Decision Theory: A Formal Philosophical Introduction SpringerLink

But perhaps more interestingly, some of the mostimportant results of decision theory—the various representationtheorems, some of which have discussed here—suggest that if aperson satisfies certain rationality requirements, then we can readher beliefs and desires, and how strong these beliefs and desires are,from her choice dispositions (or preferences). But on anoptimistic reading of these results, they assure us that we canmeaningfully talk about what goes on in other people’s mindswithout much evidence beyond information about their dispositions tochoose. Whether or notCompleteness is a plausible rationality constraint depends both onwhat sort of options are under consideration, and how we interpretpreferences over these options.

1 Allais’ paradoxes

The postulaterequires that no proposition be strictly better or worse than all ofits possible realisations, which seems to be a reasonable requirement.When \(p\) and \(q\) are mutually incompatible, \(p\cup q\) impliesthat either \(p\) or \(q\) is true, but not both. Hence, it seemsreasonable that \(p\cup q\) should be neither strictly more nor lessdesirable than both \(p\) and \(q\). Then since \(p\cup q\) is compatiblewith the truth of either the more or the less desirable of the two,\(p\cup q\)’s desirability should fall strictly between that of\(p\) and that of \(q\). However, if \(p\) and \(q\) are equallydesirable, then \(p\cup q\) should be as desirable as each of thetwo. Another important thing to notice about Jeffrey’s way ofcalculating desirability, is that it does not assume probabilisticindependence between the alternative that is being evaluated, \(p\),and the possible ways, the \(p_i\)s, that the alternative may berealised.

Alternatives to probability theory

However, the verysame people would presumably cross the street to pick up a $10 billthey had dropped. But that is just taking a gamble that has a verysmall probability of being killed by a car but a much higherprobability of gaining $10! More generally, although people rarelythink of it this way, they constantly take gambles that have minusculechances of leading to immanent death, and correspondingly very highchances of some modest reward.

1 On risk and regret attitudes

As discussed in Section 1 above, preferences that seem to violate Transitivity can be construedas consistent with this axiom so long as the options being comparedvary in their description depending on, amongst other things, theother options under consideration. The same goes for preferences thatseem to violate Separability or Independence (of the contribution ofeach outcome to the overall value of an option), discussed further in Section 5.1 below. After all, an apt model of preference issupposedly one that captures, in the description of final outcomes andoptions, everything that matters to an agent.

Broader implications of Expected Utility (EU) theory

Thenthere is an ordinal utility function that represents \(\preceq\) justin case \(\preceq\) is complete and transitive. This brings us to the Transitivity axiom, which says that if an option \(B\) is weakly preferred to \(A\), and\(C\) weakly preferred to \(B\), then \(C\) is weakly preferred to\(A\). A recent challenge to Transitivity turns on heterogeneous setsof options, as per the discussion of Completeness above.

1. A recent challenge to Transitivity turns on heterogeneous setsof options, as per the discussion of Completeness above.
2. It depicts a series of anticipated choice points, where the branchesextending from a choice point represent the options at that choicepoint.
3. The agent is not required to havepreferences over artificially constructed acts or propositions thatturn out to be nonsensical, given the interpretation of particularstates and outcomes.
4. Bradley and Stefánsson (2017) also develop a new decisiontheory partly in response to the Allais paradox.
5. In their framework, preferences satisfying some minimalconstraints are representable as dependent on the bundle of propertiesin terms of which each option is perceived by the agent in a givencontext.

Arguably, defenders of resolute choice actually have in mind adifferent interpretation of sequential decision models, whereby future“choice points” are not really points at which an agent isfree to choose according to her preferences at the time. In what follows, thes standard interpretation of sequential decision models will be assumed, and moreover, it will be assumed that rational decision theory is concerned with agents reason aboutsuch decisions in a sophisticated manner (as per Levi 1991, Maher1992, Seidenfeld 1994, amongst others). Resolute choice deviates from sophisticated choice only undercertain conditions that are not fulfilled by Ulysses, given hisinexplicable change in attitudes. Defenders of resolute choicetypically defend decision theories that violate the Independenceaxiom/Sure-thing principle (notably McClennen 1990 and Machina 1989;see also Rabinowicz 1995 for discussion), and appeal to resolutechoice to make their decision theory more palatable in the sequentialcontext. According to resolute choice, in appropriate contexts(involving preferences that are stable but which violateIndependence), the agent should count on simply sticking to thestrategy that was initially deemed best at all future choicenodes. The question is whether the resolute approach makes sense,given the standard interpretation of a sequential-decision model.

This amounts to a minimal accountof rationality, one that sets aside more substantialquestions about appropriate values and preferences, and reasonablebeliefs, given the situation at hand. The orthodox normative decisiontheory, expected utility (EU) theory, essentially says that,in situations of uncertainty, one should prefer the option withgreatest expected desirability or value. Start with the Completeness axiom, which says that an agent cancompare, in terms of the weak preference relation, all pairs ofoptions in \(S\).

Must a rationalagent have a defined preference between, say, two career options thatpull in different directions as regards opportunities for creativeself-expression versus community service (perhaps a career as a dancerversus a career as a doctor in remote regions)? Note that some ofthese challenges to EU theory are discussed in more depth in Section 5 below. To the extent that decision theory can be reconciled with the fullrange of ethical theories, should we say that there are no meaningfuldistinctions between these theories? Brown (2011) and Dietrich andList (2017) demonstrate that in fact the choice-theoreticrepresentation of ethical theories better facilitates distinctionsbetween them; terms like “(non)consequentialism” can beprecisely defined, albeit in debatable ways. More generally, we cancatalogue theories in terms of the kinds of properties (whetherintrinsic or in some sense relational) that distinguish acts/outcomesand also in terms of the nature of the ranking of acts/outcomes thatthey yield (whether transitive, complete, continuous and so on). This section expands, in turn,on the epistemological and evaluative commitments of EU theory.