Harsanyi Set Allocations

• Harsanyi set allocations are a foundational concept in cooperative game theory that decompose coalition values into unique dividends for fair distribution.
• They extend traditional methods such as Shapley values by employing flexible weight schemes and probabilistic logic to capture marginal contributions.
• Applications span from economic cost sharing and bargaining to feature attribution in machine learning, ensuring fairness with theoretical rigor.

Harsanyi set allocations are a foundational concept in cooperative game theory, economics, and more recently, interpretability in machine learning. They formalize the assignment of value, resources, or payoffs among participants (agents, features, entities) by decomposing the total outcome into incremental contributions associated with coalitions. This decomposition, derived from underlying probability logic, axiomatic principles, or combinatorial structure, provides a flexible and theoretically grounded alternative to single-point attribution methods such as Shapley values or traditional core allocations.

1. Mathematical Foundations of Harsanyi Set Allocations

At the core of Harsanyi set allocations lies the dividend decomposition, wherein the value of any coalition is partitioned into increments not attributable to any proper sub-coalition. For a cooperative game with player set $D$ and value function $v$, the Harsanyi dividend for coalition $A \subseteq D$ is

$$\phi_v(A) = \sum_{B \subseteq A} (-1)^{|A| - |B|} v(B)$$

This formula ensures that only the part of $v(A)$ not accounted for by sub-coalitions is captured. Allocations are generated by distributing these dividends among individual players using a weight scheme $\lambda$, subject to

• $\lambda_j(A) \geq 0$ for all $j \in A$
• $\sum_{j \in A} \lambda_j(A) = 1$
• $\lambda_j(A) = 0$ if $j \notin A$

The final allocation to player $j$ is

$\phi_v(j) = \sum_{A \ni j} \lambda_j(A) \phi_v(A)$

This aligns with the "weighted average of contributions" perspective of Harsanyi (Idrissi et al., 16 Jun 2025).

2. Allocation Rules, Comparison to Shapley and Weber Sets

Shapley values historically dominate efficient allocation schemes. They can be expressed within the Harsanyi framework by choosing uniform weights: $\lambda_j(A) = 1/|A|$. The allocation thus recovers Shapley values as a special case:

$Shap_v(j) = \sum_{A \ni j} \frac{1}{|A|} \phi_v(A)$

This is just one instance within the general Harsanyi family. The Weber set generalizes allocation based on marginal contributions in random feature orderings. If the expectation is taken over a non-uniform distribution on permutations, other attribution rules arise. The Harsanyi set, by focusing on dividend splitting rather than ordering, allows even greater flexibility—alternative weight systems can be used, for example, to reflect domain-specific considerations or to control the impact of spurious interactions (Idrissi et al., 16 Jun 2025).

3. Logical and Epistemic Structure in Allocation Models

The logical modeling of Harsanyi allocations is formalized in probability logic for type spaces (Zhou, 2014). Here, agent beliefs are represented using modal formulas $L_r \varphi$ ("the agent believes $\varphi$ with probability at least $r$") on a measurable state space. Allocation rules are required to be measurable and consistent with agents’ (possibly hierarchical) beliefs. For instance, if an agent has belief 1 in a property, allocations must respect this certainty.

Key results include:

• De-nesting property: For a Boolean combination of belief statements, shallow logics suffice ($\Sigma_H \vdash \phi \leftrightarrow L_r\phi$ for any $r>0$).
• Unique extension theorem: For any local finite language, maximal consistent sets have unique extensions; in single-agent cases, the canonical model is finite, and in multi-agent cases, continuum-sized.
• Definability: Knowledge (S5 operator) is implicitly but not explicitly nor reducibly defined from probabilistic beliefs—allocation rules needing knowledge cannot always derive it from belief thresholds.

Such formal properties underpin the consistency and tractability of allocation mechanisms where agents' beliefs affect equilibria or feasible allocations.

4. Algebraic and Combinatorial Approaches to Set Allocations

Matrix-based and set-system approaches expand Harsanyi allocation ideas to settings with qualitative, ordinal, or combinatorial data. Matrix frameworks integrate agent hierarchies, resource plausibility, and agent preferences (Camacho et al., 2018). Allocation rules are judged by qualitative binary relations comparing sets of “better” allocated resources, and optimality is achieved via sequences of simple (rational) deals represented by permutation matrices. Dispersion measures (e.g., local variance over partitions) allow discrimination among otherwise indistinguishable optimal allocations.

Set-system studies of house allocation problems (Gerbner et al., 2019) use extremal set theory to bound the number of distinct Pareto optimal allocations. Here, allocations correspond to injective mappings from buyers to houses, and combinatorial properties (such as disjointly representable families) determine max cardinalities for set images under Pareto optimal matchings. Improved binomial bounds deepen understanding of allocation diversity and have computational geometry implications.

5. Allocation Under Uncertainty and Stability Guarantees

Robust Harsanyi set allocations address coalition games with uncertain value functions (Pantazis et al., 2022). The “core” contains allocations immune to profitable deviations, but under uncertainty, robust stability is defined via scenario cores—allocations stable relative to maximal observed coalition values. Techniques from scenario optimization and PAC learning yield distribution-free bounds on instability probability: if the compression set (critical support samples) is small, stability under unseen uncertainty is likely.

If the robust core is empty, allocation relaxations (scenario $\zeta$-core) compensate by minimally relaxing stability constraints, and PAC-type guarantees are established for these approximate allocations. Numerical studies show that empirical instability probabilities conform to theoretical violation bounds, demonstrating practical viability.

This probabilistic approach generalizes classic Harsanyi stability to data-driven environments, maintaining fairness and incentive compatibility even when allocations must hedge against both observed and future uncertainties.

6. Cost Allocation for Combinatorial Covering: The Happy Nucleolus

Set covering problems extend Harsanyi allocations to cost sharing, with the nucleolus minimizing dissatisfaction lexicographically over all coalitions (Blauth et al., 8 Jan 2024). When the core is empty (no allocation meets all coalition rationality constraints), the “happy nucleolus” computes cost allocations maximizing the total amount allocated without group violation and maximizing the ordered excess vector.

The happy nucleolus is determined by considering at most $mn$ “simple” coalition-set pairs (those that admit trivial covers), reducing exponential coalition constraints to polynomial time computability. This allocation respects symmetry, decomposability across disjoint problem instances, and group rationality, making it suitable for large-scale public service, vehicle routing, and combinatorial cost-sharing.

Comparison with alternative cost allocations reveals the happy nucleolus’s strengths in ensuring fairness, tractability, and robust stability even beyond the core feasible region of the underlying cooperative game.

7. Axiomatic Bargaining and Weighted-Average Representations

In bargaining with side payments, Harsanyi set allocations are generalized by specifying allocation rules as weighted averages of maximal coalition claims (Curello et al., 1 Feb 2025). Solutions are uniquely characterized by additivity (allocation for compound problems is the sum of allocations for components) and inverse (marginal) monotonicity (addition of inefficient alternatives does not increase a player's allocation). Each player receives an equal share of total surplus plus an excess based on best (discounted) claims, with an inefficiency parameter controlling the benefit from “bad” options.

This approach enhances traditional Harsanyi allocations by ensuring fairness, robustness against inefficiencies, and modular additivity, supporting practical negotiation and cost-sharing settings.

8. Feature Attribution in Machine Learning: Interpretability via Harsanyi Sets

Harsanyi set allocations provide a comprehensive alternative to Shapley-based feature attribution in post-hoc model interpretation (Idrissi et al., 16 Jun 2025). The dividend-based decomposition enables flexible attribution choices, controlled by the weight system $\lambda$, tailored to domain knowledge or methodological requirements. The proposed three-step blueprint guides practitioners through:

1. Specifying the meaningful quantity (e.g., prediction or variance) to be decomposed;
2. Choosing an appropriate value function that matches this quantity when all features are present;
3. Selecting an allocation rule from the Harsanyi set (or Weber set), possibly modified for specific interpretive needs.

Concrete illustrations, such as variance decomposition in models with spurious features, demonstrate that choices of $\lambda$ (alternative to uniform) lead to attributions that avoid spurious associations—PME (Proportional Marginal Effect) correctly assigns zero attribution to features only present due to noise or confounding.

This framework supports theoretically grounded, robust, and context-sensitive interpretability across a wide range of model and value function choices.

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In summary, Harsanyi set allocations unify logic, algebraic combinatorics, bargaining axioms, and computational perspectives. They serve as a flexible foundation for fair and efficient distribution rules in games, economics, and data-driven interpretability, supporting robust, incentive-compatible, and application-tailored solutions beyond classical allocation paradigms.