Classical subjective expected utility

We consider a standard Savage setting, where (S, Σ) is a measurable state space and X is an outcome space. An act is a map f: S → X that delivers outcome f (s) in state S. Let ℱ be the set of all simple and measurable acts. [Maps such that is finite and for all .]

The DM’s preferences are represented by a binary relation ≿ over ℱ. We assume that ≿ satisfies the classic Savage axioms P.1–P.6. By his famous representation theorem, these axioms are equivalent to the existence of a utility function u: X → ℝ and a (strongly) nonatomic finitely additive probability P on S such that the SEU evaluation represents ≿. [Strong nonatomicity of P means that for each and there exists such that and . See ref. 12, p. 141–143 for the various definitions and properties of nonatomicity of finitely additive probabilities.] In this case, u is cardinally unique and P is unique.

Given any f, g ∈ ℱ and E ∈ Σ, f Eg is the act equal to f on E and to g otherwise. The conditional preference ≿_E is the binary relation on ℱ defined by f ≿_E g if and only if f Eh ≿ gEh for all h ∈ ℱ. By P.2, the sure thing principle, ≿_E is complete. An event E ∈ Σ is said to be null if ≿_E is trivial (ref. 1, p. 24); in the representation, this amounts to P(E) = 0 (E is null if and only if it is P-null).

For each nonnull event E, the conditional preference ≿_E satisfies P.1–P.6 because the primitive preference does [e.g., Kreps (ref. 7, Chap. 10)]. Hence, Savage’s theorem can be stated in conditional form by saying that ≿ satisfies P.1–P.6 if and only if there is a utility function u: X → ℝ and a nonatomic finitely additive probability P on S such that, for each nonnull event E,represents ≿_E, where P(⋅|E) is the conditional of P given E.

As usual, we denote by Δ = Δ(S, Σ) the collection of all (countably additive) probability measures on S. Unless otherwise stated, in the rest of this paper all probability measures are countably additive.

In the sequel, we consider subsets M of Δ. Each subset M of Δ is endowed with the smallest σ-algebra ℳ that makes the real valued and bounded functions on M of the form m ↦ m(E) measurable for all E ∈ Σ and that contains all singletons. In the important special case M = Δ, we write instead of ℳ.

Probability measures μ on Δ are interpreted as prior probabilities. The observation of a (non--null) event E allows us to update prior μ through the Bayes rulefor all D ∈ , thus obtaining the posterior of μ given E.

A finite subset M = {m₁, …, m_n} of Δ is linearly independent if, given any collection of scalars {α₁, …, α_n} ⊆ ℝ,

Two probability measures m and m′ in Δ are orthogonal (or singular), written m ⊥ m′, if there exists E ∈ Σ such that m(E) = 0 = m′(E^c). A collection of models M ⊆ Δ is orthogonal if its elements are pairwise orthogonal.

If E ∈ Σ and m(E) = 0 imply m′(E) = 0, m′ is absolutely continuous with respect to m and we write m′ ≪ m.

Finally, we denote by Δ_na the collection of all nonatomic probability measures. By the classical Lyapunov theorem, the range {(m₁(E), …, m_n(E)): E ∈ Σ} of a finite collection of nonatomic probability measures is a convex subset of ℝⁿ.

The first issue to consider in our normative approach is how DMs’ behavior should reflect the fact that they regard M as a datum of the decision problem. To this end, given a subset M of Δ, say that an event E is unanimous if m(E) = m′(E) for all m, m′∈ M. In other words, all models in M assign the same probability to event E.

A preference ≿ is consistent with a subset M of Δ if, given E, F ∈ Σ, with E unanimous,for all outcomes x ≻ y.

Consistency requires that the DM is indifferent among bets on events that all probability models in M classify as equally likely. The next stronger consistency property requires that DMs prefer to bet on events that are more likely according to all models.

A preference ≿ is order consistent with a subset M of Δ if, given E, F ∈ Σ, with E unanimous,for all outcomes x ≻ y.

Both these notions are minimal consistency requirements among information and preference that behaviorally reveal (to an outside observer) that the DM considers M as a datum of the decision problem. Note that order consistency implies consistency because the premise of [7] implies that also F must be unanimous (this observation also emphasizes how weak an assumption is consistency).

We can now state our basic representation result, which considers finite sets M of nonatomic models.

Let M be a finite subset of Δ_na. The following statements are equivalent:represents ≿.

Moreover, u is cardinally unique for each ≿ satisfying statement i, whereas μ is unique for each such ≿ if and only if M is linearly independent.

Although uniqueness of the utility function u is well known and well discussed in the literature, uniqueness of the prior μ is an important feature of this result. In fact, it pins down μ even though its domain is made of unobservable probability models. Because of the structure of Δ, it is the linear independence of M—not just its affine independence—that turns out to be equivalent to this uniqueness property. This simple, but useful, fact is well known [e.g., Teicher (13)].

Each prior μ: → [0, 1] induces a predictive probability on the sample space through the reduction [2]. The reduction map relates subjective probabilities on the space M of models to subjective probabilities on the sample space S, that is, prior and predictive probabilities. [Note that probability measures on S can play two conceptually altogether different roles: (subjective) predictive probabilities and (objective) probability models.] Clearly, [9] implies thatwhich is the reduced form of V, its Savage’s SEU form. As observed in the introductory section, this is the criterion that an outside observer, unaware of datum M, would be able to elicit from the DM’s behavior. It is a much weaker representation than the structural one ([9]), which can be equivalently written asbecause suppμ ⊆ M (recall that finite subsets of are measurable). This is the criterion that, instead, an outside observer aware of M would be able to elicit. In fact, denote by Δ(M) the collection of all priors μ: → [0, 1] such that suppμ ⊆ M. The informed observer would be able to focus on the restriction of the reduction map to Δ(M). If M is linearly independent, such correspondence is one-to-one and thus allows prior identification from the behaviorally elicited Savagean probability through inversion.

The structural representation [9] is a version of Savage’s representation that may be called classical SEU because it takes into account Waldean information, with its classical flavor. [Diaconis and Freedman (14) call “classical Bayesianism” the Bayesian approach that considers as a datum of the statistical problem the collection of all possible data-generating mechanisms.] In place of the usual SEU pair (u, P) the representation is now characterized by a triple (u, M, μ), with suppμ ⊆ M. According to the Bayesian paradigm, the prior μ quantifies probabilistically the DM’s uncertainty about which model in M is the true one. This kind of uncertainty is sometimes called (probabilistic) model uncertainty or parametric uncertainty.

In the introductory section, we observed that when datum M is a singleton, the classical SEU criterion [9] reduces to the von Neumann–Morgenstern expected utility criterion [4], which is thus the special case of classical SEU that corresponds to singleton data. In contrast, when M is nonsingleton but the support of a prior μ is a singleton, say suppμ = {m′} ⊆ M, then it is the DM’s personal information that prior μ reflects, which leads him to a predictive that coincides with . In this case,is a Savage’s SEU criterion.

In Proposition 1 the support of the prior is included in M; i.e., suppμ ⊆ M. In fact, because of consistency, models are assigned positive probability only if they belong to datum M. However, the DM may well decide to disregard some models in M because of some personal information. This additional information is reflected by his subjective belief μ, with strict inclusion and μ(m) = 0 for some m ∈ M. (In fact, the interpretation of μ is purely subjective, not at all logical/objective à la Carnap and Keynes.)

Next we behaviorally characterize suppμ as the smallest subset of M relative to which ≿ is consistent. These are the models that the DM believes to carry significant probabilistic information for his decision problem. In this perspective it is important to remember that M is a datum of the problem whereas suppμ is a subjective feature of the preferences.

We consider a linearly independent M in view of the identification result of Proposition 1.

Let M be a finite and linearly independent subset of Δ_na and ≿ be a preference represented as in point ii of Proposition 1. A model m ∈ M belongs to suppμ if and only if ≿ is not consistent with M\m.

Therefore, consistency arguments not only reveal the acceptance of a datum M, but also allow us to discover what elements of M are subjectively maintained or discarded.

We close by establishing the conditional and orthogonal versions of Proposition 1. We begin with the conditional version, i.e., with the counterpart of representation [5] under Waldean information.

Let M be a finite subset of Δ_na. The following statements are equivalent:represents for all non--null events .

Moreover, u is cardinally unique for each satisfying statement i, whereas μ is unique for each such if and only if M is linearly independent.

The representation of the conditional preferences thus depends on the conditional models and on the posterior probability that, respectively, update models and prior in the light of E. Criterion [11] shows how the DM currently plans to use the information he may gather through observations to update his inference on the actual data-generating process. [As Marschak (ref. 15, p. 109) remarked “to be an ‘economic man’ implies being a ‘statistical man’.” Some works of Jacob Marschak (notably refs. 15 and 16 and his classic book, ref. 17, with Roy Radner) have been a source of inspiration of our exercise, as we discuss in ref. 18. Our work addresses, inter alia, the issue that he raised in ref. 16, in which he asked how to pin down subjective beliefs on models from observables. In so doing, our analysis also shows that to study general data M, possibly linearly dependent, it is necessary to go beyond betting behavior on observables.]

The conditional predictive probability isand therefore the reduced form of [11] isThe conditional representations [11] and [13] are, respectively, induced by the primitive representations [9] and [10] via conditioning.

Orthogonality is a simple, but important, sufficient condition for linear independence that, as the next section shows, some fundamental classes of time series models satisfy. Because of its importance, the following result shows what form the classical SEU representation of Proposition 1 takes in this case.

Let M be a finite and orthogonal subset of . The following statements are equivalent:

represents .

Moreover, for each satisfying i, u is cardinally unique and μ is unique.

Note that here consistency suffices and that the prior μ is automatically unique because of the orthogonality of M. In ref. 18 we also show that a representation with an infinite M can be derived in the orthogonal case.

The reduction map between prior and predictive probabilities is easily seen to be affine. More interestingly, in the orthogonal case it also preserves orthogonality and absolute continuity.

Under the assumptions of Proposition 4, two priors μ and ν on Δ with support in M are orthogonal (resp., absolutely continuous) if and only if their predictive probabilities and on S are orthogonal (resp., absolutely continuous).

Consider a standard intertemporal decision problem where information builds up through observations generated by a sequence of random variables taking values on observation spaces . For ease of exposition, we assume that the observation spaces are finite and identical, each denoted by and endowed with the σ-algebra .

The relevant state space S for the decision problem is the sample space . Its points are the possible observation paths generated by the process . Without loss of generality, we identify with the coordinate process such that for each .

Endow with the product σ-algebra generated by the elementary cylinder sets . These sets are the observables in this intertemporal setting. In particular, the filtration , where is the algebra generated by the cylinders , records the building up of observations. Clearly, is the σ-algebra generated by the filtration .

In this intertemporal setting the pair is thus given by . The space of data-generating models Δ consists of all probability measures m on . The outcome space X has also a product structure , where is a common instant outcome space. Acts can thus be identified with the processes of their components. When such processes are adapted, the corresponding acts are called plans [here is the outcome at time t if state s obtains]. By Proposition 3, the conditional version of the classical SEU representation at iswhere and are, respectively, the conditional model and the posterior probability given the observation history . Under standard conditions, the intertemporal utility function in [14] has a classic discounted form , with subjective discount factor and bounded instantaneous utility function .

The next known result (e.g., ref. 19, p. 39) shows that models are orthogonal in the fundamental stationary and ergodic case, which includes the standard i.i.d. setup as a special case.

A finite collection M of models that make the process stationary and ergodic is orthogonal.

By Proposition 4, if satisfies P.1–P.6 and is consistent with a finite collection M of nonatomic, stationary and ergodic models, then there are a cardinally unique utility function u and a unique prior μ, with , such that [14] holds. Its reduced form features a predictive probability that is stationary (exchangeable in the special i.i.d. case).

Because a version of Proposition 6 holds also for collections of homogenous Markov chains, we can conclude that time series models widely used in applications satisfy the orthogonality conditions that ensure the uniqueness of prior μ. The Wald–Savage setup of this paper provides a statistical decision theory framework for empirical works that rely on such time series (as is often the case in the finance and macroeconomics literatures).

Under these orthogonality conditions, there is full learning. Formally, denoting bythe continuation value at of any act f and by the true model, it can be shown thatfor almost every z in . As observations build up, DMs learn and eventually behave as SEU DMs who know the true model that generates observations. The above convergence result shows how the Classical SEU framework of this paper allows to formalize, in terms of learning, the common justification of rational expectations according to which “with a long enough historical data record, statistical learning will equate objective and subjective probability distributions” [Sargent and Williams (ref. 20, p. 361)]. Further intertemporal results are studied in ref. 18 (the working paper version of this paper), which we refer the interested reader to.

Let . The following statements are equivalent: