Download A General Update Rule for Convex Capacities

A General Update Rule for Convex Capacities
Mayumi Horie
21 July, 2007
Abstract
A characterization of a general update rule for convex capacities, the G-updating rule,
is investigated. We introduce consistency properties which bridge between unconditional
and conditional preferences, and deduce an update rule for unconditional capacities. The
axiomatic basis for the G-updating rule is established through consistent counterfactual
acts, which conclusively take the form of trinary acts expressed in terms of G, an ordered
tripartition of global states.
1
Introduction
A decision maker’s subjectively uncertain situations are formally described in non-additive
probabilities or multiple priors, for which update rules are also extensively investigated in the
economic literature on the subject. Among them, the Dempster-Shafer rule(Shafer(1976))
(the DS rule) is one of the most momentous update rules for non-additive probabilities.
This rule for convex capacities is examined in Gilboa and Schmeidler(1993) by way of an
elegant, systematic method, the f -Bayesian update rule. They also showed that the DS rule
is equivalent to the maximum likelihood estimation. Moreover, the f -Bayesian update rule
concerns, when an event observed, the decision maker’s pessimistic or optimistic supposition
on the counterfactual event, which determines the relative evaluation of any conditional act.
Fagin and Halpern(1991) presented an update rule (the DFH rule) for inner/outer measures or belief/plausibility functions, which was already suggested by Dempster(1967). Tapking(2004) deals with a collection of updated preference relations de…ned on conditional acts
and proposes axioms for the DFH rule. The essence of the axiomatization is the Choquet
expected utility (CEU) representation and the fact that a convex capacity updated by the
DFH rule conforms to the lower bound of the set of conditional probabilities via Bayes’rule
in a given set of priors.
I am deeply grateful to Atsushi Kajii for detailed comments and invaluable discussions. I am also thankful
to every comment on previous version presented at RUD 2006. Remaining errors or shortcomings in this
version are, of course, my own responsibility.
1
Pires (2002) reviews the full Bayesian update rule (the FB rule) or what is called the
belief-by-belief updating rule. In fact the FB rule and the DFH rule generates the common
lower envelope of the updated belief sets, hence it is only natural for both rules to share the
same property to some extent. However, even if the updated belief set by the FB rule has
a convex capacity representing its lower probabilities, it is contained in the core of its lower
probabilities in general. Each updated belief set might be quite di¤erent.
The primary objective of this paper is to characterize a general update rule, called the Gupdating rule (Horie(2006)), which includes aforementioned three conditioning rules as special
cases. The G-updating rule enables us to deal with apparently di¤erent conditioning rules as
a single rule with each di¤erent parameter G, which takes the form of an ordered tripartition
of the global states. To achieve characterization, we introduce a consistency property which
bridge between unconditional and conditional preferences, and deduce an update rule for
unconditional capacities. The axiomatic basis for update rules is conclusively established
through counterfactual acts, which are implicitly supposed to occur on the unobserved event,
i.e. counterfactual event.
In the conventional Bayesian approach, the conditional preferences are never a¤ected by
what happens on any counterfactual event. In the context of the subjective expected utility
theory, it is implied by the independency in the sure-thing principle of Savage(1972).
The f -Bayesian update rule partially inherits the spirit of the Bayesian updating. In case
of the G-updating rule, the f -Bayesian update is extended into two directions. (i)Although
in the f -Bayesian update rule the counterfactual act (f ) has to be common among all acts,
counterfactual acts in the G-updating rule are allowed to depend on the conditional certainty
equivalence of any (binary) act. (ii)The consistency are imposed only on the set of binary
acts, not all acts. The reason is that, the G-updating rule evaluates the observed event quite
di¤erently dependent upon which subevent is taken into account. Relaxing the requirement
for all acts into binary acts enables us to conform coherently to such inconstancy.
The main result of this paper formally characterizes the G-updating rule through a property called the k-consistency. It implies the existence of a set of counterfactual acts that
formalize unconditional preferences in terms of the unconditional preferences. This set of
counterfactual acts may be chosen according to the magnitude of its conditional certainty
equivalence of any binary acts. In other words, two gambles (or, bets) that yield the same
conditional certainty equivalence are evaluated equally in terms of the same counterfactual
act.
This paper is organized as follows. The next section provides the basic de…nitions and
introduces the G-updating rule. Section 3 begins with illustrating the k-consistency and
demonstrates the main result of this paper. Section 4 is devoted to some observations on
k-consistency and the relationship between representations of preferences and update rules.
2
2
A General Update Rule
2.1
Basic Set Up
Let
be a …nite set of states with j j = n and
set in
an event. Let X
be an algebra,
= 2 . We call a non-empty
R be a set of consequences, or outcomes which is assumed that
X = [x; x] with x < x. A function f :
! X is called an act. Denote the set of all acts
by F. For the sake of simplicity, an element x in X is also designated by a constant act
which assigns x for all ! 2
if ! 2
Ec.
the act
. fE g denotes the act which yields f (!) if ! 2 E and g (!)
A convex combination of two acts f and g with respect to a
f + (1
) g where
comonotonic i¤ (f (!)
A …nite set function
f
f (!) + (1
(! 0 )) (g (!)
:
) g (!) for each ! 2
g (! 0 ))
= 0 for all ! and
! [0; 1] is called a capacity on
( ) = 1, and (ii) for every A and B in
is convex if for every A and B in
with A
a conditional or updated capacity
E
(A) = 1. For any event E in
,
. Two acts f and g are
!0
in
.
if it satis…es (i)
B, we have
(A [ B) + (A \ B) =
,
2 [0; 1], refers to
(A) 5
(?) = 0 and
(B). A capacity
(A) + (B). Given an event E,
is a capacity on E, i.e. for all A in
. When E = ,
is interpreted as
R
the unconditional capacity and we simply write it . Let f d denote a Choquet integral
E
E
has domain
with A \ E = E,
(expected value) of f with respect to .
Let
be a set of all binary relations on F. We are concerned with preference relations over
F before and after an event E is observed. Given an event E in
in
, a preference relation %E
is called a conditional preference relation contingent on E. As usual,
to asymmetric and symmetric parts of %E respectively. When E =
E
and
E
refer
, % is interpreted as
the unconditional preference relation and we simply express it as %. Throughout this paper,
we focus on preference relations that satisfy the following representation (Schmeidler (1989)
and Gilboa(1987)):
(CEU) There exist a utility function u : X ! R unique up to positive linear transformations
and a convex capacity
E
such that for all f , g in F
f %E g ()
Z
u f d
E
=
Z
u gd
E.
Since u is unique up to positive linear transformations, we normalize u such that u (x) = 1
and u (x) = 0. Let
CE
relation %E 2
is called represented by (u;
CE
be the set of binary relations satisfy CEU. A conditional preference
E ),
that is, %E is represented by a Choquet
expected utility with respected to (normalized) u and
E.
The central axiom for CEU is
called the comonotonic independence: for all pairwise comonotonic acts f , g, and h in F and
2 (0; 1), f % g , f + (1
) h % g + (1
) h.
3
2.2
The G-Updating Rule
A general update rule, called the G-updating rule, is de…ned as follows (Horie(2006)).
Suppose the set of states
is partitioned into three disjoint sets Gi , i = 1; 2; 3, where
some Gi is possibly empty but not all. Let us denote an ordered triplet of Gi , i = 1; 2; 3 by
G = hG1 ; G2 ; G3 i and let G consist of all such ordered triplets of
event E 2
, de…ne
TiG;E
E c \ Gi , i = 1; 2; 3. Although every
we denote it by Ti , i = 1; 2; 3 instead of
Given a G 2 G and an event E 2
TiG;E
. Given a G 2 G and an
TiG;E
depends on G and E,
for brevity’s sake.
, we de…ne the G-updating rule for a capacity
given
E through, for every A 2
G
E (A)
If
=
[ ((A \ E) [ T2 )
(E) > 0, then
G
E
((A \ E) [ T2 )
(T2 )
.
(T2 )] + [ (E [ T2 [ T3 )
((A \ E) [ T2 [ T3 )]
is well-de…ned. Note that the updated
G
E
has domain
(1)
.
Some properties of the G-updating rule are examined in Horie(2006). The most important
property is that the G-updating rule preserves convexity.
Characterization of G-updating rule
3
3.1
Axiom
We introduce consistency property which bridges between unconditional and conditional preferences, and deduce an update rule for unconditional capacities.
In the Bayesian approach, the conditional preferences focus only on the realized event and
are never a¤ected by what would have obtained on the unrealized events. Formally, in terms
of unconditional and conditional preferences, the Bayesian update rule is expressed as: for all
f , g, a 2 F, f %E g , fE a % gE a.
On the other hand, the f -Bayesian update rule proposed by Gilboa and Schmeidler(1993)
is de…ned through a counterfactual act a 2 F satisfying that f %E g , fE a % gE a for all
f , g 2 F. Given that % 2
CE
, %E is also in
CE ,
and a takes the form of xS x for some
S2 .
Given an act f 2 F, let R (f )
X n be the range of f . De…ne the set of k-dimensional
acts by F k = ff 2 F j dim R (f ) 5 kg. We especially call F 1 the set of constant acts, which
stands for X. When k = n, F k is equal to the set of all acts F. Since dim R (f ) 5 n for all
f 2 F, it is e¤ective to set k 5 n.
We call F 2 the set of binary acts. Any binary act f 2 F 2 is expressed by some S 2
,
f (S), and f ( nS), to be f (S)S f ( nS). We can in turn construct any binary act f 2 F 2
from some b, w 2 X, b = w and S 2 , which is denoted by bS w.
4
Similarly, the set of trinary acts is written as F 3 . Hereinafter we concentrate upon par-
ticular trinary acts a 2 F 3 , whose range R (a) consists of the best and worst outcomes, x and
x. Formally, for every x 2 X, let A (x) = a 2 F 3 j a (!) 2 fx; x; xg for all ! 2
S
A = x2X A (x).
Recall that the aforementioned G 2 G is an ordered tripartition of
is also expressed in terms of G. De…ne
:X
G!
F3
by
(x; G)
. Then,
, and so an act a 2 A
xG1 xG2 x. When x = x
or x, many di¤erent Gs designate the same (semi-degenerated) act xG1 x or x nG2 x, still every
S
(x; G) is uniquely identi…ed by x and G. Let A (G) = x2X f (x; G)g. Although there are
S
lots of overlaps in G, we have A = G2G A (G).
To characterize the G-updating rule, we now introduce a consistency property de…ned
below:
De…nition 1 Given an unconditional preferences % in
preference %E in
x
and an event E in
, a conditional
is k-consistent i¤ : for every x 2 X there exists a nonempty set of acts
F such that
ax 2
x
For each x 2 X,
if and only if x
x
E
f , xE ax
fE ax for all f 2 F k .
(2)
is called a set of consistent counterfactual acts, which is common
to k-dimensional acts that yield the same conditional certainty equivalence. The "k" in kconsistency stems from k-dimensional acts F k in (2). Note that, if %E 2
with k = 2, then it is also (k
1)-consistent.
To describe more details, we introduce
any x 2 X, E 2 , and %E , % 2
k
E
Write
k
E
S
x2X
k
E
k
E
( ; %E ; %)
E
f , xE a
,
n
a 2 F jx
(x; %E ; %)
(%E ; %)
is k-consistent
F, de…ned by the following: given
o
fE a for all f 2 F k .
(x; %E ; %).
The following lemma con…rms the relationship between the k-consistency and
Lemma 1 Given an unconditional preferences % in
preference %E in
is k-consistent if and only if
Proof. (I) Assume %E in
x
for all f 2 F k . Therefore, ax 2
k
E
, a conditional
(x; %E ; %) is nonempty for all x 2 X.
x
. By de…nition, ax satis…es x
(x; %E ; %), hence
(II) On the contrary, suppose that
Since
and an event E in
( ; %E ; %).
is k-consistent. Then, for every x 2 X there exists a nonempty
F satisfying (2). Take an ax in
k
E
k
E
k
E
k
E
k
E
E
f , xE ax
fE ax
(x; %E ; %) 6= ? for any x 2 X.
(x; %E ; %) 6= ? for all x 2 X. Take any x 2 X.
(x ; %E ; %) 6= ?, we can always …nd a 2
5
k
E
(x ; %E ; %). For this a, we have x
E
k
E
fE a for all f 2 F k by de…nition of
f , xE a
( ; %E ; %). We can set
x
=
k
E
(x ; %E ; %),
which is nonempty. It follows that %E is k-consistent.
To interpret, let xf and xg be the certainty equivalence of an act f and g in terms of %E
respectively; xf
E
f and xg
E
k
E
g. When
obtain a plain statement: for any f , g 2
F 2,
(x; %E ; %) is nonempty for all x 2 X, we
f %E g , fE axf % gE axg for axf , axg 2
k
E
(%E ; %) .
(3)
In Gilboa and Schmeidler(1993), the f -Bayesian update rule is de…ned by
f %E g , fE a % gE a for all f , g 2 F.
Based on our formulation, given that % 2
the n-consistency of %E generates the f -
CE ,
Bayesian update rule. The set of consistent counterfactual acts is reduced into a singleton:
n
E
(%E ; %) = fag where a = xS x for some S 2 .
3.2
Main Result
The main result of this paper is the following theorem that characterizes the G-updating rule:
Theorem 1 Suppose that the unconditional preference relation % in
with jEj = 2 and
(u; ). Given an event E in
equivalent:
(ii) There exists a G in G such that %E is represented by u;
f %E g ()
where
G
E
equivalent to
2
E
G 2 G such that
(x; G) 2
2
E
xS x , xE (x; G)
for any f and g in F
G
E,
there exists an
CE
is 2-consistent. By Lemma 1, 2-consistency is
(x; %E ; %) for any x 2 X.
ax
2
j ax (!) > xS
CE
E and let xS 2 X satisfy xS
xS ; %E ; %
and P (ax )
2
E
is 2-consistent. Then, for every x 2 X and S
E,
xS xEnS (x; G) for some G 2 G.
Proof. Take an arbitrary S
!2
u gd
is 2-consistent.
(x; %E ; %) 6= ? for all x 2 X. We begin with two lemmas to construct a
Lemma 2 Suppose that %E 2
E
=
Z
CE
G :
E
is de…ned as in (1).
Proof. (i))(ii) Assume that %E 2
x
u f d
G
E
is represented by
(E) > 0, the following statements are
(i) A conditional preference relation contingent on E, %E in
Z
CE
, i.e.
!2
xS
E
xS x ,
xSE ax
E
xS x. By 2-consistency,
xS xEnS ax . Let P (ax )
j ax (!) < xS . Then, P (ax ) ; P (ax ) ; n P (ax ) [ P (ax )
6
denoted by P (ax ) is an element of G. With this P (ax ), consider
xS xEnS ax , xSE
xS ; P (ax ) , and xS xEnS
xS ; P (ax ) are all pairwise comonotonic each
xS xEnS ax if and only if xSE
other. By the comonotonic independency, xSE ax
xS xEnS
xS ; P (ax ) . Therefore, we have xS
Lemma 3 Suppose that %E 2
all x 2 X,
2
E
(x; G) 2
Proof. Given %, %E 2
CE
(x; %E ; %).
CE ,
xS ; P (ax ) . Then, xSE ax ,
E
xS x , xSE
xS ; P (ax )
xS ; P (ax )
xS ; P (ax ) .
xS xEnS
is 2-consistent. Then, there exists a G 2 G such that for
a G 2 G is constructed through the following procedure (1)-(5).
(1) Take any proper subset S of E with
(S) < 1. Since
is convex and
(E) > 0, we
always …nd such an S.
(2) Given this S
E and every G 2 G, de…ne C (S; G)
C (S; G)
and C (S) =
S
G2G
n
x 2 X j xE (x; G)
X by
o
xS xEnS (x; G) ,
C (S; G). Since G is …nite and % is monotonic on X, C (S) contains
at most jGj di¤erent elements.
(3) Find an xS 2 C (S) satisfying xS
E
xS x. By Lemma 2 above, given a %E that is
2-consistent, we can always …nd such an xS in C (S). De…ne G (S)
G2G
j xS
2 C (S; G) .
G by G (S)
(3a) If jG (S)j = 1, then proceed (4a).
(3b) If there are lots of Gs which generate the same xS , i.e. jG (S)j > 1, then proceed
(4b).
G (S).
E
(4) For every G 2 G (S), compute
for every G and
G0
Note that, when jG (S)j > 1,
G (S)
E
G0
E
(S)
2 (0; 1), then set the unique element of G (S) as G (S).
(4b) If jG (S)j > 1 with G (S) $ G and
G (S)
E
G (S) and set it as G (S).
G (S)
E
=
2 G (S) by construction.
(4a) If jG (S)j = 1 and
(4c) If
G (S)
E
2 (0; 1), then pick an arbitrary G from
= 0 or 1, then put G (S) = G.
(4d) If G (S) = G, then go back to (1) and take another S 0 $ E with
(S 0 ) < 1.
(5) Repeat (1)-(4) until a unique G (S) is obtained. If for all S $ E with
G (S) = G, then this procedure ends.
7
(S) < 1,
Since
is …nite, procedure (1)-(5) terminate in …nite repetitions. There are two patterns
G (S)
E
of the end: (A) there exists a unique G (S) such that
with G (A) = G.
G (S)
E
Case A: There exists a unique G (S) such that
xS
2 (0; 1).
xS ; %E ; % . In this case,
E
E
G (S)
E
xS ; x
2 (x; x). Then, by using this event S, for every b 2
such that xS
xS
2
E
By 2-consistency, let ax 2
2 (0; 1) and (B) for all A $ E
2 (0; 1) implies that
there exists a w 2 x; xS
bS w. Similarly, for every w 2 x; xS we can choose b 2 xS ; x such that
bS w. Therefore, ax with ax (!) 2
x; xS ; x
for all ! 2 E c is a candidate for a
consistent counterfactual act. Furthermore, by construction, G (S) is uniquely determined,
so we conclude that ax =
xS ; G (S) 2
xS ; %E ; % .
2
E
Next consider x 2 X and xE (x; G (S)). In any case, for x, there is no act that generates
x other than itself, thus for any G 2 G,
2
E
(x; %E ; %). By the same argument, we have
Pick any
x0
xS ; x
2
w0 2 X such that xS
b0A w0 , hence x0
xS ; G (S) 2
ment above,
xS ; G
(x; G (S)) 2
E and let bA w 2 F 2 satisfy x0
Take any A
xSE
. Then, there exists a
(S) ,
0
b0A wEnA
2
E
xS ; G
E
(x; %E ; %), hence
2
E
(x; G) 2
2
E
(x; G (S)) 2
(x; %E ; %).
2 (0; 1) such that x0 = x + (1
) xS .
E
bA w. Then we can always …nd b0 ,
x + (1
) b0A w0 . In addition, by the argu-
xS ; %E ; % and
(x; G (S)) 2
2
E
(x; %E ; %). However,
(S) and xE (x; G (S)) are all pairwise comonotonic, so
we have
) xSE
xE (x; G (S))+(1
,
x + (1
) xS
E
x + (1
, x0E
Therefore, we conclude that
2
E
xS ; G (S)
xE (x; G (S))+(1
) xS ; G (S)
x0 ; G (S)
2
E
x0 ; G (S) 2
) b0A w0
x + (1
bA wEnA
0
) b0A wEnS
E
x + (1
x0 ; %E ; % for every x0 > xS .
x0 ; G (S) 2
x0 ; %E ; % .
(x; G (S)) 2
2
E
) xS ; G (S)
x0 ; G (S) .
In case of x0 2 (x; xS ), by the symmetric argument above, it is veri…ed that
To sum up, for G (S),
xS ; G (S)
(x; %E ; %) for all x 2 X. Although G (S) is
dependent on the choice of S, for any B $ E, we have xB
E
xB x , xB
E
xB ; G (S)
xB ; G (S) by construction. Therefore, G (S) 2 G (B) for every B $ E, so
T
G (S) 2 B$E G (B). Therefore, we can put G (S) = G .
xB xEnB
Case B: For all A $ E with G (A) = G.
Take an arbitrary G 2 G and set it G . By construction, we have xS
xE (x; G )
E
xS x ,
xS xEnS (x; G ) for every S $ E. In addition, for any S $ E, any binary
8
acts bS w and xS x are comonotonic, so xS
fore,
2
E
(x; G ) 2
(x; %E ; %) for every x 2 X.
Finally, in any case, a G 2 G satisfying
2
E
hence A (G )
bS w , xE (x; G )
E
(x; G ) 2
(%E ; %).
2
E
Lemma 4 Given any G 2 G, if A (G)
2
E
bS wEnS (x; G ). There-
(x; %E ; %) for all x 2 X is obtained,
(%E ; %), then
E
G (A)
E
(A) =
Proof. It is now to be proved that, for every G 2 G, if A (G)
for every A 2 .
(%E ; %), then
2
E
E
(A) is
as given in (1) for any A 2 . The following three cases (1)-(3) have to be examined.
Case 1: A \ E is a nonempty proper subset of E.
Consider an act xA x. Let xA satisfy xA
since xA x 2 F 2 . That is, we have xA
E
Z
u
xA
E
xA ; G
A
x ;G d =
which is calculated in terms of
Z
xA xEnA
xA ; G . Then
u xA xEnA
xA ; G d ,
2
E
xA ; %E ; % ,
as
(T2 ) + u xA [ (E [ T2 [ T3 )
(T2 )]
((A \ E) [ T2 ) + u xA [ ((A \ E) [ T2 [ T3 )
=
xA ; G 2
xA x. From Lemma 1,
E
((A \ E) [ T2 )] .
Rearranging terms, we have
u xA =
((A \ E) [ T2 )
(T2 )
.
(T2 )] + [ (E [ T2 [ T3 )
((A \ E) [ T2 [ T3 )]
[ ((A \ E) [ T2 )
On the other hand, by assumption, xA
Furthermore, for any A in
E
(E [ A)
E
G (A).
E
(A \ E)
E
(A) =
E [ A, so
E
E
=
E
we have
by convexity of . However, E
We also have
xA x, which comes to u xA
E
E
E
(E)
(E [ A) =
(A \ E) =
E
(A \ E).
E
(A \ E) ,
E
(E) = 1 by monotonicity. Hence
(A) .
(A) by monotonicity. Thus we obtain
E
(A \ E) =
E
(A) =
Case 2: A \ E = E.
In this case, xA
E
hence
E
xA ; G is reduced to xE
(A \ E) = 1 =
G (E).
E
9
xA ; G . From xA
E
xE x, we have xA = x,
Case 3: A \ E = ?.
In this case, xA xEnA
have xA = x, hence
E
xA ; G is simpli…ed to xE
xA ; G . By xA
G (?).
E
G
E (A) = E (A)
for any A 2
R
(ii))(i) Assume that %E are represented by u f d
F2
Any binary act f 2
Z
u xE (x; G) d
= 0,
A
=
(T2 [ S)
the assumption that
bA w , xE (x; G)
xA ; G , we
from (1), (2), and (3).
G
E
for some G 2 G.
is expressed in the form of bA w, where b, w 2 X with b = w and
= ( A + A)
Z
=
u xd
where
xE
(?) = 0 =
It follows that we obtain
A 2 . Then,
E
Z
u bA wEnA (x; G) d
A
G
E
u (x)
u (b) +
A+ A
Z
u bA w d G
E
(T2 ) = 0 and
A
=
A
A
+
u (w)
A
(T2 [ E [ T3 )
(E) > 0. Therefore, for any x 2 X and A
bA wEnA (x; G), i.e.
2-consistent.
(x; G) 2
2
E
(T2 [ S [ T3 ) > 0 by
E, we have x
E
(x; %E ; %). It is proved that %E is
Through Theorem 1, we conclude that there exists a G 2 G such that A (G)
2
E
(%E ; %),
which is quali…ed to represent %E via the G-updating rule.
4
Observations
4.1
On the set of counterfactual acts
The result in the previous section showed that the 2-consistency of %E generates the Gupdating rule. To elaborate in detail of the 2-consistency, the following numerical example is
illustrative of the points throughout this subsection.
Example
= f! 1 ; ! 2 ; ! 3 ; ! 4 ; ! 5 ; ! 6 g, E = f! 1 ; ! 2 ; ! 3 g
u (x) = x
(A) =
jAj2
36
for every A 2 . For simplicity, write
(k) =
k2
36 ,
where k = jAj
the updated capacities via the NB rule, DS rule, and DFH rule are calculated as in the
table below:
10
G
E
(1)
E
(2)
NB rule
DS rule
DFH rule
h ; ?; ?i
h?; ; ?i
h?; ?; i
1
9
4
9
+ 0:11
7
25
16
25
+ 0:44
1
21 + 0:048
4
13 + 0:31
= 0:28
= 0:64
Recall that 2-consistency is concerned with the set of binary acts F 2 . The relationship
in (2) may not hold for an act f if dim R (f ) > 2. It is directly caused by the sensitivity in
(x; G) to the conditional certainty equivalence (x). In fact, this happens even in the DFH
update case as in the following example.
Example 1: When dim R (f ) > 2, x
E
f , xE (x; G)
Assume the GFH rule: G = h;; ;; i and
fE (x; G) does not hold.
(x; G) = x.
1
f = 1; 31 ; 11
x
E
f ,x
fE x.
(1)] 13 + [ (5)
x
=
(1) + [ (2)
36x
=
1 + 1 + 21x + 1
1
(5)] 11
(2)] x + [1
! x = 15 .
but x
E
f , since we have
Z
f d
=
E
=
=
By Theorem 1, if
E
(1) + 13 [
E
(2)
1
1 4
1
21 + 3 13
21
1777
9009 + 0:1972.
+
E
1
11
Example 2: %E is in
CE
CE ,
E
[1
E
(2)]
4
13
but not G-updating.
but not G-updating.
Consider the following %E satisfying: for any A
xE
1
1
11
is not updated by the G-updating rule, %E is not 2-consistent. Example
2 below illustrates this point: %E is in
1
2x
(1)] +
xA xEnA
1
2x
.
E and x 2 X, x
is calculated as
E
(A) =
1
2
(A)
(E) + 12 [ (A) + 1
11
(E c [ A)]
for any A
E.
E
xA x ,
E
(?) = 0,
E
(E) = 1, and
is a convex capacity.
E
A $ E; jAj = 2
xE
1
2x
xA xEnA
1
2x
,x
(3) x + [1
E
xA x induces
1
2x
1
2x
(3)]
9x + [36
9]
E
=
=
(2) + [ (5)
4 + [25
! x=
We can …nd su¢ ciently small ",
(3) x + [1
(2) "
1
2x
=
(2)]
=
(3)]
[ (3)
E
(x + ")A (x
Z
4
11
> 0 satisfying xE
4]
(=
1
2x
(2)]
1
2x
E
1
2x
(2) (x + ") + [ (3)
(S)).
(x + ")A (x
(2)] (x
)EnA
) + [1
(3)]
1
2x
.
1
2x
0
= 45 "
!
but x
(A).
), since we have
(x + ")A (x
) d
E
=
E
(2) (x + ") + [1
= x+
= x
E
(2) "
[1
E
E
(2)] x
(2)]
4
5"
4
5"
8
55 "
Example 2 also suggests the uniqueness of the set of consistent counterfactual acts. In
general, we cannot exclude acts that does not belong to A (G) from
every x 2 X,
2
E
(%E ; %). Still, for
(x; G) is considered to be almost unique. The uniqueness is reinforced when
is completely strictly convex.
5
Concluding Remarks
We have investigated a characterization of the G-updating rule. To relax the Bayesian hypothesis, we concentrate on the set of binary acts and linearity in counterfactual acts. However,
it might also suggest the limitation in the method of unconditional-conditional preferences
approach, since further widening the sets of counterfactual acts may not generate any consistent preferences, and become disconnected with reality. Although it is quite di¢ cult to test
and measure out how to revise subjective uncertainty, it is the matter for future investigation
anticipated eagerly in a behavioral sense.
12
References
[1] Dempster, A. P. "Upper and Lower Probabilities Induced by a Multivalued Mapping,"
Annals of Mathematical Statistics, 38, 1967, pp. 205–247.
[2] Denneberg, D. "Conditioning (Updating) Non-Additive Measures," Annals of Operations
Research, vol. 52, 1994, pp. 21–42.
[3] Eichberger, J., S. Grant, and D. Kelsey. "CEU Preferences and Dynamic Consistency,"
Mathematical Social Sciences, 49, 2005, pp. 143–151.
[4] Ellsberg, E. "Risk, Ambiguity, and the Savage Axioms," Quarterly Journal of Economics,
75, pp. 643–669.
[5] Fagin, R. and J. Y. Halpern, "A New Approach to Updating Beliefs", in Uncertainty
in Arti…cial Intelligence 6, ed. by P.P. Bonissone, M. Henrion, L.N. Kanal, and J.F.
Lemmer, 1991, pp. 347–374.
[6] Gilboa, I. "Expected Utility Theory without Purely Subjective Non-Additive Probabilities," Journal of Mathematical Economics, vol. 16, 1987, pp. 65–88.
[7] Gilboa, I. "Additivizations of Non-Additive Measures," Mathematical Operations Research, vol. 14, 1989, pp. 1–17.
[8] Gilboa, I. and D. Schmeidler, "Maxmin Expected Utility with Non-Unique Prior," Journal of Mathematical Economics, vol. 18, 1989, pp. 141–153.
[9] Gilboa, I. and D. Schmeidler, "Updating Ambiguous Beliefs," Journal of Economic Theory, vol. 59, Issue 1, February, 1993, pp. 33–49.
[10] Horie, M. “A Uni…ed Representation of Conditioning Rules for Convex Capacities,”Economic Bulletin vol. 4, no.19, 2006, pp. 1-6.
[11] Ja¤ray, J. Y. "Bayesian Updating and Belief Functions", IEEE Transactions on Systems,
Man and Cybernetics, vol. 22, No. 5, September/October, 1992, pp. 1144–1152.
[12] Knight, F. H. Risk, Uncertainty and Pro…t, 1921, Chicago, University of Chicago Press
[13] Pires, C. P. "A Rule for Updating Ambiguous Beliefs," Theory and Decision, vol. 53, No.
2, September, 2002, pp. 137–152.
[14] Savage, L. J. The Foundations of Statistics, New York, Dover Publications, Inc., 1972.
[15] Schmeidler, D. "Integral Representation without Additivity," Proceedings of the American Mathematical Society, vol. 97, No. 2, June, 1986, pp. 255–261.
13
[16] Schmeidler, D. "Subjective Probability and Expected Utility without Additivity," Econometrica, vol. 57, No.3, May, 1989, pp. 571–587.
[17] Shafer, G. A Mathematical Theory of Evidence, Princeton University Press, Princeton,
N.J., 1976.
[18] Sundberg, C. and C. Wagner, "Generalized Finite Di¤erences and Bayesian Conditioning
of Choquet Capacities", Advances in Applied Mathematics, 13, 1992, pp. 262-272.
[19] Tapking, J. "Axioms for preferences revealing subjective uncertainty and uncertainty
aversion," Journal of Mathematical Economics, 40, 2004, pp. 771–797.
[20] Walley, P. Statistical Reasoning with Imprecise Probabilities, Chapman and Hall, London, 1991.
[21] Wasserman, L. A. and J. B. Kadane. "Bayes’ Theorem for Choquet Capacities," The
Annals of Statistics, 1990, vol. 18, No.3, pp. 1328–1339.
14