Nonlinear desirability theory

TL;DR

This paper extends desirability theory to nonlinear utility scales using closure operators, allowing for more realistic modeling of decision making and resolving classical paradoxes like Allais.

Contribution

It introduces a nonlinear desirability framework that generalizes traditional linear models by employing closure operators, enabling better representation of actual decision-making behavior.

Findings

The new theory models rewards in nonlinear currency.

It provides a solution to the Allais paradox.

The role of probability sets is discussed in the nonlinear context.

Abstract

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.