Coordinate-free utility theory
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912876485148672 |
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| author | Aryal, Safal Raman |
| author_facet | Aryal, Safal Raman |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_07991 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Coordinate-free utility theory Aryal, Safal Raman Theoretical Economics 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. |
| title | Coordinate-free utility theory |
| topic | Theoretical Economics |
| url | https://arxiv.org/abs/2512.07991 |