Directory UMM :Data Elmu:jurnal:M:Mathematical Social Sciences:Vol38.Issue1.Jul1999:

Qualitative probabilities on

l

-systems

* Jiankang Zhang

Department of Economics, Social Science Centre, University of Western Ontario, London, N6A 5C2, Canada

Received 1 March 1998; received in revised form 1 July 1998; accepted 1 August 1998

Abstract

The class of unambiguous events in a state space is naturally modeled as al-system. Given a binary ‘likelihood’ relationKdefined on al-system, this paper provides necessary and sufficient conditions such that Kcan be represented numerically by a convex-ranged, countably additive probability measure. 1999 Elsevier Science B.V. All rights reserved.

Keywords: Qualitative; l-systems

1. Introduction

Given a state space S and a binary relationK_{on a class}!_{of events or subsets of S, a}

number of papers have described necessary and sufficient conditions in order that K

admit numerical representation by a probability measure. For a finite state space see Kraft et al. (1959); Scott (1964); for an infinite state space, see Savage (1954); Villegas (1964); Fishburn (1970, 1986); Chateauneuf (1985) where the representing probability measure is convex-ranged. The cited results help to axiomatize decision theories in which preference is based on probabilities and these represent beliefs about likelihoods of events.

In all of the above studies, it is assumed that!_{is a}s-algebra. This a priori restriction is problematic for the following reason: The Ellsberg (1961) Paradox and related evidence show that many decision-makers do not attach probabilities to some events, namely to ‘ambiguous’ ones. In other words, probabilities are assigned only to ‘unambiguous’ events. But the collection of such events is often not a s-algebra because, as illustrated shortly, it is not closed with respect to intersections. On the other hand, it is intuitive that the collection of unambiguous events is closed with respect to

*E-mail address: jzhang2@julian.uwo.ca (J. Zhang)

complements and disjoint unions. Thus, as argued in Zhang (1997); Epstein and Zhang (1997), a l-system is a more appropriate mathematical structure for modeling the collection of unambiguous events.

This paper provides necessary and sufficient conditions such that the binary relation

K_{on a}l_{-system can be represented by a convex-ranged, countably additive probability}

measure.

To illustrate the failure of ! _{to be an algebra, consider the following example taken}

from Zhang (1997): There are 100 balls in an urn and a ball’s color may be black (B), red (R), grey (G) or white (W). The sum of black and red ball is 50 and the sum of black and grey ball is also 50. One ball will be drawn at random. It is intuitive that the unambiguous events are given by

D

!_{5hf, B,R,G,W , B,G , R,W , B,R , G,W ,h j h j h j h j h jj (1.1)}

where each of these events has the obvious probability. Observe thathB,GjandhB,Rjare in!_{, but that hBj5hB,Gj}>hB,Rjis not in !_.

2. Preliminaries

2.1. l-systems

Let S be a nonempty set. Say that a nonempty class of subsets!_{of S is a}l-system, if

l.1 S[!_; c

l.2 A[!⇒_A [!_{; and}

l.3 A [!_{, n51,2, . . . and A} >A 55,;i±_j⇒< A [!_.

n i j n n

These properties are intuitive if we interpret! _{as the collection of all events that are}

assigned probability by the decision-maker. For example, if she can assign probability to

c

A, then it is natural for her to assign the complementary probability to A . Forl.3, if she can assign probabilities to the disjoint events A and B, then it is natural for her to assign the sum of these probabilities to A<B. On the other hand, when these events have a

nonempty intersection, the union may be ambiguous. Equivalently, as we have seen, the intersection of two unambiguous events may be ambiguous.

We have the following lemma (see Billingsley, 1986):

Lemma 2.1. A nonempty class of subsets ! _{of S is al-system if and only if}

l.19 5, S[!_;

l.29 A, B[! _{and A}#B⇒B\A[!_{; and}

l.39 A [! _{and A} #A , n51,2, . . . ,⇒< A [!_. n n n11 n n

p: A→[0,1]

is a countably additive probability measure over! _if:

P.1 p(5₎50, p(S )51; and

P.2 p(< A )5o p(A ),;A >A 55_{, for all i}±_j.

n n n n i j

Denote by (S,!_{, p) al-system probability space. A probability p is convex-ranged if}

for all A[! _{and 0,r,1, there exists B},A, B[!_{, such that p(B )5rp(A). A}

probability measure p is nonatomic if for all A[! _{with p(A).0, there exists B},A, B[!_{such that 0,p(B ),p(A). It is well known that p is convex-ranged if and only if it}

is nonatomic when ! _{is a s-algebra. However, it is an open question whether such}

equivalence is valid on a l-system.

2.2. Qualitative probabilities

Let K _{be a binary relation} !_. K_{is a qualitative probability if}

Q.1 K_{is a weak order (reflexive, complete and transitive);}

Q.2 AK5_{for all A}[!_;

Q.3 Ss5_{; and}

Q.4 A>C5B>C55 _{implies [A}K_B⇔_A<CK_B<C].

A countably additive probability measure p on ! _represents K_if

AK_B⇔_{p(A)$p(B ),}

for all A and B in!_.

`

The sequencehA j in!K_{-converges to A}[!_{if for any events A , A* in}!_with

n n51 *

A a_Aa_{A*, there exists an integer N such that}

*

A a_A a_{A*, whenever n$N.}

* n

3. Main theorem

Given a qualitative probability K _{on a} l_-system !_{, can we find a representing}

convex-ranged probability measure? Necessary and sufficient conditions are given next. Denote by

1₍5)5hA[!_:A|5j.

n

A partition hAj of S in ! _{is a uniform partition (u. p) if A} |_A |_{? ? ?}|_{A .}

i i51 1 2 n

Theorem 3.1. Let ! _{be a l-system and} K _{a qualitative probability on} !_{. Then there}

exists a convex-ranged, countably additive probability measure p on! _representingK

n

Moreover, under Conditions (a) –(c), the representing measure p is unique.

Conditions (a)(i) and (a)(ii) are similar to Savage’s fine and tight ones, respectively. Nonatomness is derived directly from Condition (a)(i). Villegas (1964) employs a monotone continuity condition that applies both to increasing and decreasing sequences of events. His condition is equivalent to our Condition (b) when! _{is a}s-algebra. That

`

However, the Eq. (3.1) is not true generally if ! _{is only a l-system. For example,}

consider a qualitative probability K _{on Eq. (1.1) as follows:}

B,R,G,W s _B,G s_B,R s _R,W s _G,W s5.

h j h j h j h j h j

c c

Then,hB,Gjsh_B,Rj_{, but} h_B,Gj ₅h_R,Wjsh_G,Wj₅h_B,Rj _{. This illustrates that Villegas’s} argument does not apply when the domain is only al-system. Lemma 4.2 in Section 4 gives a sufficient condition for Eq. (3.1) on al-system!_{. The additional axiom (c) is}

adopted here to compensate for the fact that!_{is not a}s-algebra. Obviously, Condition (c) follows from Q.4 if ! _{is an algebra.}

4. Proof of Theorem 3.1

We prove Theorem 3.1 in this section. Because necessity is routine, we provide only the proof of sufficiency. Throughout,K_{is a qualitative probability on the} l_-system!_.

By Q.4,

The following two lemmas are similar to the ones in (Villegas, 1964, p. 1790). For completeness, we provide them here.

Lemma 4.3. LetK _{be a qualitative probabilityl-system}! _{satisfying Condition(b). If}

hA[!_:i[Ijis an infinite collection of disjoint events, then, given any event A[!_with i

9

As5, there is only a finite number of A ’s such that As_A.

i i

Proof. Assume to the contrary, that there is a sequence of disjoint events hA :ni_n 5

`

1,2, . . .j such that Ai_ns_{A for all n. Consider the sequence S}m₅<n5m A . Evidently,i_n

`

S o> S 55and S K_A s_{A contradicts Condition (b).}

m m51 m m i_m

Lemma 4.4. LetK _{be a qualitative probability}l_-system! _{satisfying Condition(b). If}

hA jis a decreasing sequence in ! _{such that A} A_{A \A , then}h_A jK_-converges

n n11 n n11 n

to5.

Proof. Because the sequencehA \An n11:n51,2, . . .jis a collection of disjoint events in

!_{, the conclusion follows from Lemma 4.3. j}

`

Lemma 4.5. Let K _{satisfy Conditions (a) –(c) and let} h_A j _{be an increasing}

n n51 sequence of events in !_{. Then:}

`

The following lemma is adapted from (Fishburn, 1970, pp. 195–198).

1

Lemma 4.6. Let K _{satisfy Conditions (a) –(c). Then it also satisfies the following:}

(C1) If B,C[! _{and B}#C, then 5A_BA_CA_S.

(C4) (A_{) If A, B, C, D, B}<C[!_{, A}>C5B>D55 and AA_{B, C}A_{D, then A}<C

A_B<D.

(C4) (|_{) If A, B, C, D, B}<C[!_{, A}>C5B>D55, A|_{B and C}|_{D, then A}<C|

B<D.

(C5) If5a_A[!_{, then A can be partitioned into two events B and C in}! _{such that}

5a_{B and} 5a_C.

(C6) If A, B, C[!_\1₍5) are pairwise disjoint, AA_{B and B}a_A<C, then there

exists D[!_{, D},C such that 5a_{D and B}<Da_A<(C\D).

(C7) If A, B[!_\1₍5) and A>B55, then B can be partitioned into C and D in ! such that CA_DA_A<C.

(C8) If5a_A[!_{, then A can be partitioned into B and C in} ! _{such that B}|_C.

n

(C9) If5a_A[!_{, then for any positive integer n there is a 2 part partition of A in} ! _{such that} | _{holds between each two events in the partition.}

Proof.

C1: Let B, C[! and B#C. By l.29, C\B[!. By Q.2, 5A_{C\B. By Q.4, B}55< BA_{(C\B )}<B5C. Clearly, 5A_{B and C}A_{S.C4 (}A_{): B}<C[!⇒B\C, C\B[!_{. A}A_B

and Q.4⇒

A<(C•B )A_B<(C•B )5B<C. CA_{D and Q.4}⇒

B<C5C<(B•C )A_D<(B•C ).

Thus,

A<(C•B )A_D<(B•C ).

Because B>C is disjoint from A<D,

A<C5A<(C•B )<(B>C )A_D<(B•C )<(B>C )5D<B. C4 (|_{): Implied by C4(}A_).

Therefore,5a_{C. By Condition (a)(ii), there exists a partition}h_{D ,D , . . . ,D} j_{of C in}!

1 2 n

such that Dia_{C, i51,2, . . . ,n and B<D}ia_{A<C, i51,2, . . . ,n. If all D are in}i 1₍5), then C5<D_i,CD is also ini 1₍5) by Lemma 4.1, a contradiction. Therefore, there exists

Di,C such that5a_{D and B}i <Dia_A<C. By C5, there exist a partitionhD9,D0jof D ini ! _{such that} 5a_D9AD0_{. Thus,}

B<D9<D0 5B<Dia_A<C5A<(C•D9)<D9.

By Q.4, B<D0a_A<(C\D9) and, using also D9AD0 _{and B}>D95B>D055,

B<D9A_B<D0a_A<(C•D9).

C7: Let A, B be two disjoint events in !_,5a_{A and} 5a_{B. If B}A_{A, the conclusion}

follows easily from C5; take any partition of B into C and D with 5a_CA_D.

Assume that Aa_{B. By Condition (a)(ii), there is a partitionhG 1,G , . . . ,G j of B in}

h 2 n

! _{such that G}a_{A, i51,2, . . . ,n. Proceed precisely as in (Fishburn, 1970, p. 196).}

i

C8. By the arguments in (Fishburn, 1970, pp. 196–197), there exists a sequence

hB ,C ,D j,hB ,C ,D j, . . . ,hB ,C ,D j, . . . of three part partitions of A in!_{, such that}

1 1 1 2 2 2 n n n

for each n$1,

(a) Bna_Cn<D and Cn na_Bn<D ,n

(b) Bn#Bn11, Cn#Cn11, Dn11#D ,n

(c) D_n11A_{D \Dn n11.}

It follows that > D [!_,5a_{D and that D contains the two disjoint and equally likely}

n n n n

events Dn11 and D \Dn n11. From Condition (b) and Lemma 4.4, >nDn|5. Let

` ` `

S D S D

B5

<

B and Cn 5

<

Cn <

>

Dn .

n51 n51 n51

2

Because B>G55 and BA_{B, we have}

Conclude from the claim and Eqs. (4.1)–(4.4) that

` `

S D S

D

B <D a_B <GA_B<Ga

<

_C •GA

<

_C•_D a_{C .}

n n n n i n n

n51 i51

This contradicts the construction C a_{B <D from Condition (a).}

n n n

Proof of Sufficiency in Theorem 3.1. Let Conditions (a)–(b) hold. Let

n n

Prove sufficiency through a series of steps following Fishburn. The proofs of Steps 3–5 are as in (Fishburn, 1970, pp. 198–199).

n

u.p. of S for which there exist A and B , unions of elements of this partition, such thatn n n n n n

A >B 55_{, A} [C(k(A,2 ),2 ), B [C(k(B,2 ), 2 ), A A_{A and B}A_{B. The key is to}

n n n n n n

show that

An<BnA_A<B. (4.7)

Given this, the proof may be completed as in Fishburn. To prove Eq. (4.7), one might try to invoke C4, but it is inapplicable because it is not certain that A<B [!_{. This}

n

added hypothesis, absent in the standard case where! _{is as-algebra, requires a slight}

variation of the argument in Fishburn.

n ₂n

Prove that there exists another 2 uniform partition Ch ji i51 such that

<

C |_{A ,}

<

_C |_{B and}

<

_C <A [!_,

uniform partition of S; and similarly for B .) Now C4 may be applied to deduce Eq.n

Step 7: p is countably additive: It follows from Condition (b) and the convex range of p

on !_.

Step 8: p is convex-ranged: The proof is identical to (Fishburn, 1970, p. 199).

Acknowledgements

I especially thank Larry Epstein for his valuable suggestions, continuous encourage-ment, intellectual stimulation, time and interest which he gave me during the course of writing this paper. I wish to thank the anonymous referee of this journal for suggesting substantial improvements, particularly in Condition (a) of Theorem 3.1. All errors are my responsibility.

References

Billingsley, P., 1986. Probability and Measure, John Wiley and Sons.

Chateauneuf, A., 1985. On the existence of a probability measure compatible with a total preorder on a Boolean algebra. Journal of Mathematical Economics 14, 43–52.

Ellsberg, D., 1961. Risk, ambiguity and the savage axioms. Quarterly Journal of Economics 75, 643–669. Epstein, L., Zhang, J., 1997. Subjective probabilities on subjectively unambiguous events, Mimeo, University

of Toronto.

Fishburn, P.C., 1970. Utility Theory for Decision Making, Wiley, New York.

Fishburn, P.C., 1986. The axioms of subjective probability. Statistical Science 1, 335–358.

Kraft, C., Pratt, J.W., Seidenberg, A., 1959. Intuitive probability on finite sets, Annals of Mathematical Statistics 30.

Savage, J.L., 1954. The Foundations of Statistics, John Wiley, New York.

Scott, D., 1964. Measurement structures and linear inequalities. Journal of Mathematical Psychology 1, 233–247.

Villegas, C., 1964. On qualitative probabilitys-algebras. Annals of Mathematical Statistics 36, 1787–1796. Zhang, J., 1997. Subjective ambiguity, expected utility and Choquet expected utility, Mimeo, University of