∨ u(f (s)) = u(f (s)) ∨ u(g(s)) P
∗-a.e. ∀P ∈ C =⇒
f ∨ g ∼ (f ∨ g) ∨ f
0.
(i) implies (ii): Assume that % satisﬁes GK but that not all measures in C are
mutually absolutely continuous. Then there exist an event E and P1,P 2 ∈ C such that
3P1 (E)=0<P 2 (E).L e tP ∈ C be such that P (Ec) ≤ P (Ec) for all P ∈ C.T h e n
P (E) ≥ P2 (E) > 0,a n ds oP (Ec) < 1.
Wlog let x and y be outcomes such that u(x)=1and u(y)=0 .S i n c e(xEy)∨y =
xEy,w eh a v eV (xEy ∨ y)=V (xEy)=P1 (E)=0 . Hence, xEy ∨ y ∼ y.O n t h e
other hand, xEy ∨ y ∨ xEcy = x and y ∨ xEcy = xEcy,s ot h a t
V (xEy ∨ y ∨ xE
cy)=1> P (E
c)=V (y ∨ xE
cy),
and xEy ∨ y ∨ xEcy Â y ∨ xEcy,c o n t r a d i c t i n gG K . ¥
Remark 1 Partial versions of Theorem 1 holds for more general classes of preferences.
For example, (i) implies (iii) for the biseparable preferences of [3], while (ii) implies
(i) for the variational preferences of [6] (in this case condition (ii) becomes the mutual
absolute continuity of the probabilities belonging to the eﬀective domain of the function
c used in their representation).
It is natural to understand GK as follows: suppose that f ∼ f ∨ f0.T h e n i t
must be that the set of states where f0 (s) is strictly preferred to f (s) is “impossible,”
and therefore that the event {s : f (s) % f0 (s)} is “certain.” Then the larger event
{s : f (s) ∨ g(s) % f0 (s) ∨ g(s)} must also be “certain.” Conclude that f∨g ∼ f∨g∨f0,
as required by GK. The next result makes precise this intuitive connection between GK,
“impossibility” and “certainty,” at least for MP preferences.
Theorem 2 Let % be a MP preference. Given any f and f0 in L0, consider the
following properties:
(i) f ∼ f ∨ f0;
(ii) the event {s : f0 (s) Â f (s)} is null;
(iii) the event {s : f (s) % f0 (s)} is universal.
Then (iii)= ⇒ (i)= ⇒ (ii); and the three properties are equivalent for any f and f0
in L0 iﬀ % satisﬁes GK.
Proof. (i) implies (ii): If f ∼ f ∨ g,t h e nt h e r ei sP∗ ∈ C such that
min
P∈C
Z
u(f)dP =m i n
P∈C
Z
u(f ∨ f
0)dP =
Z
u(f ∨ f
0)dP
∗ =
Z
(u(f) ∨ u(f
0))dP
∗
≥
Z
u(f)dP
∗ ≥ min
P∈C
Z
u(f)dP,
so that
R
u(f)dP ∗ =
R
(u(f) ∨ u(f0))dP ∗.T h e r e f o r e , P∗ ({s : f0 (s) Â f (s)})=0 ,
which implies that {s : f0 (s) Â f (s)} is null.
(iii) implies (i): If {s : f (s) % f0 (s)} is universal, then P ({s : f (s) % g(s)})=1
∀P ∈ C,a n du(f0 (s)) ≤ u(f (s)) P-a.e. ∀P ∈ C. It follows that f ∼ f ∨ g.
4Let % satisfy GK. By Theorem 1, {s : f0 (s) Â f (s)} is null iﬀ {s : f (s) % f0 (s)}
is universal. Thus, given GK, (ii) implies (iii) and (i)-(iii) are equivalent. Con-
versely, assume (i)-(iii) are equivalent for any f and f0 in L0.T h e n f ∼ f ∨ f0 =⇒
P ({s : f (s) % f0 (s)})=1∀P ∈ C =⇒ P ({s :( f ∨ g)(s) % (f0 ∨ g)(s)})=1∀P ∈ C
=⇒ f ∨ g ∼ (f ∨ g) ∨ f0. ¥
We conclude with a characterization of GK for all monotone (not necessarily MP)
preferences. Say that the preference % on L0 is monotone if f % g whenever f (s) %
g(s) for each s ∈ S.
Inspired by [5, p. 568], deﬁne a relation >∗ on L0 by
f >
∗ f
0 if f ∼ f ∨ f
0.
For monotone preferences, GK is equivalent to transitivity of >∗. In particular, under
GK the symmetric part of >∗ partitions L0 into equivalence classes of acts that are
indistinguishable according to %.
Theorem 3 Let % be monotone. Then % satisﬁes GK iﬀ >∗ is transitive.
Proof. Suppose >∗ is transitive. Then, by monotonicity, f ∨g >∗ f. Therefore, since
>∗ is transitive, f >∗ g =⇒ f ∨ g >∗ g,w h i c hp r o v e sG K .
Conversely, assume GK.2 Suppose f >∗ g and g >∗ h,t h a ti s ,f ∼ f ∨ g and
g ∼ g ∨h.W eh a v ef ∼ f ∨g % g ∼ g ∨h by monotonicity. By GK, g ∼ g ∨h implies
g ∨ f ∼ g ∨ h ∨ f, and, by monotonicity, g ∨ h ∨ f º h ∨ f. Therefore,
f ∼ f ∨ g ∼ g ∨ h ∨ f º h ∨ f > f,
which implies f ∼ h ∨ f,t h a ti s ,f >∗ h. ¥
References
[1] P. Artzner, F. Delbaen, J.M. Eber, D. Heath, and H. Ku, Coherent multiperiod
risk adjusted values and Bellman’s principle, Annals Oper. Res.,f o r t h c o m i n g .
[2] L.G. Epstein and M. Schneider, Recursive multiple-priors, J. Econ. Theory 113
(2003), 1—31.
[3] P. Ghirardato and M. Marinacci, Risk, ambiguity, and the separation of utility and
beliefs, Math. Oper. Research 26 (2001), 864-890.
2We use an argument of Kreps [5, p. 569].
5[4] I. Gilboa and D. Schmeidler, Maxmin expected utility with a non-unique prior, J.
Math. Econ. 18 (1989), 141—153.
[5] D. Kreps, A representation theorem for ‘preference for ﬂexibility’, Econometrica 47
(1979), 565-577.
[6] F. Maccheroni, M. Marinacci, and A. Rustichini, Ambiguity aversion, robustness,
and the variational representation of preferences, Econometrica, forthcoming.
6

A representation theorem for ‘preference for ﬂexibility’,

Ambiguity aversion, robustness, and the variational representation of preferences,

and the separation of utility and beliefs,

Coherent multiperiod risk adjusted values and Bellman’s principle, Annals Oper. Res.,f o r t h c o m i n g .

Recursive multiple-priors,

Mutual Absolute Continuity of Multiple Priors

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Mutual Absolute Continuity of Multiple Priors

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