Rational Choice Using Imprecise Probabilities and Utilities

Abstract

Suppose I am deliberating whether I should live on a boat and sail the Caribbean for a year. This is a decision not to be taken lightly. Many factors will matter for my decision. Several of these depend on uncertain states of the world. Will I be able to make a living? Is my boat really seaworthy? Will I miss my friends? How bad will the next winter be in my home town?1 I understand decision-making under 'uncertainty' here to refer to any case where an agent is not certain what the consequences of her actions will be, or what state will come about. A distinction is sometimes made between risk, uncertainty, ignorance and ambiguity, where 'risk' refers to the case where objective probabilities are known, 'uncertainty' refers to the case where an agent can make a subjective judgement about probabilities, an agent is in a state of 'ignorance' if she cannot make such probability assignments, and 'ambiguity' occurs when an agent can make probability assignments for some states, but not others.12 Risk aversion is further discussed in Section 5.3. Also see Mas-Colell, Whinston, and Green (1995), chapter 6 for more detail on expected utility theory's treatment of risk aversion. 13 This is why von Neumann and Morgenstern's theory is sometimes referred to as a theory of decision-making under risk, and Savage's is referred to as a theory of decision-making under uncertainty. In the former, probabilities are already known, in the latter, subjective probabilities can be assigned by the agent. However, note that, as we pointed out above, von Neumann and Morgenstern's theory can also be applied when probabilities are subjective.49 See, for instance, Wakker (1988). 50 See McClennen (1990) for a characterisation of different dynamic choice rules.

The received view of framing

“…A common principle for such decision problems classifies an option as rational if it is a top-preference according to a possible completion of the agent's all-things-considered preference-ranking of options (see, e.g., Weirich, 2021). Framing may suggest a possible completion of a preference-ranking of options and an option at the top of this possible completion.…”