2.1. Open-mindedness and symmetry

Appendix A includes an example of a probability measure defined for all subsets of the natural numbers which exhibits the following main properties advocated by de Finetti: (a) it is fa but not ca; (b) it assigns measure 0 to all singletons; and (c) it is uniformly distributed (cf. Example A.1). Opinions, however, differ widely as to why each of these properties constitutes a rational constraint on the notion of subjective probability, especially in view of the operationalist account of probability we owe to writers like Ramsey, Savage and, surely, de Finetti himself.Footnote ⁶ For our purposes, let us focus on uniformity.

De Finetti’s insistence on the inclusion of uniform distributions is by and large based on the consideration of open-mindedness – rational agents should not have prior prejudices over different types of distributions, all distributions should at least be considered permissible. In addition, the assumption of uniformity is often justified on the ground of certain symmetry considerations, that is, in the absence of any existing choice algorithm, each member of a set over which a probability distribution is defined can be seen as having equal possibilities of being chosen. Hence, the argument goes, if there is any incompatibility between uniformity and ca, the latter should yield to the former because of these ‘higher’ considerations.Footnote ⁷

Admittedly, as a plea for inclusiveness, it is no doubt quite an attractive idea to be all-embracing – Savage’s own demand for μ to be definable for all subsets of S is one such example (more discussion on this below), so is the demand for all distributions to be considered permissible if not mandatory. This call for openness may resonate positively at first, yet upon a closer examination it is unclear on what grounds this claimed liberalism principally constrains rational decision making. To put it plainly, in making a decision why does anyone have to be subject to the constraints of open-mindedness and symmetry at all? Advocates who appeal to this line of justification for the use of uniform distributions in argument (†) hence face the problem of explaining why a rule for rational action should always respect these ‘higher’ mandates.

In fact, in the same spirit of liberty, if one is not dogmatic about the criteria of open-mindedness and symmetry – i.e. they are only permissible but not mandatory, to use the same terminologies – then it is easy to see that there is ample room for casting doubts on these principles. Howson and Urbach (Reference Howson and Urbach1993), for instance, challenge the basis on which a decision maker randomly chooses an integer and treats the choices as being equal: ‘any process [of selecting a number]’, they say, ‘would inevitably be biased towards the ‘front end’ of the sequence of positive integers’. Indeed, consider, for instance, 1729 and 2^77,232,917 −1, both are famous numbers by now.Footnote ⁸ However, before the latter was discovered, it would take a considerable stretch of imagination to envisage a situation where the two numbers are treated as equal citizens of the number kingdom: for one thing, the second is a number with 23, 249, 425 digits, it would take about 6,685 pages of A4 paper to print it out fully. It is not difficulty to imagine that, before its discovery, this number hardly appeared anywhere on the surface of planet earth, let alone being considered as equally selectable by some choice procedure.

In practice, it has become a matter of taste, so to speak, for theorists to either endorse or reject one or both of these principles, based on their intuitions as well as on what technical details their models require. In fact, Savage is among those who contest the assumption of uniformity. He says:

As seen, for Savage, only a thin line separates the assumption of uniformity from a spell of good faith. There is yet another and more technical reason why he refrains from making this assumption as it is alluded to in the quote above. This will take a little unpacking.

In deriving probability measures, both de Finetti and Savage invoke a notion of qualitative probability, which is a binary relation $\succeq$ defined over events: For events E, F, E $\succeq$ F says that ‘E is weakly more probable than F ’ (or ‘E is no less probable than F ’). E and F are said to be equally probable, written E ≡ F, if both E $\succeq$ F and F $\succeq$ E. A qualitative probability satisfies a set of intuitive properties.Footnote ⁹ The goal is to impose some additional assumption(s) on $\succeq$ so that it can be represented by a unique quantitative probability, μ, that is, E $\succeq$ F if and only if μ(E) ≥ μ(F).

The approach adopted by de Finetti (Reference de Finetti1937b) and Koopman (Reference Koopman1940a, Reference Koopman1940b, Reference Koopman1941) was to assume that any event can be partitioned into arbitrarily small and equally probable sub-events – i.e. the assumption of uniform partitions (UP):

As noted by Savage, when added to the list of properties of qualitative probabilities, UP is deductively sufficient in delivering a numeric representation (UP is a simple version of the Archimedean condition normally required in a representation theorem). Thus, it is out of both the intuitive appeal to symmetry and mathematical necessity that de Finetti comes to endorse UP.

Savage, however, is not tied to either of these two considerations. Given his view on uniform distributions – they are being ‘flagrantly ad hoc’ – Savage needs to find an alternative way of arriving at a representation theorem without invoking UP. To this end, he introduces a concept of almost uniform partition (AUP):

The idea is to partition an event into arbitrarily ‘fine’ sub-events but without asking them to be equally probable. It is plain that a uniform partition is an almost uniform partition, but not vice versa.

The genius of Savage’s proof was thus to show (1) that his proposed axioms are sufficient in defining AUP in his system and (2) that AUP is all that is needed in order to arrive at a numeric representation of qualitative probability. Technical details of how numeric probability measures are derived in Savage’s theory do not concern us here.Footnote ¹⁰ But what is clear is that, in view of Savage’s take on uniformity it would be misplaced to invoke argument (†) against ca within his system.

2.2. Money pump and infinite bets

Even if we grant that uniformly distributed measures be permissible in Savage-type decision models, it can be shown that the admission of such a measure together with fa may subject an agent to a Dutch book. Appendix B contains an example of Adams’ money pump, where the agent’s subjective probability is fa but not ca and is defined in terms of the uniform distribution λ introduced in Example A.1. As shown there, a series of bets can be arranged such that the bettor suffers a sure loss.

Adams’ money pump is surprising, because it results in a set of incoherent choices made by the bettor, a result that is precisely what subjectivist systems like Savage’s are devised to avoid. Indeed, if there is a single principle that unites Bayesian subjectivists it would arguably be coherence. One important reason this principle is fundamental to Bayesian subjectivism is that this coherence-oriented approach to rationality democratizes, so to speak, legitimate reasons for acting: different agents might be moved by different (probabilistic or utility) considerations that lead them to behave differently (or similarly), but as long as their behaviours are consistent, they are equally rational in the subjectivist sense. For instance, when both picking up a slice of cheesecake, I doubt my five-year-old nephew had to endure the same kind of inner struggles that I had to deal with – he, for one thing, certainly would not think of going for the second largest slice for a quite nuanced reason. But as long as we both are consistent in our choices we are equally rational in the eyes of subjectivists. For subjectivists, the notion of rationality is based on a rather weak logical principle, namely coherence.Footnote ¹¹ Adams’ example reveals a conflict with this basic principle.

When faced with this difficulty, advocates of fa often argue that given that a subjective decision theory is a systematization of coherent decision making by rational agents it is unclear what it means for a bettor to fulfil the task of coherently betting infinitely many times. On this view, the challenge from Adams’ money pump, which requires the bettor to accept infinitely many gambles, is a non-starter, for it envisages a situation that is not operationally feasible.Footnote ¹²

Perhaps, this is another place where a theorist needs to exercise their acquired taste in order to discern whether or not it is conceptually salient to entertain infinite bets. I, nonetheless, want to point out that there is already a great deal of idealization built into the subjectivist theory that requires us to be somewhat imaginative. As a starter, it is a common practice to assume that the agents being modelled are logically omniscient. One reason for upholding such an assumption is that, in a model of decision making under uncertainties, uncertainties may stem from different sources: they may be due to the lack of empirical knowledge or the result of computational failures. To separate different types of uncertainties, it is often assumed, like in de Finetti’s and Savage’s systems, that the agents are equipped with unlimited deductive capacities in logic.

However, it would be quite a double standard to insist in these systems that agents are infinitely capable when it comes to computational or inferential performances, on the one hand; but to appeal to certain physical or psychological realism when it comes to accepting bets, on the other. This, nevertheless, does not mean that there can be no limits to the amount of idealizations that one injects into a model. In what follows we will explore some measures for constraining idealism that may lead to a better understanding of theoretic modelling.

3.1. Logical omniscience and measurability

As seen, Savage’s set-theoretic argument for not imposing ca was given in ZFC + CH, where it is known that, in the uncountable case, there is no non-trivial measure that simultaneously satisfies conditions (i)–(iii) above; and that Savage’s immediate reaction was to replace the third, i.e. the ca condition, with fa.Footnote ¹⁵ The particular set-theoretic argument Savage relied on – namely the existence of the Ulam matrix which leads to the non-existence of a measure over ${{\aleph }_{1}}$ (cf. fn 13) – uses AC in an essential way.

One unavoidable consequence of this approach is that it introduces non-measurable sets in ZFC. Now, if one insists on imposing condition (i) for defining subjective probability – that is, a subjective probability measure be defined for all subsets of the states – this amounts to introducing non-measurable sets into Savage’s decision model. Yet, it is unclear what one may gain from making such a high demand.

Non-measurable sets are meaningful only insofar as we have a good understanding of the contexts in which they apply. These sets are highly interesting within certain branches of mathematics largely because the introduction of these sets reveals in a deep way the complex structures of the underlying mathematical systems. However, this does not mean that these peculiar set-theoretic constructs should be carried over to a system that is primarily devised to model rational decision making.

It might be objected that the subjectivist theory we are concerned with is, after all, highly idealized – the assumption of logical omniscience, for instance, is adopted in almost all classical decision models. By that extension, one may as well assume that the agent being modelled is mathematically omniscient (an idealized perfect mathematician). Then, it should be within the purview of this super agent to contemplate non-measurable sets/events in decision situations. Besides, if we exclude non-measurable sets from decision theory, then why stop there? In other words, precisely where shall we draw the line between what can and cannot be idealized in such models?

This is a welcome point, for it highlights the issue of how much idealism one can instil into a theoretic model. It is no secret that decision theorists are constantly torn between the desire to aspire to the perfect rationality prescribed in the classical theories, on the one hand; and the need for a more realistic approach to practical rationality, on the other. Admittedly, this ‘line’ between idealism and realism is sometimes a moving target – it often depends on the goal of modelling and, well, the fine taste of the theorist. An appropriate amount of idealism allows us to simplify the configuration of some aspects of a theoretic model so that we can focus on some other aspects of interest. An overdose, however, might be counter-effective as it may introduce unnecessary mysteries into the underlying system.

Elsewhere (Gaifman and Liu Reference Gaifman and Liu2018), we introduced a concept of conceptual realism in an attempt to start setting up some boundaries between idealism and realism in decision theory. We argued that, as a minimum requirement, the idealizations one entertains in a theoretic model should be justifiable within the confines of one’s conceivability. And, importantly, what is taken to be conceivable should be anchored from the theorist’s point of view, instead of delegating it to some imaginary super agent. Let me elaborate with the example at hand.

As noted above, it is a common practice to adopt the assumption of logical omniscience in Bayesian models. This step of idealization, I stress, is not a blind leap of faith. Rather, it is grounded in our understanding of the basic logical and computational apparatuses involved. We, as actual reasoners, acknowledge that our inferential performances are bounded by various physical and psychological limitations. Yet a good grasp of the underlying logical machinery gives rise to the conceivable picture as to what it means for a logically omniscient agent to fulfil, at least in principle, the task of, say, drawing all the logical consequences from a given proposition.Footnote ¹⁶

This justificatory picture – which apparently is based on certain inductive reasoning – becomes increasingly blurry when we start contemplating how our super agent comes to entertain non-measurable events in the context of rational decision making. The Banach–Tarski paradox sets just the example of how much we lack geometric and physical intuitions when it comes to non-measurable sets. This means, unlike logical omniscience, we don’t have any clue of what we are asking from our super agent: beyond any specific set-theoretic context there is just no good intuitive basis for conceiving non-measurable sets. Yes, it might be comforting that we can shift the burden of conceiving the inconceivables to an imaginary super agent, but in the end such a delegated job is either meaningless or purely fictional. So it seems that if there is any set-theoretic oddity to be avoided here it should be non-measurable sets.Footnote ¹⁷

On this matter, I should add that Savage himself is fully aware that the set-theoretic context in which his decision model is developed exceeds what one can expect from a rational decision maker (Savage Reference Savage1967). He also cites the Banach–Tarski paradox as an example to show the extent to which highly abstract set theory can contradict common sense intuitions. However, it seems that Savage’s readiness to embrace all-inclusiveness of defining the background algebra as ‘all sets of states’ overwhelms his willingness to avoid this set-theoretic oddity.

3.2. The standard approach

In fact, the situation can be largely simplified if we choose to work, instead of with all subsets of the state space S, but with a sufficiently rich collection of subsets of S (for instance, the Borel sets ${\scr B}$ in the case where $S={\BB R}$ ) where, as a well established theory, ca is in perfectly good health. That is, instead of (i), we require that

(i*) μ is defined on (Lebesgue) measurable sets of ${\BB R}$ .

Note that the price of forfeiting the demand from condition (i) is a rather small one to pay. It amounts to disregarding all those events that are defined by Lebesgue non-measurable sets. Indeed, even Savage himself conceded that

It shall be emphasized that this modification of the definition of events from, say, the set of all subsets of (0,1] to the Borel set ${\scr B}$ of (0,1] is not carried out at the expense of dismissing a large collection of events that are otherwise representable. As noted by Billingsley (Reference Billingsley2012: 23), ‘[i]n fact, ${\scr B}$ contains all the subsets of (0,1] actually encountered in ordinary analysis and probability. It is large enough for all ‘practical’ purposes.’ By ‘practical purposes’ I take it to mean that all events and measurable functions considered in economic theories in particular are definable using only measurable sets, and, consequently, there is no need to appeal to non-measurable sets.

To summarize, I have shown that Savage’s mathematical argument against ca is inconclusive – its validity depends on the background set-theoretic set-ups – and that ca can in fact form a consistent theory under appropriate arrangements. This brought us to a fine point concerning how one understands background mathematical details and how best to incorporate idealism into a theoretic model. As we have seen, a healthy amount of conceptual realism may keep us from many highly idealized fantasies.

4.1. Quantitative and qualitative continuities

In Savage’s representation theorem, it is assumed that the background algebra is a σ-algebra (i.e. closed under infinite unions and intersections). Savage remarks that

This ‘peculiar’ feature of the system reveals a certain ‘imbalance’ between the finite adding speed, so to speak, of fa, on the one hand; and countable unions in a σ-algebra, on the other. There are two intuitive ways to restore the balance. One is to reduce the size of the background algebra. This is the approach adopted in Gaifman and Liu (Reference Gaifman and Liu2018), where we invented a new technique of ‘tri-partition trees’ and used it to prove a representation theorem in Savage’s system without requiring the background algebra to be a σ-algebra.

The other option is to tune up the adding speed and bring ‘continuity’ between probability measures and algebraic operations. To be more precise, let $(S,{\cal F},\mu )$ be a measure space, μ is said to be continuous from below, if, for any sequence of events $\{{{A}_{n}}\}_{n=1}^{\infty }$ and event A in ${\cal F}$ , A_n ↑ A implies that μ(A_n ) ↑ μ(A); it is continuous from above if A_n ↓ A implies μ(A_n ) ↓ μ(A), and it is continuous if it is continuous from both above and below.Footnote ¹⁸ It can be shown that continuity fails in general, if μ is merely fa.Footnote ¹⁹ In fact, it can be proved that continuity holds if and only if μ is ca.

One way to introduce continuity to Savage’s system is to impose a stronger condition on qualitative probability. Let $\succeq$ be a qualitative probability, defined on a σ-algebra ${\cal F}$ of the state space S. Following Villegas (Reference Villegas1964), $\succeq$ is said to be monotonously continuous if, given any sequence of events A_n ↑ A (A_n ↓ A) and any event B,

$${{A}_{n}}\preceq B({{A}_{n}}\succeq B)\ {\rm {for \ all \,}} }n\,\Rightarrow \,A\preceq B(A\succeq B).$$ (3)

Moreover, Villegas showed that if a qualitative probability $\succeq$ is atomless and monotonously continuous then the numerical probability μ that agrees with $\succeq$ is unique and ca.Footnote ²⁰ Since the qualitative probability in Savage’s system is non-atomic, it is sufficient to introduce the the property of monotone continuity in order to bring in ca. We thus can add the following postulate, P8, to Savage’s P1–P7 (in Appendix C), which is a reformulation of (3) in terms of preferences among Savage acts.

P8: For any a, $b\in X$ and for any event B and any sequence of events $\{ {A_n}\} $ ,

1. (i) if A_n ↑ A and ${{{\frak c}}_{a}}|{{A}_{n}}+{{{\frak c}}_{b}}|\overline{{{A}_{n}}}\preccurlyeq {{{\frak c}}_{a}}|B+{{{\frak c}}_{b}}|\overline{B}$ for all n then ${{{\frak c}}_{a}}|A+{{{\frak c}}_{b}}|\overline{A}\preccurlyeq {{{\frak c}}_{a}}|B+{{{\frak c}}_{b}}|\overline{B}$ ;

2. (ii) if A_n ↓ A and ${{{\frak c}}_{a}}|{{A}_{n}}+{{{\frak c}}_{b}}|\overline{{{A}_{n}}}\succcurlyeq {{{\frak c}}_{a}}|B+{{{\frak c}}_{b}}|\overline{B}$ for all n then ${{{{\frak c}}_{a}}|A}+{{\frak c}}_{b}}|\overline{A}\succcurlyeq {{{\frak c}}_{a}}|B+{{{\frak c}}_{b}}|\overline{B}$ .

This allows us to introduce ca into Savage’s decision model as an added postulate. P8 (and hence ca), however, cannot replace the role played by P7. This is a delicate matter, we investigate it in the next section.

4.2. Countable additivity and P7

In Savage’s theory, a representation theorem for simple acts (i.e. acts that may potentially lead to only finitely many consequences) can be given under P1–P6. Savage’s P7 plays the sole role of extending utility representations from simple acts to general acts (i.e. acts that may potentially lead to infinitely many different outcomes).

Savage (Reference Savage1972: 78) gave an example which satisfies the first six of his seven postulates but not the last one. This is intended to show that the seventh postulate (P7) is independent of other postulates in Savage’s original system. Upon showing the independence of P7, Savage remarked that ‘[f ]inite, as opposed to countable, additivity seems to be essential to this example’, and he conjectured that ‘perhaps, if the theory were worked out in a countably additive spirit from the start, little or no counterparts of P7 would be necessary’. This section is aimed at taking a closer look at Savage’s remark on the relation between ca and utility extension under various versions of P7.

In a footnote to the remark above Savage adds: ‘Fishburn (Reference Fishburn1970, Exercise 21, p. 213) has suggested an appropriate weakening of P7’. It turned out that this is inaccurate. To wit, the following is Fishburn’s suggestion (expressed using our notation).

• P7b: For any event $E \in {\cal F}$ and $a \in {X}$ , if ${{{\frak c}}_{a}}{{\succcurlyeq }_{E}}\,{{{\frak c}}_{g(s)}}$ for all $s \in {E}$ then ${{{\frak c}}_{a}}{{\succcurlyeq }_{E}}\,g$ ; and if ${{{\frak c}}_{a}}{{\preccurlyeq }_{E}}\,{{{\frak c}}_{g(s)}}$ for all $s \in {E}$ then ${{{\frak c}}_{a}}{{\preccurlyeq }_{E}}\,g$ .

P7b is weaker than P7 in that it compares act g with a constant act instead of another general act f. Note that Fishburn’s P7b is derived from the following condition A4b that appeared in his discussion on preferential axioms and bounded utilities (§10.4).

• A4b: Let X be a set of prizes/consequences and Δ(X) be the set of all probability measures defined on X, then for any $P\!\in \!\Delta(X)$ and any A ⊆ X if P(A) = 1 and, for all $x\!\in \!A$ , ${{\delta }_{x}}\succcurlyeq (\!\!\preccurlyeq\!){{\delta }_{y}}$ for some $y\!\in \!X$ then $P\succcurlyeq\!(\!\!\preccurlyeq\!){{\delta }_{y}}$ , where δ_x denotes the probability that degenerates at x.

A4b, together with other preference axioms discussed in the same section, are used to illustrate, among other things, the differences between measures that are countably additive and those are not. It was proved by Fishburn that the expected utility hypothesis holds under A4b, that is, if Δ(X) contains only ca measure, then

$$P\succ Q\,\Longleftrightarrow \,E(u,P)>E(u,Q),\quad {\rm{for \ all }}\,\,\,P,Q\in \Delta (X).$$ (4)

Fishburn then showed, by way of a counterexample, that the hypothesis fails if the set of probability measures contains also merely fa ones. Because of its direct relevancy to our discussion on the additivity condition below, we reproduce this example (Fishburn Reference Fishburn1970: Theorem 10.2) here.

Example 4.1

Let $X={{{\BB N}}^{+}}$ with u(x) = x/(1 + x) for all x ∈ X. Let Δ(X) be the set of all probability measures on the set of all subsets of X and defined u on Δ(X) by

$$u(P)=E(u,P)+\inf \big\{P[u(x)\ge 1-\epsilon ]:0 \lt \epsilon \le 1\big\}.$$ (5)

Define ≻ on Δ(X) by P ≻ Q iff u(P) > u(Q). It is easy to show that A4b holds under this definition. However if one takes P to be the measure in Appendix A, i.e. a finitely but not countably additivity probability measure, then we have u(λ) = 1 + 1 = 2. Hence u(λ) ≠ E(u,λ) = 1. This shows the expected utility hypothesis fails under this example.⊲

However, as pointed out by Seidenfeld and Schervish (Reference Seidenfeld and Schervish1983: Appendix), Fishburn’s proof of (4) using A4b was given under the assumption that Δ(X) is closed under countable convex combination (condition S4 in Fishburn Reference Fishburn1970: 137), which in fact is not derivable in Savage’s system. They show through the following example (see Example 2.3, p. 404) that the expected hypothesis fails under the weakened P7b (together with P1–P6) and this is so even when the underlying probability is ca.

Example 4.2

Let S be [0,1) and X be the set of rational numbers in [0,1). Let μ be uniform probability on measurable subsets of S and let all measurable function f from S to X satisfying $V[\;f]=\mathop{\lim }_{i\to\infty }\mu [\;f(s)\ge 1-{{2}^{-i}}]$ be acts. For any act f, let $U[\;f]=\int_{S}{}u(\;f)d\lambda$ where u(x) = x is a utility function on X and define

$$W[\;f]=\, U[\;f]+V[\;f] \\ =\int_S u(\;f(s)) d\mu(s)+\lim_{i\to\infty}\mu\bigg[f(s)\ge 1-{{1}\over{2^i}}\bigg].$$ (6)

Further, define f ≻ g if W[ f] > W[g]. It is easy to see that P1–P6 are satisfied. To see that W satisfies P7b, note that if for any event E and any $a \in X$ , ${{{\frak c}}_{a}}{{\succcurlyeq }_{E}}\,{{{\frak c}}_{g(s)}}$ for all $s \in E$ , then by (6), we have 1 > u(a) ≥ u(g(s)) for any $s \in E$ . Note that 1 > u(g(s)) implies V[gχ_E ] = 0 where χ is the indicator function. Thus, $W[{{{\frak c}}_{a}}{{\chi }_{E}}]=\int_{E}{}u(a)d\mu \ge \int_{E}{}u(g(s))d\mu (s)=W[g{{\chi }_{E}}]$ . The case ${{{\frak c}}_{a}}{{\preccurlyeq }_{E}}\,{{{\frak c}}_{g(s)}}$ can be similarly shown.⊲

In other words, contrary to what Savage had thought, P7b is in fact insufficient in bringing about a full utility representation theorem even in the presence of ca.Footnote ²¹ This shows, a fortiori, that ca alone is insufficient in carrying the utility function derived from P1–P6 from simple acts to general acts. Seidenfeld and Schervish (Reference Seidenfeld and Schervish1983: Example 2.2) also showed that this remains the case even if the set of probabilities measure is taken to be closed under countable convex combination.

As seen, Savage’s last postulate which plays the role of extending the utilities from simple acts to general acts cannot be easily weakened even in the presence of ca. Yet, on the other hand, it is clear that ca is a stronger condition than fa originally adopted in Savage’s theory. So, for future work, it might be of interest to find an appropriate weakening of P7 in a Savage-style system in the presence of ca that is still sufficient in extending utilities from simple acts to general acts.