Symmetries of Quasi-Values

According to Shapley’s game-theoretical result, there exists a unique game value of finite cooperative games that satisfies axioms on additivity, efficiency, null-player property and symmetry. The original setting requires symmetry with respect to arbitrary permutations of players. We analyze the consequences of weakening the symmetry axioms and study quasi-values that are symmetric with respect to permutations from a group $G\leq S_n$. We classify all the permutation groups $G$ that are large enough to assure a unique $G$-symmetric quasi-value, as well as the structure and dimension of the space of all such quasi-values for a general permutation group $G$. We show how to construct $G$-symmetric quasi-values algorithmically by averaging certain basic quasi-values (marginal operators).

💡 Research Summary

The paper revisits the foundational Shapley value theorem, which uniquely characterizes a game value for finite cooperative games by imposing four axioms: efficiency, additivity, the null‑player property, and full symmetry with respect to all permutations of the player set. The authors ask what happens if the symmetry axiom is weakened so that the value need only be invariant under a proper subgroup G of the full symmetric group Sₙ. They introduce the notion of a G‑symmetric quasi‑value: a linear operator φ from the space of cooperative games to ℝⁿ that satisfies efficiency, additivity, the null‑player property, and symmetry only for permutations belonging to G.

The first major contribution is a classification of those subgroups G that are “large enough” to guarantee a unique G‑symmetric quasi‑value. By exploiting the orbit structure of G acting on the player set, the authors show that if G is transitive (i.e., there is a single orbit), the quasi‑value space collapses to a one‑dimensional affine subspace, and the additional requirement of additivity forces a unique solution. When G is 2‑transitive, the unique G‑symmetric quasi‑value coincides exactly with the classical Shapley value, thereby recovering Shapley’s original result as a special case.

For an arbitrary subgroup G the paper provides a complete description of the linear space QV_G of all G‑symmetric quasi‑values. Let 𝒪₁,…,𝒪_k be the distinct G‑orbits on the set of players. For each orbit 𝒪_i the authors construct basic marginal operators (also called marginal contributions or “marginal operators”) associated with permutations in G. They prove that the set of all linear combinations of these orbit‑specific marginal operators spans QV_G, and that the dimension of QV_G is given by the simple formula

 dim QV_G = Σ_{i=1}^k (|𝒪_i| − 1).

Thus the dimension is determined entirely by the orbit sizes: a single orbit of size n yields dimension n − 1 (the space of all efficient, additive, null‑player operators), while multiple orbits increase the dimension accordingly. In particular, if G is not transitive, the space is non‑trivial, and many distinct G‑symmetric quasi‑values exist.

The authors then turn to constructive methods. They define a “group‑averaging” operator

 Φ_G = (1/|G|) ∑_{π∈G} φ_π,

where φ_π is a basic marginal operator corresponding to the permutation π. They prove that Φ_G is always a G‑symmetric quasi‑value, and that every G‑symmetric quasi‑value can be expressed as a convex combination of such averaged operators. This yields an algorithmic recipe: pick a generating set for G, compute the associated marginal operators, and average them (or a weighted average) to obtain any desired G‑symmetric quasi‑value. Because the averaging respects the group action, the resulting operator automatically satisfies the required symmetry, and the computational cost scales linearly with the size of the generating set rather than with |G|, making the approach feasible even for large groups.

The paper also discusses several illustrative scenarios where G‑symmetry is natural. In networked environments, only subsets of players (e.g., nodes within the same geographic region) may be interchangeable, leading to a subgroup G that permutes players within each region but not across regions. In hierarchical organizations, symmetry may be limited to members of the same department. In these contexts, imposing full Shapley symmetry would be unrealistic, whereas a G‑symmetric quasi‑value captures the genuine symmetry constraints while preserving efficiency and fairness properties.

Finally, the authors outline future research directions. They suggest investigating the economic interpretation of G‑symmetric quasi‑values, especially how they relate to concepts of fairness and bargaining power when symmetry is partial. Extending the theory to infinite player sets or to games with continuous strategy spaces is another promising avenue. From a computational perspective, they propose developing parallel implementations of the averaging algorithm and exploring connections with representation theory to further reduce the complexity of constructing quasi‑values for large or highly structured groups.

In summary, by weakening the symmetry axiom to a subgroup G and systematically analysing the resulting quasi‑values, the paper provides a complete classification of when uniqueness holds, an explicit formula for the dimension of the quasi‑value space for any G, and a practical, group‑theoretic algorithm for constructing all such values. This work bridges cooperative game theory with permutation group theory, opening up new possibilities for modeling and solving allocation problems where only partial symmetry is justified.