Brian Skyrms (Reference Skyrms1987, Reference Skyrms1990, Reference Skyrms1993, Reference Skyrms1997) has discussed the role of dynamic coherence arguments in the theory of personal or subjective probability. In particular, Skyrms (Reference Skyrms1997) both reviews and discusses the utility of martingale arguments in establishing the convergence of beliefs within the context of radical probabilism. The classical martingale converence theorem, however, assumes the countable additivity of the underlying probability measure; an assumption rejected by some subjectivists such as Bruno de Finetti (see, e.g., de Finetti Reference de Finetti1930 and Reference de Finetti1972). This brief note has a very modest goal: to briefly consider the extent to which Skyrms's argument can be extended to the finitely additive case.
1. Dynamic Coherence
Let us begin by briefly reviewing Skyrms's argument. Let (Ω, 
, P) be a probability space; that is, Ω is a set (the sample space), 
is a σ-algebra of subsets of Ω, and P is a countably additive probability measure on 
. By definition, a random variable X on (Ω, 
, P) is an 
-measurable function X: Ω → R.
Let 
1 ⊆ 
2 ⊆ 
3 … be an increasing sequence of sub-σ-algebras of 
; intuitively 
n represents the information available to us at time n. Let X 1, X 2, X 3, … be a sequence of random variables on the probability space. One says that X 1, X 2, X 3, … is a martingale with respect to {
n : n ≥ 1} provided:
1. X n is
n— measurable;2. E [ | X n| ] < ∞;
3. E [ X n+1>|
n] = X n almost surely.
n— measurable;
n] = X n almost surely. The celebrated Doob martingale convergence theorem (see, for example, either Billingsley, Reference Billingsley1995 or Durrett, Reference Durrett1996) assures us that if the expectations E[| X n|] are bounded, then such a sequence of random variables converges almost surely to an 
-measurable random variable X:
Skyrms's clever observation is the following. Suppose that X n represents your subjective probability regarding an uncertain outcome on the basis of the information available to you at time n; and let 
n = σ(X 1, X 2, … , X n), the σ-field generated by the random variables X 1, X 2, … . , X n. (No particular model such as Bayesian updating is being assumed here, the revision has taken place in a “black box”.) Then the resulting sequence 
1, 
2, 
3, … is an increasing sequence of sub-σ-algebras of 
and each of the three conditions above in the definition of a martingale must be satisfied: the first because a random variable is (by definition) measurable with respect to the σ-algebra it generates; the second because X n is a probability (and therefore 0 ≤ X n ≤ 1); and the third is an instance of the “principal principle”,
and can be justified on the grounds of a Dutch book coherence argument (see Skyrms, Reference Skyrms1997, Section 5).
Thus the sequence X 1, X 2, X 3, … is a martingale. Because of the very special nature of the random variables in question (they are nonnegative and bounded), the martingale convergence theorem applies to the sequence, and therefore (subjective) probability one is necessarily assigned to the event that the sequence of probability assessments will converge to a limiting value.
What are the potential vulnerabilities in such an argument? There are two obvious points of attack. First, there is the key martingale assumption that E[ X n+1 || X 1, X 2, … , X n] = X n almost surely. It is certainly possible, given the black box nature of the process, that this might not hold but, as Skyrms discusses, this would violate a natural coherence condition.
That leaves the assumption of countable additivity, implicitly invoked in the use of the MCT (martingale convergence theorem). Is it really necessary? That question is the focus of this note.
2. The Finitely Additive Version of the MCT
In an important paper, Purves and Sudderth (Reference Purves and Sudderth1976) have discussed the extension of many of the classical limit theorems of probability to the finitely additive setting. Their approach is based on the concept of a strategy. Here are the basic definitions.
Let X and Y be nonempty sets. From now on, a probability on a set means a finitely additive probability measure defined on all the subsets of the set; and a conditional probability on Y given X is a function that assigns to each element x ∊ X a probability (in this sense) on Y. A strategy σ on X × Y is then a pair (σ0, σ1) consisting of a probability σ0 on X and a conditional probability σ1 on Y given X. Every such strategy in turn determines a probability on X × Y: given a set S ⊆ X × Y, if S x = { y ∊ Y: (x, y) ∊ S}, then
(The existence and uniqueness of this integral is a standard topic in finitely additive integration theory.)
Probability measures on X × Y arising in this way are said to be strategic; and it is an unexpected feature of finitely additive probability measures on a product space X × Y that not all such probabilities are strategic.
In the martingale convergence theorem, probability statements are made about infinite sequences of observations. Such probabilities are built up from probabilities defined on more basic objects, sets of finite sequences. Thus, given a set X, let X* denote the set of all finite sequences of elements in X (including the null sequence). A strategy σ in this extended context is then defined to be a conditional probability on X given X*. (The earlier definition of strategy only assigned conditional probabilities given sequences of length zero and one.)
Let σ0 denote the conditional probability on X that σ assigns to the empty sequence (that is, the unconditional probability on X). It is apparent that one can, in the manner discussed earlier, use σ (in conjunction with σ0) to obtain a strategic measure on X n, the Cartesian product of X with itself n times, for all n ≥ 1. It is less obvious, but a fundamental result of Dubins and Savage (Reference Dubins and Savage1965, 7–21) that such a strategy determines a positive linear functional on the bounded real-valued functions on X ∞, the space of infinite sequences of elements of X, endowed with the discrete topology. This in turn means that the strategy determines a probability on certain special subsets of X ∞, the sets that are at once both open and closed in the discrete topology. Purves and Sudderth (Reference Purves and Sudderth1976) prove that the probability can in fact be extended to a much larger class of sets 
(σ); this class is an algebra (but not a σ-algebra) and is quite rich: it contains the Borel sets.
This extension of σ in turn permits the establishment of a martingale convergence theorem in the finitely additive setting (Purves and Sudderth Reference Purves and Sudderth1976, Theorem 7.3). The set of sequences for which convergence does not occur is contained in an 
(σ)-set that has (extended) σ probability zero. (Purves and Sudderth (Reference Purves and Sudderth1983) discuss some other examples of zero-one laws in the finitely additive setting.)
3. Strategic Measures and Coherence
One remaining question is the justification for considering strategies and strategic measures. Lane and Sudderth (Reference Lane and Sudderth1985) have advanced an attractive notion of coherence that singles out precisely the strategic measures. This goes briefly as follows. Suppose P is a probability on X × Y, and q is a conditional probability on Y given X. If A ⊆ X × Y is a set, let I A denote the indicator function of the set (that is, I A(x) = 1, x ∊ A, I A (x) = 0, x ∉ A), and let A x = { y ∊ Y: (x, y) ∊ A}. Interpreting P and q as prices for gambles on (x, y) and y given x, respectively, a bet of c on A yields a net return of
and, letting S ⊆ X and B = {(x, y): y ∊ B x}, a bet of d on B x conditional on x ∊ S yields a net return of
The Lane and Sudderth notion of coherence is then very simple: viewing the posted odds as those of a bookie, if a gambler is limited to a finite number of transactions of each of these two types, then the bookie is coherent provided the gambler cannot ensure a positive return. That is, one cannot find functions φ 1, … , φ m and ψ 1, … , ψ n such that
Lane and Sudderth then prove that a pair (P,q) is coherent in this sense if and only if P is a finitely additive probability and (letting P 0 denote the marginal of P on X)
for every subset A ⊆ X × Y.
4. Concluding Comments
Strategic probability measures are also sometimes termed disintegrable. Dubins (Reference Dubins1975, Theorem 1) demonstrated that this property is equivalent to another, apparently different one, the earlier property of conglomerability, discovered by de Finetti (Reference de Finetti1930 and Reference de Finetti1972, 98). The Lane-Sudderth notion of coherence is closely related to an earlier notion of coherence introduced by Freedman and Purves (Reference Freedman and Purves1969) and later studied by Heath and Sudderth (Reference Heath and Sudderth1978) and Lane and Sudderth (Reference Lane and Sudderth1985). This earlier notion (and implicitly the later Lane-Sudderth notion) has been criticized by Kadane, Schervish, and Seidenfeld (Reference Kadane, Schervish and Seidenfeld1996), appealing to a result of Schervish, Seidenfeld, and Kadane (Reference Schervish, Teddy and Kadane1984). This important result states that every finitely additive probability that is not countably additive fails to be disintegrable (or, as discussed above, conglomerable) with respect to some countable partition.
The comments of Lane and Sudderth, however, appear to adequately address this point. They point out that this argument, which appears to argue that a probability must be countably additive in order to be coherent, runs afoul of a simple objection: if countable partitions are permitted, then it seems arbitrary to exclude uncountable partitions. But then even countably additive probabilities can fail to satisfy this restriction.
In a later survey paper, Sudderth (1994) returned to this issue, and his brief comment seems very much to the point:
Several authors (see, for example, Heath and Sudderth (Reference Heath and Sudderth1972) or Skyrms (Reference Skyrms1984)) have remarked that if a gambler is allowed to make countably many bets, then P must be countably additive to avoid a sure loss. However, de Finetti (Reference de Finetti1972, p. 91) considered such arguments to be circular because they rely on the usual conventions about infinite sums which are tantamount to an assumption of countable additivity.
Acknowledgements. I would like to thank Brian Skyrms for suggesting this topic and providing several references. Thanks also to Persi Diaconis and Teddy Seidenfeld for helpful conversations.
