Nonlinear desirability theory
This work addresses foundational issues in decision theory for researchers and practitioners, offering a more general framework that weakens assumptions to better model real-world nonlinear rewards like money, though it appears incremental as it builds on existing desirability theory.
The paper tackles the limitation of linear utility scales in desirability theory, which conflicts with rational decision-making as highlighted by Allais' paradox, by extending the theory to nonlinear scales using closure operators, resulting in a solution to Allais' paradox and a characterization of properties from gamble sets and prevision perspectives.
Desirability can be understood as an extension of Anscombe and Aumann's Bayesian decision theory to sets of expected utilities. At the core of desirability lies an assumption of linearity of the scale in which rewards are measured. It is a traditional assumption used to derive the expected utility model, which clashes with a general representation of rational decision making, though. Allais has, in particular, pointed this out in 1953 with his famous paradox. We note that the utility scale plays the role of a closure operator when we regard desirability as a logical theory. This observation enables us to extend desirability to the nonlinear case by letting the utility scale be represented via a general closure operator. The new theory directly expresses rewards in actual nonlinear currency (money), much in Savage's spirit, while arguably weakening the founding assumptions to a minimum. We characterise the main properties of the new theory both from the perspective of sets of gambles and of their lower and upper prices (previsions). We show how Allais paradox finds a solution in the new theory, and discuss the role of sets of probabilities in the theory.