Lexicographic choice functions

TL;DR

This paper explores a generalized framework for convex choice functions on vector spaces, linking them to lexicographic probabilities and sets of desirable gambles, without requiring the Archimedean property.

Contribution

It introduces lexicographic choice functions as a new class satisfying convexity but not Archimedeanity, connecting them to sets of desirable gambles and lexicographic probabilities.

Findings

Choice functions based on sets of desirable options relate to lexicographic probabilities.

Lexicographic choice functions are uniquely determined by certain sets of desirable options.

They can identify the most conservative convex choice function for a given binary relation.

Abstract

We investigate a generalisation of the coherent choice functions considered by Seidenfeld et al. (2010), by sticking to the convexity axiom but imposing no Archimedeanity condition. We define our choice functions on vector spaces of options, which allows us to incorporate as special cases both Seidenfeld et al.'s (2010) choice functions on horse lotteries and sets of desirable gambles (Quaeghebeur, 2014), and to investigate their connections. We show that choice functions based on sets of desirable options (gambles) satisfy Seidenfeld's convexity axiom only for very particular types of sets of desirable options, which are in a one-to-one relationship with the lexicographic probabilities. We call them lexicographic choice functions. Finally, we prove that these choice functions can be used to determine the most conservative convex choice function associated with a given binary relation.