Every time we go for a walk,drive our car, fly somewhere, and so on, there is some chance of ushaving a fatal accident. But since the probabilities of theseaccidents are sufficiently small, we decide to take our chances. As noted above, preferenceconcerns decision theory is concerned with the comparison of options; it is a relation betweenoptions.

Onthe value side, many contend that a rational agent may simply find twooptions incomparable due to their incommensurablequalities. (Here a prominent usage of these terms will be followed,whereby particular options may be described as incomparable in value,while general properties or dimensions of value may be described asincommensurable.) As in, the agent’s evaluations of thedesirability of sure options may not be representable by any preciseutility function. Likewise, on the belief side, some contend (notably,Joyce 2010 and Bradley 2017) that the evidence may be such that itdoes not commit a rational agent to precise degrees of beliefmeasurable by a unique probability function. The worry that EU theory is too permissive with respect to desire isrelated to the worry that the theory is unfalsifiable. Theworry is that apparently irrational preferences by the lights of EUtheory can always be construed as rational, under a suitabledescription of the options under consideration.

1. (Here a prominent usage of these terms will be followed,whereby particular options may be described as incomparable in value,while general properties or dimensions of value may be described asincommensurable.) As in, the agent’s evaluations of thedesirability of sure options may not be representable by any preciseutility function.
2. Where \(x_1Ax_2Bf\) denotes the act that yields \(x_1\) for all \(s\inA\), outcome \(x_2\) for all \(s\in B\) and \(f(s)\) for all other\(s\).
3. Suppose you are offered a choice between two lotteries,one that results in you winning a nice prize if a coin comes up headsbut getting nothing if the coin comes up tails, another that resultsin you winning the same prize if the coin comes up tails but gettingnothing if the coin comes up heads.
4. The EU characterisation of rational desire is in many ways highlypermissive, but it may be more controversial than first meets theeye.
5. But here a different interpretation of preference is brought tobear on the comparison of options.

Broader implications of Expected Utility (EU) theory

Wesay that alternative \(f\)“agrees with” \(g\) inevent \(E\) if, for any state inevent \(E\), \(f\) and \(g\) yieldthe same outcome. Since the late 1970s, a considerable number of generalisations of SEUhave been devised to accommodate the problematic preference patterns.A brief survey of these is provided in the following subsection. In this fifth article, we venture into the world of Decision Theory, the academic cornerstone for understanding how choices are made. Whether in the labyrinth of personal dilemmas or at the crossroads of corporate strategies, Decision Theory provides the map to navigate the maze of choice.

2 On rational desire

Thus we see why the agent can describe her decision problemjust as she sees it; there is no requirement that she identify a setof states (in Jeffrey’s case, this would be a partition of theproposition space that is orthogonal to the act partition) such thatthe states are appropriately fine-grained and probabilisticallyindependent of the acts. Like the Continuity axiom of vNM, Non-Atomicity implies that nomatter how bad an outcome \(X\) is,if \(g\) is already preferredto \(f\), then if weadd \(X\) as one of the possibleoutcomes of \(f\)—therebyconstructing a new alternative \(f’\)—\(g\)will still be preferred to the modified alternative as long as theprobability of \(X\) is sufficientlysmall. In effect, Non-Atomicity implies that \(\bS\)contains events of arbitrarily small probability.

The EU characterisation of rational desire is in many ways highlypermissive, but it may be more controversial than first meets theeye. Here the focus will be on the compatibility of EU theory withprominent ethical positions regarding the choice-worthiness of acts,as well as meta-ethical positions regarding the nature of value andits relationship to belief. As noted, a special case is when the content of\(p\) is such that it is recognisably something the agent can chooseto make true, i.e., an act.

Descriptive vs Normative Decision Theory

On the values side, manycontend that a rational agent may simply find twooptions incomparable due to their incommensurablequalities. Likewise, on the belief side, some contend (notably,Joyce 2010) that the evidence may be such that it does not commit arational agent to precise beliefs measurable by a unique probabilityfunction. Sure, the theory has much to say about assessing therationality of an agent’s preference attitudes once theseattitudes are suitably expressed relative to a particularrepresentation of the agent’s decision situation. The concern,however, is that EU theory does not itself address important priorquestions of representation/decision modelling. As such, one mightdoubt whether the theory is adequate for assessing the rationality ofreal-world choices, or whether it has any substantial content atall. This disanalogy is due to the fact that there is nosense in which the \(p_i\)sthat \(p\) is evaluated in terms ofneed to be ultimate outcomes; they can themselves be thought of asuncertain prospects that are evaluated in terms of their differentpossible realisations.

1 Savage’s theory

For instance, theaforementioned authors considered and characterised preferences thatexhibit exponential time discounting. Either way, it may yet be argued that EU theory does not go far enoughin structuring an agent’s preference attitudes so that we mayunderstand the reasons for these preference attitudes.Dietrich and List (2013 & 2016a) have proposed a more generalframework that fills this lacuna. In their framework, preferencessatisfying some minimal constraints are representable as dependent onthe bundle of properties in terms of which each option is perceived bythe agent in a given context.

Then if it turns out thatyou are indifferent between \(p\)joined with \(r\)and \(q\) joinedwith \(r\), that must be because youfind \(p\) and \(q\) equally probable. Otherwise, you would preferthe union that contains the one of \(p\) and \(q\) thatyou find less probable, since that gives you a higher chance of themore desirable proposition \(r\). Itthen follows that for any other proposition \(s\) that satisfies the aforementionedconditions \(r\) satisfies, youshould also be indifferent between \(p\cups\) and \(q\cup s\), since,again, the two unions are equally likely to resultin \(s\). These are intended as constraints on an agent’s preferencerelation, \(\preceq\), over a set ofacts, \(\bF\), as described above.