Coordinate-free utility theory

TL;DR

This paper introduces a geometric, coordinate-free approach to utility theory, representing preferences with a family of functions via a canonical structure called the ledger group, applicable to incomplete or intransitive preferences.

Contribution

It provides a geometric characterization of multi-utility representations using the ledger group and dual cone, shifting utility theory from specific functions to the measurement space structure.

Findings

Reconstructed standard multi-attribute utility as dual cone intersection.

Demonstrated the framework's applicability to incomplete preferences.

Showed impossibility of representation for lexicographic preferences.

Abstract

Standard decision theory seeks conditions under which a preference relation can be compressed into a single real-valued function. However, when preferences are incomplete or intransitive, a single function fails to capture the agent's evaluative structure. Recent literature on multi-utility representations suggests that such preferences are better represented by families of functions. This paper provides a canonical and intrinsic geometric characterization of this family. We construct the \textit{ledger group} $U(P)$, a partially ordered group that faithfully encodes the native structure of the agent's preferences in terms of trade-offs. We show that the set of all admissible utility functions is precisely the \textit{dual cone} $U^*$ of this structure. This perspective shifts the focus of utility theory from the existence of a specific map to the geometry of the measurement space itself. We demonstrate the power of this framework by explicitly reconstructing the standard multi-attribute utility representation as the intersection of the abstract dual cone with a subspace of continuous functionals, and showing the impossibility of this for a set of lexicographic preferences.