Multi-Utility Representations

• Multi-utility representations are rigorous frameworks that parameterize all admissible utility functions to model incomplete or intransitive preferences using advanced geometric constructions.
• They integrate cone-duality, nonparametric inference, and robust optimization to facilitate analysis in economics, risk measurement, and decision theory.
• These frameworks extend to diverse settings, enabling applications in multi-task learning, privacy gradients, and logical reasoning with practical implications in finance and optimization.

Multi-utility representations provide the principled mathematical infrastructure for encoding, analyzing, and operationalizing incomplete, intransitive, or multi-criteria preference relations in economic theory, decision analysis, risk measurement, and learning. When standard single-utility theory fails to deliver a unique or even well-posed numeric representation—whether due to incompleteness, branching, symmetry breaking, or multiple sources of evaluative uncertainty—multi-utility frameworks parameterize all admissible utility functions (or utility vectors) compatible with the observed data or axioms. These frameworks incorporate universal identification regimes, cone-duality and geometric constructions, nonparametric inference, partial and lattice-theoretic generalizations, and robust optimization architectures. Recent research further extends multi-utility theory to dynamic, statistical, and computational settings including privacy gradients, risk aggregation, logical reasoning, and universal representation learning.

1. Mathematical Foundations and Definitions

Let $X$ denote a set of alternatives and $\succeq$ a reflexive, transitive binary relation (a preorder). Classical utility theory seeks $u:X\to\mathbb{R}$ with $x\succeq y \iff u(x)\ge u(y)$. For incomplete or intransitive $\succeq$, such a single $u$ rarely exists. Multi-utility representation, as formalized by Richter and Peleg, introduces a family $\mathcal{U}=\{u_i: X\to\mathbb{R}\}$ such that $x\succeq y\iff u_i(x)\ge u_i(y)$ for all $i$ (Bosi et al., 24 Jan 2024). Extensions to partial orderings (semiorders, interval orders, etc.) replace total functions with partial functions, allowing vastly greater generality and, for some classes, reducing the required number of utility-like objects for exact recovery.

In convex and probabilistic models, expected multi-utility representations over lotteries—families $U$ of von Neumann–Morgenstern utilities $u:Z\to\mathbb{R}$ such that $p\succeq q\iff \mathbb{E}_p[u]\ge\mathbb{E}_q[u]$ for all $u\in U$—characterize incomplete, independence-satisfying preference relations in mixture and lottery spaces (Leonetti, 2022). Similar structures arise in mixture spaces when strong independence and mixture continuity/multidimensional countable domination are imposed (McCarthy et al., 2021).

In coordinate-free geometric terms, the set of admissible utilities is precisely the dual cone $U^*$ of the ledger group $U(P)$ constructed from all trade-offs asserted by the preference relation (Aryal, 8 Dec 2025).

2. Identification and Uniqueness in the Random Utility Model

The identification problem in random utility models (RUM) exemplifies the emergence of multi-utility representations. With a finite set $X$ and the space $\Pi$ of all linear orders, a random utility representation comprises a probability measure $\nu\in\Delta(\Pi)$ rationalizing choice probabilities $P_A(x)$. Turansick provides two characterizations for uniqueness:

• Graphical test: The probability-flow diagram's supported paths map to linear orders; uniqueness fails if distinct paths "branch," facilitating swap of probability inflow/outflow and admitting multiple $\nu$'s (Turansick, 2021).
• Algebraic test: The existence of pairs of orders $(\pi,\pi')$ and three alternatives $(x,y,z)$ violating specific contour-set conditions certifies non-uniqueness.

Support identification is equivalent to overall identification: all rationalizing $\nu$'s share the same support iff $\nu$ is unique. Consequently, multi-utility appears whenever observed choices are compatible with several distinct distributions over preferences.

3. Duality, Cone Representations, and Geometric Structures

Multi-utility theory is tightly bound to cone-duality and order geometry. For preferences defined by a cone $C$ in a vector space (alternatives, lotteries), the dual cone $C'$ parameterizes all admissible utility functionals:

• Cone-duality: $C$ is the set of "directions" $(p-q)$ with $p\succeq q$. Its dual $C'$ is the family $U$ such that $E_p[u]\ge E_q[u]$ for all $u\in U$ (Leonetti, 2022).
• Geometric characterization: The ledger group construction, together with passage to its real-linear extension, enables universal factorization of any order-preserving utility through the convex cone structure (Aryal, 8 Dec 2025).
• Uniqueness: Any two multi-utility representations generate the same convex cone (closure under positive combination and limits), guaranteeing essential uniqueness up to extremal rays and closure.

These constructions encompass classical multi-attribute utility theory, expected utility over mixture spaces, and more general convex-preference systems.

4. Extensions: Partial, Continuous, and Bitopological Representations

Partial multi-utility representations, via partial functions, extend classical theory to incomplete, intransitive, and discontinuous orderings. Notable technical advances include:

• Partial function families: Any preorder admits a multi-utility representation (possibly infinite), and partial functions allow representation of semiorders, interval orders, and other complex structures not attainable via total functions (Bosi et al., 24 Jan 2024).
• Continuity and bitopology: Existence of maximal semicontinuous Richter–Peleg representations is characterized by the precontinuity condition (Erné) and quasicompactness in bitopological spaces (Andrikopoulos, 2020). Families of Scott- and $\omega$-continuous utilities arise as canonical multi-utility sets exactly when precontinuity holds.

When restrictions to boundedness or countable support are imposed, representations persist under suitable continuity and net conditions (Leonetti, 2022).

5. Applications: Risk Measures, Optimization, and Data Utility

Multi-utility is central in robust optimization, risk theory, privacy, and multi-task learning:

• Financial risk measures: Set-valued risk measures induce incomplete preferences represented via a convex cone of dual parameters. The multi-utility representation is derived from scalarizations $U_T(x)=-P_T(x)$ and general dual extensions $u^*_T(x)$. Regularity (concavity, monotonicity, semicontinuity) and parsimony follow from properties of the acceptance sets, leading to tractable multi-criteria optimization (Munari, 2020).
• Robust optimization: In multi-attribute UPRO, the ambiguity set for utility functions is constrained by pairwise comparisons (via Lebesgue–Stieltjes integrals); piecewise-linear approximation techniques yield LP/MILP reformulations for computing maximin-optimal policies under risk and utility ambiguity (Wu et al., 2023).
• Differential privacy with multi-utility tiers: ML-DPCS achieves multi-level utility tradeoffs by publishing a single perturbed measurement; access controls (encoding keys, knowledge sets) partition users into utility tiers with provable privacy and reconstruction error gradients, integrating compressive sensing and linearly encoded impulsive noise (Jiang et al., 2021).
• Universal representations in multi-task/domain learning: Joint distillation from multiple task/domain teachers into a unified deep representation is operationalized with alignment adapters and multi-loss optimization, yielding competitive performance across segmentation, classification, and few-shot tasks (Li et al., 2022).

6. Logical, Semantic, and Network Interpretations

Multi-utility extends deeply into logical semantics, probabilistic reasoning, and network theory:

• Logical synthesis: Constraint-driven desirability measures can be aggregated as truth-values via t-norms, t-conorms, and residua; differential preference relations, modal ignorance bounds, and similarity metrics are directly tied to multi-utility constructs (Ruspini, 2013).
• Utility networks: Factorized utility distributions model conditional utility and utility independence analogously to Bayesian networks, with chain rule, graphoid axioms, and compact factorization schemes enabling efficient elicitation and inference (Shoham, 2013).

These frameworks allow for modular, scalable, and interpretable modeling of decision problems subject to competing objectives, multi-layered constraints, and epistemic uncertainty.

7. Limitations, Non-Representability, and Open Questions

Key limitations and impossibility results shape the current boundaries of multi-utility theory:

• Non-representability: Lexicographic or highly discontinuous orderings require continuum-sized multi-utility families and cannot be encoded by finite or countable utility sets (Aryal, 8 Dec 2025, Bosi et al., 24 Jan 2024).
• Failure under weak continuity or insufficient topological control: In infinite-dimensional or uncountable-outcome spaces, sequential closure of cones is insufficient for multi-utility existence unless strong continuity or countable domination conditions are invoked (Leonetti, 2022, McCarthy et al., 2021).
• Trade-offs between continuity and completeness: Finite continuous partial multi-utility families imply totality; infinite families are unavoidable for non-regular semiorders (Bosi et al., 24 Jan 2024).

Open questions remain regarding the exact sufficiency of closed graph + mixture continuity for representation existence in general mixture spaces (McCarthy et al., 2021).

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In summary, multi-utility representations anchor the rigorous mathematical analysis of complex, incomplete, and multi-criteria preference structures. They leverage cone-duality, order geometry, algebraic topology, partial and lattice-theoretic extensions, and robust optimization architectures, subsuming classical utility models and providing a unifying methodological foundation for diverse applications in decision theory, risk measurement, privacy-utility gradients, logical reasoning, and universal representation learning.