Valuation-Theoretic Framework

• Valuation-theoretic frameworks are rigorous methods for assigning quantitative or qualitative values to objects using axiomatic principles and efficient approximations.
• They integrate concepts from game theory, algebra, and probability to quantify contributions, balance risk, and ensure fairness across diverse applications.
• Scalable computational algorithms enable practical approximations, supporting applications in machine learning, financial pricing, and geometric analysis.

A valuation-theoretic framework provides a rigorous, often axiomatic, foundation for assigning consistent quantitative or qualitative values to objects, data, cash flows, or mathematical structures. Across domains including machine learning, financial mathematics, insurance, cooperative games, algebraic geometry, and combinatorics, valuation-theoretic methods formalize the processes of attribution, aggregation, and decomposition of value, uniquely characterizing both underlying symmetries and tradeoffs between risk, information, or fairness.

1. Game-Theoretic, Probabilistic, and Algebraic Foundations

Several valuation-theoretic frameworks are grounded in cooperative game theory, where a value function $v:2^N\to\mathbb{R}$ (with $N$ a finite set of "players" such as data points, features, or agents) specifies utility or performance for each coalition. Central solution concepts include the Shapley value and Banzhaf value, each characterized by axioms such as efficiency, symmetry, dummy/null player, and additivity. In machine learning and data valuation, Shapley-based approaches are used to measure the marginal utility of each datum to a model's predictive accuracy or loss (Jia et al., 12 Feb 2025, Bian et al., 2021).

Energy-based frameworks reinterpret the value allocation problem via maximum entropy principles, introducing a Gibbs distribution over coalitions and characterizing value indices as variational solutions to convex optimization (KL-minimization). This approach justifies mean-field approximations to classical indices and yields new variational valuations that interpolate between Banzhaf, Shapley, and optimal decoupling criteria (Bian et al., 2021).

In algebraic settings, valuation theory defines valuations $v: K^\times\to G$ on fields $K$, imposing order and compatibility with field operations, and classifying valuations via invariants such as rank and value group. This theory underpins results in essential dimension, birational geometry, and singularity theory (Meyer, 2012, Günther, 2016, Kuhlmann, 2010).

In functional analysis and combinatorics, "valuation" refers to finitely additive mappings on posets or lattices of sets or geometric objects—e.g., simple translation-invariant valuations on polytopes, which are classified by decomposition theorems and play a role in equidecomposability (Kusejko et al., 2015).

2. Axiomatic Properties and Uniqueness

Key valuation-theoretic criteria are formalized via axioms:

• Efficiency: Total value is distributed, e.g., $\sum_i \phi_i = v(N) - v(\emptyset)$.
• Symmetry: Indistinguishable objects/players receive the same value.
• Null/Dummy: Objects with zero marginal contribution receive zero value.
• Additivity: Values respect sums of value functions.
• Translation-invariance and Monotonicity: Particularly in financial contexts, allowing aggregation or shift-invariance in valuations.

These axioms yield uniqueness characterizations—e.g., Shapley value is the unique function satisfying symmetry, efficiency, dummy, and additivity. Similar uniqueness holds in 2D generalizations for fragmented data (Liu et al., 2023), marginalization properties for valuation algebras (Kohlas, 2016), and intersectional data valuation (Garrido-Merchán, 19 Jul 2025) using mutual information.

Axiomatic extensions also classify the structure of valuation algebras—regular, cancellative, separative—depending on whether division (removal), inversion, and conditional operators exist internally, in extensions, or only partially (Kohlas, 2016).

3. Computational and Algorithmic Realizations

Naïve evaluation of axiomatic valuation indices is often computationally intractable owing to exponential scaling. Multiple frameworks develop scalable surrogates:

• HarsanyiNet and Interaction-based Networks: Use neural architectures to learn interaction effects and efficiently collapse Shapley computations from $\mathcal{O}(2^{|D|})$ to $\mathcal{O}(n)$ per batch via structured aggregation over learned neuron interactions (Jia et al., 12 Feb 2025).
• Variational Mean-Field Methods: Fixed-point or iterative updates on mean-field approximations to the Gibbs measure yield efficient approximations and new "Variational Indices" with provably lower decoupling error (Bian et al., 2021).
• Permutation and Out-of-Bag Estimators: For data fragmentation, joint sample–feature valuations via 2D-Shapley or 2D-OOB use permutation sampling, bagging, or nearest-neighbor surrogates to approximate marginal contributions tractably (Sun et al., 2024, Liu et al., 2023).
• Monte Carlo and Histogram-based Methods: For intersectional data valuation, mutual information is estimated via plug-in histogram or kernel density estimators, allowing flexible, model-independent valuation of privacy risks (Garrido-Merchán, 19 Jul 2025).

4. Integration with Statistical, Financial, and Risk Models

Valuation-theoretic frameworks underpin the rigorous pricing of assets and liabilities under uncertainty:

• Dynamic Asset Pricing and Insurance: The Fundamental Theorem of Asset Pricing (FTAP) ensures that no-arbitrage and completeness lead to unique stochastic discount factors and martingale valuation. Life-contingent insurance claims receive valuation via expectation under a risk-neutral measure and pricing kernel, with explicit closed-form solutions for annuities and insurances (Ling, 27 Mar 2025).
• Production-Cost and Capital Margin: For insurance, valuation-theoretic approaches reconcile regulatory capital requirements (Solvency II, SST), capital cost, and fulfillment probability via risk-margin decompositions and convex programs enforcing financiability and fulfillment conditions (Moehr, 2023).
• Cash-Flow Subject to Capital, Model Uncertainty: Multiple-prior frameworks handle model uncertainty by optimizing over sets of equivalent martingale measures, decomposing liability value into replicating portfolio plus risk margin interpretable as a worst-case optimal stopping or American option (Engsner et al., 2021).
• Credit Valuation Adjustment (CVA): Distance-to-default models embed default arrival, survival, and credit exposure in structural or reduced-form SDEs. Risky valuations are recast as discounted martingale processes, and the aggregated effect of credit risk is folded into an adjusted discount rate, allowing consistent Monte Carlo or PDE implementation (Xiao, 2023).
• Bayesian Triangulation for Financial Valuation: Noisy estimates (market, DCF, comparables) are pooled using Bayesian posterior mean estimation, yielding analytic weights optimized for precision and dependence structure (0707.3482).

5. Applications in Data Attribution, Fairness, and Privacy

Modern valuation-theoretic frameworks extend to quantifying value in large-scale data and societal systems:

• Explainable Data Valuation in Machine Learning: Cooperative-game-theoretic (Shapley value) and interaction-learning approaches assess the importance of data points for model performance, guiding sample selection, debiasing, and data purification. The integration of metric adaptors employing reinforcement learning accommodates arbitrary (including non-differentiable) utility criteria such as fairness and diversity (Jia et al., 12 Feb 2025).
• Fine-Grained and Fragmented Data Attribution: 2D-Shapley and 2D-OOB framework generalize classical Shapley to attribute value at the (sample, feature) or cell level, revealing the microstructure of data quality and supporting outlier detection, issue localization, and privacy for high-dimensional data (Liu et al., 2023, Sun et al., 2024).
• Information-Theoretic Data Valuation: Pricing of data externalities is formalized via mutual information, with surcharges set as linear functions of the privacy leakage measured by entropy reduction. This model-agnostic approach supports consistent societal and regulatory calibration (Pigouvian pricing) and quantifies intersectional privacy costs (Garrido-Merchán, 19 Jul 2025).

6. Geometric and Algebraic Valuation Theories

Valuation-theoretic perspectives produce structural results in pure mathematics:

• Birational Geometry and Essential Dimension: Valuation theory yields bounds on transcendence degree via the Abhyankar inequality and rank of value groups, informing lower bounds on essential dimension, orbits of algebraic group actions, and uniformization theorems in algebraic variety theory (Meyer, 2012, Günther, 2016, Kuhlmann, 2010).
• Polytope Decomposition and Equidecomposability: Classification results for translation-invariant valuations on polytopes facilitate equidecomposability characterizations and encode intrinsic geometric invariants in additive functions on geometric decompositions (Kusejko et al., 2015).
• General Valuation Algebras: The separative valuation algebra framework axiomatizes compositionality, marginalization, and removal, unifying the analysis of probability distributions, Gaussian densities, and Dempster–Shafer belief functions under a common abstract algebraic architecture (Kohlas, 2016).

7. Unification, Extensions, and Ongoing Research

Valuation-theoretic frameworks continue to unify and extend foundational concepts across disciplines. Notable themes include:

• Unified Algebraic, Probabilistic, and Decision-Theoretic Models: Frameworks such as Bayesian triangulation, atomless Boolean algebra decision systems, and model-theoretic approaches to valuation support unification of epistemic, axiological, and algebraic value refinement processes (London et al., 21 Nov 2025).
• Extension to Multi-Agent, Adversarial, and Social Settings: Multi-agent extensions yield mechanisms for mutual value refinement that guarantee Pareto improvements and positive-sum transformation of games initially labeled as zero-sum (London et al., 21 Nov 2025).
• Robustness and Model-Agnostic Applicability: Many frameworks are explicitly robust to modeling assumptions, operating via axiomatic mutual information, game-theoretic axioms, or surrogates generalizable to diverse stochastic, nonparametric, and learning-theoretic settings (Garrido-Merchán, 19 Jul 2025, Bian et al., 2021, Jia et al., 12 Feb 2025).

Valuation-theoretic methods thus form the mathematical backbone for equitable, interpretable, and robust value assignment in data science, economics, risk management, and pure mathematics, with rigorous axioms and computable surrogates enabling wide-ranging application and ongoing theoretical development.