Download The Annals of Mathematical Statistics, 38, 1967, pp. 780-786.

The Annals of Mathematical Statistics, 38, 1967, pp. 780-786.
SUFFICIENT CONDITIONS FOR THE EXISTENCE OF A FINITELY
ADDITIVE PROBABILITY MEASURE'
University of Pennsylvania
1. Introduction. Suppose that we have given a qualitative relation, which is
to be interpreted as "at least as probable as," over a family of events, then under
what conditions can we construct an order preserving, additive probability measure? A counter-example du'e to Kraft, Pratt, and Seidenberg [9] shows that certain obvious necessary conditions, which define what has been called a qualitative
probability (see, e.g., p. 295 of [lo] or p. 32 of [Ill), arg insuficient to guarantee
the existence of such a measure. Sufficient conditions for a finitely additive
measure have been presented by de Finetti [4], Koopman [5], [B], [7], and Savage
[ll]. They all involve, either explicitly or implicitly, the strong property that for
each integer n the universal event can be partitioned into n events that are
equally probable under the given ordering. Among other things, this forces the
family of events to be infinite. The existence of countably additive measures has
been studied by Villegas [14].
This paper provides still another axiomatization for the finitely additive representation; it is of interest because it does not demand arbitrarily fine partitions
into equally, likely events-in fact, finite models of the axioms exist. The proof
is also of some inherent interest because it depends upon a theorem from the
theory of extensive measurement. The parallel between the additivity of probability and of extensive measures has always been apparent, but to my knowledge
no intimate connection between them has been previously shown. The reason
is that the additivity of probability only holds for certain pairs of events-disjoint ones-whereas in the classical theory of extensive measurement (see [2],
[3], [12], and [13]) additivity holds over all pairs of entities without any restriction. However, Behrend [I] and Luce and Marley [9], in attempts to make extensive measurement more realistic, have shown that an additive representation
can still be constructed when only certain pairs of entities can be concatenated.
Their result, which is stated fully in Section 3, is used to prove the present probability theorem.
2. The axioms and representation theorem. A preliminary definition is
needed. It states, in essence, what we shall mean by an event having, qualitatively, a probability equal to that of n disjoint copies of a given event.
DEFINITION
1. Let X be a non-empty set, A a family of subsets of X that
an
includes @ and that is closed under complementation and union, and
Received 12 December 1966.
This work was carried out while the author was a National Science Foundation Senior
Postdoctoral Fellow a t the Center for Advanced Study in the Behavioral Sciences. The comments of David Krantz, Patrick Suppes, and Amos Tversky have been most helpful.
780
EXISTENCE OF A FINITELY
ADDITIVE
PROBABILITY MEASURE
781
equivalence relation on A. Let a E A. A sequence al , az , . . , ai , . . . , where
ai E A, is a standard series relative to a if for i = 1, 2,
there exist bi and ci
E A such that:
a,
(i) al = bl and bl
(ii) bi n ci = @,
(iii) ai+l = bi u c, ,
(iv) bi
ai ,
(v) C j
a.
To gain an intuitive idea of the meaning of a standard series and the role it
will play, note that
-
-
a? = bl u cl
,
where bl
a i + ~= bi u c i , where bi
- - a, c1
a, and bl n c1 = @,
a i , cc
a, and b i n ci
=
@.
Thus, ai+l is equal to an event that is the disjoint union of one that is indifferent
to ai and another that is indifferent to a. So, crudely, ai is an event that acts like
the union of i mutually disjoint events each of which is indifferent to a; however,
since i such events may not exist, the definition has to be somewhat indirect.
If a finitely additive, order preserving probability measure p exists, then by
induction it is easily seen that p(ai) = @(a). Therefore, if p(a) > 0, the standard
series must be finite. The qualitative restatement of this is one of the two new
axioms in the following definition.
DEFIEITION2. Let X be a non-empty set, A family of subsets of X that includes @ and that is closed under complementation and union, and >, a binary
relation on A. The triple (X, A, >,) is called a regular system of qualitative
probability if, for all a, b, c, d e A, the following five axioms hold?
( 1) 2 is a weak ordering of A.
(2) a 2 @ and X > @.
( 3 ) If a n c = b n c = @, then a 2 b if and only if a u c 2 b u c.
( 4 ) If a n b = @, a > c, and b 2 d, then there exist c', d', e E A such that
e a u b, c'- C,d'
d, e 2 c'u d', and c ' n d' = @.
(5) If a > @, then any standard series relative to a is finite.
The first three axioms, which are necessary whenever a non-trivial finitely
additive representation exists, define what is called a qualitative probability
structure. The fourth axiom is a somewhat weaker, and so more acceptable,
version of the assert,ion that if a and b are disjoint and dominate c and d, respectively, then there are disjoint subsets of a u b that are equivalent in probability to c and cl. I t is not a necessary condition. The last axiom, which is really
an Archimedian property, states in essence that any event that is strictly more
prqbable than the null event behaves as if it has non-zero probability. It is a
-
N
We define
> and
-
in terms of
2 in the usual way.
782
R. DUNCAN LUCE
necessary condition when an order preserving, finitely additive probability
measure exists.
The following is to be proved.
1. If (X, A, 2 ) is. a..regular system of qualitative probability, then
THEOREM
there exists a unique, jinitely additive probability measure p over A that preserves
the order of 2 , i.e., for all a, b E A:
(i) a 2 b if and only if p(a) 2 p(b).
(ii) 0 $ p(a) 5 1.
(iii) p(@) = 0 and p ( X ) = 1.
(iv) If a n b = @, then p ( a u b) = p(a)
p(b).
As was indicated in the introduction, the proof involves reducing this assertion to a result in the theory of estensive measurement. This theorem is stated
next.
+
3. A result from the theory of extensive measurement.
DEFINITION
3. Let A be a non-empty set, B a non-empty subset of A X A,
R a binary relation on A, and 0 a binary function from B into A. The quadruple
(A, B, R, 0) is called an extensive system without a maximal element3if, for all x,
Y, 2, €4
( 1) R is a weak ordering of A.
(2) If (x, y) E B and (x 0 y, x) E B, then (y, X ) E B, (x, y 0 2 ) E B, and
( x o y)oxRxo ( y o z ) .
( 3 ) If (x, z) E B and xRy, then (x, y) E B and x 0 zRx 0 y.
(4) If not zRy, then there exists y - x E A such that (x, y - x) e B,
yRxo (y - x), and xo ( y - x)Ry.
(5) If (x, y) E B, then not xRx 0 y.
(6) Let n be a positive integer. For n = 1,define l x = x. For n > 1,if ( n - 1)z
is defined and ( ( n - l ) x , x) E B, then define nx = ( n - 1 ) s 0 x. For all x, y E A,
the set
{ n I n is a positive integer and yRnxj
is jinite.
I n this system, Axiom 2 captures associativity; Axiom 3 both insures commutativity and that inequalities are preserved when the same element is concatenated with both terms; Axiom 4 asserts that the system is complete in the
sense that certain equations can be solved; Axiom 5 excludes both zero and negative elements; and Axiom 6 is a suitable formulation of the Archimedean property when only some pairs of elements can be concatenated.
2. If (A, B, R, 0) is an extensive system without a maximal element,
THEOREM
then there exists a positive real-valued function (p on A such that
The theory in [9] also deals with extensive systems that have maximal elements in the
f,ollowing special sense: a E A is maximal relative t o R and o if for all x E A, aRx, and if for
some x E A, (a, x) E B. As the proof of Theorem 1 does not require the results for the case
when there is a maximal element, I do not state them here.
784
R. DUNCAN LUCE
Suppose that there is no a for which both a
implies either a E pi or a E X, and
p(a)
=
1 i f EX
=
Oifa~pi
clearly fulfills the assertions of the theorem.
Henceforth we assume that a > @ and d
B
=
( (a, b ) I a
>
@ and
ci
> @ for some
> @, b > @, and there exist a'
E a, b' E
> @. Then a E A
a E A and define:
b such that a' n b'
=
@].
B is nonempty since we have assumed that an a exists for which a > @ and
ci > @. If (a, b ) E B and, with no loss of generality, a n b = @, then we define
the binary operation 0 by: a 0 b = a u b. Two applications of Corollary 1 of
Lemma 3 show that o is well defined.
We now prove that (A*, B, R, o), where A* = (A/-) - 6 and R is the restriction of >N/- to A*, is an extensive system without a maximal element
(Definition 3). Note that A* excludes events of qualitative probability zero;
they are reinstated later.
(1) R is obviously a strict ordering of A* since, by Axiom 1, 2 is a weak
ordering of A .
(2) Suppose that (a, b ) E B, a n b = @, and ( a 0 b, c) E B. By definition of
B, there exist d E a u b and c' E c such that C' n d = @. Since a u b 2 b and
a u b > b. This, together with c'
c and
a > @, Lemma 5 implies that d
c' n d = @, implies that (Axiom 4) there exist b', cNE A for which b' E b, cNE c,
and b' n ? = @. Thus, (b, C ) E B.
b', and a n b = @,
Next, we establish that (a, b 0 c) E B. Since a > @, b
Corollary 2 of Lemma 3 yields d a u b > b'. But c'
c" and c' n d = @, so by
Axiom 4 there exist bN E b, c" E C, and e E A such that e d u c', e 2 bN u c"', and
b" n c" = @. Suppose that d > (e - c'"). Since c' cmand c' n d = @, Corollary 2
of em ma 3 yields d u c' > (e - c") u c" = e (d u c'), which is impossible by
Axiom 1. The supposition (e - cm) > d and c"'
c' leads to a similar'contradiction. So d (e - c").NOW,suppose a' > a, where a' = e - (b" u c"'). Since
'11
bN
b and a' n b" = @, Corollary 2 of Lemma 3 implies that d
(e - c ) =
(a' u bN) > ( a u b)
d, which is impossible. And if a > a', then d
( a u b) >
(a' u b")
d, which is also impossible. So, a
a'. Since a' E a, b" u c" E b 0 c,
and a' n (b" u c") = @, the assertion is proved. Moreover,
-
-
-- ---
-
- - -
-
a
0
--
(b 0 c) = a' u b" u crr'
=
( a 0 b ) 0 c.
" (3) Suppose that (a, c) E B and aRb, where with no loss of generality,
a n c = @ . If a = b, it follows immediately that a 0 c = c 0 b. So we assume
EXISTENCE OF A FINITELY ADDITIVE
-
PROBABILITY
MEASURE
785
n > b. Silicc c
e, Axion1 4 t~ssertstllc cxistelice of 6' E b and d E c such that
6' n c' = @. So (c, b ) E B, and by Corollary 2 of L e n l ~ n3,
t ~a u e > b' u e', whence,
by definition, not c 0 bRa 0 c.
(4) Suppose that not aRb. Thus, b > a and so by Lemma 4 there exist a' E a,
b' E b such thas b' I> a'. By Lemma 5, ( 6' - a') > @, a > @ since a E A*, and
a ' n (b' - a') = @ , s o b = a 0 (b' - a').
( 5 ) Suppose that (a, b ) E B, where a n b = @. Since b > @, it follow from
Lemma 5 that a u b > a, and so not aRa 0 b.
( 6 ) Finally, we show that { n 1 bRna) is finite, where na = ( n - l ) a 0 a and
l a = a. We do this by showing that the existence of na implies the existence of a
standard series relative to a that hasn elements. Because a E A*, a > @, and so
by Axiom 5 any such series must he finite; therefore there exists some integer n z
such that na is not defined for n > m. Suppose that 2a kxists, then by definition
of 0 there exist al = bl E a and s E a such that bl n c1 = @ and az = 61 u c~E 2a.
Suppose that na exists and that, for i 5 n - 1, we have constructed a standard
series a , relative to a, with auxiliary b,-l and ci-1, and that ai E ia. We extend it to
i = n. By definition of 0, there exist 6,-1 E ( n - 1)a and c,,-1 e a such that
6,-1 n en-1 = @ and 6,-1 u c,-1 E na. Thus, if we set a,, = b,-1 u en-1 , the series is
extended.
By Theorem 2, there is a positive real-valued function cp on A* such that
aRb if and only if
cp(a) 2 cp(b),
and
if
(a, b )
E
B, then cp(ao b )
Select that cp for which cp(X)
=
=
cp(a)
+ cp(b).
1 and, for a E A, define
Using the properties of cp and Lemma 6, it is easy to see that p fulfills the assertions
of the theorem. A/loreoverp is unique since if another such function existed there
would be an additive measure in the extensive system not related to cp by a multiplicative constant, thus violating part (iii) of Theorem 2. Q.E.D.
6. Relation to Savage's system. In [Ill, Savage has shown that Axioms 1-3
plus the condition that 2 is fine and tight are sufficient to prove Theorem 1. Recall that 2 is fine if for every a > @, there exists a partition ( b l , . . . , b,] of X
such that for i = 1, . . . , n, a 2 bi ; that a and b are alfnost equivalent, denoted
a * b, if for every c, d such that c > $5, d > @, and a n c = b n cl = @,
then a u c 2 b and b u cl 2 u ;and that 2 is tight if a * b implies a
b.
THEOREM
3. If 2 is a qualitative probability on 2X that is fine and tight, then
(X, 2 X , 2 )is a regular system of qualitative probability.
PROOF.Since Axioms 1-3 are assumed, it is sufficient to prove 4 and 5.
Suppose that a n b = @, a > c, and b 2 d. Since 2 is fine, Theorem 3, p. 37
-
-
-
780
R. DUNCAN LUCE
of [ll]states that there exists a unique probability 1ne:tsure 11 tlmt almost agrees
wit11 ,> (i.e., a ,> 6 irnpliesp(a) 2 p ( b ) ) , that a -* b if and only if p(a) = p(6),
that for any a and any number p, 0 $ p $ 1, there exists b G a such that
p(b) = pp(a), and that if a > @, p ( a ) > 0. Since a > c and b ,> cl, p ( a ) I p(c)
and p(b) 2 p(d). Let p, p' be such that pp(a) = p(c) and prp(b) = p(d). So
pp(a) = p(c) and p(d') =
there exist c' 5 a and d' 5 b such that p(c')
plp(b) = p(d). Therefore, c' N* c and d' N* d, but since ,> is tight, c'
c and
d , thereby proving Axiom 4 with e = a u 6.
d'
Axiom 5 is immediate. Q.E.D.
It is obvious that the converse of Theorem 3 is false sincesavage's assumptions
imply that X is infinite whereas the ordering induced by the uniform distribution
on a finite set satisfies Definition 2.
-
-
N
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