A simple sufficient condition for the unique representability of a finite qualitative probability by a probability measure

JOURNAL

OF MATHEMATICAL

PSYCHOLOGY

33, 91-98 (1989)

Theoretical

Note

A Simple Sufficient Condition for the Unique Representability of a Finite Qualitative Probability by a Probability Measure L. VAN LIER C.E.M.E.

Universitk

Libre

de Bruxelles

Let X = { .r, , xz,.... x,} be a finite set and Q be an algebra of subsets of X called events. Let 2 be a qualitative probability relation on Q x Q. A probability measure p is said to uniquely agree with 2 if, for all A and B in 52, A 2 B if and only if p(A) > p(B), and p is the only probability measure with that property. We give a sufficient condition for the existence of a uniquely agreeing probability measure which is signiticantly simpler and more general than 0 1989 Academic Lute’s (1967, Annals of Mathematical Stutistics, 38, 780-786) condition. Press. Inc.

1. INTRODUCTION Given a finite set X= (x1, x2,..., xn), and an algebra B of subsets of X which are called events, a binary relation 2 on 52 x Q is a qualitative probability if it satisfies Axioms l-3 below. For A and B in Q, A 2 B is then interpreted as meaning that A is not less probable than B. The asymetric and symetric parts of 2 are denoted by > and m, respectively. AXIOM

(1) (2)

1. (Q, 2 ) is a weak order, i.e.: connectedness: A, BEG!*AkB or B2.A transitivity: A 2 B and B 2 C =- A 2 C.

AXIOM 2. AXIOM

X> @ and AZ@, for aN A in Q.

3. AnB=AnC=@=sBZC$andonlyifAvBkAvC.

A probability measure p on (X, 52) is said to represent the qualitative probability 2, or to agree with that relation, if for A, BE 52, AZ B if and only if p(A) 3 p(B). Clearly, any probability measure p on (X, 0) defines a qualitative Requests for reprints should be addressed to L. Van Lier, Rue General Mac Arthur, Bruxelles, Belgium.

20, 1180

91 0022-2496/89 $3.00 480/33/l-7

Copyright Q 1989 by Academtc Press. Inc. All rights of reproduction in any form reserved

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probability relation that agrees with it. But it has been established (Kraft, Prat, & Seidenberg, 1959) that Axioms 1-3 are not sufficient for the existence of an agreeing probability measure. Sufficient and necessary conditions due to these authors do not ensure unicity and are quite complex. Hence, the question remains how to find simpler conditions that are easier to justify intuitively, especially when the qualitative probability is thought of as being subjective. If an individual is presented with uncertain events, but has little or no knowledge about the mechanisms that generate them, it cannot be assumed that this person has access to their objective probabilities. However, in certain circumstances, it can be hypothesized that the individual evaluates the relative uncertainties of events rationally, that is, in accordance with Axioms 1-3. This is what is referred to as a subjective (de Finetti, 1937) or personal (Savage, 1954) qualitative probability. This paper is concerned with the additional condition under which the qualitative probability can be represented by a unique probability meaure. In their book on measurement theory, Krantz, Lute, Suppes, and Tversky (1971) state two alternative simple conditions (Axioms 4’ and 4” in this paper) which, when combined with Axioms 1-3, ensure the existence of a unique probability measure. The first of these conditions (Axiom 4’) is due to Lute (1967) and is not very intuitive. And the second condition (Axiom 4”), which is due to Suppes (1969), has the implication that all atoms are equivalent. If X= { 1,2,..., n}, Q is the set of all subsets of X, and (i) > 0 for all i, then Suppes’s condition implies that (i> - {if for all i, j. Hence, p( ( i} ) = l/n. We propose a new condition, namely Axiom 4, which is more general than Lute’s (1967) condition and at the same time is comparable in simplicity to Suppes’s (1969) condition. In Section 2 we present the axioms and the proof of our main result. Then in Section 3 we use results of Fishburn and Odlyzko (1987) to evaluate, for small sets, the relative generality of the alternative axioms presented in the previous section.

2. THE AXIOMS AND THE RESULTS Lute’s (1967) condition

(Axiom 5 in Krantz ef al., 1971, p. 207) follows:

Suppose that (X, 0, 2 ) is a structure of qualitative probability. A > C, and B 2 D, then there exist C’, D’, E E D such that: AXIOM

4’.

If A, B, C, DE Sz are such that A n B = 0,

(i) (ii) (iii) (iv)

E-AuB C’ n D’ = 0 E contains C’v D’ C“- C and D’ -D.

FINITE QUALITATIVE

A much simpler condition 1971, p. 217): AXIOM

4”.

93

PROBABILITY

is Suppes’s (1969) condition

(Axiom 6 in Krantz et al.,

If A 2 B, then there exists C E Sz such that A w B v C.

As mentioned earlier, this axiom leads to an obvious structure in the presence of Axioms l-3: all atoms are equivalent. An event is an atom if it is non-null (i.e., not w@) and such that any other event that is contained in it is either a null event or just as probable as that event: A E Q is an atom if A > 0 and for any BE Sz, if B is contained in A, then either B- 0 or B-A. Following Krantz et al. (1971, p. 220), it is then easy to show that Axiom 4” indeed has the implication that all atoms are equivalent. However, Axiom 4” can be generalized in a straightforward manner to our Axiom 4 which is also more general, in the finite case, than Lute’s axiom. AXIOM

4.

If A and B are atoms and A > B, then there exists CE Q such that

A-BvC.’ PROPOSITION. If (X, Sz, 2 > is a finite structure of qualitative probability that satisfies Axiom 4, then there exists a unique order-preserving function p such that (X, Q, p ) is a probability space.

Proof. To simplify the exposition, we start by eliminating the null events: let N be the union of all’null events, and define a new set X’= x\N, a new algebra Q’= {A\N 1 A E Q}, and a new relation 2 such that A\N 2’ B\N if and only if A 2 B. Also, let p’(A\N) = p(A). The structure (X’, Q’, 2 ‘) satisfies the assumptions of the proposition; besides p’ is an order-preserving probability measure on (X’, a’) if and only if p is an order-preserving probability measure on (X, 8). Hence, there is no loss in generality if we assume that the only null event in (X, Sz, 2 ) is the empty set 0. Then, any nonempty event has a unique decomposition into pairwise disjoint atoms. Let A, be a minimal atom, i.e., an atom such that any other non-null event is qualitatively not less probable than Al. Denote by A r the indifference class of A,. Let A, be the indifference class of an atom A, which is minimal in the set of atoms qualitatively more probable than AI-if there is any such A,. Define in the same way A, and A,,..., A, and A,, provided they exist. ’ The original formulation of this axiom was Axiom 4*: if A and B are atoms and A > B, then there exist events A’, B’, and C* such that A’ - A, B’ - E, and A’ _ E’ u C*. I am grateful to a referee who pointed out to me that Axiom 4, which is simpler than Axiom 4 *, is tantamount to it in the presence of Axioms l-3. Indeed, that Axiom 4 implies Axiom 4* is obvious. So assuming that Axiom 4’ is satistied, one can write A -A’ - B’ v C*, where C* n B’ = 0, with no loss of generality. Rewriting, one has: A - B’ v (C*\B) u (C* n B). Since B is an atom, then either C* n B - 0 or C* n B - B. In the first case, A - B’u (C*\E), and Axiom 3 says that B’- E=B’u(C*\B)-Bu(C*\B). Hence, A-BuC, where C=C*\B. In the second case, B\C*-0 and A-(B’uC*)u(B\C*) or A- (B’u ((C*\B)u (C* n B)))u (B\C*). Rearranging, A-Bu C, with C=B’u(C*\B).

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LIER

Thus, the atoms can be ordered according to (2ltA,(-A;-A;...)<

. . .
or A,

with

J2 1.

We put w(A i ) = 1, the probability of any atom in A, before normalization. If A, is not empty, there exist A, > A i and CE IR, with A2 N A i u C. Without loss of generality, C n A, = 0. Besides, C < A, and C 2 A, _ Therefore, C is an event that decomposes in a unique way into a finite number of pairwise disjoint atoms which are necessarily in A,. We put w(A,)= w(A,)+I’(C), where l’(C) is the number of atoms in A, that are in C. If A, is not empty, there exist A, > A, and DE 52, with A3 N A2 u D. Without loss of generality, D n A, = 0. Besides, D < A,. Hence, D has a unique decomposition into a finite number of pairwise disjoint atoms of which I’(D) are in A, and Z’(D) are in AI; l’(D) 2 0, Z’(D) 2 0, l’(D) + 12(D) > 0. We put w(A,) = 1’(D) w(A,) + (1 + I’(D)) w(A,). Clearly, the argument can be repeated until w(A,) is defined. Given that any event EE Q can be represented in a unique way as a union of pairwise disjoint atoms from A, u ... u A,, we put j=J

w(E)=

c l’(E) w(A,) j=l

It is then easy to verify that p(E) = w(E)/w(X) is a probability measure. It is also clear that if p agrees with 2, then it is the only probability measure with that property. It remains to prove that p agrees with 2. Let E, FE 52, with E-F. We prove that p(E) = p(F). This is immediate if E= F= 0. So we assume that E and F are nonempty. They each have a unique decomposition into pairwise disjoint atoms from A, u ... u A,. Whenever there are an atom contained in E and an atom contained in F which are members of the same equivalence class Aj, we remove these atoms from E and F, respectively. Hence, we obtain E’ and F’, with E’ N F’. Besides, of the atoms removed p(E) = PW + r’, where r’ is the sum of the probabilities from E. Clearly, we have also p(F) = p(F’) + r’. If E’ = F’ = 0, then p(E) = p(F). Otherwise, we consider the atoms A and B which are minimal among the atoms in E’ and F’, respectively. Then, either A > B or B > A. We retain the case with A > B. By Axiom 4, there exists C such that A - BuC, and BnC=IZ( with no loss in generality. The atom B is disjoint from E’\A; also, C is disjoint from E’\A, since it has a unique decomposition into pairwise disjoint atoms which are necessarily in E’ with respect to 2. So, by Axiom 3,
FINITE QUALITATIVE

PROBABILITY

95

and

r” > r’. The process can be repeated a finite number of times until F@‘) = 0, p(E) = p(0) + r(“‘), p(F) = p(0) + r(“‘). Hence, p(E) = p(F). If E > F, the same procedure gives E@“) > F’““, F’“” = 0, p(E) = p(E’““) + P’), p(F) = p(0) + r@“). Hence p(E) > p(F). Finally, that p(E) 2 p(F) implies Ek F follows, by contraposition, from the fact that F> E implies p(F) > p(E). 1 E’“‘-

That Axiom 4 is more general than Axiom 4” is obvious. So we turn to examine the relation between Axiom 4’ and Axiom 4. First, Axiom 4’ implies Axiom 4: for A > B and 0 N 0, there exist E and B’ such that E-A, B’- B, and E contains B’. Thus A N B’v C*, with C* = E\B’, and Axiom 4* (see Note 1) is satisfied. According to the result in Note 1, there exists C such that A N B v C. Hence, Axiom 4 is satisfied. Second, for X= {x1, x2, x3, x4}, and Q the algebra of all subsets of X, let 2 be the qualitative probability relation associated with the probability measure defined by p({x,))=& p({x,})=i, p({x,})=& p({x,})=$ The condition of Axiom4 is met. However, Axiom 4’ is not satisfied when A = {x4}, B = D = {x2, xX} and C= (x,, x3}. Indeed, there do not exist C’- C and D’ ND such that C’ n D’ = 0.

3. A COMPARISON OF em AXIOMS FOR SMALL SETS

Further insight into the relations between Axioms 4, 4’, and 4” may be gained by using the concept of a regular probability measurerecently introduced by Fishburn and Odlyzko (1987). For X,, = {x1 ,..., x,}, 9, the algebra of all subsets of X,, and p( {xi}) = pj, a probability measure p is said to be regular if (a)

p uniquely

(b)

O 1 is the sum of one or more pi for i < j.

(c)

agrees with some 2 on 52,

In fact, it is clear that every qualitative probability relation that satisfies Axiom 4 has a uniquely agreeing probability measure which is regular in the above sense. Since Fishburn and Odlyzko have established the numerical pattern of regular probability measures, the gain in generality that can be associated with Axiom 4 compared to 4’ can be computed directly at least for small values of n. Using Fishburn and Odlyzko’s (1987) convention for unique agreeing probability measures, a vector of probabilities (p,, pz,.,., p,) is written without normalization as a sequence of integers with greatest common divisor equal to 1. Thus, e.g., the sequence 1123 represents (pl = $, p2 = f, p3 = f, p4 = $). Denoting by R, the set of all regular measures on X,, and using the former convention, the only element in R, is 1, while the only element in R, is 11. In both cases the associated relation 2 satisfies Suppes’s (1969) Axiom 4”. The two elements of R3 are 111 and 112, and the associated relations 2 satisfy Lute’s (1967) Axiom 4’, while only the relation associated with 111 satisfies Suppes’s (1969) Axiom 4”. Continuing, R, has six elements 1111, 1112, 1113, 1122, 1123, and 1124.

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All relations 2 associated with these measures satisfy our Axiom 4, but Lute’s (1967) Axiom 4’ fails for the relation 2 which is associated with 1124 (see Section 2), and also for the relation associated with 1113: for A = (x4}, B= {x1, x,}, C={x,,x,},D=B,itistruethatA>C,AnB=0,B~D,buttheredonotexist C’ N C and D’ ND such that C’ n D’ = 0. Only the relation associated with 1111 satisfies Suppes’s axiom. In the case of R5, there are 27 regular measures which range from 11111 to 11248 (Fishburn & Odlyzko, 1987). And there already appears one measure such that the associated relation 2 does not satisfy our Axiom 4: for 11245 there exists no event C such that {x5} N {x3} u C.’ To examine the failure of Axiom 4’ we use the convention that the events are represented by the sequence of indices of their elements written in bold type: e.g., 124 represents the event (x1, -x2, x4}. The list of the 17 regular measures of which the associated relation 2 does not satisfy Axiom 4’ is given in the Appendix. We repeat: of the regular measures in R,, only one, 11111, satisfies Axiom 4”. Thus, even for small sets, Axiom 4 represents a significant gain in generality compared to Axiom 4’. As regards Axiom 4”, the question is in fact of another nature. The relevance of this axiom depends on a fair coin argument similar to the argument which has been invoked (e.g., Savage, 1954) to justify de Finetti’s (1937) assumption for an infinite set X such that for any n there exists a partition of X into n equivalent events. If the person’s algebra Q includes all finite sequences of heads and tails generated by what the person thinks is a fair coin, then de Finetti’s assumption must be fulfilled. Similarly, in the finite case, by allowing for conditional events consisting of finite sequences of heads and tails of a fair coin, a finite qualitative probability structure (X, Sz, 2 ) which satisfies Axiom 4 or Axiom 4’ may be extended to one which satisfies Axiom 4”. Considering, for example, the relation 2 associated with 1113, and denoting a sequence (head, tail,..., head) by ht ... h, the atom {x4} can be partitioned into three equivalent parts xl,=x, x;=x‘$ x:=x,

and and and

(hh 1hh or ht or th) (ht 1hh or ht or th) (th 1hh or ht or th).

And 1113 becomes 111111. Thus it is only when the fair coin argument cannot be accepted that a genuine increase in generality is provided by axioms richer than Axiom 4”. 4. DISCUSSION

We have provided a sufficient condition for the existence of a uniquely agreeing probability measure which is more general than Lute’s (1967) condition and com’ This is developed

in detail

in a subsequent

paper

by Fishburn

and Roberts

(1988).

FINITE

QUALITATIVE

97

PROBABILITY

parable in simplicity to Suppes’s (1969) condition. If the well-known fair coin argument cannot be invoked to make the assumption of equivalent atoms acceptable or to change the problem to one in an infinite algebra, the gain in generality associated with our new condition, which Section 3 has illustrated for small sets, is genuine. However, as the comparison with Fishburn and Odlyzko’s (1987) regular probability measures on finite sets reveals, our new condition does not completely characterize all simple structures.

APPENDIX

We list the measures in R, that do not satisfy Axiom 4’: indeed A > C, A n B = 0, Bk D and there do not exist C’ - C and D’ - D such that C’ n D’ = 0. All except 11245 fulfill Axiom 4. Notations are those of Section 3. As in Section 2, w denotes the unnormalized probability. The first column is a selection of Fishburn and Odlyzko’s (1987) Appendix.

wi

11114 11124 11125 11133 11134 11135 11136 11225 11226 11244 11235 11236 11237 11245 11246 11247

11248

Cwi

8 9

10 9

10 11 12 11 12 12 12 13 14 13 14 15 16

A

4-4)

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

4 4 5 3 4 5 6 5 6 4 5 6 7 5 6 7 8

C

123 123 123 12 12 12 12 124 124 13 14 14 14 13 14 14 14

w(C)

3 3 3 2 2 2 2 4 4 3 4 4 4 3 5 5 5

D=B 234 34 34 23 23 23 23 123 123 23 123 123 123 23 24 24 24

0) 3 3 3 2 2 2 2 4 4 3 4 4 4 3 5 5 5

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REFERENCES (1937). La prevision: ses lois logiques, ses sources subjectives. Annales de l’lnstitut Henri 168. FISHBURN, P. C., & ODLYZKO, A. M. (1987). Unique subjective probability on finite sets. Preprint, AT&T Bell Laboratories, Murray Hill, NJ. FISHBURN, P. C., & ROBERTS,F. S. (1988). Axioms for unique subjective probability on finite sets. Preprint, AT&T Bell Laboratories, Murray Hill, NJ. KRAFT, C. H. PRATT, J. W., & SEIDENBERG, A. (1959). Intuitive probability on finite sets. Annals of Mathematics and Statistics, 30, 408419. KRANTZ, D. H., LUCE, R. D., SUPPES,P., & TVERSKY, A. (1971). Foundations of measurement. 1. New York: Academic Press. LUCE, R. D. (1967). Suflicient conditions for the existence of a tinitely additive probability measure. DE FINETTI,

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of Mathematical

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38, 78Ck786.

L. J. (1954). The foundations of sfatisfics. New York: Wiley. SUPPES,P. (1969). Sludies in the methodology and foundations of science. Dordrecht: Reidel. SAVAGE,

RECEIVED: July 1, 1987