Information and capacities - Springer Link

Statistical Papers 43, 111-125(2002)

Statistical Papers 9 Springer-Verlag

2002

Information and capacities Jean-Christophe Vergnaud EUREQUA-CNRS, University Paris 1, Maison des Sciences Economiques, 106-112 Bld de l'HSpital, 75647 Paris Cedex 13, France (e-mail: vergnaud @univ-paris I .fr) Received: November 2000; revised version: July 2001

Abstract Most economists consider that the cases of negative information value that non-Bayesian decision makers seem to exhibit, clearly show that these models are not models representing rational behavior. We consider this issue for Choquet Expected Utility maximizers in a simple framework, that is the problem of choosing on which event to bet. First, we find a necessary condition to prevent negative information value that we call Separative Monotonicity. This is a weaker condition than Savage Sure thing Principle and it appears that necessity and possibility measures satisfy it and t h a t we can find conditioning rules such that the information value is always positive. In a second part, we question the way information value is usually measured and suggest that negative information values are merely resulting from an inadequate formula. Yet, we suggest to impose what appears as a weaker requirement, that is, the betting strategy should not b e Statistically Dominated. We show for classical updating rules applied to belief functions that this requirement is violated. We consider a class of conditioning rules and exhibit a necessary and sufficient condition in order to satisfy the Statistical Dominance criterion in the case of belief functions. K e y w o r d s : information value, non additive measure, u p d a t i n g rule, statistical dominance J . E . L c l a s s i f i c a t i o n n u m b e r : D81-D83

* We wish to thank Alain Chateauneuf, Mich~le Cohen, Jean-Yves JaffTay, JeanMarc Tallon and an anonymous referee for helpful discussions and comments. ** EUREQUA-CNRS-University Paris 1, Maison des Sciences Economiques, 106112 Bld de l'HSpital, 75647 Paris Cedex 13 E-mail : vergnaud~univ-parisl.fr

112

Jean-Christophe Vergnaud

1 Introduction While non-Bayesian decision making models such as the Choquet Expected Utility model or the Multi Prior model are intuitively very appealing for representing behavior under uncertainty, they have not yet succeeded in challenging the standard Subjective Expected Utility model in economic theory. One main reason is the difficulties encountered when applying these models in sequential decision making problems which are frequently encountered in economics. A very simple but representative problem is the information value problem considered in decision theory. A decision maker has to make a choice under uncertainty (i.e to choose an act) but can postpone the time of his choice after he gets a new piece of information. The information process enables the decision maker to condition his decision on the piece of information received. From an e x a n t e point of view, this amounts to enlarging the choice set which seems to put the decision maker in a more favourable situation. Indeed, in the Bayesian model where the optimal strategy is defined by using dynamic optimisation techniques that is, by rolling back the decision tree, the optimal strategy is always to accept information and thus the information value is always positive. Yet, if we use the same backward induction technique for a non-Bayesian decision maker, we may find negative information values. Since for many economists, positive information value in decision making is nearly a rationality axiom, such a result has been used to dismiss non-Bayesian models. In this paper, we reconsider this negative result. Previous attempts at restoring a positive information value have essentially focused on backward induction. Backward induction was criticized on several grounds and models of "resolute choice" (cf [7], [8]) have been developed. Yet backward induction has fervent supporters. If one adopt a multi-selves interpretation, that is, if one considers that a sequential decision making is a sequential game among the successive selves of the decision maker, then the backward induction solution is no more than the subgame perfect equilibrium. Backward induction is also very convenient to solve sequential decision problem. Indeed, to find the optimal strategy of a "resolute decision maker", one needs to consider the "normal form tree" which embodies all possible strategies and can be quite large. For this practical reason, we choose here to keep backward induction. The paper is divided into two parts. In the first part, we reconsider the information value problem for Choquet expected utility maximizers and exhibit a necessary condition on the beliefs to get a positive information value. This condition, called Separative Monotonicity, is weaker than Savage's Sure Thing Principle. It appears that possibility and necessity measures satisfy it and indeed, we show that for a certain conditioning rules, the information value is always positive. In the second part, we question the information value criteria and argue that the way this information is evaluated can be reconsidered. Finally, we also argue that a weaker requirement called Statistical Dominance, should be considered. In the belief functions frame-

Information and capacities

113

work and a particular class of conditioning rules, we exhibit necessary and sufficient conditions for positive information value. It appears t h a t these conditions are very strict: the Dempster-Shafer rule and the Bayes rule fail to satisfy it. Section 2 develops the betting framework. In section 3, we examine the information value problem. In section 4 we introduce the Statistical Dominance criteria and show t h a t the usual conditioning rules fail to satisfy it. In Section 5, we define a class of conditioning rules and exhibit necessary and sufficient conditions to get positive information value for belief functions. Section 6 conclude.

2 T h e d e c i s i o n m a k i n g set u p Let ~ be a finite set of states of the world. D e f i n i t i o n 1 A capacity is a function # : 2 ~ -* [0, 1] such that lz (0) = O, # (~) = 1 and VA, B E 2 ~, A C_ B ::~ IJ(A) _ p ( S U C) < ~/~(A U D) _> #(B U n ) . The following result shows that the "separative monotonicity" condition is a necessary condition to get positive value of information. P r o p o s i t i o n 1 If the capacity Iz does not satisfy separative monotonicity,

then for any conditioning rule, there exists a betting problem where the information value is negative. Proof Assume there exists A, B C_ J?, C, D C_ J?\ (A U B) such that #(A U C) > p(B U C), and #(A u D) < #(B U D). Consider q3 = {A U C, B O C}, q3' = {A U D, B U D } , and the partition {A U B, 12\ (A U B)}. First, note that


115

~{AuB,a\(AwB)} ~- ~ a n d ~t{AtgB,a\(AoB)} -~ ~ t Indeed, for the two bet sets, the partition does not enlarge them. We now show that a negative information value necessarily occurs in one of the two betting problems. Indeed, whatever the conditioning rule, there are three exclusive possible cases: either (i)#AuB(A) < IZAuB(B) or (ii)#AuB(A) = tZAuB(B) or (iii) IZAuB(A) > DAuB(B). In case (i) I V ( ~ , {A U B, [2\ (A U B)}) = #(B U C) - #(A U C) < 0 In case (ii), I V (~, {A O B, ~ \ (A U B)}) = M i n {/z(A U C),/~(B U C)} - / ~ ( A U C) < 0 In case (iii) I V ( ~ ' , {A U B, ~ \ (A u B)}) = #(A u D) - # ( B U D) < 0 Separative Monotonicity is a weaker condition t h a n the Sure Thing Principle and there is a quite large class of capacities t h a t satisfy the separative monotonicity condition while failing to satisfy the Sure Thing Principle. It is the case for the possibility and necessity measures. D e f i n i t i o n 5 A possibility measure is a capacity 11 such that VA, S C 1"2,11(A U B) = M a x ( H ( A ) , 1-I(B)) A necessity measure is a capacity N such that VA, B c ~ , N ( A A B) = M i n ( g ( A ) , N ( B ) ) Consider the following examples. Example 1 Let [2 = {wl,w2,w3,wa} and consider /1 and N which are respectively a possibility and a necessity measure. /7 ({031)) ~-~ . 5 , / / ( { 0 3 2 } ) ---~0,/-7 ({033}) = 1, 17 ({034}) :

.S

and for all other E, 17 (E) = Max~e ~ (17 ({w})) Consider the necessity measure N which is the dual of 17 (i.e: VE, N ( E ) = 1 - H(E~))

For all E C {031,w2,w4}, N(E) = 0 N ({033}) • N ({w2,033}) --'~g ({031,033}) ---~N ({031,032,033}) ~-~ .2 g ({033,034}) -~--N ({032,033,034}) ~--- .5

(1)


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N ({Wl,W3,w4}) = N ({Wl,W2,W3,w4}) = 1 The fact t h a t n

= .5 > n ( { , , , 2 } ) = o

while //({Wl,W3}) = / 7 ({w2,w3}) = 1 and the fact t h a t N ( { w a } ) = NC{wx,Wa}) = .2 while N ({w3,w4}) = .5 < N ({Wl,Wa,w4}) = 1 are violations of the Sure Thing Principle. The following results show that the Sure Thing Principle is not a necessary condition for positive value of information in our betting problems 2. 2 (a) If the capacity on the events is a possibility measure II and if the conditioning rule is the Bayes updating rule, then for all betting problems (~, {A1, ..., An}),

Proposition

I V (~, {A1, ..., An}) _> 0 (b) If the capacity on the events is a necessity measure N and if the conditioning rule is the Dempster-Shafer rule, then for all betting problems ( ~ , {A1, ..., A , } ) ,

I V (~, {A1, ..., A , } ) >_ 0 Proof (a) Let B~ be a bet that solves Max~e~TIA~(E) and consider B* = Ui=l,..,n (Ai n B~)3. Let B E ~ . Let us show that / / ( B * ) > / / ( B ) . Define Bi = Ai n B. If / / ( A i ) > 0, the Bayes updating rule gives HA, (E) = ~ . If //(A~) = 0, then VE C A~, / / ( E ) = 0. Thus, since Vi B~ e A r g M a x E e v I I A , ( E ) , ll(B~) > / / ( B i ) and finally, II(B*) = M a x (//(B~))i=l,.., n > M a x (//(Bi))i=l,..,n = II(B). (b) Let B~, B*, B, Bi be defined similarly as in case (a). First, note that we have B* = ni=l,..,n ((Ai n B*) u A~) and B = ni ((Ai n B) u A~). Since N is a necessary measure

N(B*) = Uin~=l,..,n ( N ((Ai n B*) u A~)) and

N ( B ) = Mini=i,..,,, ( N ((Ai n B*) o A~)) By definition of the Dempster-Shafer rule, we have Vi = 1, .., n

g ( ( A i N B;) U A~) = N((Ai n S*) u AT) > N ( B i u A~) = N((Ai n B) u AT) and thus N(B*) > N(B). 2 This result proves also that the possibility and the necessity measures satisfy the separative monotonicity condition. a It is easy to see that with the Bayes updating rule, I1A~ is also a possibility measure.


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The choice of the conditioning rule is crucial. Indeed, we show in the example below t h a t using the Dempster-Shafer rule m a y lead to negative information value for possibility measures while the the Bayes updating rule may lead to negative information value for necessity measures. E x a m p l e ~ Consider again t h e possibility and necessity measures H and N defined in example 1. Let us consider the partition {{wl, w2, w3}, {0-)4}} 9 First, consider ~ = {{Wl}, {w2}} and the Dempster-Shafer rule to update H. Since / / ( { w 1 } ) = .5 > /-/({0)2} ) = 0 and H{~l,wa,ws} ({wx}) = = 0,

I V (~, { { w 1 , 0 ) 2 , r M i n (17 ({wl } ) , H ( ( W 2 } ) )

, { 0 ) 4 } } ) ---~ -- /-/ ((W 1}) = - - . 5

Second, consider ~ = ( ( w l, wa, w4}, {0)2, W3,034}}and the Bayes updating rule to u p d a t e N. Since N((Wl,Wa,w4})

-- 1 > g ((to2,wa,w4}) ----.5

and /{Wl,W2,w3}

({0')1'0")3'0")4}) : N{wl,o)2,wa} ( { w 2 , ( M 3 , w 4 } )

:

1

then

IV

{{w1, w2,033} , {034}}) = M i n ( N ( { w l , w 3 , w 4 } ) , N ({w2,w3, 0.)4})) 4 Statistically

undominated

--

N ({w2,w3, 0.)4} )

=

--.5

strategy

Negative information value is a very puzzling problem. We have insisted on the fact that information should be valuable since it enlarges the choice set. Yet, negative information value suggests that a non-Bayesian decision maker may refuse to be informed. W h y a non-Bayesian decision maker may be information averse is hardly intuitive. The Choquet Expected Utility model allows to c a p t u r e ambiguity aversion in one shot decision problems such as the Ellsberg paradox. Is there a link between ambiguity aversion and information aversion? It is hard to give an intuitive answer and experimental works m a y help to clarify that point. Our opinion is t h a t an ambiguity averse decision-maker may well be an information lover and t h a t the negative information value is only a technical problem: t h e formula we give in definition 3 does not measure correctly how a non-Bayesian decision maker evaluates information a. In the following, we consider t h a t a decision maker will always accept information as soon as it is freely proposed to him. 4 For instance, let us consider the following formula. First for any event B and B partition {A1, ..., An}, note V{A 1..... An} : s --* R such that V~A l ..... An) (W) ---/~Ai(B) if w E Ai (it is the act that gives the conditional Choquet expected utility). Note also v ~{A1..... Am) : ~ --* R such t h a t v ~{At ..... An} (w) =MaX, BE~I~Ai (B)

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To determine the strategy chosen, we use backward induction. Yet, from a prescriptive point of view, we doubt t h a t all behaviors are optimal. Consider the following example.

Example 3 Let ~ = {w l, w2, w3, w4} and consider the belief function # for which the Moebius inverse is m ((Wl, w3)) = .4, m ((w2}) = m ((w4}) = .3, and m(E) = 0 for all other E. Consider ~ = ((O~l,td3},{o~2,w4}}, the partition ({w l, w2 }, (w3, w4 }} and the Dempster-Shafer rule. Computations lead to 4 3 and similarly to 4

3

Thus, there is a unique backward induction strategy B* = {wl, w3}. Remark t h a t there is no ambiguity about the events (Wl, w3} and (w2, wa}. Indeed, ~ ( { 0 ~ l , ~ d 3 } ) -{- ~ ({~d2, 0J4} ) = .4 -b .6 ~-~ 1

and we are sure t h a t there is a .4 probability for {0J1,0.~3} and a .6 probability for (w2,w4}. Thus, the strategy B* = {wl,w3} is statistically dominated by (it gives less chance of winning than) the strategy {w2,w4} which is avalaible in ~3. We may complain t h a t it is a suboptimal strategy 5 Let us define more precisely a criterion of statistical dominance. For t h a t , we concentrate on the case of belief functions (i.e c o - m o n o t o n e capacities) and we adopt a usual interpretation: the core of the belief function is the set of probability distributions containing the real but unknown probability distribution. For any event E, ~(E) = Minpecor~(~,)p(E ) < Maxpecor~(~,)p(E) = 1 - p(EC). Thus, for two events A, B, if #(A) > 1 - #(BC),we are sure t h a t A has a higher probability t h a n B. D e f i n i t i o n 6 A bet B statistically dominates a bet B' if ~(B) > 1 -p(B'C). if w E A i . f v~A ` ..... A,,}dD is the Choquet expected utility of conditional Choquet expected utility and it differs from p(B)(contrary to the additive case where the expected utility of the conditional expected utility is equal to the ex ante expected utility). Consider now IV* (~3, (A1,...,An}) = f v~A` .....A,,}dp-

MaxBev f v~A1..... A , ~ ) d ~ . The idea is to use the dynamic agregator to evaluate ex ante the bets. Since Choquet expected utility preserves dominance, it is easy to see that this value will always be positive. 5 To developp a similar example of a statistically dominated backward induction strategy using the Bayes updating rule or the full Bayesian updating rule, just consider the same example with a slight modification of the Moebius weights: m ({o.~1, td3}) -~- .6, m ({oJ2}) = m ({o~4}) = .2.


119

To ask that the backward induction strategies in a b e t t i n g problem ( ~ , (A1, ..., An}) should not be statistically dominated by a b e t t i n g strategy available in ~{AI ..... An} is a minimal requirement. In a sense, it is a weakening of the positive information value criterion. 3 Let (~3, {Ax, ..., Am}) be a betting problem. I f ~ is a belief function and if there exists B 9 ~j{A1 ..... A,,} which statistically dominates a backward induction strategy B*, then there exists a negative information value in that betting problem. Proposition

Proof Let B 9 ~{AI ..... A.) statistically dominates B*: # ( B ) > 1 - #(B'C). Consider ~3' = ( B , B*}. By definition, B* is also a backward induction strategy in the betting problem ( ~ ' , (A1, ..., An}). Since/z is a belief function, 1 - / ~ ( B *c) _>/~(B*) and it is easy to see that I V (~, {A1, ..., A n } ) < O. Example 3 show that this minimal requirement fail to be satisfied for the classical updating rule we have considered till now.

5 Dempster-type

rules and Statistical dominance

In this section, we consider a certain class of updating rules, t h a t we call Dempster-type rules, and examine whether there exist some rules for which the backward induction strategies are never statistically dominated by any other available strategy. To introduce these rules, let us first consider w h a t is going wrong with the Dempster-Shafer rule for instance. The problem of conditioning a capacity on an event A can be thought in terms of reallocating the Moebius weights to the events included in A. The Dempster-Shafer rule does the following: - the Moebius weights already allocated to events included in A remains allocated to these events, - the Moebius weights allocated to events which are not included in A b u t have a non-empty intersection with A are reallocated to this intersection, - the others Moebius weights are dropped (that is, the weights allocated to events included in A c) - the reallocated weights are normalized. Thus, the Moebius inverse mA of the capacity/~A is such that m A (B) = 0 ~.~EC_Ac m( BU E)

if B ~ A and m A ( B ) = 1-~(A~) if B C A. In example 3, t h e weight m ({wl,w3}) was reallocated to {Wl} when conditioning on {Wl, w2} while it was reallocated to {w3} when conditioning on {wa, w4}. Thus, in the backward induction process, this weight plays a role twice while t h e weights m ({w2}) and m ({w4}) play only once. Let us examine this reallocation question more generally. W e consider a general class of updating rules. Let us first describe how these rules handle the Moebius weight reallocation:

120


the Moebius weights already allocated to events included in A remain allocated to these events, a part 45(A, B) of the Moebius weights allocated to events B which are not included in A b u t have a non-empty intersection with A are reallocated to A A B, the others Moebius weights are dropped (that is, the weights allocated to events included in A ~) - the reallocated weights axe normalized. -

-

-

More formally, the definition is the following: D e f i n i t i o n 7 An updating rule is a Dempster-type rule if (a) there exists ~5: ( 2 ~ \ 0 ) • 2 ~ --~ [0, 1] such that (i) q~(A, B) = 1 if

B c_ A, (ii) qS(A, B) = 0 if B C A c (b) the Moebius weights are such that mA (B) = 0 if B ~ A, mA(B) =

EEC_Aeq~(A, B U E ) . m ( B U E) ECC.A(EECA e ,~(A, C U E).m(CU E))

if B C_ A and if the denominator is non null. The Dempster-Shafer rule is the special case where 45(A,B) = 1 if B N A ~ @ while the Bayes updating rule corresponds to q~(A, B ) = 0 if B ~ A. 6 The next proposition shows t h a t in order to fulfill the statistical non-dominance requirement, a very strict condition is needed. 4 Let # be a belief function and consider a Dempster-type rule. The following assertions are equivalent. (i) For all belief functions p and for all betting problems (~, {A1, ..., An}) there is no backward induction strategy which is statistically dominated by a betting strategy in ~(Aa ..... A.}, (ii) For all partition {A1, ..., An}, for all B C ~2, ~']4/A,~B q~(Ai, B) < 1 and ifVi Ai ~= B, ~-]4=l,..,n ~(Ai, B) = 1.

Proposition

Proof (i)~(ii) First, suppose there exists a partition {A1, ..., A , } and B C_ f2, such that ~-]4/A,~:B qS(Ai, B) = ~o > 1. Let us exhibit a betting problem with a statistically dominated backward induction strategy. There exists e > 0 such that ~ > ~.~--r _ Let us consider the belief function # associated to the following Moebius inverse: Vi s.t. Ai ~ B and ~5(Ai, B) > 0, m ( A i \ B ) = 9 (A,,B) (89 + e), m(B) = 89- e and for all other E C / 2 , re(E) = O. Let

v =

(A, n B)}

o The Pull Bayesian updating rule is not a Dempster-type rule since a weight may be reallocated to several events.


121

Thus we have p (tOi/A, gS & o(Ai,B)>oA,\B) = 89+ e while 1

l*((U,/a,~s,,,,a,,o,>o(A'NB)))=~-e For i s.t. Ai ~ B,

ma,(AikB) =

~o

(89+

O(A,,B). (89 + e) + # ( A , , B ) . (89 - e)

= PA, ( A i \ B )

while

mA~ (A~ n B) =

#(A,,B).( 89

O(A,,B). (1 + e) + #(A,,B). (89 - e) ~o

= #At (Ai N B)

By definition of e,

pA,(Ai N B) > #A,(AiXB) and thus [Ji/Ai~B & ff'(Ai,B)>0 (Ai N B) is the backward induction strategy which is statistically dominated by Oi/A~gB & O(A~,B)>oAi\B. Suppose there exists a partition {A1, ..., An} and B C_ Y2, such t h a t Vi A~ ~ B and Y]~i=I,..,n #(Ai, B) # 1. If ~i=l,..,n #(Ai, B) > 1 we come back to the previous situation. Let us suppose that ~i=l,..,n #(Ai, B) = ~o < 1. There exists e > 1 such that ~o < L ~ . Let us consider the following Moebius inverse:

Vi m ( A i \ B ) = #(Ai, B).e +

(1 - e

-

ecp)

m(B) = e

?2

and for all other E C J2, m(E) = 0. Let ~ = {o~A~\B, B}. Thus we have

# (U,=I,..,nAi\B) = 1 - p ((Oi=l,..,nAi\B) c) = 1 while #( B ) = For all i,

mA,(Ai\B) =

#(Ai, B).e + (1-,-,~,) n #(Ai, B).e + ( 1 - ~ o ) + #(Ai, B).E

= #A, ( A i \ B )

while

mA, (A~ r B) =

9 (A,, B).~ = #a, (Ai n B) #(Ai, B).e + (1-,~,e) + 4'(Ai, B).e

By definition of e, lZA,(Ai\B) > IZA,(Ai f) B) and thus Ui=l,..,nAi\B is the backward induction strategy which is statistically dominated by B. (ii)=~(i) Let us consider a partition {A1, ..., An}. Let B* and B be such that Vi/ZA, (A~ N B*) = #A, (S~) _> #A, (Ai N B) = PA, (Bi).

122


Let us show that 1 - # (B *c) > p ( B ) . This result implies that if B* is a backward induction strategy for the betting problem (~, {A1, ..., A , } ) there is no event in q3(AI .... ,A.} t h a t statistically dominates B*. Consider ~' : {A1, ..., An} x 2 n --* [0, 1] such t h a t qS'(Ai, B) = ~(A,, B) VAi ~ B, and 4"(Ai, B) = 1-EJ/A#~s~(A~'B) #{j/A~C_B} We have that VB, ~-]i=1,..,, r B) = 1. For i such t h a t Vi IZA, (t3[) >_tZA, (Bi) is equivalent to

(E~C_A$r Ai'C u E)'m(C U E)) CO_B"

Let us consider the following 3 cases: (a) B~, Bi ~ Ai. T h e n VC C_ B~ #(A~, C tA E) = 4t'(A~, C tA E ) ( i d e a for C C B~) and #A, (B~) > #A, (Bi) is also equivalent to

z (ZO',A, o

ccs. \EC_A7 > m

~_, ( ~_,or CC_B~

E):m(CU E))

%%EC_A i

(b) B~ = A~. It is straightforward t h a t we have:

cc_B. >_ ~-~ ,(EC_A~-~ ~ ~'(A,,CUE).m(Cu (c) B~ ~ A,, Bi = Ai. T h e n #m, (B$) > IZA,(Bi) is equivalent to

cc_B.


123

and in particular we have that

m(C U E ) ) /

~'~CC_B,\B7( ~EC_A~4'(A,,C t3 E).

= O. Since B~ r 0,

CC_BAB" =

(E~C_A74~(Ai'C UE)'m(C OE)) (Zc~"(Ai, CUE).m(CUE)) =0

~

CC_Bi\B; kEC_Ai

O n the whole, we have that

~_, ( E ~'(A',CU E)'m(CU E)) C/CC_B,,CnB;#0 kEgA~

Summing for all i this inequalites, we get that

i/B~=Ai

C/CC_Bi,CnB~O,CnB~#O

~.

Reorganizing the left t e r m of this last inequality, we observe that it is equal to

CC~/CnB*#O

i/CnAiCB i

i/B~=Ai,CnB~#O,CnB~ c

By definition of qS', 9 '(A,, i/O N A i C_B ~

C) +

E

4"(A,,C) < 1

i/ Bi = Ai ,ONB~ ~ , C N B ~ c

for all C. Since # is a belief function, the left terni is inferior to

mCC) ccn/cns*#o

= 1- #(B "c)

124

Jean-Christophe Vergnaud Reorganizing the right term, we get

CC_B

i/CNA~C_BI

_> Cc_T2/CnB#O,CnBe#$

i/CnA~CB~

since/~ is a belief function. Thus, we prove t h a t 1 - # (B *c) > #(B). We see t h a t the Dempster-Shafer rule and the Bayes rule are far from fulfilling condition (ii) of proposition 4. 4i such t h a t 4~ (A, B) = #(AnB) #B gives a rule t h a t satisfies this condition. Condition (ii) has much to do with additivity and we may feel that the weak statistical non-dominance requirement still implies a lot.

6 Concluding remarks The results we find show that we can be mildly optimistic for non-Bayesian models. First, we proved that contrary to a c o m m o n opinion, probabilistic beliefs are not necessary to get positive information values. Yet, the class of capacities for which the information value is positive is relatively small: most of the convex capacities fail to satisfy Separative Monotonicity. However, we suggested that, by reconsidering the way information is evaluated, it may be possible to restore positive information value. T h e criterion we introduce, Statistical Dominance, seems to be the minimal prescriptive condition one has to impose on chosen strategies. This requirement is equivalent to a technical condition for the class of conditioning rules we have considered. This condition can be interpreted as a global consistency condition: the way a decision maker updates after he gets a message should not only depend on the message but should also take into account counterfactual updating, that is, the way he would have updated had he received another message. In the literature about updating, this issue is ignored. Our feeling is t h a t this issue is also conceptually important for updating: updating does not only depend on the initial beliefs and the particular message received but also on the counterfactual messages.


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