Capacities and probabilistic beliefs: a precarious coexistence
* Klaus Nehring
Department of Economics, University of California, Davis, CA 95616-8578, USA
Abstract
This paper raises the problem of how to define revealed probabilistic beliefs in the context of the capacity / Choquet Expected Utility model. At the center of the analysis is a decision-theoretically axiomatized definition of ‘‘revealed unambiguous events’’. The definition is shown to impose surprisingly strong restrictions on the underlying capacity and on the set of unambiguous events; in particular, the latter is always an algebra. Alternative weaker definitions violate even minimal criteria of adequacy. Rather than finding fault with the proposed definition, we argue that our results indicate that the CEU model is epistemically restrictive, and point out that analogous problems do not arise within the Maximin Expected Utility model. 1999 Elsevier Science B.V. All rights reserved
1. Introduction
Following Ellsberg (1961) classical experiments, it has become widely accepted that the preferences of empirical decision-makers often violate the consistency conditions characteristic of classical Subjective Expected Utility theory, and in particular that they fail to reveal a well-defined subjective probability measure. By now, there exists a variety of axiomatic models designed to accommodate Ellsbergian behavior; the two most frequently studied are the Choquet (1953) and Maximin Expected Utility Models (CEU respectively MMEU) due to Schmeidler (1989) respectively Gilboa and Schmei-dler (1989).
While on a heuristic and rhetorical level the epistemic distinction between risk and uncertainty has been important in stimulating an interest in such non-standard models, little work has been done in determining their epistemic content, i.e. in relating preferences to appropriate notions of belief (see Epstein and Zhang (1996); Sarin and
*e-mail: [email protected]
Wakker (1998), and Nehring (1994), as well as Ghirardato (1996); Mukerjee (1995), and Nehring (1991) from rather different perspectives).
This paper addresses a particular issue within this general problematics: when can one legitimately attribute to an agent an unambiguous probabilistic belief about an event or set of events? And, in a related vein: which conditions must preferences satisfy in order to reflect / be consistent with a set of given (‘‘objective’’) probabilities?
A satisfactory answer to these basic questions seems not only essential to an adequate understanding of models of non-probabilistic uncertainty, it also promises to have significant value in applications. By allowing to ‘‘localize’’ ambiguous beliefs, it should yield models with more specific predictions and sharper comparisons to traditional ‘‘global’’ expected-utility models. For example, in a game-theoretic context, one may want to describe the extensive-form game itself (in particular the ‘‘moves of Nature’’) in standard Bayesian manner in terms of unambiguous probabilities, while allowing at the same time for ambiguity in players’ beliefs about other players’ strategic choices (‘‘strategic uncertainty’’).
We will conduct the analysis in the context of the CEU or ‘‘capacity’’ model as does most of the existing epistemic literature. The first thing to note is that, as simple and as elementary as they look, the questions raised do not have an obvious answer. Indeed, it will be seen that it is not even clear that any satisfactory answer exists within the CEU model.
The non-triviality of the issue becomes clear through the following preliminary consideration. For an agent to believe in the occurrence of some event A with subjective probability a, not only must the capacity of A, n(A), be equal to a, but that of the complement 12n (A) must be equal to its probability 12a as well. Yet more is required. If in addition the agent believes in the occurrence of the disjoint set B with subjective probability b, then he also believes (of conceptual necessity) that the probability of the event A<B is equal to a 1 b, hence n(A<B) must be equal to
a 1 b 5 n(A)1n(B). Probability judgements have a ‘‘logical syntax’’ that needs to be accounted for.
In the literature, only the very recent and thorough contribution by Zhang (1997) has taken up the issue of defining revealed probabilistic beliefs explicitly in the context of an axiomatization of CEU preferences for capacities that can be represented as ‘‘inner
1
measures’’. Otherwise, the special case of probability one beliefs has received quite a bit of recent interest (see Haller (1995); Morris (1995); Sarin and Wakker (1998)); the issue has also connections with that of defining independent product capacities (see Hendon et al. (1996); Ghirardato (1997) and Eichberger and Kelsey (1996); cf. Section 5).
The plan for the remainder of the paper is as follows.
Section 2 sets out the issue of defining ‘‘revealed unambiguous events’’ from a capacity, and establishes criteria for the ‘‘soundness’’ of any proposed definition. These criteria are violated by the simplest natural definitions (Section 3).
In Section 4, capacities are interpreted as ‘‘rank-dependent probability assignments’’;
1
this suggests a definition of unambiguous events with a canonical look to it. It is derived from conditions on preferences whose applicability and appeal are not restricted to the CEU model. All proposed definitions are shown to coincide for the class of convex capacities.
Section 5 characterizes the surprisingly strong implications of unambiguous events for the underlying capacity, and shows that the class of unambiguous events is always an algebra. The latter implies for example that whenever a decision-maker has unambigu-ous beliefs about the marginal distributions of each of a collection of random variables, he has unambiguous beliefs about their joint distribution as well. For convex capacities, this result takes a particularly striking form: if a convex capacity has additive marginals on a product space, it must be a probability measure.
These apparently overly restrictive implications might be accounted for in two ways: they may indicate that the adopted definition is too strong; alternatively, they may show that the CEU model is applicable only when an agent’s probabilistic beliefs take a certain form. In the concluding Section 6, we argue for the latter as the more plausible interpretation.
2. Criteria for the definition of unambiguous events
S
Let S be a finite set of states with[S5n, and letD denote the probability-simplex on S.
S
A capacityn is a mapping from the power set 2 of S into [0, 1] such thatn(5)50,
S
n(S)51, and n(A)$n(B) whenever A$B. It is convex if for all A, B[2 : n(A)1 n(B)#n(A>B)1n(A<B).
The expectation of a random-variable f : S→R with respect to the capacity n is defined as its Choquet-integral
n
E
f dn:5O
f(s )k ?(n(hs , . . . ,s1 kj)2n(hs , . . . ,s1 k21j)),k51
2
withhsk k5j 1, . . . ,n chosen such that f(s )j $f(s ) whenever jk #k.
Let C denote a set of consequences. An act x maps states to consequences, x: S→C,
S S
or, in equivalent notation, x[C . A preference ordering K on C has a ‘‘Choquet
Expected Utility’’ (CEU) representation if there exist a capacityn and a utility-function
u: C→R such that xKy if and only if eu+xdn $eu+ydn.
To simplify argument and notation, we will focus on ‘‘risk-neutral’’ decision-makers with C5R and u5id. As long as the ‘‘true’’ utility-function u is defined on a connected domain C and is continuous, this is without effective loss of generality. Under risk-neutrality, a capacity induces a unique CEU preference-ordering Kn according to
the condition: xKny if and only if exdn $e ydn.
n21 2
Equivalently, this can be written asef dn:5f(s )n 1o ( f(s )k 2f(sk11))?n(hs , . . . ,s1 kj).
The task is to define from a given preference-relation K a collection of ‘‘revealed
n
ua
unambiguous’’ events ! for which the agent is understood to have probabilistic
n
beliefs. Within the CEU-model (which is assumed throughout), this is equivalent to
ua
defining!n directly in terms of the associated capacityndue to the one-to-one relation between the two. Conceptually, a primitive definition of unambiguous events should be in terms of the preference relation as the primitive entity; this point of view is adopted in Section 4 which attempts to provide ‘‘the right’’ definition. On the other hand, the implications of any given definition are more easily described in terms of the capacity representation; likewise, the set of possible definitions is more easily surveyed in terms of the representation.
ua
Thus, we will take a definition of revealed unambiguous events to be a mapping!? :
ua
n∞! . To be satisfactory, it should have the property that for any three events A, B, C
n
such that the value of a probability measure on C is uniquely determined by its values on
ua
A and B, C should be in !n whenever both A and B are, for any capacity n.
ua
Specifically, ! should be closed with respect to disjoint union as well as
com-n
plementation. In the measure-theoretic terminology introduced by Zhang (1997) into
ua
decision-theory, !n must be al-system.
S
Definition 1. A collection![2 is al-system if it has the following three properties:
i) 5, S[!,
ii) A, B[!, A>B55⇒A<B[!.
c iii) A[!⇒A [!.
! is an algebra if it satisfies in addition
iv) A, B[!⇒A>B[!. has ‘‘unambiguous’’, ‘‘probabilistic’’ marginal beliefs about each component of the state, but ‘‘non-probabilistic’’, ‘‘ambiguous’’ beliefs about their joint distribution. In this
ua
case,!n is a l-system but not an algebra.
ua
Furthermore, one will want n on ! to be ‘‘coherently interpretable’’ as a
n
probability; this is captured by
Definition 2.nis probabilistically coherent on! if there exists a probability measure p
S
on 2 that agrees with non !.
S
Note that, due to the requirement that p be defined on all of 2 , ‘‘probabilistic coherence’’ implies additivity of n on ! but is typically stronger, even if ! is a
l-system (see fact 2 below). As shown by the following example, requiring n to be
ua
probabilistically coherent on! is not quite enough.
n
that 90 balls are white or yellow, and that 90 balls are white or red. This is naturally modelled by setting S5hW, Y, R, Bj, and definining a probability measure on the
l-system #5hhW, Yj, hR, Bj, hW, Rj, hY, Bj, 5, Sj by f(hW, Yj)5f(h W, Rj)50.9,
f(hR, Bj)5f(hY, Bj)50.1, f(5)50, f(S)51. Consider the capacity n(A)
ua
5suphf(E)uE[h#, E#Aj, the ‘‘inner measure’’ off(Zhang (1997)). Suppose that!?
ua ua
is such that ! 5#. Then ! satisfies the desiderata mentioned above: it is a
n n
ua 3
l-system, and n on! is probabilistically coherent.
n
Nonetheless, n is not ‘‘truly consistent’’ with the information given. In particular,
n(hWj)50, whilen(hY, R, Bj)50.1; in terms of decision making, betting on the draw of a white ball is dispreferred to betting on the draw of a non-white ball, i.e. 1 s
hY,R,Bj n
1hWj, with 1A denoting the indicator-function of the event A. Since there are at least four times as many white balls in the urn as there are non-white ones, this seems hardly acceptable: it is materially irrational for the decision-maker to prefer to bet on the event
4
that is unambiguously less likely in view of his information. Thus, the capacityn does not fully incorporate the probabilistic information about the events in#. In other words,
ua ua
on the correct definition of ! ,# should not be contained in ! .
n n
Motivated by the above discussion, the requirements on a minimally satisfactory definition of unambiguous events are summarized in the following notion of ‘‘sound-ness’’.
ua ua
Definition 3. A definition of revealed unambiguous events! :n∞! is sound iff, for
? n
To illustrate the force of clause ii), consider again example 1. Here n fails to be
ua ua
probabilistically coherent on! <hWj, whenever! $#. To be sound,nwould need
n n
to satisfyn(hWj)$0.8 and n(hY, R, Bj)#0.2.
ua
If ! is an algebra rather than merely a l-system, the second clause simplifies.
n
ua
Fact 1. If ! is an algebra, the following two statements are equivalent:
n
A trivial example of a sound definition of revealed unambiguous events is the constant
3
This example is similar to example 1.1 of Zhang (1997).
4
mapping n∞h5, Sj for all n. Thus ‘‘soundness’’ of the definition says only that the
ua
events given by ! can be thought of as ‘‘genuinely unambiguous / probabilistic’’; it
n
ua
does not address the issue whether! comprises all ‘‘genuinely probabilistic’’ events.
n
3. Weak definitions don’t work
A particularly simple and straightforward definition of unambiguous events is given by
3 S c
! :5hA[2 un(A)1n(A )51j.
n
3
This however fails miserably: !n is generally not closed under disjoint unions, thus
3
failing to qualify as al-system. Moreover, even if!n happens to be an algebra,n may
3
The example suggests that ! fails to ‘‘build in’’ additivity with respect to events
n
c
outside the partition hA, Aj. A natural move is to strengthen the definition to
2 S
!n:5hA[2 un(A<B )2n(B)5n(A) for all B such that A>B55j.
2 2
! seems on the right track; for instance, it ensures additivity ofn on! whenever the
n n
2
latter is an algebra.! has been adopted with reservations by Zhang (1997), who gives
n
a preference-based characterization of it and notes that it may fail to be a l-system, violating closure under complementation (condition iii) as for instance in example
2 S
example 2, where!n5hA[2 u[A$2j. He responds to this by simply imposing closure
2
under complementation on! ; note that this is in effect a restriction on the domain of
n
2
capacities to which the definition n∞! is applied.
n
2
Yet even if this domain-restriction is accepted, ! is unsound. In example 1, for
n
2 2
instance,! 5#, which makes! unsound as shown above. Indeed,n may even fail to
n n
2
be probabilistically coherent on! .
n
2
Fact 2. There exist capacitiesnsuch that!nis al-system andnis not probabilistically
2
coherent on ! ; in particular, not every q that is additive on a l-system ! can be
n
S
extended to a probability-measure on 2 .
4. A preference-based definition of unambiguous events
5
Consider a risk-neutral decision-maker who has to decide between two acts x and y
c c
such that x –y is hA, Aj-measurable (i.e. constant within A and A ) and such that
xA.y . A decision in favor of x over y can be viewed as accepting the incremental betA
x –y on A. If the decision-maker assigns an unambiguous subjective probability to the
event A, the incremental bet has an unambiguous expectation, and it seems highly reasonable that he should accept this incremental bet if and only if its expectation is positive. Conversely, this condition yields a natural criterion for the non-ambiguity of an event based on preferences over acts.
Definition 4. The event A is K-unambiguous if, for all x, y such that x –y is hA,
c
Aj-measurable, xKy⇔x –yK0.
Example 3. Consider an Ellsbergian four-color urn with 100 balls analogous to example 1. Now, the decision-maker knows that 50 balls are white or yellow, and that 50 balls are white or red. Letting #5hhW, Yj, hR, Bj, hW, Rj, hY, Bj, 5, Sj, and f(hW, Yj)5f(hW, Rj)5f(hR, Bj)5f(hY, Bj)50.5, f(5)50, f(S)51, define the capacity
n*(A)5suphf (E)uE[#, E#Aj.
Consider the preference relationK induced by the capacityn* and the acts x5(21,
n*
9, 9, 29) and y5(0, 10, 0, 20). Then x2y5(21, 21, 9, 9) is hhW, Yj, hR, Bjj -measurable, with e(x2y)dn*54 (which is equal to the intuitively unambiguous expectation of x2y), and thus x2ys 0. For the event hW, Yjto be K
-unambigu-n* n*
ous, it must be the case that xsn*y; however, sinceexdn*54,eydn*55, in fact the converse holds. In other words, the capacity n* does not fully incorporate the given probabilistic information. Indeed, the assignment of a certainty-equivalent of 4 to the act
x seems unacceptable, since, conditional on being informed of the composition of the
urn, the expected value of x is at least 9, whatever the true composition is.
The task of this section is to characterizeK -unambiguous events directly in terms of
n
the capacity; for this, it proves helpful to interpret capacities as ‘‘rank dependent probability assignments’’.
A ranking of states is a one-to-one mappingr: S→h1, . . . ,nj, let5denote the set of
such rankings. The ranking r is a neighbour ofr9(‘‘rNr9’’) iff, for at most two states
S
s[S:r(s)±r9(s), and, for all s[S,ur(s)2r9(s)u#1. A mapping p: 5→D is called a
rank-dependent probability assignment (RDPA) iff for allr,r9such that rNr9, and all
s[S such thatr(s)5r9(s):pr(hsj)5pr 9(hsj).
n S n
For any capacity n, define a mapping p : 5→D by pr(htj)5n(hsur(s)#r(t)j)2 n n(hsur(s),r(t)j). When there is no ambiguity, we will often drop the superscript inp . There is a one-to-one relation between capacities and RDPAs.
5
S
Proposition 1. A mappingp:5→D is a rank-dependent probability assignment if and
n
only if there is a (unique) capacity nsuch thatp5p .
Proof. The if-part is immediate from the definition of an RDPA.
For the converse, in view of the following lemma, one can setn(A)5pr(A) for any r n
such that A5hs[Sur(s)#[Aj. This yields a capacity n with the property that p 5p.
S
Lemma 1. For all A[2 and r,r[5 such that A5hs[Sur(s)#[Aj5hs[Sur9(s)#
[Aj: pr(A)5pr 9(A).
Proof of lemma. Note first that the claim of the lemma is straightforward from the definition of an RDPA for all r, r9 such that A5hs[Sur(s)#[Aj5hs[Sur9(s)#[Aj and such that rNr9.
Now take arbitrary r, r9[5. It is clear that there exists a sequence of rankings hrjjj#k such thatr 5r0 , r 5r9k andrjNrj11for all j,k, and such that A5hs[Surj(s)#
[Aj. Since prj(A)5prj11(A) for all j from the above, one obtains pr(A)5pr 9(A) as desired. j
S
Say thatris comonotonic with x[R if, for all s and t,r(s)$r(t) implies xs#x . It ist
easily verified that Choquet-integration of x amounts to ordinary integration with respect to the appropriate rank-dependent probability measure pr, i.e. thate xdn 5exdpr for any r that is comonotonic to x.
An interpretation of the capacity model and of Choquet-integration along similar lines has recently been advocated by Sarin and Wakker (1998). It also arises naturally from within the classic contribution of Schmeidler (1989), in that his Comonotonic In-dependence axiom is simply the InIn-dependence axiom restricted to comonotonic equivalence classes (classes of acts comonotonic to the same ranking r).
On an RDPA interpretation of a capacity, ambiguity of an event is naturally associated with dependence of the assigned probability on the ranking. Correspondingly, an event is naturally defined as unambiguous if its rank-dependent probability does not depend on the ranking:
1 n
! :5hAup (A)5n(A) for allr[5j.
n r
1
Note that it follows directly from the definition that !n is a l-system and that the
1
definitionn∞! is sound.
n
Remark: Say that A is connected with respect to r if, for all s, s9, s0 such that
2
r(s),r(s9),r(s0), Ałs9[A whenever A$hs, s0j. Then ! can be written as follows:
n
2 n
! 5hAup (A)5n(A) for allr[5 such that A is connected with respect torj.
n r
2 1
From a rank-dependent point-of-view,! looks like an ad-hoc-restricted version of! .
1
That ! is the right definition of unambiguous events is confirmed by the following
n
result.
Theorem 1. The following three statements are equivalent:
1
i) A[! .
n
ii) A is Kn-unambiguous.
c
iii) For all x, y such that y is hA, Aj-measurable, e(x1y) dn 5exdn 1e ydn.
Proof. The implications iii)⇒ii) and ii)⇒i) are verified without much difficulty; by contrast, the implication i)⇒iii) is non-trivial.
S
Definition 5. For A[2 , let ¯ denote the following equivalence relation on 5:
A
c
r¯Ar9 iff, for all s, t such that hs, tj#A or hs, tj#A :
r(s),r(t)⇔r9(s),r9(t).
S
Also, define for r[5 and A[2 an associated ranking r [5 uniquely by the
A
following two conditions:
c
i) for all s[A, t[A : rA(s),rA(t), and ii)rA¯Ar.
The key to the proof is the following lemma.
1
Lemma 2. If A[! , then, for all r, r9 such thatr¯ r9: p 5p .
n A r r 9
Proof of the lemma. Note first that it suffices to prove validity of the claim for neighbouring rankings r and r9, since any two r and r9 satisfying r¯Ar9 can be connected by a chain of neighbouring rankings r1, . . . ,rk satisfying rj¯Arj11.
1 S
Assume thus A[! and rNr9, take any B[2 , and let n(A)5a.
n
The following table describes the rank-dependent probabilities for the events in
c c c c
@:5hA>B, A>B , A>B, A>B j,
E pr(E) pr 9(E)
A>B pr(A>B) pr 9(A>B)
c
A>B a 2pr(A>B) a 2pr 9(A>B)
c c c
A>B pr(A>B) pr 9(A >B)
c c c c
A>B 12a 2pr(A>B) 12a 2pr 9(A>B)
c
From r¯Ar9 and rNr9, it follows that for exactly one s[A and exactly one s[A ,
c c
r(s)±r9(s). Hence pr(A>B)5pr 9(A>B) or pr(A>B )5pr 9(A>B ), as well as
c c c c c c
c c
ranking that is comonotonic with x. Then by the hA, Aj-measurability of y, x1y is
comonotonic with some r9 such that r9¯Ar. By the lemma, p 5pr r 9. Note that
It is also of interest to note that the proper definition of unambiguous events is a live issue only for non-convex capacities; for convex capacities, all proposed definitions coincide.
It is well known that any convex capacity has the following representation:
S
Unambiguous events turn out both to have a surprising amount of structure themselves, and entail surprisingly strong restrictions on the capacity that hosts them.
1
Theorem 2. For any capacityn, ! is an algebra.
n
1 1
Proof. We need to show that ! is intersection-closed. Thus, take A, B[! , and let
n n
Considerr,r9such that r is a neighbour ofr9. From the definitional property of an RDPA it follows thatp(E)5p (E) for at least two E[@. However, in view of (1), this
r r 9
implies that the rank-dependent probability of all four events in@stays the same, and in
particular, thatpr(A>B)5pr 9(A>B).
Now take arbitrary r, r9[5. It is clear that there always exist a sequence of
rankings hrjjj#k such that r 5r0 , r 5r9k and rjNrj11 for all j,k. Sinceprj(A>B)5 p (A>B) for all j from the above, one obtainsp(A>B)5p (A>B) as desired. j
rj11 r r 9
For convex capacities, theorem 2 has a particularly striking consequence: if a convex capacity has additive marginals on a product space, it must be a probability measure.
Corollary 1. Suppose thatnis a convex capacity on S5S13S that is additive on each2 Si
marginal algebra !i5hS2i3AuA[2 j; thenn itself is additive.
3 3 1
Proof. Under the assumptions onn,! $! <! . By proposition 2,! 5! . Since by
n 1 2 n n
1 1 S
theorem 2,!n is an algebra, in fact!n52 . The claim follows, sincen is additive on
1
! . j
n
Theorem 2 is not all; in addition, a capacity is always ‘‘additively separable’’ across its unambiguous events.
For a capacity n, define the set of its ‘‘separating events’’
4 S c S
! :5hA[2 un(B)5n(B>A)1n(B>A ) for all B[2 j.
n
1 4
Theorem 3. For any capacityn, ! 5! .
n n
1 4 1 S
Proof.! #! : Take any A[! and B[2 . Letr be any ranking such that, for all
n n n
c c
s1[A>B, s2[A>B and s3[B : r(s )1 ,r(s )2 ,r(s ).3
By construction,
pr(B)5n(B).
Since r¯ArA by definition, one obtains from lemma 2,
pr(B)5prA(B).
From the interdefinition ofp andn and the definition of rA, one obtains
c
prA(B)5n(A>B)1[n(A<(A >B))2n(A)].
1 2
Finally, since !n#!n,
c c
n(A<(A <B))2n(A)5n(A >B).
c
1 4 4
!n$!n: Take any A[!n, and arbitrary r, r9[5; we have to show thatpr(A)5 pr 9(A).
The key is the following lemma.
4
The claim of the theorem is now easily established.
We have pr(A)5prA(A) (by lemma 3)5n(A) (by definition)5pr 9A(A) (by definition),5p (A) (by lemma 3 again). j
r 9
Remark. Zhang (1997) shows that
4 5 S c
!n5!n:5hA[2 un(A1<B)5n(A )1 1n(B) for all A1#A and B#Aj,
5
considers (and rejects)! as a possible definition of unambiguous events, and gives a
n
4 5
decision-theoretic (almost-) characterization. The intuitive content of ! or ! as
n n
capturing the events to which the agent assigns an unambiguous subjective probability is however not clear. And indeed, as pointed out in Section 6, the decision-theoretic
1 4
definitions of unambiguous events underlying ! and ! diverge outside the CEU
n n
model.
A successful definition of ‘‘revealed unambiguous belief’’ makes it possible to express formally the notion that an agent’s beliefs incorporate a set of ‘‘given’’ probabilities (‘‘set of probabilistic constraints’’), which may be thought of as information about objective probabilities. In the following definition, # describes the set of events about
whose probability the agent is informed of.
S
Definition 6. A probabilistic constraint set is a pair (#,f), where##2 andf:#→[0,
1] is probabilistically coherent on #.
ua
i) n(A)5f(A) for all A[#, and
ua
ii) ! $#.
n
Theorems 2 and 3 yield as a corollary a characterization of the class of capacities consistent with a given set of probabilistic constraints.
S
For #[2 , let #* denote the algebra generated by #, #*:5>h@$#u@ is an
algebraj, and let ^* denote the atoms of that algebra which form a partition of S, ^*:5hF[#*ufor no G,F: G[#*j.
1
Corollary 2.n is ! -consistent with the constraints (#, f) if and only if
n
i) for all A[#,n(A)5f(A), and
S
ii) for all A[2 :n(A)5 o n(A>F).
F[^*
1 1
Proof. ‘‘If’’: By theorem 3 and ii),! $#*$#; hence n is! -consistent with (#,f)
n n
by i).
‘‘Only if’’: i) is obvious.
1
ii) Let^*5hFi i#kj and define Bj5< F . By theorem 2,i !n$^*. Since Bj115Bj>
i$j
c
F , it follows from theorem 3 thatj
n(A>B )j 5n(A>F )j 1n(A>Bj11) for all j: 1#j#k21.
Repeated substitutions yield immediately
n(A)5n(A>B )5
O
n(A>F ). j1 j
j#k
Corollary 2 suggests a natural definition of the independent product of a capacity and
7
a probability measure, for what it is worth. Suppose that S5S13S , A2 25hS13AuA[
S2
2 j. Let a probability f on! be given, as well as a ‘‘marginal capacity’’n on !
2 2 1 1
analogously defined.
Proposition 3. There exists a unique product capacity n (5:n1^f2) such that
1
i) n is !n-consistent with (!2,f2), and
ii) for all A[S , B[1 S :2 n(A3B)5n(A3S )2 ?f(S13B).
Proof. Uniqueness: For s[S , let E2 s5ht[S1u(t, s)[Ej. By corollary 2 and i),n(E)5
o n(Es3hsj), hence by ii),n(E) is uniquely determined by
s[S2
n(E)5
O
n1(Es3S )s ?f2(S13hsj). (2)s[S2
Existence: n defined by (2) clearly satisfies i) and ii).j
7
The charm of proposition 3 lies in the fact that the consistency requirement i) uniquely singles out the product capacity n ^f which has been considered (and compared to
1 2
alternative definitions) by Hendon et al. (1996) and Ghirardato (1997), and also appears in Eichberger and Kelsey (1996). Being a consequence of theorem 3, it critically hinges
1
on the strong definition of unambiguous events! .
n
6. Discussion
The results of Section 5 indicate that a capacity-representation of preferences and probabilistic constraints on beliefs do not live together very harmoniously; in many situations, one will have to give. Which of the two will depend on one’s judgement about which is more fundamental. To us, it seems evident that probabilistic constraints are the more fundamental notion; indeed, it seems hard to even imagine what kind of argument might be adduced that could render probabilistic constraints defeasible.
This judgment is confirmed by the fact that it takes very little to obtain consistency with probabilistic constraints on preferences and beliefs in a satisfactory way. In particular, consistency can be achieved in the MMEU model in which capacities are replaced by closed convex sets of probabilitiesP, and Choquet integration by ‘‘maximin integration’’ exdP:5min exdp.
p[P
In the MMEU-model, an event A is naturally defined as P-unambiguous if p(A)5 p9(A) for all p, p9[P; note that this definition coincides with the one given for capacities whenever the two integration-functionals coincide (i.e. for convex capacitiesn
and their core, cf. proposition 2). Under this definition, it can be shown that the preference-based characterization of unambiguous events in the manner of theorem 1 is preserved, while none of the adverse consequences are entailed.
The latter can be verified by reconsidering example 3. In the MMEU model (but not in the CEU model, as shown above), the specified constraints are consistent with ‘‘complete ignorance’’ with respect to the missing information, i.e.. with setting e1hW,BjdP 5e1hY,RjdP 50. This is uniquely achieved by the set of priors P*5hp[
S 1
]
D up(hW, Yj)5p(hW, Rj)52j; note thatP* is the core of the non-convex capacity n*
8
defined in example 3. It is easily verified that the set of P*-unambiguous events is exactly the l-system #. Note also that the analogue to the problematic separability
condition for unambiguous events as in theorem 3 is not entailed; for instance, for
1
]
A5hW, Yj and B5hW, Rj, we have e1 dB P*52±05e1B>AdP*1e1B>AcdP*, while A isP*-unambiguous.
1
If! is accepted as the correct definition of unambiguous events in the CEU model
n
(for instance on the basis of its equivalence with the class ofK -unambiguous events),
n
theorems 2 and 3 are naturally read as describing epistemic presuppositions of the CEU model. In particular, for the CEU-model to be applicable, the decision maker’s probabilistic beliefs must range over an algebra, possibly the trivial one h5, Sj. It may seem hard to imagine how capacities could possibly be epistemically restrictive, since their definition seems to involve only trivial assumptions (essentially monotonicity).
8
Such an intuition forgets, however, that capacities acquire decision-theoretic meaning only as parameters of Choquet integrals x∞exdn, a point argued extensively in Sarin and Wakker (1998). The class of Choquet integrals, as well as the class of preference orders it serves to represent, is characterized by non-trivial properties which a priori might well be restrictive.
Acknowledgements
The paper develops some of the ideas that have been presented at the Second Conference on Logic and the Foundations of the Theory of Games and Decisions, Torino, Italy, December 1996, in a talk entitled ‘‘Preference and Beliefs without Additivity’’; I thank Giacomo Bonanno, Jian-Kang Zhang and one anonymous referee for valuable comments.
Appendix
Proof of Fact 2
By complexifying example 1, this can be shown with the help of the following lemma.
S
Lemma 4. Suppose !#2 has the following three properties:
i) 5[!,
c
ii) A[! implies A[!,
c
iii) A, B[!•h5j and A>B55imply B5A.
Suppose also that q: !→[0, 1] satisfies, for all A[!:
i) q(5)50
ii) q(A).0 if A±5, and
c
iii) q(A)1q(A )51.
2
Then ! is a l-system, and q can be extended to a capacity n such that ! 5!.
n
Proof of lemma
S
It is straightforward to verify that ! is a l-system. Define n on 2 by n(A)5
suphq(E)uE[!, E#Aj; following Zhang (1997),n may be called the ‘‘inner measure’’ of q. The set-function n is evidently a well-defined capacity; it has the following two properties:
c
c
Verification: i) The assumptions imply A[!, hence, for no E#B, E[!. ii) Similarly,
the assumptions imply: if E#B and E[! then E5A.
Consider A[! and B disjoint from A.
c
If B5A , then n(A<B)5n(A)1n(B) by assumption ii) on q.
c
If B,A , then n(A<B)5n(A)5n(A)1n(B) by properties i) and ii) ofn.
2
This shows that !#! .
n
c
Consider now A[⁄ !. By the assumptions on!, at most one ofhA, Ajcontains some
E[!.
c
Hence by properties i) and ii) ofn, and assumption ii) on q:n(A)1n(A ),1, which
2
shows that ! #!.h
n
Consider now ! and q given by the following table, letting S5T3T with T5ha, b,
cj.
A[! q(A)
ha, bj3T a
hcj3T 12a
T3ha, bj b
T3hcj 12b
hb, cj3hb, cj g
(haj3T)<(T3haj) 12g
5 0
T3T 1
!is easily checked to satisfy the assumptions of the lemma; q satisfies the assumptions
as well whenevera, b,g[(0, 1). Letn denote the inner measure induced by!and q.
2
Then n is probabilistically incoherent on ! 5! whenever a 1b 1g ,1.
n
S
This is seen as follows. Suppose q (5n on!) has an additive extension p on 2 .
Then p(h(c, c)j)$12p(ha, bj3T)2p(T3ha, bj)512a 2b, but also p(h(c, c)j)#g, which implies 1#a 1b 1g.j
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