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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 REFERENCES [I] BEHREND, F. A. (1956). A contribution to the theory of magnitudes and the foundation of analysis. Math. Zeit 63 345-362. [2] CAMPBELL, N. R. (1920). Physics: the elements. Cambridge University Press, Cambridge. Reprinted (1957) as Foundations of Science: the philosophy of theory and experiment. Dover, New York. [3] CAMPBELL, N. R. (1928). An account of the principles of measurement and calculation. Longmans, Green, London. [4] DE FINETTI,B. (1937). La prbvision: ses lois logiques, ses sources subjectives. A n n . Inst. Poincard. 7 1-68. English translation in H. E . Kyburg, Jr., & H. E . Smokler (Eds.) (1964) Studies in subjective probability. Wiley, New York. 93-158. [5] KOOPMAN, B. 0. (1940). The bases of probability. Bull. Amer. Math Soc. 46 763-774. [6] KOOPMAN, B. 0. (1940). The axioms and algebra of intuitive probability. Annals of Math. 41 269-292. B. 0. (1941). Intuitive probabilities and sequences. Annals of iVath. 42 [7] KOOPMAN, 169-187. A. (1959). Intuitive probability on [8] KRAFT,C. H., PRATT,J. W., and SEIDENBERG, finite sets. A n n . Math. Stat. 30 408-419. [9] LUCE,R . 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