Peterson's way of formally representing a decision problem also seems more helpful to me than the ways proposed by Jeffrey and Savage. Peterson (2008) explains:

I shall conceive of a formal representation as an ordered quadruple π =<
A,S,P,U>. The intended interpretation of its elements is as follows.

A = {a1,a2, . . .} is a non-empty set of acts.
S ={s1, s2, . . .} is a non-empty set of states.
P = {p1 : A×S→[0,1], p2 : A×S→[0,1], . . .} is a set of probability functions.
U = {u1 : A×S→Re,u2 : A×S→Re, . . .} is a set of utility functions.

An act is, intuitively speaking, an uncertain prospect. I shall use the term ‘uncertain prospect’ when I speak of alternatives in a loose sense. The term ‘act’ is used when more precision is required.

By introducing the quadruple
<A,S,P,U>, a substantial assumption is made. The assumption implies that everything that matters in rational decision making can be represented by the four sets A,S,P,U. Hence, nothing except acts, states, and numerical representations of partial beliefs and desires is allowed to be of any relevance. This is a standard assumption in decision theory.

A formal decision problem under risk is a quadruple π =
<A,S,P,U> in which each set P and U has exactly one element. A formal decision problem under uncertainty is a quadruple π =
A,S,P,U in which P = /0 and U has exactly one element. Since P and U are sets of functions rather than single functions, agents are allowed to consider several alternative probability and utility measures in a formal decision problem. This set-up can thus model what is sometimes referred to as ‘epistemic risks’, i.e. cases in which there are several alternative probability and utility functions that describe a given situation. Note that the concept of an ‘outcome’ or ‘consequence’ of an act is not explicitly employed in a formal decision problem. Instead, utilities are assigned to ordered pairs of acts and states. Also note that it is not taken for granted that the elements in A and S (which may be either finite or countable infinite sets) must be jointly exhaustive and mutually exclusive. To determine whether such requirements ought to be levied or not is a normative issue that will be analysed in greater detail in subsequent chapters.

It is easy to jump to the conclusion that the framework employed here presupposes that the probability and utility functions have been derived ex ante (i.e. before any preferences over alternatives have been stated), rather than ex post (i.e. after the agent has stated his preferences over the available acts). However, the sets P and U are allowed to be empty at the beginning of a representation process, and can be successively expanded with functions obtained at a later stage.

The formal set-up outlined above is one of many alternatives. In Savage’s theory, the fundamental elements of a formal decision problem are taken to be a set S of states of the world and a set F of consequences. Acts are then defined as functions from S to F. No probability or utility functions are included in the formal decision problem. These are instead derived ‘within’ the theory by using the agent’s preferences over uncertain prospects (that is, risky acts).

Jeffrey’s theory is, in contrast to Savage’s, homogenous in the sense that all elements of a formal decision problem—e.g. acts, outcomes, probabilities, and utilities—are defined on the same set of entities, namely a set of propositions. For instance, ‘[a]n act is . . . a proposition which it is within the agent’s power to make true if he pleases’, and to hold it probable that it will rain tomorrow is ‘to have a particular attitude toward the proposition that it will rain tomorrow’. In line with this, the conjunction B∧C of the propositions B and C is interpreted as the set-theoretic intersection of the possible worlds in which B and C are true, and so on. Note that Jeffrey’s way of conceiving an act implies that all consequences of an act in a decision problem under certainty are acts themselves, since the agent can make those propositions true if he pleases. Therefore, the distinction between acts on the one hand and consequences on the other cannot be upheld in his terminology, which appears to be a drawback.

However, irrespective of this, the homogenous character of Jeffrey’s set-up is no decisive reason for preferring it to Savage’s, since the latter can easily be reconstructed as a homogenous theory by widening the concept of a state. The consequence of having a six-egg omelette can, for example, be conceived as a state in which the agent enjoys a six-egg omelette; acts can then be defined as functions from states to states. A similar manoeuvre can, mutatis mutandis, be carried out for the quadruple <
A,S,P,U>.

I think it is more reasonable to take states of the world, rather than propositions, to be the basic building blocks of formal decision problems. States are what ultimately matter for agents, and states are less opaque from a metaphysical point of view. Propositions have no spatio-temporal location, and one cannot get into direct acquaintance with them. Arguably, the main reason for preferring Jeffrey’s set-up would be that things then become more convenient from a technical point of view, since it is easy to perform logical operations on propositions. In my humble opinion, however, technical convenience is not the right kind of reason for making metaphysical choices.

An additional reason for conceiving of a formal decision problem as a quadruple <
A,S,P,U>, rather than in the way proposed by Savage or Jeffrey, is that it is neutral with regard to the controversy over causal and evidential decision theory. To take a stand on that issue would be beyond the scope of the present work. In Savage’s theory, which has been claimed to be ‘the leading example of a causal decision theory’, it is explicitly assumed that states are probabilistically independent of acts, since acts are conceived of as functions from states to consequences. This requirement makes sense only if one thinks that agents should take beliefs about causal relations into account: Acts and states are, in this type of theory, two independent entities that together cause outcomes. In an evidential theory such as Jeffrey’s, the probability of a consequence is allowed to be affected by what act is chosen; the agent’s beliefs about causal relations play no role. Evidential and causal decision theories come to different conclusions in Newcomb-style problems. For a realistic example, consider the smoking-caused-by-genetic-defect problem: Suppose that there is some genetic defect that is known to cause both lung cancer and the drive to smoke. In this case, the fact that 80 percent of all smokers suffer from lung cancer should not prevent a causal decision theorist from starting to smoke, since (i) one either has that genetic defect or not, and (ii) there is a small enjoyment associated with smoking, and (iii) the probability of lung cancer is not affected by one’s choice. An evidential decision theorist would, on the contrary, conclude (incorrectly) that if you start to smoke, there is an 80 percent risk that you will contract lung cancer.

It seems obvious that causal decision theory, but not its evidential rival (in its most na¨ıve version), comes to the right conclusion in the smoking-caused-bygenetic-defect problem. Some authors have proposed that for precisely this reason, evidential decision theory should be interpreted as a theory of valuation rather than as a theory of decision. A theory of valuation ranks a set of alternative acts with regard to how good or bad they are in some relevant sense, but it does not prescribe any acts. Valuation should, arguably, be ultimately linked to decision, but there is no conceptual mistake involved in separating the two questions.

A significant advantage of e.g. Jeffrey’s evidential theory, both when interpreted as a theory of decision and as a theory of valuation, is that it does not require that we understand what causality is. The concept of causality plays no role in this theory.

Since there are significant pros and cons for either the causal or evidential approach, it seems reasonable to opt for a set-up that allows for both kinds of theories, until the dispute has been resolved...