A Conflict between Finite Additivity and Avoiding Dutch Book Teddy Seidenfeld; Mark J. Schervish Philosophy of Science, Vol. 50, No. 3. (Sep., 1983), pp. 398-412. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28198309%2950%3A3%3C398%3AACBFAA%3E2.0.CO%3B2-3 Philosophy of Science is currently published by The University of Chicago Press. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/ucpress.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact support@jstor.org. http://www.jstor.org Tue Mar 4 10:40:36 2008 http://links.jstor.org/sici?sici=0031-8248%28198309%2950%3A3%3C398%3AACBFAA%3E2.0.CO%3B2-3 http://www.jstor.org/about/terms.html http://www.jstor.org/journals/ucpress.html A CONFLICT BETWEEN FINITE ADDITIVITY AND AVOIDING DUTCH BOOK* TEDDY SEIDENFELD Departntent of Philosophy Washington University, St. Louis MARK J . SCHERVISH? Department of Statistics Carnegie Mellon University, Pittsburgh For Savage (1954) as for de Finetti (1974), the existence of subjective (per- sonal) probability is a consequence of the normative theory of preference. (De Finetti achieves the reduction of belief to desire with his generalized Dutch-Book argument for previsions.) Both Savage and de Finetti rebel against legislating countable additivity for subjective probability. They require merely that prob- ability be finitely additive. Simultaneously, they insist that their theories of pref- erence are weak, accommodating all but self-defeating desires. In this paper we dispute these claims by showing that the following three cannot simultaneously hold: (i) Coherent belief is reducible to rational preference, i.e. the generalized Dutch-Book argument fixes standards of coherence. (ii) Finitely additive probability is coherent. (iii) Admissible preference structures may be free of consequences, i.e. they may lack prizes whose values are robust against all contingencies. 1. Introduction. One of the most important results of the subjectivist theories of Savage and de Finetti is the thesis that, normatively, prefer- ence circumscribes belief. Specifically, these authors argue that the the- ory of subjective probability is reducible to the theory of reasonable pref- erence, i.e. coherent belief is a consequence of rational desire. In Savage's (1954) axiomatic treatment of preference, the existence of a quantitative subjective probability is assured once the postulates governing preference are granted. In de Finetti's (1974) discussion of prevision, avoidance of a (uniform) loss for certain is thought to guarantee agreement with the requirements of subjective probability (sometimes called the avoidance of "Dutch Book"). Obviously, the significance of these results depends upon the avowed *Received December 1982; revised March 1983. ?We would like to thank P. C. Fishburn, Jay Kadane. Isaac Levi, Patrick Maher and a referee for their helpful comments. Also, we have benefited from discussions about con- sequences with E. F. McClennen. Philosophy of Science, 50 (1983) pp. 398-412. Copyright Q 1983 by the Philosophy of Science Association 399 FINITE ADDITIVITY AND AVOIDING DUTCH BOOK liberalism regarding the range of preferences and beliefs the theories are said to tolerate. Both Savage and de Finetti are explicit in their opposition to the stipulation of countable additivity for probability, and, of course, each insists that the constraints imposed on reasonable preference are weak, permitting all but self-defeating desires. It is our purpose in this paper to challenge these claims. We aim to show that the reduction of belief to preference cannot be carried off as Savage and de Finetti suggest without contracting the range of admissible states of preference and belief. In particular, we argue that the purported reduction fails unless (i) subjective probability is countably additive, or (ii) each agent is required to acknowledge the existence of a rich supply of consequences, i.e. prizes whose values are robust against the contingencies of nature. As we see in section 3, Savage recognized that consequences serve as expedients in his theory for constructing "constant acts" and should not be essential to subjectivism. We find no convenient stockpile of conse- quences. In fact, it seems reasonable to deny that there are consequences in practical decisions. Thus, our position is that, lacking consequences, expected utility theory must treat subjective probability distributions as extraneous (Fishburn 1970, § 12.2 and chapter 13). Otherwise, proba- bilities which are not countably additive cannot be sanctioned. In light of our findings in section 4, the problem is deeply rooted indeed. The expected utility hypothesis fails for acts with denumerably many out- comes, when probability (extraneous or otherwise) is merely finitely ad- ditive and consequences are absent. Fortunately, Savage's theory is axiomatized so that the first six of his seven postulates deal with the structure of preference for gambles, i.e. acts which produce only finitely many different outcomes almost surely. It is the seventh postulate, P7, which carries the extension of expected utility theory to acts in general. Thus, our focus in section 2 is on the final axiom. We examine several conjectures about the conditions under which P7 remains independent of P1-P6 and demonstrate that the inde- pendence is not a matter of the additivity of the probability. Hence, one may satisfy PI-P6 with a countably additive probability but violate the expected utility hypothesis for acts that are not gambles. In other words, compliance with PI-P6 fails to guarantee the expected utility hypothesis for random variables in general, even on the condition that probability (based on PI-P6) is countably additive. Readers unfamiliar with Savage's theory may wish to skip section 2 on a first reading. In section 3, we analyze P7 and show that its role in extending utility to acts in general trades on an undesirable feature consequences are con- ceded to require. For instance, when P7 is reformulated to avoid this 400 TEDDY SEIDENFELD AND MARK J . SCHERVISH feature of consequences, the resulting theory precludes all but countably additive subjective probability. In our discussion of de Finetti's argument against "Dutch Book" in section 4, we grant his working hypothesis that there is a linear utility function for outcomes (as when he assumes dollars linear in utility), then show that his standards for coherence of previsions prohibit merely finitely additive probability. In parallel with Savage's the- ory, if consequences are introduced and coherence confined to previsions involving consequences exclusively, then, as desired, finite additivity is all that follows from avoidance of "Dutch Book". Thus, the dilemma is between mandating consequences and denying the admissibility of merely finitely additive distributions. 2. On the Independence of P7. In his classic The Foundations of Sta- tistics (1954), L. J . Savage constructs a theory of utility, axiomatized in seven postulates. The first six of Savage's axioms yield a theory of ex- pected utility for gambles, i.e. acts which produce at most finitely many consequences almost surely. The seventh postulate (P7) extends the the- ory to acts in general. Immediately following the introduction of P7, Sav- age demonstrates its independence from the first six with the aid of a finitely, but not countably additive probability. He concludes the dem- onstration with this terse remark: Finite, as opposed to countable, additivity seems to be essential to this example; perhaps, if the theory were worked out in a countably additive spirit from the start, little or no counterpart of P7 would be necessary (1954, p. 78).' Our purpose in this section is to demonstrate that the conjecture implicit in the above remark is not accurate. That is, we will produce examples involving only countably additive probabilities for which PI-P6 are sat- isfied but P7 is not. This means, on the condition that the expected utility hypothesis is valid for acts in general, some replacement for P7 is nec- essary even if the theory is worked out in a countably additive spirit. We will assume that the reader either is familiar with Savage's pos- tulate system or else has a copy of Savage (1954) readily available. Ad- 'Savage had misgivings about this comment. In a letter to P. C. Fishburn (dated 30 June 1965) he wrote: You suggest that I review the last sentence on page 78 of F , of S. [Foundations of Statistics] It is hard for me now to feel sure what I meant by that sentence, and I have serious doubts that it is defensible. But what it seems to say is not that something stronger than P7 would be needed in a countably additive context, but rather something weaker might suffice. And in another letter to Fishburn (dated 9 September 1966): Once you convince yourself, with Zorn's lemma, that the Blackwell-Girshick theorem cannot be had without some counterpart of P7, you will have shown that the conjecture at the bottom of page 78 of F, of S . is more or less incorrect. We thank Professor Fishburn for bringing these to our attention. FINITE ADDITIVITY AND AVOIDING DUTCH BOOK 40 1 ditionally, we recommend Fishburn (1970, Ch. 14) and Fishburn (1981) for helpful discussions of Savage's theory. In any event, P1-P4 are stated in the proof of Lemma 1 below, a lemma useful for the investigation of conditions under which P7 remains independent of PI-P6. The remaining three postulates, P5-P7, are stated following Lemma 1. Savage's postulates concern states (elements of a set S), events (subsets of S), consequences (elements of a set F ) , acts (functions from S to F ) , and a relation between acts 5 (read "is not preferred to"). Iff 5 g and g 5 f, we say g and f are equivalent. Iff is an act, the consequence of f occurring in the state s is denoted f(s). To avoid additional notation and with only slight encumbrance on the reader, we often identify a conse- quence with the act which produces that consequence in all states, that is, the constant act. If B is an event, we will use the notation I, to denote the indicator of B, that is, the function which is 1 if B occurs and 0 if not. Following Savage, we will denote the complement of B, -B. Lemma 1: Let S be a measurable set, and let F be a subset of the real numbers containing zero and closed under division (by non-zero elements) and multiplication. Assume all acts are measurable func- tions from S to F, and assume that whenever f and g are acts, and B is an event, fIB+ gI-, is an act. Assume all constant functions are acts, and denote the act which is constantly 0 as 0 . Let W be a mapping from the set of all acts to the finite real numbers which statisfies W(0) = 0, whenever fg = 0, and for all events B and all consequences f, c E F . Define f 5 g if and only if W(f) 5 W(g). Then P1-P4 are satisfied. Proof: PI: The relation 5 is a simple ordering. This is trivial and needs no proof. P2: Iff, g , and f ', g ' are acts and B is an event such that: 1. for s E -B, f(s) = g(s), and f l ( s ) = gl(s), 2. for s E B, f(s) = f t ( s ) , and g(s) = gr(s), 3 . f 5 g; then f ' 5 g ' . The conditions of P2 say that fl-, = gI-,, f 'I-, = g'I-,, f I B = f 'I,, and gIB = g t I B . Since h = hl, + hI-,, for every act h , and = 0, it follows from (2.1) that f ' 5 g' iff 5 g . 402 TEDDY SEIDENFELD AND MARK J . SCHERVISH P3: Iff - g, f ' = g', and B is not null; then f ff ' given B, if and only if g 5 g ' (as constant acts). Savage defines "f ff ' given B" to mean g C g' for every pair of acts g , g' satisfying gI, = fI, , glIB= f 'I,, and gl-, = g'I, . Under (2.1) and the conditions of P3, this can only happen if g 5 g ' (as constant acts). P4: Iff, f ', g , g ' are consequences, A , B are events, and f ,,f, , g, , g, are acts such that 1. f ' < f, g ' < g (as constant acts), 2. f,(s) = f, g,(s) = g for s E A , 3. f,(s) = f ' , g,(s) = g ' for s E - A , 4. fB(s) = f, g,(s) = g for s E B, 5. f,(s) = f ' , gB(s) = g' for s E - B , 6. f, ff,, then g, 5 g, . The conditions of P4 say that f, = fIA+ f 'I-, ,f, = fIB+ f 'I-, , g, = gl, + g'l,, and g, = gIB + g'I-, . Condition 6 together with (2.1) and (2.2) yields for some positive c E F. Condition 1 implies that (f - f ') and (g - g') have the same sign. Hence It follows from (2.2) and (2.3) that g, f g, The final three postulates are: P5: There is at least one pair of consequences f , f ' such that f ' ) g(s), in the antecedent. Our examples 2 . 2 and 2 . 3 apply to that form of P7 as well. 403 FINITE ADDITIVITY AND AVOIDING DUTCH BOOK The following lemma is stated without proof because it is so straight- forward. We then proceed to Savage's example. Lemma 2: P5 will be satisfied if W assigns different values to at least two different constant acts. Under the conditions of Lemma 1, P6 will be satisfied if for each act g, each consequence f , and each E > 0 there exists a finite partition B , , . . . , B,, such that lw(gzB,) - W(fzBt)l < € 9 for all i. Example 2.1: (Savage 1954) Let S be the set of positive integers and F the interval [0.0, 1.0). Let P be any finitely additive proba- bility on S which assigns probability 0 to each integer, assigns prob- ability to the even integers, and admits a (finite) partition of S into events of arbitrarily small probability. Any limit point (as n -t m) of the sequence of discrete uniform distributions over the first n integers will do. Define W( f ) = 1.f (s)dP(s) + lim ,,, P{f (s) r 1 - E). It is easy to see that W satisfies the conditions of Lemmas 1 and 2. Note that i f f is a gamble, i.e. having finitely many consequences almost surely, lim,,o P{f (s) r 1 - E) = 0. Thus, for gamble f, W( f ) is a utility, with W( f ) = f for a constant act f 3 f . TO see that P7 is violated, let f equal 1 - l / n for even n and 0 for odd n, and let g(s) equal the larger of and f(s). We now have W( f ) = 1 , W(g) = and W(g(s)) < 1, for all s E S . So f 1 g(s) given S for all s , but f < g . The following is a similar example which uses countably additive probabilities. Example 2.2: Let S = F be the interval of real numbers [0.0, 1.01, and let P be uniform probability on Lebesgue measurable sub- sets of S . Let all measurable functions from S to F be acts, and define a ( f ) = inf{P(E) : f (s) assumes only finitely many values on -E). Note that a(f + g ) = a(f) + a(g) whenever fg = 0. For each act f define 404 TEDDY SEIDENFELD AND MARK J . SCHERVISH It is easy to see that W satisfies the conditions of Lemmas 1 and 2. I f f is a gamble, a(f) = 0 , hence W(f) is a utility. To see that P7 is violated, let g(s) = s for all s except s = 1, and let g(1) = 0 . Let f(s) = 1, for all s . Then W(g) = 1.5, W(f) = 1, W(g(s)) = s for all s except s = 1, and W(g(1)) = 0. So f > g(s) for all s , while f < g . The feature that drives Example 2.2 is the fact that the "worth" W of an act is increased from its expected value by the extent to which the act produces uncountably many consequences. Savage proves that (given P1- P6) P7 holds for gambles (effectively Theorem 2.7.3 of Savage 1954). His example (2.1, above) shows that P7 need not hold for acts that as- sume countably many consequences. The following example shows that this remains the case even when the probability is countably additive (un- like example 2.1). Example 2.3: Let S be the half-open interval [0.0, 1.0), and F the rational numbers in S . Let P be uniform probability on Lebesgue measurable subsets of S . Let all measurable functions f from S to F satisfying a ( f ) = lim,, P{f(s) 2 1 - 2-'12' < a be acts (all subsets of F being measurable). Define Once again, W satisfies the conditions of Lemmas 1 and 2. I f f is a gamble, a ( f ) = 0 , hence W(f) is a utility. To see that P7 is violated, I/;+' for 1 - l / Z k + l > s 2let f(s) = '1: -1 and let g(s) equal 1 - '1;. Then W(f) = l/3 + 1 =1 - > W(g(s)), for each s E S . Yet W(g) = + 2 = 8 / 3 , SO that g > f. What examples 2.2 and 2.3 illustrate is the independence of the rela- tionship between PI-P6 and P7 from the degree of additivity which per- sonal probability possesses. What we hope to show in the next section is that this independence follows from the special role that consequences play in P7. If we deny the existence of consequences, attempts to refor- mulate P7 lead to the exclusion of merely finitely additive probabilities.3 3. Dominance and Conglomerabiiity of Probability. Savage's seventh postulate contrasts acts in general through a comparison of one with the 3Fishburn (1970) has also studied the relationship between P7 and countable additivity. See the Appendix to this paper for a discussion of the connection between his results and those of the present paper. 405 FINITE ADDITIVITY AND AVOIDING DUTCH BOOK consequences of the other (on some non-null event). This is possible be- cause for each consequence, say g ( ~ ) , there is an act, the constant act g* = g(s), which serves naturally as the counterpart for the consequence. However, the reader is reminded that constant acts (consequences) have, in virtue of P3, rather distinguished properties. To wit: as stipulated by P3, the relative values of consequences are unaffected given non-null events, i.e. their values are invariant under different states. Thus, a consequence must behave like a prize whose value is robust against whatever (non- null) information we might acquire.4 In practical terms, P3 prohibits approximating a consequence by an award of, e.g. stock options where the relative attractiveness of two stocks may be a function of the state of the economy. Are there good candidates for consequences? As Savage argues in his typically evenhanded style, . . . what are often thought of as consequences (that is, sure ex- periences of the deciding person) in isolated decision situations typ- ically are in reality highly uncertain. Indeed, in the final analysis, a consequence is an idealization that can perhaps never be well ap- proximated. I therefore suggest that we must expect acts with actually uncertain consequences to play the role of sure consequences in typ- ical isolated decision situations (1954, p. 84). If we concede that consequences, in the sense required by P3, are not the entities typically viewed as outcomes in familiar decisions, what can we offer in place of P7 if we ignore the relativization to contrast by con- sequences? We can recast the question this way. Let .rrB = {h,,. . .) be a, possibly infinite, partition of the event B by non-null elements hi.Can we make sense of a comparison, given B , between act f and each outcome gIhzeven though gIhpis not a consequence in the fashion of P3? (Unfor- tunately, we are forced to consider non-null hi exclusively because in Savage's theory all acts are equivalent given a null event. That is, Sav- age's program cannot generate probability conditional upon an event of 0 probability.) Suppose we attempt to avoid consequences entirely. Instead of com- paring acts f and g through consequences, we might contrast the outcomes off and g given hi(i = 1 , . . .) directly. Thus, we have P8: Iff 5 (2)g given hifor all i , then f 5 (2)g given B . 4This feature of consequences is separate from the requirement discussed by Fishburn (1970, p. 166 and 1981, p. 162) that each consequence be "relevant", i.e. an outcome for some act, for each state. In other words, the entire class of Savage-type consequences F is needed to exhaust the range of outcomes for each state. This restriction prohibits the strategy of adopting outcomes under fine descriptions for consequences, since for different states the descriptions are contraries, in violation of the clause that each consequence be "relevant" to each state. We do not know of an axiomatic approach that avoids completely the existence of (at least a pair of) Savage-type consequences "relevant" for each state. 406 TEDDY SEIDENFELD AND MARK J. SCHERVISH Of course, P8 does not suffice as a replacement for P7 if the goal is to extend expected utility theory to acts in general. To wit: the preference structures W(.) of examples 2.2 and 2.3 satisfy PI-P6 and P8, yet W(.) does not admit a ranking of acts by their expected utility of consequences. Thus, a substitute for P7 must do more (or other) than P8 to reach the goal of the expected utility hypothesis. Ideally, we would formulate a rule like P8 without the restriction that h, be non-null, i.e. to permit strict preferences conditional upon an event of zero probability. To repeat, this move is not available to us within Savage's theory because of his definition of null events. Nonetheless, we find it productive to analyze the relation between P8 and PI-P7. Our investigation uncovers a hidden tie to countable additivity, a tie that, per- haps, underpins Savage's (1954, p.78) statement about P7. P8 trades for its plausibility on a dominance principle, extended to in- finite partitions by non-null events. That is, the tacit assumption behind P8 is this: since f is not preferred to g on each element of T ~ ,f should not be preferred to g given B. Surprisingly, P8 is inconsistent with PI-P7 unless finitely, but not countably additive probability is precluded. That is, P8 fails, though PI- P7 do not, whenever the agent's subjective probability P(.) is not con- glomerable in T ~ ,or equivalently (Dubins 1975) whenever P ( . ) is not disintegrable in r B ,Without loss of generality, hereafter we assume B = S , the sure event ." Non-conglomerability (de Finetti 1972, $5.30) of a probability P(.) oc- curs in a partition T = {hl, . . .) if, for some event E and constants k1 and kZ, kl 5 ~ ( ~ l h , )k2 for each h, E T , yet P(E) < k, or P(E) > k,.5 For example (due to Dubins, see de Finetti 1972, p. 205), let the sure- event be the union of the events (i,j) where i = 1, 2, . . . is an integer and j = 0 or 1. Let I stand for the first coordinate, and let J stand for the second coordinate. Let T = {h,lh, = {(i,O),(i,1))) be a partition. Con- sider a finitely, but not countably additive probability P ( . ) such that P ( J = 0) = P ( J = 1) = and P(I = i l ~= j) = 2-"+J'. There are many such P(.). Each has probability "adherent" along the sequence of ( i , l ) events. That is, P ( J = 1) = P(U,{(i,l))) = ' 1 2 , but Z , P ( i , l ) = l/4. By finite additivity, P(h,) = so each element of 3 ~ 2 - " + ~ ' , T is non-null. Bayes' Theorem entails that P ( J = j:h,) = (2 - j ) / 3 . Hence, for each element of T , P ( J = llh,) = 1/3, however P ( J = 1) = l/z. P(.) is not conglomerable in T . Failure of P8 follows directly from non-conglomerablity of P(.) in T . 5The convenience, B = S, is justified by the result that conglomerability of P(.) fails in a partition a of the sure-event just in case it fails for P ( @ ) , given some non-null B , in the restricted partition T" (Kadane, Schervish, and Seidenfeld 1981, $4). FINITE ADDITIVITY AND AVOIDING DUTCH BOOK 407 To show that P(.) does not satisfy P8, let the dollar symbol $ denote utiles, and assume that prizes worth any desired number of utiles exist. (These prizes need not be consequences in Savage's sense.) Consider the acts f: an even odds bet on E , the event that J = 1, with stake of $2, that is, the agent places $1 on E against an opponent who places $1 on -E and g : the bet on -E {J= 0) at odds of 3:2 with a stake of $2, that is, the agent places $1.2 on -E against an opponent who places $0.80 on E . Since P(E) = = l / 3 , forg 0) he bets against E. We separate de Finetti's thesis, that there exist previsions when prizes are measured in the appropriate scale, i.e. utiles, from his working hy- pothesis, that small dollar amounts are utiles. Is the thesis neutral re- garding disputes in inductive inference? Kyburg (1978) argues in the neg- ative. For our purposes, fortunately, we do not need to enter this debate. We are prepared to grant both the thesis and working hypothesis, since our goal is to show that the reduction of belief to preference does not follow from the standard of coherence alone and our criticism is com- patible with both assumptions. However, to grant the working hypothesis is not to concede that dollar prizes (or whatever) are consequences (in Savage's sense). For construct- ing a lottery for a prevision of X, we need only have available prizes that can be awarded in cx units when X = x , subject to the thesis that the prevision is independent of the sign and magnitude of c . The unreason- ableness of viewing prizes as Savage-type consequences is as apparent to de Finetti as to Savage. When discussing the dispensibility of his working hypothesis (to cover the familiar problem of "risk aversion"), he says It would be more appropriate, instead of considering the variable x representing the gain, to take f + x, where f is the individual's 'for- tune' (in order to avoid splitting hairs, inappropriate in this context, one could think of the value of his estate). Anyway, it would be convenient to choose a less arbitrary origin to take into account the possibility that judgments may alter because in the meantime vari- ations have occurred in one's fortune, or risks have been taken, and in order not to preclude for oneself the possibility of taking these things into account, should the need arise. Indeed, as a recognition of the fact that the situation will always involve risks, it would be more appropriate to denote the fortune itself by F (considering it as a random quantity), instead of with f (a definite value) (de Finetti 1974, p.79). What de Finetti says of the agent's 'fortune' applies mutatis mutandis to the payoff of the lottery. That is, given X = x , one augments the agent's capital reserve by an amount, cx units. But the prize itself is no different in kind from the fortune (F), each of which the agent owns outright and neither of which (normatively) need be a full-blooded con- sequence. In light of our discussion in section 3, it should come as no surprise to learn that unless lottery prizes are consequences, de Finetti's criterion 410 TEDDY SEIDENFELD AND MARK J. SCHERVISH of coherence precludes all but countably additive distributions. As before, failure of conglomerability entails a (uniform) failure of dominance. Co- herence, after all, requires avoidance of previsions that induce a failure of dominance with respect to the alternative: no bet. But conglomerability in denumerable partitions is equivalent to countable additivity. Thus, for previsions of random variables assuming more than finitely many out- comes, coherence entails countable additivity. The following (typical) construction illustrates the incoherence of merely finitely additive distri- butions. For simplicity, let P(.) be a finitely additive probability assuming in- finitely many different values. It follows from Theorem 3.1 of Schervish, Seidenfeld, and Kadane (1981) that there exist an event E , a positive number d, and a partition n = {h, . . .) such both that P(h,) > 0 and P(E) - > d for all i. That is, conglomerability fails in n~ ( ~ l h , ) with regard to the event E . Let x, = ~ ( ~ l h , ) = supixi so that P(E) 2and k k + d. Consider, next, a wager W that yields a $1 (one utile) prize in case E occurs and $0 otherwise. W is worth at least k + d. However, given h,, W is worth at most k. So, the prevision of W, P(W) 2 k + d, and ~ ( ~ l h , ) = x, just < k for all i . Define the random variable X so that X in case h, obtains. That is Then X is the (conditional expected) utility of any lottery whose prizes have (conditional expected) utility given h, equal to xi for all i. W is such a lottery. Define the lottery Y by saying that Y awards prize WIht if h, obtains for i = 1, 2 , . . . . It is clear that Y = W; however, note that the prizes WIht awarded by Y are not consequences. Since 0 5 xi 5 k for all i , the prevision for any lottery whose prizes are worth xi under hi should be between 0 and k. Y is such a lottery, hence y' should be between 0 and k. But, W = Y so - ( W - y + ) should be considered fair. However, w+ r k + d, so -(W - y + ) is worth no more than -d, and hence is unfair. On the other hand, if y + is chosen equal to w', then y+ is not the expected utility over the partition n. In summary, we see that de Finetti's criterion of coherence for previ- sion rules out merely finitely additive distributions unless lotteries are restricted to prizes which are consequences in the sense of Savage. In the construction of the previous paragraph, WI,, serves as a prize whose value, given h , , equals xi; however WI,, is not a consequence, since its value is not independent of, e . g . , the event E . The dilemma is, of course, that consequences are hard to come by and, it would seem, beyond what is required by consideration of rational preference. Can we not argue, like Savage and de Finetti, that there always are risks? The lesson is clear. We will not have all three of the following: FINITE ADDITIVITY AND AVOIDING DUTCH BOOK (i) reduction of coherent belief to rational preference; (ii) coherence of finitely additive probability; (iii) admissible preference structures free of consequences. In light of the highly questionable character of consequences, it seems best to us to dispose of either (i) or (ii). Since many classical statistical procedures require finitely additive "priors" in order to be Bayesian (see e.g. Heath and Sudderth 1978), there is good reason to resist abandoning (ii). However, given (ii) and (iii), the expected utility hypothesis fails (as shown above). Without consequences to fall back upon, non-conglom- erability of finitely additive distributions cannot be squared with a re- quirement of (uniform) dominance for acts. This leaves statistical deci- sion theory devoid of a formidable criterion: admissibility (see Savage 1954, p. 114). We do not find this an easy choice to make. Perhaps further discussion will suggest a solution. APPENDIX In this appendix we examine several results and suggestions of Fishburn (1970) which are related to our discussion in section 2 . First, Fishburn (1970, C h . 10) considers the relationship between countable additivity and a set of postulates not equivalent to Savage's PI-P6. For example, Fishburn's (1970, p.137) postulate S4, that the set of probability measures be closed under countable convex combination, is not implied by Savage's sys- tem. In fact, this postulate rules out example 2 . 3 but not example 2 . 2 . Second, Savage (1954 [second edition 19721, p . 7 8 n . ) references a suggestion by Fish- burn (1970, p.213, ex. 21) for weakening P7 to accommodate acts in general, subject to the constraint that probabilities are countably additive. Fishburn's suggested version, called P7b, requires that i f f 5 (2)g(s) given A , for each s E A , then f 5 ( 2 )g given A , for constant act f = f P7b is less demanding than P7 in that it contrasts acts in general with constant acts solely, and not with other acts in general. However, P7b is not sufficient for extending utility theory to the class of acts in general even when probability is countably additive. In ex- ample 2 . 3 , W(.) satisfies P7b though acts are not ranked by W i n accord with their expected utilities (as fixed by PI-P6). T o see that W of example 2 . 3 satisfies P7b it is sufficient to verify that: (case 1) i f f 5 g(s) given A , for each s E A , then fIA5 Jg(s)IAdP(s). Since a ( . ) is non-negative, it follows that W(fIA) 5 W(gI,); hence f 5 g given A as required. (case 2) iff 2 g(s) given A , for all s E A , then as f < 1 , a ( g I A ) = 0 . Thus, W(f1,) 2 Jg(s)I,dP(s) = W(gI,), and f 2 g given A as required. It is to be observed that there can be no counterexample to the conjecture that P7b suffices for extending utility theory to acts in general if F is closed under limits of utility (as fixed by P1-P6). Since P1-P6 entail that consequences have finite utility (Savage 1954, p.81), the assumption of "closure" yields bounded utilities for gambles. However, subject to "closure" of F under preference, P7b entails P7, given P1-P6, regardless of the ad- ditivity of probability. This is seen as follows: Assume the antecedent of P7, that is f 5 ( 2 )g(s) given A , for each s E A . By hypothesis of "closure", there exist constant acts g, and g * which are the infemum and supremum of the consequences of g for s E A . Clearly f 5 g, ( 2 g*) given A . Then by P7b, g* 5 (g*2 ) g . By transitivity, f 5 ( 2 ) g , as required by P7. 412 TEDDY SEIDENFELD AND MARK J. SCHERVISH REFERENCES de Finetti, B . (1972), Probability, Induction and Statistics. New York: Wiley. de Finetti, B . (1974), Theory of Probability (2 vols.). New York: Wiley. Dubins, L. (1975), "Finitely additive conditional probabilities, conglomerability and dis- integrations", Annals of Probability 3: 89-99. Fishbum, P. C. (1970), Utility Theory for Decision Making. New York: Wiley. Fishburn, P. C. (1981), "Subjective Expected Utility: A Review of Normative Theories", Theory and Decision 13: 139-199. Heath, D. and Sudderth, W. (1978), "On finitely additive priors, coherence, and extended admissibility", Annals of Statistics 6: 333-345 Kadane, J . B . , Schervish, M . J . , and Seidenfeld, T . (1981), Statistical Implications of Finitely Additive Probability. Technical Report 206. Department of Statistics, Pitts- burgh: Carnegie-Mellon University, July. Kyburg, H. E . (1978), "Subjective probability: considerations, reflections, and problems", Journal of Philosophical Logic 7 : 157-180. Savage, L. J. (1954), Foundations of Statistics. New York: Wiley. Savage, L. J . (1971), "Elicitation of Personal Probabilities and Expectations", Journal of the American Statistical Association 66: 783-801. Schervish, M . J . , Seidenfeld, T . , and Kadane, J . B . (1981), The Extent of Non-conglom- erability of Finitely Additive Probabilities. Technical Report 198, Department of Sta- tistics, Pittsburgh: Carnegie-Mellon University, March. Shimony, A . (1955), "Coherence and the axioms of confirmation", Journal of Symbolic Logic 20: 1-28. You have printed the following article: A Conflict between Finite Additivity and Avoiding Dutch Book Teddy Seidenfeld; Mark J. Schervish Philosophy of Science, Vol. 50, No. 3. (Sep., 1983), pp. 398-412. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28198309%2950%3A3%3C398%3AACBFAA%3E2.0.CO%3B2-3 This article references the following linked citations. If you are trying to access articles from an off-campus location, you may be required to first logon via your library web site to access JSTOR. Please visit your library's website or contact a librarian to learn about options for remote access to JSTOR. [Footnotes] 7 Elicitation of Personal Probabilities and Expectations Leonard J. Savage Journal of the American Statistical Association, Vol. 66, No. 336. (Dec., 1971), pp. 783-801. Stable URL: http://links.jstor.org/sici?sici=0162-1459%28197112%2966%3A336%3C783%3AEOPPAE%3E2.0.CO%3B2-V References Finitely Additive Conditional Probabilities, Conglomerability and Disintegrations Lester E. Dubins The Annals of Probability, Vol. 3, No. 1. (Feb., 1975), pp. 89-99. Stable URL: http://links.jstor.org/sici?sici=0091-1798%28197502%293%3A1%3C89%3AFACPCA%3E2.0.CO%3B2-Q On Finitely Additive Priors, Coherence, and Extended Admissibility David Heath; William Sudderth The Annals of Statistics, Vol. 6, No. 2. (Mar., 1978), pp. 333-345. Stable URL: http://links.jstor.org/sici?sici=0090-5364%28197803%296%3A2%3C333%3AOFAPCA%3E2.0.CO%3B2-5 http://www.jstor.org LINKED CITATIONS - Page 1 of 2 - NOTE: The reference numbering from the original has been maintained in this citation list. http://links.jstor.org/sici?sici=0031-8248%28198309%2950%3A3%3C398%3AACBFAA%3E2.0.CO%3B2-3&origin=JSTOR-pdf http://links.jstor.org/sici?sici=0162-1459%28197112%2966%3A336%3C783%3AEOPPAE%3E2.0.CO%3B2-V&origin=JSTOR-pdf http://links.jstor.org/sici?sici=0091-1798%28197502%293%3A1%3C89%3AFACPCA%3E2.0.CO%3B2-Q&origin=JSTOR-pdf http://links.jstor.org/sici?sici=0090-5364%28197803%296%3A2%3C333%3AOFAPCA%3E2.0.CO%3B2-5&origin=JSTOR-pdf Elicitation of Personal Probabilities and Expectations Leonard J. Savage Journal of the American Statistical Association, Vol. 66, No. 336. (Dec., 1971), pp. 783-801. Stable URL: http://links.jstor.org/sici?sici=0162-1459%28197112%2966%3A336%3C783%3AEOPPAE%3E2.0.CO%3B2-V Coherence and the Axioms of Confirmation Abner Shimony The Journal of Symbolic Logic, Vol. 20, No. 1. (Mar., 1955), pp. 1-28. Stable URL: http://links.jstor.org/sici?sici=0022-4812%28195503%2920%3A1%3C1%3ACATAOC%3E2.0.CO%3B2-T http://www.jstor.org LINKED CITATIONS - Page 2 of 2 - NOTE: The reference numbering from the original has been maintained in this citation list. http://links.jstor.org/sici?sici=0162-1459%28197112%2966%3A336%3C783%3AEOPPAE%3E2.0.CO%3B2-V&origin=JSTOR-pdf http://links.jstor.org/sici?sici=0022-4812%28195503%2920%3A1%3C1%3ACATAOC%3E2.0.CO%3B2-T&origin=JSTOR-pdf