The Shapley Value of Coalitions to Other Coalitions

The paper asks how to quantify the value of one coalition to another within cooperative games. Building on the Shapley value, which assigns each player a fair share interpreted as expected marginal contribution, the study generalizes the concept to coalition-to-coalition valuations. Prior work decomposed player Shapley values into a player-to-player matrix; here, the author introduces a symmetric 2^n x 2^n matrix where each entry gives the value of row coalition I to column coalition J, allowing assessments whether coalitions are disjoint, overlapping, or identical. The aim is to derive properties, axioms, and a constructive formula that link coalition-to-coalition values to underlying player-to-player values, and to explore implications, usefulness, and weighted extensions.

The study connects indirectly to Myerson’s balanced contributions and Hart–Mas-Colell’s potential, both showing symmetries consistent with Shapley value properties. Myerson extended Shapley’s axioms to partition function form and conference structures, yielding unique fair allocation rules consistent with balanced contributions. Hart and Mas-Colell showed the uniqueness of a potential whose marginal contributions coincide with the Shapley value. Casajus and Huettner decomposed solutions assigning to a player the difference between the grand coalition’s worth and its worth after the player exits, showing the Shapley value is a unique decomposable decomposer. Classic coalition structure literature (Maschler; Aumann and Dreze; Shenoy; Kurz; Aumann and Myerson) and recent extensions (Hu and Li; Skibski et al.) have not explicitly analyzed the value of one coalition to another. The present work fills that gap by formalizing coalition-to-coalition values while retaining Shapley-style axiomatic roots.

• Setup and basic definitions: A cooperative TU game (N, v) has characteristic function v: 2^N → R with v(∅)=0. The standard Shapley value φ_i(N, v) is recalled. Definition 1 defines the Shapley value of a coalition I as Φ_I(N, v) = Σ_{i∈I} Φ_i(N, v).
• Player-to-player decomposition: Following Hausken and Mohr (2001), φ_i(N, v) is decomposed into n components φ_ij(N, v) (value of i to j), with lemmas showing (i) φ_i(N, v) = Σ_j φ_ij(N, v), (ii) symmetry φ_ij(N, v) = φ_ji(N, v), and (iii) φ_j(N, v) = Σ_i φ_ij(N, v). A closed-form expression links φ_ij to second-order differences of v.
• Axioms for coalition-to-coalition values φ_IJ(N, v):
  1. Symmetry under permutations (coalition names irrelevant). 2a) Efficiency carrier: for any carrier S of v and any partitions of N, summed coalition-to-coalition values distribute v(N). 2b) Null coalition carrier: if I and/or J is null in v, then φ_IJ(N, v)=0.
  2. Additivity: values add across games with the same player set.
• Non-uniqueness under axioms 1–3: An example with parameters a and b for within- and cross-player cells shows efficiency s(s−1)a + sb = 1 allows multiple solutions. Two illustrative alternatives for a game with n=3 produce distinct but axiom-consistent matrices (Tables 1 and 2): • Alternative 1: a=0, b=1/s yields positive diagonal entries at player-to-player level and zero off-diagonals (except as aggregated to coalitions). • Alternative 2: a=1/(s(s−1)), b=0 yields zero diagonals and redistributed off-diagonals (scaled by n−1). This shows coalition-to-coalition values are not uniquely pinned down by standard axioms, unlike individual Shapley values.
• Proposition (core construction): Define φ_IJ(N, v) = Σ_{i∈I} Σ_{j∈J} φ_ij(N, v). This implies φ_IJ = φ_JI and extends to overlapping, nested, or disjoint I and J. Proof parallels Shapley’s and uses player-level symmetry and additivity. Consequences include: (i) coalition self-value equals sum of member self-values plus twice all cross-member pair values; (ii) φ_IJ, φ_IN, φ_NJ are all ≤ v(N); and partition-based aggregation identities: Σ_{I in a partition of N} φ_Ij = φ_j and Σ_{J in a partition of N} φ_iJ = φ_i.
• Example (numeric): With N={1,2,3}, v(1)=180, v(2)=v(3)=v(2,3)=0, v(1,2)=360, v(1,3)=v(1,2,3)=540, the Shapley vector is [390, 30, 120]^T. The 3x3 φ_ij matrix is [[295,25,70],[25,25,−20],[−20,70,70]]. Using the proposition, the full 2^3 x 2^3 coalition matrix (Table 3) is constructed; entries illustrate symmetry and aggregation (e.g., φ_{ {1,3}, {2,3} }=145 from summing four relevant φ_ij cells; φ_{N,N}=v(N)=540).
• Weighted Shapley extensions: Two standard formulations are incorporated. • Dragan (2009) weights: φ^ω_i(v,N,λ) = λ_i Σ_{S⊆N} γ_S (v(S)−v(S{i})), with γ_S depending on weights within S. Lemmas analogous to the unweighted case show decomposition into φ^ω_ij and symmetry. Coalition values follow φ^ω_IJ = Σ_{i∈I} Σ_{j∈J} φ^ω_ij. • Kalai–Samet (1987) lexicographic weights: φ^ω_i equals expected marginal contribution under an order distribution induced by weights. Decomposition lemmas and the coalition proposition carry over, preserving symmetry and additivity.
• Interpretation: φ_ij can be read as value, power, interest, or dependence of i relative to j; φ_IJ extends these interpretations to coalitions, regardless of overlap, nesting, or disjointness. The matrix representation is 2^n x 2^n with player block in the upper-left, null coalition row/column zero, and φ_{N,N}=v(N).

• The coalition-to-coalition value can be consistently defined as φ_IJ(N, v) = Σ_{i∈I} Σ_{j∈J} φ_ij(N, v), yielding a symmetric 2^n x 2^n value matrix across all coalitions.
• Symmetry: φ_IJ = φ_JI for all coalitions I, J; at the player level φ_ij = φ_ji.
• Aggregation properties: Summing over a partition of N recovers individual Shapley values, and φ_IN = Φ_I and φ_NJ = Φ_J; self-coalition values decompose into self and cross-member terms.
• Boundedness: All coalition-to-coalition values involving strict subsets are ≤ v(N); φ_{N,N} = v(N).
• Non-uniqueness under standard axioms: Axioms analogous to Shapley’s (symmetry/permutation invariance, efficiency with carriers and partitions, null coalitions, additivity) do not uniquely determine φ_IJ; multiple axiom-consistent coalition matrices exist (Tables 1 and 2).
• Example data points (n=3): Shapley vector [390, 30, 120]; φ_ij matrix [[295,25,70],[25,25,−20],[−20,70,70]]; selected coalition values: φ_{ {1,3},{2,3} }=145, φ_{ {2,3},3 }=50, φ_{N,N}=540.
• Weighted Shapley values: The decomposition and coalition-sum proposition extend to weighted settings (Dragan; Kalai–Samet), preserving symmetry and additivity.

The research question—how to quantify the value of one coalition to another—is addressed by constructing a coalition-to-coalition value matrix grounded in the player-level Shapley decomposition and extending it via a summation proposition. This representation adds perspective beyond the characteristic function v(S), which only reports a coalition’s collective payoff, by specifying bilateral valuations across all coalition pairs, including overlapping and nested coalitions. Symmetry and aggregation properties ensure internal coherence with standard Shapley values. The framework aids analysis of coalition behavior: it quantifies mutual benefits or harms between coalitions, informs potential mergers, and clarifies contributions within nested structures. Extending the construction to weighted Shapley values shows robustness to heterogeneous bargaining power or importance. Overall, the results enhance interpretability and applicability of cooperative game solutions by providing a fine-grained, relational view among coalitions.

The paper generalizes the Shapley value to a symmetric 2^n x 2^n matrix of coalition-to-coalition values. It proves that (1) the value of a coalition to a player equals the sum of its members’ values to that player, and by symmetry equals the player’s value to the coalition; (2) the value of a coalition to another coalition equals the sum of member-to-member values across the two coalitions, regardless of overlap; (3) the value of a coalition to all players equals the coalition’s Shapley value, i.e., the sum of member Shapley values; (4) for any partition of players, summed coalition-to-player values equal the player’s Shapley value; (5) summed player-to-coalition values across a partition equal the player’s Shapley value. The approach yields substantially more information than the characteristic function alone. The framework also extends to weighted Shapley values. The issue of uniqueness under coalition-level axioms is left open, as different axiom-consistent matrices may be desirable depending on context.

• Non-uniqueness: Standard Shapley-style axioms (permutation invariance, efficiency with carriers/partitions, null coalitions, additivity) are insufficient to uniquely determine coalition-to-coalition values; the paper leaves additional axioms or the desirability of uniqueness as an open question.
• Model scope: The analysis operates in characteristic function (transferable utility) games; externalities and partition function forms are not modeled in the main results (though related literature is discussed).
• Interpretational assumptions: Coalition values are defined via sums of player-to-player values; alternative definitions might be conceivable depending on application needs.
• Practical overlap constraints: While values for overlapping coalitions are defined, simultaneous formation of overlapping coalitions may not be feasible; strategic and informational frictions (e.g., incomplete information) are suggested for future research rather than resolved here.