Weber and Harsanyi Sets

Updated 30 June 2025

Weber and Harsanyi sets are key constructs in game theory and epistemic logic that aggregate individual beliefs, preferences, and utilities into coherent collective outcomes.
The Harsanyi set comprises all linear, Pareto-respecting aggregations based on individual utilities and belief hierarchies, useful in incomplete information games.
The Weber set is defined as the convex hull of marginal vectors from sequential player contributions, generalizing the Shapley value for flexible and Pareto-efficient allocations.

Weber and Harsanyi sets are fundamental constructs in cooperative game theory, epistemic logic, and social choice, representing structured approaches for aggregating player beliefs, values, or preferences into coherent solution sets for games or societal decision processes. They encapsulate, via convex analytic or modal-logical methods, the spectrum of consistent ways to combine individual agent contributions, beliefs, or utilities into collective outcomes. Both concepts provide bridges between allocation, belief hierarchies, and aggregation principles, with far-reaching implications for economic modeling, welfare analysis, and epistemic game theory.

1. Definitional Overview

A Harsanyi set typically refers to the class of all admissible aggregations of individual preferences or beliefs consistent with strong independence and Pareto-like axioms, where aggregation is linear (usually, expectation-based) over a population of agents or types. In games of incomplete information, Harsanyi type spaces formalize infinite hierarchies of beliefs and types.

The Weber set arises in cooperative game theory as the convex hull of all marginal vectors corresponding to the increments accrued by players when joining coalitions in every possible order. It generalizes from the Shapley value (the centroid of this set), relaxing symmetry to permit all convex combinations of marginal allocations, and admits non-symmetric, yet Pareto-efficient, value assignments.

Structurally, both sets are tied to the operationalization of fairness, rationality, and consistency constraints, but each emphasizes different aspects: Harsanyi sets encode all (weighted) utilitarian aggregators, while Weber sets reflect all feasible allocations built from sequential marginal contributions.

2. Mathematical Structure and Characterizations

The Harsanyi set is defined by all additive, Pareto-respecting mappings from the space of profiles (beliefs, values, or utilities) to a collective or social evaluation. For a finite population, every representative in the Harsanyi set can be written as

$u = \sum_{i=1}^n \alpha_i u_i,$

with $\alpha_i \geq 0$ , $\sum \alpha_i = 1$ , where each $u_i$ is an individual utility or belief function. Under axioms such as strong independence, convexity (or co-convexity for infinite populations), and Pareto indifference, the entire set of admissible social aggregators forms a convex set of linear combinations, potentially extended to vector-valued or lexicographically ordered representations for incomplete or discontinuous preferences (Aggregation for potentially infinite populations without continuity or completeness, 2019).

The Weber set, for a cooperative game $v: 2^N \rightarrow \mathbb{R}$ , is given by

$\mathcal{W}(v) = \operatorname{conv}\{ m^{\sigma}(v) \mid \sigma \in S_n \},$

where $m^{\sigma}(v)$ is the marginal vector associated with permutation $\sigma$ , assigning to each player $i$ the incremental value

$m_i^{\sigma}(v) = v(\{ j \mid \sigma(j) < \sigma(i) \} \cup \{i\}) - v(\{ j \mid \sigma(j) < \sigma(i) \}).$

All convex combinations of such allocations—i.e., all random-order (probabilistic) values à la Weber—constitute the Weber set (Shapley-like values without symmetry, 2018). The Krein-Milman theorem is explicitly invoked to characterize the Weber set as the convex hull of its extreme points—the collection of all special allocations (marginal vectors from each permutation).

3. Relationships to Other Solution Concepts

The core, Weber set, and intermediate set form a hierarchy of solution sets in coalition games:

Core: The set of all allocations where no coalition can improve upon their prescribed share—precisely, the Fréchet superdifferential of the Lovász extension at the grand coalition.
Intermediate Set: The limiting superdifferential of the Lovász extension, consisting of allocations justified by some chain (but not all) of nested coalitions, and containing the core (The Intermediate Set and Limiting Superdifferential for Coalition Games: Between the Core and the Weber Set, 2015).
Weber Set: The Clarke superdifferential at the grand coalition; the convex hull of all marginal allocations, containing both the core and intermediate set.

For supermodular (convex) games, these sets coincide; for general games, inclusions are strict, and the Weber set may be strictly larger and less selective than the core or intermediate set.

Solution Concept	Definition	Properties
Core	Allocations stable against coalitional deviations	Convex, may be empty, Pareto optimal
Intermediate Set	Limiting superdifferential, chain-based	Union of convex polytopes, non-convex
Weber Set	Convex hull of all marginal vectors	Convex, always nonempty, Pareto optimal

4. Logical and Epistemic Foundations

In epistemic game theory, the Harsanyi set is underpinned by type spaces that encode all belief hierarchies via probabilistic modalities. The modal logic $\Sigma_H$ (Probability Logic for Harsanyi Type Spaces, 2014) captures the introspection properties and unique extension conditions ensuring that, in a single-agent setting, belief hierarchies yield a finite canonical space (and thus a finite Harsanyi set of types); in contrast, multi-agent interactive epistemology exhibits the continuum cardinality of the type space, with uncountably many belief hierarchies.

Key modal axioms include: $L_r \phi \to L_1 L_r \phi, \quad \neg L_r \phi \to L_1 \neg L_r \phi,$ articulating that agents are certain about their own belief levels. The unique extension theorem ensures that, for each consistent belief set at depth $d$ , there is a unique extension to depth $d+1$ without branching, reflecting tight control over possible belief hierarchies.

Completeness (via the infinitary Archimedean rule) ensures all valid belief statements are provable, aligning syntactic reasoning with the structure of the Harsanyi set.

5. Generalizations in Social Aggregation and Welfare Analysis

Research on aggregation in infinite or incomplete settings (Aggregation for potentially infinite populations without continuity or completeness, 2019) has generalized the Harsanyi set to encompass all linear, order-preserving maps from the joint representation of individual valuations or beliefs—whether scalar, vector, or lexicographically ordered—subject to convexity conditions and various (strong/weak) Pareto axioms. This framework allows for representations in partially ordered vector spaces, inclusion of discontinuous or incomplete preferences, and infinite populations.

In welfare economics, the Harsanyi set is characterized as the set of all utilitarian social welfare functions (weighted averages of individual utilities) consistent with strong Pareto and independence. Median or quantile-based distributional welfare measures, which do not fall within the Harsanyi set, require alternative choice-functional justifications. For social choice functions, mixtures of individual stochastic choice rules form a convex "utilitarian set" paralleling the Weber set for allocations (Utilitarian Social Choice and Distributional Welfare Analysis, 2 Nov 2024).

Aggregation Procedure	Social Welfare Function?	Justified by Harsanyi/Weber Set?
Mean/expected utility (utilitarian)	Yes	Yes
Median/quantile compensating variation	No (for welfare), Yes (choice rule)	No (for welfare), Yes (choice rule)

6. Alternative Axiomatic Foundations and Non-Symmetric Allocations

Dropping symmetry in the allocation rules, but retaining efficiency and additivity, still yields the Weber set as the set of all reasonable allocations (Shapley-like values without symmetry, 2018). The "reasonableness" axiom requires that each player's allocation is within the range of their marginal contributions, opening the way for biased or priority-based value assignments within the convex hull of all marginal vectors. The Krein-Milman theorem provides the foundational convex-analytic machinery showing that any allocation satisfying efficiency and reasonableness is a convex combination of special allocations (i.e., marginal vectors of all orderings).

7. Applications and Implications

The properties of Weber and Harsanyi sets illuminate practical modeling and policy analysis in game theory, welfare economics, and epistemic logic:

In games and mechanism design, the tractability of type or allocation spaces is essential; completeness, unique extension, and denesting results determine whether belief hierarchies (Harsanyi sets) or allocation options (Weber/intermediate sets) are finite or continuum-sized.
For welfare and distributional analysis, the Harsanyi set constrains aggregation to utilitarian functionals, while the Weber set and its generalizations support analysis of value assignments under weaker or asymmetric principles.
In logical epistemology, the distinction between knowledge and probabilistic certainty (S5 not reducible to belief operators) delineates the expressive boundary of probabilistic versus modal reasoning.

These sets form the mathematical and conceptual backbone of reasoning about aggregation, fairness, and consistency in multi-agent scenarios, guiding the analysis and engineering of allocation, belief, and decision systems in both theoretical and applied domains.

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