Unique Preference Aggregation

Updated 29 January 2026

Unique preference aggregation is the process of combining diverse preferences into a single, affine-invariant ranking that is mathematically principled and unambiguous.
It relies on measurement-theoretic foundations and canonical models, such as weighted centroid aggregation of z-normalized scores, to enforce uniqueness and comparability.
This method is applied in multi-objective design, social choice theory, federated learning, and economic modeling, offering robust and computationally efficient decision support.

Unique preference aggregation is the process of combining diverse individual or group preferences into a single criterion or ranking that is mathematically principled, interpretable, and free from ambiguity in representation or invariance. Ensuring uniqueness in aggregation is critical for robust multi-objective design optimization, multi-criteria decision analysis, social choice theory, federated learning, categorical data analysis, and economic modeling. This article systematically presents the theoretical foundations, canonical models, representation theorems, algorithmic methods, and boundaries of unique preference aggregation, referencing key results and frameworks from contemporary research.

1. Measurement-Theoretic Foundations and Axioms

Unique aggregation fundamentally demands that the aggregation operator respect the interval structure and affine invariance intrinsic to preference data. Preferences, whether individual scores, criteria values, or utility functions, reside in affine spaces: only differences and relative units are meaningful (M. et al., 27 Jan 2026). Barzilai’s Preference Function Modelling (PFM) establishes the required axioms:

Preference Preservation / Δ-meaningfulness: Aggregation must be invariant under affine transformations $p \mapsto a p + b$ , $a>0$ .
Comparable Criteria: All criteria must be commensurate; affine normalization ensures units and origins are standardized.
Meaningful Zero Reference: A common zero is required among criteria to permit linear combination of differences.
Uniqueness / Affine Invariance: The aggregation function must be unique (modulo global affine scaling) and must never produce spurious ties or reversals under valid transformations.

The only valid transformations for raw preference scores are affine, and any aggregator must generate the same ordering regardless of the particular affine representation chosen.

2. Unique Aggregators: Canonical Representations and Theorems

Given these axioms, there is a unique admissible aggregator: the weighted centroid in linear preference space (LPS) of z-normalized differences (M. et al., 27 Jan 2026). Formally, for alternatives $A_i$ and criteria $C_j$ , with scores $p_{i,j}$ , weights $w_j$ , means $\mu_j$ , and standard deviations $\sigma_j$ ,

$z_{i,j} = \frac{p_{i,j} - \mu_j}{\sigma_j}$

$P_i^* = \sum_{j=1}^J w_j\,z_{i,j}$

This aggregation is linear, zero-referenced, and affine-invariant. The corresponding ranking is induced by sorting $P_i^*$ over all alternatives.

A representation theorem establishes that any other form (e.g., nonlinear, distance-based, or geometric mean aggregators) violates one or more axioms and cannot yield unique, transformation-invariant aggregation (M. et al., 27 Jan 2026). Weighted arithmetic means (WAM), weighted geometric means (WGM), and distance-based measures such as Topkis or Manhattan metrics demonstrably fail to respect affine invariance or meaningful zero, leading to inconsistencies under common scale manipulations.

Unique aggregation rules are also characterized in economic environments. The main result of Bajgiran–Owhadi (Bajgiran et al., 2021) proves that any rational aggregation (consistent with a generalized Extended Pareto axiom) must be of the following form:

Define a weak order $\ge$ over experts and assign each a positive weight $w$ .
For a coalition $A$ , identify the highest-ranked set $H(A)$ and aggregate only over members of $H(A)$ : $u_A = \sum_{i \in H(A)} \frac{w(i)}{\sum_{j \in H(A)} w(j)} u_{\{i\}}$ Uniqueness is assured by normalization (e.g., $\sum w(i) = 1$ ) and—in strict cases—by collapsing the weak order to a single equivalence class, resulting in a pure weighted average over all experts’ utilities.

Classical rules such as dictatorship, weighted averaging (utilitarianism), lexicographic aggregation, and two-stage (tiered) rules are unified under this representation. Imposing further regularity (continuity, permutation invariance) forces uniqueness as a strict weighted average (Bajgiran et al., 2021).

4. Algorithmic and Combinatorial Models of Unique Aggregation

For large-scale or complex preference structures, uniqueness is addressed with combinatorial or probabilistic models:

Pairwise Comparison Models: Hybrid-MST (Li et al., 2018) applies the Bradley-Terry likelihood, latent scores, and EIG-driven active sampling, enforcing uniqueness (up to an additive constant fixed by $\sum s_i = 0$ ). The use of a uniquely determined Minimum Spanning Tree ensures that every run yields the same ranking regardless of sampling order.
Conditional Preference Aggregation: In CP-net aggregation (Ali et al., 2023), uniqueness is approached via minimizing swap disagreement using majority-context aggregation on Conditional Preference Tables (CPTs). The improved polynomial-time algorithm refines CPT aggregation to always outperform trivial selection, offering unique or near-unique consensus under certain symmetry conditions.
Lattice-theoretic Aggregation: When preferences admit a lattice structure under single-crossing dominance (Curello et al., 2019), every collection possesses a unique join (least upper bound), calculated via combinatorial chain construction. This mechanism is applied in comparative statics and maxmin uncertainty-averse preferences.

5. Pareto-Based, Categorical, and Geometric Aggregation

Unique preference aggregation can be realized by objective Pareto rules, especially for group preferences over categorical attributes (Bikakis et al., 2015). Here, per-user matching vectors induce partial orders, and group-level Pareto-optimal sets are extracted without any numeric scoring or arbitrary weighting. Weak ordering via relaxed unanimity (minimal supporting subset size) resolves ties without introducing subjectivity; final rankings are unique modulo tie-breaking conventions.

In economic modeling under homothetic preferences, unique aggregation reduces to weighted averaging in log-expenditure space (Sandomirskiy et al., 2024). Aggregation-invariant domains correspond to convex sets in log-expenditure space, and uniqueness obtains when this set is a simplex (i.e., a Choquet simplex). Indecomposable preferences—extreme points of the convex set—serve as atoms; the aggregate preference is uniquely determined by its mixing measure on these atoms.

6. Practical Implementation and Limitations

The practical computation of unique aggregation is linear in the number of alternatives and criteria for z-score centroid aggregation (M. et al., 27 Jan 2026). Categorical Pareto-based methods and R-tree–accelerated skyline extraction scale to thousands of users and millions of alternatives (Bikakis et al., 2015). Key limitations are:

Requirement of interval-scale data or transformability from ordinal/pairwise sources.
Static weight assignments; dynamic group aggregation needs further protocol development.
Potentially large Pareto frontiers (“curse of dimensionality”), addressed by further index refinement or penalty functions.

7. Applications and Implications

Unique preference aggregation methods are critical in:

Multi-objective design and MCDA (robust, reproducible rankings) (M. et al., 27 Jan 2026).
Social choice and welfare economics (representation of heterogeneous expert groups, utilitarian consensus, lexicographic rules) (Bajgiran et al., 2021).
Federated RLHF for pluralistic LLM alignment, with adaptive schemes balancing fairness and alignment (Srewa et al., 9 Dec 2025).
Information design and discrete choice (mixture identification over preference atoms) (Sandomirskiy et al., 2024).
Categorical multi-user decision settings (objective group choices, indexable selection) (Bikakis et al., 2015).
Crowdsourced ranking and accuracy-aware consensus (heterogeneous Thurstone models with provable convergence guarantees) (Jin et al., 2019).
Lattice-structured preference reasoning and maxmin uncertainty-aversion (Curello et al., 2019).

A plausible implication is that the principled enforcement of uniqueness not only averts inconsistencies and representation-dependent artifacts in ranking but also facilitates transparency, compositionality, and robust decision support in complex, multi-agent, and high-dimensional systems.

Table: Canonical Unique Aggregators

Context / Model	Aggregator Formula / Principle	Uniqueness Guaranteed By
MCDA/Design (PFM)	$P_i^* = \sum_j w_j z_{i,j}$	Affine-invariant axioms (M. et al., 27 Jan 2026)
vNM Preference Aggregation	Weighted average of utilities over top tier $H(A)$	Consistency + normalization (Bajgiran et al., 2021)
Categorical Group Choice	Pareto skyline, weak tie-order via subsets	Pareto objectivity (Bikakis et al., 2015)
CP-nets, Swaps	Majority-context CPT aggregation, minimization	Polynomial-time refinement (Ali et al., 2023)
Single-crossing Lattice	Combinatorial join via P-chains	Lattice theory (Curello et al., 2019)
Economic Homothetic Domains	Log-expenditure weighted average, simplex structure	Convex geometry + Choquet (Sandomirskiy et al., 2024)
Hybrid-MST, Bradley-Terry	MLE of latent scores, fixed centering	Gaussian approximation, MST (Li et al., 2018)
Thurstone Heterogeneous Model	Alternating gradient MLE over accuracy-parameter space	Blockwise convexity (Jin et al., 2019)

In sum, unique preference aggregation—anchored in rigorous measurement theory, combinatorial and convex geometry, and axiomatic rationality—ensures interpretability, reproducibility, and robustness in the synthesis of multi-perspective judgments. Its methods unify approaches across disciplines and provide computational and theoretical guarantees critical for advanced decision systems.