Finite Linear Qualitative Probability

Journal of Mathematical Psychology  1105 journal of mathematical psychology 40, 6477 (1996) article no. 0004

Finite Linear Qualitative Probability Peter C. Fishburn AT 6T Bell Laboratories, Murray Hill, New Jersey 07974

We say that (2 S, o ) is representable by an order-preserving probability measure, or in Savage's (1954) terms has an agreeing measure, if there is a probability measure P on 2 S such that, for all A, B # 2 S,

Many years ago, Bruno de Finetti asked whether certain axioms for a comparative probability ordering relation o on the subsets of an nelement set that are necessary for the existence of an order-preserving probability measure are also sufficient. This was answered in the negative for n5 by Kraft, Pratt, and Seidenberg [Annals of Mathematical Statistics 30 (1959), 408419]. The present paper extends their analysis of comparative probabilities that satisfy de Finetti's axioms but lack order-preserving measures when it is assumed also that o is a linear order. We refer to a linear order on the subsets of [1, 2, ..., n] that satisfies de Finetti's axioms as an LQP (linear qualitative probability), and say that it is an NLQP (nonrepresentable LQP) if it has no order-preserving probability measure. The paper characterizes all NLQPs for n=5, shows that every n5 has an NLQP that violates the simplest extension of de Finetti's additivity axiom yet has an order-preserving measure on the family of subsets of every (n&1)-element set, and proves that as n   there is no upper bound on the number of o comparisons needed to verify the nonexistence of an order-preserving measure. Special examples for small n illustrate other facets of NLQPs, including the necessity of considering multiplicities of an o comparison in testing whether an LQP has an orderpreserving measure. ] 1996 Academic Press, Inc.

Ao B  P(A)>P(B).

The modern history of representability for comparative probability on finite sets begins with de Finetti's (1931) observation that the following axioms are necessary for (2 S, o ) to be representable: A1 (Order). o on 2 S is a weak order; A2 (Nonnegativity). A-< for every A # 2 S; A3 (Nontriviality). So p 1 +p 4 p 1 +p 5 >p 2 +p 3 p 3 +p 4 >p 2 +p 5 p 2 >p 3 +p 5 , 64

65

FINITE LINEAR QUALITATIVE PROBABILITY

and summation and cancellation in these inequalities gives 0>0. KPS also shows what must be added to de Finetti's axioms so that the resulting axioms are sufficient as well as necessary for representability. Let (A 1 , ..., A M )=0 (B 1 , ..., B M ) mean that A j , B j # 2 S for all j and, for every i # [1, ..., n], |[ j : i # A j ]| = |[ j : i # B j ]|. The preceding example has (235, 15, 34, 2)=0 (14, 23, 25, 35). The new axiom can be stated as follows. A5 (Strong Additivity). For all M2 and all A j , B j # 2 S, if (A 1 , ..., A M )=0 (B 1 , ..., B M ) and A j -B j for all j1. It follows for this case that a 1 a 3 = 12 and z o 0 by Lemma 2.2. If a 1 >a 2 , then z 1 =1, hence a 1 &a 2 =a 3 and z=(a 1 a 3 )(x 1 +x 2 )+x 2. Because x 1 +x 2 {0, this case also has x 1i =x 2i {0 for some i>1, and this and z # T n force the contradiction that a 1 =0. A similar contradiction obtains if a 2 >a 1 . This completes the proof. Our hypotheses imply that &x 3 is x 1 +x 2, 2x 1 +x 2, x 1 +2x 2, or (x 1 +x 2 )2, and in each case we get &x 3 o 0 by L4 or Lemma 2.2. K In terms of T n, an LQP (T n, o ) is representable if and only if there is an :=(: 1 , : 2 , ..., : n ) in R n with : i >0 for all i such that, for all x # T n, xo 0  ( :, x) >0.

and x 1 o 0 and x 2 o 0. Then &x 3 o0.

z=

Since a 2 >0 and z # T n, we never have x 1i =x 2i {0. Hence x 1 +x 2 # T n and x 1 +x 2 o0 by L4. If there is no i for which x 1i = &x 2i {0 then, because x 2 {0, some i has x 2i {0 and x 1i =0. This implies a 2 a 3 =1, so z=x 1 +x 2 and zo 0. Suppose, however, that x 1i = &x 2i {0 for some i, in which case z i equals 1&a 2 a 3 or &1+a 2 a 3 . If z i =1&a 2 a 3 , we require either z i =0 (hence a 2 =a 3 and z=x 1 +x 2 ) or z i = &1 (hence a 2 =2a 3 and z=x 1 +2x 2 ). Similarly, z i = &1+a 2 a 3 implies either z=x 1 +x 2 or z=x 1 +2x 2. We noted above that x 1 +x 2 o 0 and x 1 +x 2 # T n, so if z=x 1 +2x 2 =(x 1 +x 2 )+x 2, another application of L4 gives z o 0. Suppose henceforth that no i has |x 1i +x 2i | =1, so 1 x i =0  x 2i =0 for every i. If x 1 =x 2 then z=x 1, so z o 0. Assume henceforth that x 1 {x 2. Assume also without loss of generality that x 11 =1 and x 21 = &1, so

\a + x . 2

3

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We could normalize the : i to sum to 1, so p i =: i 7: i , but this is unnecessary and it is convenient not to do so. Note also that a given positive : yields an LQP (T n, o ) when o is defined by the preceding correspondence if, and only if, ( :, x) {0 whenever x # T n "[0]. We consider two renditions of strong additivity for our linear formulation that emphasize slightly different aspects of the condition. The first version is based on sequences x 1, x 2, ..., x M of members of T n that can contain replications of the same x, and uses condition C n(M ) defined as follows: C n(M ): For all sequences x 1, ..., x M of terms from T n, j j if  M j=1 x =0, then it is false that x o 0 for j=1, ..., M.

The second accounts directly for multiplicities in the definition of C *(m): n

68

PETER C. FISHBURN

C n*(m): For all distinct x 1, ..., x m # T n and all positive integers a 1 , ..., a m , if 

m j=1

j

a j x =0 then it is false

j

that x o0 for j=1, ..., m. Given n, C n(M ) emphasizes the number M of terms in the x sequence, which corresponds to 7a j in C n*(m), and C n*(m) emphasizes the number m of distinct x j in the x sequence. The distinction plays a role later when we consider how large m or M may have to be to discern a violation of strong additivity for every LQP that is not representable. Our two renditions of strong additivity are as follows. L5.

C n(M ) holds for all M4.

L5*. C *(m) holds for all m4. n Because a violation of L5* for some m4 gives a violation of L5 for an Mm, we have L5 O L5*. We also have L5* O % L5 because there are violations of L5 for M4 which use fewer than four distinct x j and have no other x k o 0. However, when (T n, o ) is assumed to be an LQP, it follows easily from preceding results that L5* O L5. Hence L5 and L5* are equivalent in the LQP setting. We note later that the weaker condition that arises when the x j in C n(M ) are assumed to be distinct, or a 1 = } } } = a m =1 is presumed in C n*(m), is not sufficient for representability of an LQP for larger values of n. Theorem 2.4. An LQP (T n, o ) is representable if and only if L5 or L5* holds. Proof. The theorem is an easy corollary of Lemmas 2.1 and 2.3 and Theorem 1.1. In the application of Theorem 1.1, ( :, x k ) =0 of part (i) is not involved because of linear ordering, and K=N=(3 n &1)2. Each of the K ( :, x k ) >0 inequalities corresponds to an x k o0. These include e i o 0 for i=1, ..., n, so if an : solution exists, it has : i >0 for all i. If part (ii) holds for an LQP, we have a violation of C *(m) n for some m. By Lemma 2.3, the violation cannot occur for m=2 (take x 1 =x 2 ) or m=3, so it must occur for m4. In other words, an LQP is not representable if and only if L5* is violated. K We refer to an LQP that is not representable as an NLQP. The KPS example for n=5 in Section 1 uses a linear order on 2 S that satisfies A1A4 and is therefore an example of an NLQP. Many others will be noted later, but all require n5. Lemma 2.5.

Every LQP for n4 is representable.

Proof. The lemma is a corollary of a similar result for weak orders noted informally in KPS. It follows immediately for n=3 from Theorem 2.4. When n=4, it is holds for every LQP (T 4, o ), espeeasily seen that C *(m) 4 cially when results in the next section are used. One can also enumerate the n=4 LQPs when 1o 2 o 3 o 4 (there are

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14) and observe that each has an order-preserving probability measure. K Lemma 2.5 completes our preliminary results. We conclude this section with an overview of the rest of the paper phrased in terms of the formulation developed in this section. Section 3 shows how a natural dominance relation r on T n based on the ordering 1 o 2 o } } } o n of atoms affects LQPs. In particular, if xr0, then every LQP has x o 0, and x is never part of a minimal sequence x 1, x 2, ..., x M that violates L5. Section 4 identifies all NLQPs for n=5. Given 1 o 2 o 3 o4 o 5, there are precisely 30 NLQPs, and each of these violates C 5(4) with four x j o 0 that belong to one of several simple patterns. Section 5 considers how large M in C n(M ) or m in C n*(m) must be to encompass some violation of L5 or L5* for every NLQP on n atoms. We also discuss the notion of an irreducible pattern, which is an m-tuple (a 1 , ..., a m ) of nondecreasing positive integers with no common divisor greater than 1 such that some NLQP for some n violates C *(m) n with multiplicities a 1 , ..., a m for distinct x 1, ..., x m, respectively, and the NLQP has no smaller violation of L5*. The only irreducible pattern for m=4 is (1, 1, 1, 1), and m=5 has irreducible patterns (1, 1, 1, 1, 1) and (1, 1, 1, 1, 2) but no others. An example for (1, 1, 1, 1, 2) with n=9 shows that some NLQPs violate L5* only when different multiplicities a j are used for the x j o0 for which 7a j x j =0. Section 6 presents a theorem and design principles that are useful in constructing and verifying examples of LQPs which violate specific instances of C n(M ) or C n*(m) and have no smaller case violations of L5 or L5*. The theorem and principles are then illustrated by the n=9 example for irreducible pattern (1, 1, 1, 1, 2). Section 7 concludes the paper with additional applications of the theorem in Section 6. It shows that each n # [6, 7, 8] has an NLQP whose minimum violation of strong additivity uses n&1 x j # T n, that every n5 has an NLQP which violates C n(4) and is representable on every subset of n&1 atoms, and that as n   there is no upper bound for NLQPs on the number of distinct x j involved in minimum-m violations of C *(m). n 3. LINEAR DOMINANCE

This section describes effects of the linear ordering of atoms (or singletons) on LQPs. We assume for convenience that [1] o [2] o } } } o [n], i.e., L6. e i &e i+1 o 0 for i=1, ..., n&1. We define a dominance partial order r on T n using partial sums as follows:

FINITE LINEAR QUALITATIVE PROBABILITY j

xr0

if : x i 0

for all jn,

i=1 j

: x i >0

for some

jn.

i=1

If xr0 then x{0, the first nonzero x i is positive, and successive partial sums never go negative. Thus (1, &1, 1, &1)r0, (0, 1, 0, &1)r0, and not [(1, 0, &1, &1)r0]. Dominance affects all LQPs similarly. Lemma 3.1. For all x # T n, x o 0 for every LQP (T n, o ) that satisfies L6 if, and only if, xr0. Proof.

If xr0, we can write it as x=: e i +: (e j1 &e j2 ) I

J

where I[1, ..., n], J is a set of ordered pairs ( j 1 , j 2 ) with 1 j 1 < j 2 n, I _ J{ } } } >: n >0 of (d). Proof. Given the hypotheses, let x 0 denote an x that satisfies (c). Theorem 1.1 then implies that there is an ==(= 1 , ..., = n ) in R n (or Z n ) such that for all x # X "[x 0 ].

( =, x) >0

Division by a suitably large number allows us to presume that n

The primary concerns of this section for f, F, and Am are important because they impinge directly on conditions that need to be tested to determine whether an LQP is representable. An auxiliary concern that arose is the question of whether a proposed N matrix does in fact identify an irreducible pattern. We address this in the next two sections. 6. A CONSTRUCTION THEOREM

N(M+1) : |= i | M=*M( ;, x) >0, where (6.1) is used for the second inequality. In addition, &=*(;, x 0 ) 0.

j

x =0,

no nonempty sequence y 1, ..., y v of members of X has y =0 when vM=* for this t j, and because &=*( ;, x0 ) 5.

Theorem 7.2. Every n5 has an NLQP that violates C n(4) and is representable on every subset of n&1 atoms. FIG. 2.

N matrices for irreducible patterns (1, 1, ..., 1).

which is ( :, x) =0 for the final row of the n=8 matrix in Fig. 2. Equations in : 3 through : 8 that involve : 3 are : 3 +: 6 =: 4 +: 5 : 3 =: 4 +: 6 &: 7 which correspond to rows 5 and 6 of the N matrix. Those in : 2 through : 8 that involve : 2 must involve : 3 also because the &11 for : 2 can not be balanced by the constants in : 4 through : 8 . We have : 2 &: 3 =c&4, which is obtained from : 4 through : 8 only by : 4 &: 5 &: 8 and : 5 &: 6 &: 7 . This gives : 2 +: 5 +: 8 =: 3 +: 4 and : 2 +: 6 +: 7 =: 3 +: 5 for rows 3 and 4 of the N matrix. The &22 in : 1 implies that an equation in : 1 through : 8 that involves : 1 also involves : 2 and : 3 with : 1 &: 2 &: 3 = &4. There are exactly two ways to get &4 from : 4 through : 8 , namely : 4 &: 5 &: 6 &: 8 and : 4 &: 5 &: 6 &: 7 +: 8 , and these two correspond to the first two rows of the N matrix. Because the hypotheses of Theorem 6.1 hold for each case in Fig. 2, the conclusion of the theorem shows that each yields an irreducible pattern. K The importance for part (d) of Theorem 6.1 of verifying that an : solution for an N matrix has no z in T n "[[0] _ X _ (&X )] for which ( :, z) =0 cannot be overemphasized. Experience indicates that many N matrices proposed for irreducible patterns turn out to have such z's, and subsequently reduce to smaller N matrices. The design

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Proof. The theorem follows easily from Theorem 6.1 by way of the N matrix 1

2

3

4 1 4 2 } } } 4K

5

x 0 =x 1

+ & & + + }}} + &

x2 x3

& 0 + + + }}} + + 0 + & & & }}} & +

x4

0

0

+ & & }}} & &

A oB 14 o235 345 o1 25 o34 3 o45

where we use the dual of IV in Theorem 4.1 and replace [4] by [4 1 , 4 2 , ..., 4 K ]=4 so that n=4+K. The ( :, x) =0 equations for (d) of Theorem 6.1 have solution : 5 =1 K

: 4k =; k , k=1, ..., K,

with ;= : ; k , k=1

: 3 =;+1 : 2 =2; : 1 =2;+2, where 1 : 5 >0, and by taking (1, ; K , ; K&1 , ..., ; 1 ) to be superincreasing we ensure that ( :, x) =0 for nonzero x only when x or &x is in the preceding matrix. We use x 1 in (c) of Theorem 6.1 for the representable subsets conclusion. K Theorem 7.3. For every m4 there is an NLQP with a 1 =a 2 = } } } =a m =1 (T n, o ) that violates C *(m) n for all 4m$