Preference for Flexibility and Dynamic Consistency
with Incomplete Preferences
Fernanda Senra de Moura
Gil Riella
yJanuary 2013
Preliminary and Incomplete
Abstract
We generalize the analysis in Riella (2012) to the case where preferences are not necessarily complete. Formally, we show that Flexibility Consistency has the same consequences in the incomplete preferences case that it has in the complete preferences one. That is, we show that Flexibility Consistency is equivalent to a subjective state space version of Dynamic Consistency and, as it happens when we have an objective state space, this subjective version of Dynamic Consistency is equivalent to a Bayesian updating result in the preferences over menus world.
JEL Classi…cation: D11, D81.
Keywords: Incomplete Preferences, Preference for Flexibility, Dynamic Consis-tency, Bayesian Updating, Subjective State Space.
1
Introduction
Riella (2012) provides an updating result in the choice over menus framework. The analy-sis there assumes that, although the decision maker makes use of a subjective—therefore unobservable—state space, there are objective—therefore observable—signals that may af-fect her behavior. The main di¢cult in such a situation is that, although the signals are
Department of Economics, University of Rochester. Email: fernandasenra@gmail.com.
observable, we have no means to understand how the individual interprets these signals, in terms of her subjective state space. The main result in Riella (2012) characterizes when a given signal is interpreted by the decision maker as an event in her subjective state space and when, upon learning this event, she acts in a dynamically consistent way.
The analysis in Riella (2012) is performed under the assumption that the individual’s preferences admit a…nite Positive Additive Expected Utility representation and, consequently, they are complete. In the present work we investigate what happens with Riella (2012)’s result if we drop the assumption that the individual’s preferences are complete. Our main result shows that Riella (2012)’s main result remains true when we do that. Our main axiom is a slight generalization of Riella (2012)’s main postulate, which basically add the possibility of incomparability to the original property.
Preferences over menus of lotteries that admit a Positive Additive Expected Utility repre-sentation were axiomatized by Dekel, Lipman, and Rustichini (2001)—henceforth DLR—and Dekel, Lipman, Rustichini, and Sarver (2007). Dekel, Lipman, and Rustichini (2009) intro-duced an axiom that guarantees that the state space used in the representation of a given Positive Additive Expected Utility representation is …nite. (See also Kopylov (2009).) The incomplete version of Dekel et al. (2001)’s model we will work with here was axiomatized by Kochov (2007). (See also Chandrasekher (2012) and Galaabaatar (2011).)
In the next section we introduce our framework. In section 3 we discuss Riella (2012)’s main result, while we present our main result in section 4. The proof of our main result appears in the appendix.
2
Preliminaries
We work within the setup of DLR. In what follows X stands for a …nite set of alternatives and (X) for the set of probability measures on X. We view (X) as a metric subspace of RjXj and represent its elements by p; q; r, etc.. Let X represent the space of nonempty
closed subsets of (X). The elements of X are represented by capital letters A; B; C; etc., and are calledmenus. We writeint(X)to represent the subset of the elements ofX that are included in the relative interior of (X). That is, int(X) is the set of all nonempty closed subsets of (X) that include only lotteries with full support.
De…nition 1. We say that a relation % onX admits a …nite Incomplete Positive Ad-ditive expected utility (IPAEU) representation if there exists a …nite setS, a setM
of probability measures onS and a function U :S (X)!R such that
1. For any two menus A and B,
A%B () X
s2S
(s) max
p2A U(s; p)
X
s2S
(s) max
p2B U(s; p), for every 2 M;
2. For each s2S, there exists a nonconstant u2RX such that
U(s; p) = X
x2X
p(x)u(x);
for every p2 (X);
3. [
2M
supp( ) = S and, for each distinct s and s0 in S, U(s; :) and U(s0; :) are not
positive a¢ne transformations of each other.1
When a relation % admits a …nite IPAEU representation we call % a …nite IPAEU preference. IPAEU preferences without the restriction of a …nite state space were axiomatized by Kochov (2007). The property that characterizes when an IPAEU preference admits a …nite IPAEU representation is the following:
Finiteness. For every menu A2 X, there exists a …nite subsetB of A such thatA B.
Formally, an IPAEU preference admits a …nite IPAEU representation if and only if it satis…es Finiteness.
The usual interpretation for a …nite IPAEU representation is that the agent solves a two stages decision problem. In the …rst stage the individual chooses a menu of options, knowing that in the second stage she will have to choose an option from that menu. This individual is uncertain about the future and, in particular, she is uncertain about what her tastes will be when she …nally has to make a choice from a given menu—where each state of the world represents a di¤erent taste. She also does not have a precise description of her uncertainty, in the sense that she holds multiple priors over her state space. Given her set of priors, she acts in a very parsimonious way, considering one menu better than the other only if the …rst menu has an ex ante expected utility higher than the second one for all her priors.
In a …nite IPAEU representation the set S is only an index set and it is not directly relevant. The relevant aspect is the set of ex post preferences induced by fU(s; :) :s 2Sg. Condition 2 in the de…nition of a …nite IPAEU representation says that all these ex post
preferences admit an expected utility representation, while condition 3 requires that the index set S contains no redundant states, in the sense that each state is associated with a di¤erent ex post preference, and that it contains no trivial states, in the sense that every state s has positive probability for at least one prior. Following DLR, we refer to the set of expected utility preferences induced by fU(s; :) : s 2 Sg as the subjective state space. Kochov (2007) shows that the subjective state space of a given …nite IPAEU preference is unique, in the sense that any two …nite IPAEU representations of this relation have the same subjective state space. Given condition 3, we might assume, without loss of generality, that the index set S in any …nite IPAEU representation coincides with its subjective state space. From now on, whenever we write a state space S it is understood that S is the set of expected utility preference relations on (X) that corresponds to %’s subjective state space. Also, for a given subjective state s, we write <s to represent the expected utility
preference associated with s.
3
The Complete Preferences Case
When%is a complete …nite IPAEU preference the set of priors in any IPAEU representation of % becomes a singleton and we call % a …nite Positive Additive expected utility (PAEU)
preference only. Now suppose % and % are two …nite PAEU preferences. Riella (2012) introduced the following condition, linking % and % :
Flexibility Consistency. For any menuA2 X and menuB 2int(X),A B andB %A
imply that there exists a menu C such that A[B[C A[C, but A[B[C A[C.
a strict improvement. Riella (2012) shows the following result:
Theorem 1 (Riella (2012)). Let % and % be …nite PAEU preferences. The following statements are equivalent:
1. % and % satisfy Flexibility Consistency;
2. Let S and S be the unique subjective state spaces of % and % , respectively, and let
U :S[S !R be such that, for anys 2S[S ,U(s; :) is an expected utility function that represents <s. For any two menusA and B with
max
p2A U(s; p) = maxp2B U(s; p) for all s2SnS ;
B %A () B % A;
3. For every …nite PAEU representation (S; ; U) of %, there exists T S such that (T; T; U) represents % , where T is the Bayesian update of after the observation
of T:2
The result above shows that, for two …nite PAEU preferences, Flexibility Consistency is equivalent to a subjective version of Dynamic Consistency, and that this subjective version of Dynamic Consistency is equivalent to a Bayesian updating result, similarly to what happens in the objective state space case. The main result of the present paper generalizes the result above to the case where% and % are not necessarily complete.
4
The Incomplete Preferences Case
The goal of the present section is to generalize Riella (2012)’s result to …nite IPAEU prefer-ences. For that, we need to work with a slightly di¤erent version of Flexibility Consistency, that incorporates the possibility of incomparability.
Flexibility Consistency II. For any menu A 2 X and menu B 2 int(X), A % B
and not A % B or not B % A and B % A imply that there exists a menu C such that
A[B[C A[C, but A[B[C A[C.
We can now state the main result of this paper:
2It is implicit in the statement above that (T)>0, so that
Theorem 2. Let % and % be …nite IPAEU preferences. The following statements are equivalent:
1. % and % satisfy Flexibility Consistency II;
2. Let S and S be the unique subjective state spaces of % and % , respectively, and let
U :S[S !R be such that, for anys 2S[S ,U(s; :) is an expected utility function that represents <s. For any two menusA and B with
max
p2A U(s; p) = maxp2B U(s; p) for all s2SnS ;
B %A () B % A;
3. For every …nite IPAEU representation (S;M; U) of %, there exists T S such that (T;MT; U) represents % , where MT is the set of Bayesian updates of each prior
2 M such that (T)>0.3
The theorem above is a precise generalization of the result in Riella (2012) to …nite IPAEU preferences. As in that theorem, it shows that Flexibility Consistency II is equivalent to a subjective version of Dynamic Consistency, and that, as in the objective state space case, this subjective version of Dynamic Consistency is equivalent to updating all priors using Baye’s rule.
A
Appendix
A.1
Preliminaries
In this section we collect some results that will be useful for the proof of the main theorem. De…ne
U := 0
@u2RjXj :
jXj
X
i=1
ui = 0 and jXj
X
i=1
u2i = 1
1
A:
We call U the set of normalized expected utility functions over X. Each function u 2 U
represents a di¤erent expected utility preference over (X), and every nontrivial expected utility preference over (X)can be represented by exactly one functionu2 U.4 Given that,
we will often writeus to represent the function us2 U that represents <s.
3It is implicit in the statement above thatM
T 6=;. That is, (T)>0for some 2 M.
4That is, for every nontrivial expected utility preference< (X) (X)there exists a unique function
u2 U such that, for everyp; q2 (X), p<q () P
x2Xp(x)u(x) P
For each menu A, we de…ne the function A:U !Rby
A(u) := max
p2A Ep(u), for each u2 U.
For a given menu A, the function A is called the support function of A. It can be shown
that if a relation% has a …nite IPAEU representation, then there exists a unique closed and convex set of priors P% over %’s subjective state space S such that, for any two menus A
and B in X,
A%B () X
s2S
(s) A(us)
X
s2S
(s) B(us);
for every 2 P%. We call P% the canonical …nite IPAEU representation of%. We can check
that support functions satisfy the following:
Lemma 1. For any pair of menus A and B, the following is true:
1. A+(1 )B = A+ (1 ) B for all 2[0;1] ;
2. A[B= maxf A(u); B(u)g, for every u2 U;
Now, let C(U) be the space of continuous functions overU endowed with the supnorm distance. It is well known that
C :=f A :A is a menug C(U):
We are also going to work with the following subsets of C(U) :
H := S
r 0
rC
and
H :=H H:
DLR proves thatH satis…es the following properties:
Lemma 2. H satis…es the following conditions:
1. H is a linear subspace of C(U) ;
2. For any f 2H , there exists r >0 and 1; 2 2C such thatf =r( 1 2) ;
A.2
Proof of Theorem 2
Let P% and P% be the canonical …nite IPAEU representations of % and % , respectively.
We will also consider the following property:
Property *. LetPS% be the Bayesian update ofP% after the observation of the subjective
state space of % , S . Then,co(cl(PS%)) =P% .
We will now prove Theorem 2.
[1 =) 2] Let S and S be the unique subjective state spaces of %and % , respectively, and let U : S [S ! R be such that, for any s 2 S [S , U(s; :) is an expected utility representation of <s. Now …x any two menus A and B with
max
p2A U(s; p) = maxp2B U(s; p) for all s2SnS .
We can assume, without loss of generality, that A; B 2 int(X).5 Suppose now that C is a
menu such that A[B[C A[C. From the representation of % it is clear that
max
p2A[B[CU(s; p) = maxp2A[CU(s; p) for all s 2S .
But it is also clear that
max
p2A[B[CU(s; p) = maxp2A[CU(s; p) for all s2SnS ;
which implies that A[B [C A[C. That is, for no menu C it can be the case that
A[B [C A[C, but A[B[C A[C. Now Flexibility Consistency II implies that
A%B () A% B.
[2 =) Property *] Let S and S be the unique subjective state spaces of % and % , respectively. Suppose co(cl(PS%))6=P% . Fix any sphere E 2int(X). That is, E 2int(X)
can be written as E := fq 2 (X) :d(p ; q) g, for some p 2 (X) and > 0. We note that, for every s 2 S [S , E(:)(us) has a unique maximizer, p, in E. Moreover, if
s 6= s0, then the maximizer of E(:)(us) is di¤erent from the maximizer ofE(:)(us0). Suppose
5If this is not true, we can simply mix A and B with a lottery with full support and use the fact that both %and % satisfy Independence to …nish the proof. That is, we can de…ne menus A~ and B~ by
~
A:= A+ (1 )fpg andB~ := B+ (1 )fpg, for some 2(0;1) and some lotteryp2 (X) with full support. By the representations of%and% we haveA%B () A~%B~ ,A% B () A~% B~, and
max
p2A~
U(s; p) = max
p2B~
…rst that there exists 2 P% nco(cl(P%
S )). If P
%
S 6= ;, use the separating hyperplane
theorem to …nd a continuous function f : U ! R such that P
s2S (s)f(us) > 0 for every
2 co(cl(PS%)) and P
s2S (s)f(us) < 0. If P
%
S = ;, let f be any continuous function
such that P
s2S (s)f(us)<0. We need the following claim:
Claim 1. There exist menus A and B in X such that P
s2S (s)( A(us) B(us))>0 for
every 2 co(cl(PS%)), P
s2S (s)( A(us) B(us)) < 0 and A(us) = B(us) for every
s2SnS .6
Proof of Claim. By Lemma 2, there exist menus A~ and B~ such that P
s2S (s)( A~(us)
~
B(us))>0for every 2co(cl(P
%
S )), and
P
s2S (s)( A~(us) B~(us))<0. LetE be any
sphere in int(X)and …x 2(0;1)large enough so that, for everys2SnS and s 2S , if
pmaximizes E(:)(us) in( E+ (1 ) ~A)[( E+ (1 ) ~B), thenEp(us )< E+(1 ) ~A(us )
and Ep(us )< E+(1 ) ~B(us ). Now, for each s2SnS , letps be a maximizer ofE()(us)in
( E+ (1 ) ~A)[( E+ (1 ) ~B). De…neA:= ( E+ (1 ) ~A)[ fps :s 2SnS gandB :=
( E+ (1 ) ~B)[ fps :s2SnS g. By Lemma 1, we havePs2S (s)( A(us) B(us))>0
for every 2co(cl(PS%)),P
s2S (s)( A(us) B(us))<0, and A(us) = B(us)for every
s2SnS . k
Now pick menus A and B as in the claim above. Notice that, since A(us) = B(us)
for every s 2 S nS , P
s2S (s)( A(us) B(us)) > 0 for every 2 co(cl(P
%
S )) implies
that P
s2S (s)( A(us) B(us)) 0 for every 2 P%. That is, A % B. Since we
also have that it is not true that A % B and A(us) = B(us) for every s 2 S nS ,
this contradicts 2. We conclude that P% co(cl(P%
S )). Suppose now that there exists
2 PS% n P% . A reasoning similar to the one above allows us to …nd menusA and B such
thatP
s2S (s)( A(us) B(us))>0for every 2 P% ,
P
s2S (s)( A(us) B(us))<0
and A(us) = B(us) for every s 2SnS . For such menus A and B we have A B, it is
not true that A %B, and A(us) = B(us) for every s 2 SnS . This again contradicts2.
We conclude thatPS% P% , which implies that co(cl(P%
S )) P% .
[Property * =) 3] Fix any …nite IPAEU representation, hS;M; Ui, of %. By the uniqueness properties of expected utility representations, for each s2S, there exist unique
s 2 R++ and s 2 R such that U(s; p) = sEp(us) + s for every p 2 (X). For each
prior over S, de…ne a prior over S by (s ) := P s (s )
s2S s (s), for every s 2S. Now, let PM :=f : 2 Mg. By construction, hS;PM;E
(:)(u(:))i is a …nite IPAEU representation
of %. By the uniqueness properties of canonical …nite IPAEU representations, this implies
6Notice that we are allowing for the possibility that co(cl(P%
S )) = ;, which makes the condition P
s2S (us)( A(us) B(us))>0for every 2co(cl(P %
that co(cl(PM)) = P%. Now let PM
S be the set of Bayesian updates of the priors in PM
after the observation of S . We need the following claim:
Claim 2. hS ;PM
S ;E(:)(u(:))i is a …nite IPAEU representation of % .
Proof of Claim. By assumption,P% =co(cl(P%
S )). This implies that hS ;P
%
S ;E(:)(u(:))i is
a …nite IPAEU representation of % . Suppose now that hS ;PM
S ;E(:)(u(:))i is not a …nite
IPAEU representation of % . SincePM S P
%
S , this means that there exist menus A and B
such that P
s2S (s)( A(us) B(us)) 0 for every 2 PSM, but
P
s2S (s)( A(us) B(us)) < 0 for some 2 P
%
S . Using a similar reasoning to the one used in the proof
of Claim 1, we can …nd such menus that in addition satisfy that A(us) = B(us) for
every s 2 SnS . This implies that P
s2S (s)( A(us) B(us)) 0 for every 2 PM,
but there exists 2 P% such that P
s2S (s)( A(us) B(us)) < 0, which contradicts
the fact that hS;PM;E
(:)(u(:))i is a …nite IPAEU representation of %. We conclude that
hS ;PM
S ;E(:)(u(:))i is a …nite IPAEU representation of % . k
Now, for each 2 M, let S be the Bayesian update of after the observation of S
and S be the Bayesian update of after the observation of S . It can be checked that
S (s ) =
s S (s ) P
s2S s S (s), for every s 2 S . This now implies that hS ;P
M
S ;E(:)(u(:))i and
hS ;MS ; Ui are …nite IPAEU representations of the same relation. Since, by the previous
claim, we know that hS ;PM
S ;E(:)(u(:))i represents % , we conclude that hS ;MS ; Ui also
represents % .
[3 =) 1] Let (S;M; U) and (T;MT; U) be …nite IPAEU representations of % and
% , respectively, where MT is the Bayesian update of M after the observation of T. Now
suppose thatA 2 X and B 2int(X) are such that B %A, but it is not true thatB % A, orA% B, but it is not true that A%B. It is clear that this can happen only if there exists
s 2SnS such that
max
p2B U(s ; p)>maxp2A U(s ; p):
We now show that this implies that there exists a menuC such that
max
p2A[B[CU(s; p) = maxp2A[CU(s; p), for every s2S , (1)
but
max
p2A[B[CU(s ; p)>p2A[Cmax U(s ; p): (2)
For each s 2 S , de…ne qs as follows: if maxp2AU(s; p) maxp2BU(s; p), let qs be
q 2 arg maxp2BU(s; p) with U(s ; q) < maxp2BU(s ; p), let qs := q. We are left with
the case where maxp2BU(s; p) > maxp2AU(s; p), but U(s ; q) = maxp2BU(s ; p) for all
q 2 arg maxp2BU(s; p). We …rst note that this implies that it cannot be the case that
U(s; :) is cardinally equivalent to U(s ; :). If this was the case, we would necessarily have maxp2BU(s ; p) < maxp2AU(s ; p). This implies that there exist lotteries p and
p0 such that U(s; p) = U(s; p0), but U(s ; p) < U(s ; p0). Fix q 2 arg maxp2BU(s; p)
and let := q +p p0. Note that, for every 2 (0;1), U(s; q+ (1 ) ) = U(s; q),
but U(s ; q+ (1 ) ) < U(s ; q) = maxp2BU(s ; p). Since B 2 int(X), we have
q + (1 ) 2 (X) when is large enough. In this case, let qs := q + (1 )
for some 2 (0;1) such that q+ (1 ) 2 (X). Now de…ne C := fqs:s2S g. It is
clear that (1) and (2) hold for such menu C and, consequently, A[B [C A[C, but
A[B[C A[C. That is, % and % satisfy Flexibility Consistency II.
References
Chandrasekher, M. (2012). Incomplete menu preferences and ambiguity. Manuscript.
Dekel, E., B. L. Lipman, and A. Rustichini (2001). Representing preferences with a unique subjective state space. Econometrica 69(4), 891–934.
Dekel, E., B. L. Lipman, and A. Rustichini (2009). Temptation-driven preferences. Review of Economic Studies 76(3), 937–971.
Dekel, E., B. L. Lipman, A. Rustichini, and T. Sarver (2007). Representing preferences with a unique subjective state space: a corrigendum. Econometrica 75(2), 591–600.
Galaabaatar, T. (2011). Menu choice without completeness. Manuscript.
Kochov, A. (2007). Subjective states without the completeness axiom. Manuscript, Univer-sity of Rochester.
Kopylov, I. (2009). Finite additive utility representations for preferences over menus. Journal of Economic Theory 144, 354–374.
