The Geometry of Cooperative Game Solutions: Stratified Egalitarian Shapley Values

Frank M. V. Feys

arxiv: 2605.22847 · v1 · pith:R6AS2FFNnew · submitted 2026-05-15 · 💻 cs.GT · econ.TH· math.CO

The Geometry of Cooperative Game Solutions: Stratified Egalitarian Shapley Values

Frank M. V. Feys This is my paper

Pith reviewed 2026-05-25 00:28 UTC · model grok-4.3

classification 💻 cs.GT econ.THmath.CO

keywords cooperative gameslinear valuesShapley valueegalitarian ShapleyHarsanyi dividendsorthogonal decompositionR-squaredefficiency

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The pith

The space of efficient symmetric linear cooperative game values is isomorphic to R^{n-1} through orthogonal stratification by coalition size.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the space of linear value maps on finite-player cooperative games carries a canonical inner product induced by the Harsanyi dividend decomposition of the underlying game space. This inner product is intrinsic, so the induced geometry on the subspace of efficient symmetric linear values is independent of basis choice. Within that geometry an orthogonal stratification by coalition size produces a canonical isomorphism from the value space to R^{n-1}, with each value uniquely represented by n-1 stratified epsilons. The classical egalitarian Shapley family occupies exactly the diagonal line in these coordinates. Projection of any other value onto the diagonal supplies both an optimal egalitarian parameter, given by a weighted mean of the stratified epsilons, and an R-squared goodness-of-fit equal to one minus the relative weighted variance of those epsilons.

Core claim

The induced orthogonal stratification of L by coalition size yields a canonical linear isomorphism L^{ESL} = R^{n-1}, under which every efficient symmetric linear value map decomposes uniquely into n-1 stratified epsilons, one per coalition size. The classical egalitarian Shapley family is precisely the diagonal slice of this R^{n-1}. The orthogonal projection of any Psi in L^{ESL} onto this diagonal yields an optimal parameter eps*(Psi) equal to the weighted mean of the stratified epsilons under an explicit probability distribution {w_a} over coalition sizes, and the goodness-of-fit R^2(Psi) equals one minus the relative weighted variance of those epsilons.

What carries the argument

The orthogonal stratification by coalition size of the subspace L^{ESL} of efficient symmetric linear value maps, which induces the isomorphism to R^{n-1} and the unique decomposition into one stratified epsilon per coalition size.

If this is right

The egalitarian Shapley family occupies exactly the diagonal line inside the R^{n-1} coordinate system.
For any efficient symmetric linear value the optimal egalitarian parameter equals the weighted mean of its stratified epsilons under the explicit weights {w_a}.
The R^2 measure for any such value equals one minus the relative weighted variance of its stratified epsilons.
At n=4 the Banzhaf value has R^2 approximately 1 percent, the equal-surplus-division value 38 percent, and the solidarity value 99.6 percent.
As n tends to infinity, R^2 for equal-surplus-division and solidarity both tend to 1 while R^2 for Banzhaf tends to 1/2.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The stratified-epsilon coordinates could be used to define new families of solution concepts by choosing arbitrary sequences in R^{n-1}.
The regression analogy suggests treating the selection of a cooperative solution as a statistical model-fitting task whose residuals measure deviation from egalitarianism.
Analogous orthogonal stratifications might be constructed for subspaces that drop efficiency or symmetry, yielding comparable decompositions outside L^{ESL}.
The explicit weights {w_a} and variance formula may admit closed-form extensions to continuous or infinite-player games.

Load-bearing premise

The inner product on the space of games induced by the Harsanyi dividend decomposition is intrinsic, so the same numerical value is obtained from any orthonormal basis.

What would settle it

An explicit computation for n=3 that produces two different numerical values for the inner product of the same pair of value maps when two distinct orthonormal bases of G^N are used with respect to the Harsanyi inner product.

Figures

Figures reproduced from arXiv: 2605.22847 by Frank M. V. Feys.

**Figure 1.** Figure 1: The Pythagorean decomposition Ψ − Sh = ε ∗ (Ψ) · (ED − Sh) + Φ⊥(Ψ). The optimal F ε ∗(Ψ) is the orthogonal projection of Ψ onto the affine line of egalitarian Shapley values. The residual Φ⊥(Ψ) is orthogonal to the line. Definition 7.1. For Ψ ∈ L, the egalitarian Shapley residual of Ψ is Φ⊥(Ψ) := Ψ − F ε ∗(Ψ) . Lemma 7.2. For all Ψ ∈ L, ⟨Φ⊥(Ψ),ED − Sh⟩L = 0. Proof. Write Ψ − Sh = ε ∗ (Ψ)(ED − Sh) + Φ⊥(Ψ). … view at source ↗

read the original abstract

The space L of linear value maps on a finite-player cooperative game G^N is finite-dimensional, and admits a canonical inner product induced by the Harsanyi-dividend decomposition of G^N. We show that this inner product is intrinsic: the same value arises from any orthonormal basis of G^N with respect to the Harsanyi inner product. Within this geometry, the subspace L^{ESL} of efficient, symmetric, linear value maps admits a clean structure theorem. The induced orthogonal stratification of L by coalition size yields a canonical linear isomorphism L^{ESL} = R^{n-1}, under which every efficient symmetric linear value map decomposes uniquely into n-1 stratified epsilons, one per coalition size. The classical egalitarian Shapley family of Joosten (1996) is precisely the diagonal slice of this R^{n-1}. The orthogonal projection of any Psi in L^{ESL} onto this diagonal yields an optimal parameter eps*(Psi) equal to the weighted mean of the stratified epsilons under an explicit probability distribution {w_a} over coalition sizes, and the goodness-of-fit R^2(Psi) equals one minus the relative weighted variance of those epsilons. The framework is a literal regression-statistics analogue of the coefficient of determination. At n=4 it produces a clean three-way classification of the standard alternatives to the Shapley value: the Banzhaf value is nearly orthogonal to the egalitarian Shapley axis (R^2 ~ 1%); the equal-surplus-division value is moderately aligned (R^2 ~ 38%); the solidarity value is almost entirely aligned (R^2 ~ 99.6%). Asymptotically R^2(ESD) -> 1, R^2(So) -> 1, and R^2(Bz) -> 1/2, the last reflecting a structural identity between the efficiency defect and the egalitarian-Shapley deviation of the Banzhaf value at every coalition size.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper organizes efficient symmetric linear values into an (n-1)-dimensional space with an orthogonal stratification by coalition size, placing egalitarian Shapley on the diagonal and giving an R^2 alignment score for other maps.

read the letter

The paper's main move is to put an inner product on linear value maps coming from the Harsanyi dividend decomposition of the game space, prove that this inner product is the same no matter which orthonormal basis you pick, and then restrict to the efficient symmetric subspace. That subspace gets an orthogonal stratification by coalition size, which gives a canonical isomorphism to R^{n-1}. The classical egalitarian Shapley family sits exactly on the diagonal of that space. Any other efficient symmetric map then has a projection onto the diagonal that defines an optimal epsilon and an R^2 value equal to one minus the weighted variance of its stratified components across coalition sizes. The n=4 numbers and the asymptotic limits for Banzhaf, equal-surplus-division, and solidarity are the concrete output that makes the setup usable right away. The stratification and the resulting R^2 formula look like the genuinely new pieces; the 1996 reference is only for the family itself, not for this decomposition. The basis-independence claim is the load-bearing step, and the abstract states that it is shown, so the geometry is presented as canonical rather than coordinate-dependent. Once that holds, the rest is linear algebra plus the standard regression interpretation of R^2, which is tautological by construction but still gives a clean quantitative comparison tool. No obvious gaps in the chain from inner product to the three-way classification at n=4. This is for readers already inside cooperative game theory who want a geometric way to line up different solution concepts rather than just list axioms. The framework is self-contained enough and produces explicit numerical claims, so it deserves a serious referee even if the geometric language is not to everyone's taste. I would send it out for review.

Referee Report

2 major / 2 minor

Summary. The paper claims that the space L of linear value maps on cooperative games admits a canonical inner product induced by the Harsanyi-dividend decomposition of G^N, which is shown to be intrinsic (independent of orthonormal basis choice). Within this geometry the subspace L^{ESL} of efficient symmetric linear values admits an orthogonal stratification by coalition size, yielding a canonical isomorphism L^{ESL} ≅ R^{n-1} under which every such value decomposes uniquely into n-1 stratified epsilons. The egalitarian Shapley family is the diagonal slice; the orthogonal projection of any Psi onto this diagonal gives an optimal eps*(Psi) as the weighted mean of the stratified epsilons under explicit weights {w_a}, with R^2(Psi) equal to one minus the relative weighted variance. Concrete n=4 classifications and asymptotic limits for Banzhaf, equal-surplus-division and solidarity values are provided.

Significance. If the intrinsicness of the inner product holds, the work supplies a regression-style geometric decomposition that classifies how closely standard efficient symmetric values align with the egalitarian Shapley family, together with explicit R^2 figures and asymptotic statements. The concrete n=4 calculations and the explicit probability weights {w_a} constitute reproducible content that could be checked independently.

major comments (2)

[inner-product definition and intrinsicness claim] The section establishing intrinsicness of the Harsanyi-induced inner product: the argument that the same inner-product value on L arises for any orthonormal basis of G^N must be fully explicit, because this invariance is required for the orthogonality of the stratification, the uniqueness of the decomposition into stratified epsilons, and the invariance of eps* and R^2.
[structure theorem and isomorphism] The structure theorem for L^{ESL}: the derivation of the linear isomorphism L^{ESL} = R^{n-1} and the identification of the egalitarian Shapley diagonal both rest on the claimed orthogonality of the coalition-size stratification; any dependence on basis choice would render the decomposition and the subsequent projection non-canonical.

minor comments (2)

Explicit formulas for the weights w_a and the stratified epsilons should be stated in the main text rather than left implicit, to permit direct verification of the reported R^2 values at n=4.
The n=4 calculations would benefit from a short table listing the stratified epsilons for each of the three values (Banzhaf, ESD, solidarity) so that the R^2 figures (~1%, ~38%, ~99.6%) can be reproduced from the weighted-variance formula.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and the emphasis on explicitness in the foundational sections. We address the major comments point by point below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

Referee: [inner-product definition and intrinsicness claim] The section establishing intrinsicness of the Harsanyi-induced inner product: the argument that the same inner-product value on L arises for any orthonormal basis of G^N must be fully explicit, because this invariance is required for the orthogonality of the stratification, the uniqueness of the decomposition into stratified epsilons, and the invariance of eps* and R^2.

Authors: We agree that a fully explicit argument is necessary to establish the basis-independence of the inner product. In the revised version, we will provide a complete proof in the relevant section, deriving the inner product explicitly from the Harsanyi dividend representation and showing invariance under change of orthonormal basis without assuming any particular basis. This will include the explicit formula for the inner product on L. revision: yes
Referee: [structure theorem and isomorphism] The structure theorem for L^{ESL}: the derivation of the linear isomorphism L^{ESL} = R^{n-1} and the identification of the egalitarian Shapley diagonal both rest on the claimed orthogonality of the coalition-size stratification; any dependence on basis choice would render the decomposition and the subsequent projection non-canonical.

Authors: The orthogonality of the stratification follows directly from the intrinsic inner product once its basis-independence is established. We will expand the proof of the structure theorem to explicitly verify the orthogonality using the revised inner-product definition, thereby confirming the canonical nature of the isomorphism and the uniqueness of the decomposition into stratified epsilons. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from proved inner-product properties

full rationale

The paper defines an inner product on L via the Harsanyi-dividend decomposition and explicitly states that it proves this inner product is intrinsic (independent of orthonormal basis choice). This proved property underpins the orthogonal stratification and the isomorphism L^{ESL} ≅ R^{n-1} without reducing to a self-citation or fitted input. The R^2(Psi) is openly presented as the standard regression coefficient of determination for the orthogonal projection onto the diagonal slice, which is definitional rather than a hidden circularity. No load-bearing self-citations, ansatzes smuggled via citation, or predictions that reduce by construction to inputs appear in the provided text. The framework is therefore a direct geometric consequence of the stated inner-product structure.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the Harsanyi dividend decomposition supplying a basis-independent inner product and on the resulting stratification being orthogonal; these are domain assumptions of cooperative game theory rather than new postulates.

axioms (2)

domain assumption The Harsanyi-dividend decomposition induces a canonical inner product on the space of cooperative games that is independent of the choice of orthonormal basis.
Invoked to define the geometry on L and to guarantee that the orthogonal stratification is well-defined.
standard math Finite-dimensional vector space structure on the set of cooperative games with n players.
Standard linear-algebra background used throughout.

invented entities (1)

stratified epsilon no independent evidence
purpose: Coordinate that captures the component of an efficient symmetric linear value along the direction associated with coalitions of a fixed size.
Introduced by the structure theorem as the unique coordinates in the isomorphism L^{ESL} ≅ R^{n-1}.

pith-pipeline@v0.9.0 · 5901 in / 1827 out tokens · 34194 ms · 2026-05-25T00:28:44.948961+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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Relation between the paper passage and the cited Recognition theorem.

The space L of linear value maps … admits a canonical inner product induced by the Harsanyi-dividend decomposition of G^N. We show that this inner product is intrinsic … The induced orthogonal stratification of L by coalition size yields a canonical linear isomorphism L^ESL ≅ R^{n-1}
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

⟨Ψ,Φ⟩_L = ∑_{∅≠A⊆N} ⟨Ψ(u_A),Φ(u_A)⟩_RN … Theorem 5.1 (Basis Independence)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

J. F. Banzhaf. Weighted voting doesn’t work: a mathematical analysis.Rutgers Law Review 19(1965), 317–343

work page 1965
[2]

Casajus and F

A. Casajus and F. Huettner. Null players, solidarity, and the egalitarian Shapley values. Journal of Mathematical Economics49(2013), 58–61

work page 2013
[3]

Casajus and F

A. Casajus and F. Huettner. Null, nullifying, or dummifying players: the difference between the Shapley value, the equal division value, and the equal surplus division value.Economics Letters122(2014), 167–169

work page 2014
[4]

T. S. H. Driessen and Y. Funaki. Coincidence of and collinearity between game theoretic solutions.OR Spektrum13(1991), 15–30

work page 1991
[5]

J. C. Harsanyi. A bargaining model for the cooperativen-person game. InContributions to the Theory of Games IV, ed. A. W. Tucker and R. D. Luce, pp. 325–355, Princeton University Press, 1959

work page 1959
[6]

Hart and A

S. Hart and A. Mas-Colell. Potential, value, and consistency.Econometrica57(1989), 589–614

work page 1989
[7]

Joosten.Dynamics, Equilibria and Values

R. Joosten.Dynamics, Equilibria and Values. PhD thesis, Maastricht University, 1996

work page 1996
[8]

Kalai and D

E. Kalai and D. Samet. On weighted Shapley values.International Journal of Game Theory 16(1987), 205–222

work page 1987
[9]

Kamijo and T

Y. Kamijo and T. Kongo. Whose deletion does not affect your payoff? The difference between the Shapley value, the egalitarian value, the solidarity value, and the Banzhaf value.European Journal of Operational Research216(2012), 638–646

work page 2012
[10]

A. S. Nowak and T. Radzik. A solidarity value forn-person transferable utility games. International Journal of Game Theory23(1994), 43–48

work page 1994
[11]

G. Owen. Values of games with a priori unions. InEssays in Mathematical Economics and Game Theory, ed. R. Henn and O. Moeschlin, pp. 76–88, Springer, 1977

work page 1977
[12]

T. Radzik. Is the solidarity value close to the equal split value?Mathematical Social Sciences 65(2013), 195–202

work page 2013
[13]

L. S. Shapley. A value forn-person games. InContributions to the Theory of Games II, ed. H. W. Kuhn and A. W. Tucker, pp. 307–317, Princeton University Press, 1953

work page 1953
[14]

van den Brink, Y

R. van den Brink, Y. Funaki and Y. Ju. Reconciling marginalism with egalitarianism: consistency, monotonicity, and implementation of egalitarian Shapley values.Social Choice and Welfare40(2013), 693–714

work page 2013
[15]

H. P. Young. Monotonic solutions of cooperative games.International Journal of Game Theory14(1985), 65–72. 34

work page 1985

[1] [1]

J. F. Banzhaf. Weighted voting doesn’t work: a mathematical analysis.Rutgers Law Review 19(1965), 317–343

work page 1965

[2] [2]

Casajus and F

A. Casajus and F. Huettner. Null players, solidarity, and the egalitarian Shapley values. Journal of Mathematical Economics49(2013), 58–61

work page 2013

[3] [3]

Casajus and F

A. Casajus and F. Huettner. Null, nullifying, or dummifying players: the difference between the Shapley value, the equal division value, and the equal surplus division value.Economics Letters122(2014), 167–169

work page 2014

[4] [4]

T. S. H. Driessen and Y. Funaki. Coincidence of and collinearity between game theoretic solutions.OR Spektrum13(1991), 15–30

work page 1991

[5] [5]

J. C. Harsanyi. A bargaining model for the cooperativen-person game. InContributions to the Theory of Games IV, ed. A. W. Tucker and R. D. Luce, pp. 325–355, Princeton University Press, 1959

work page 1959

[6] [6]

Hart and A

S. Hart and A. Mas-Colell. Potential, value, and consistency.Econometrica57(1989), 589–614

work page 1989

[7] [7]

Joosten.Dynamics, Equilibria and Values

R. Joosten.Dynamics, Equilibria and Values. PhD thesis, Maastricht University, 1996

work page 1996

[8] [8]

Kalai and D

E. Kalai and D. Samet. On weighted Shapley values.International Journal of Game Theory 16(1987), 205–222

work page 1987

[9] [9]

Kamijo and T

Y. Kamijo and T. Kongo. Whose deletion does not affect your payoff? The difference between the Shapley value, the egalitarian value, the solidarity value, and the Banzhaf value.European Journal of Operational Research216(2012), 638–646

work page 2012

[10] [10]

A. S. Nowak and T. Radzik. A solidarity value forn-person transferable utility games. International Journal of Game Theory23(1994), 43–48

work page 1994

[11] [11]

G. Owen. Values of games with a priori unions. InEssays in Mathematical Economics and Game Theory, ed. R. Henn and O. Moeschlin, pp. 76–88, Springer, 1977

work page 1977

[12] [12]

T. Radzik. Is the solidarity value close to the equal split value?Mathematical Social Sciences 65(2013), 195–202

work page 2013

[13] [13]

L. S. Shapley. A value forn-person games. InContributions to the Theory of Games II, ed. H. W. Kuhn and A. W. Tucker, pp. 307–317, Princeton University Press, 1953

work page 1953

[14] [14]

van den Brink, Y

R. van den Brink, Y. Funaki and Y. Ju. Reconciling marginalism with egalitarianism: consistency, monotonicity, and implementation of egalitarian Shapley values.Social Choice and Welfare40(2013), 693–714

work page 2013

[15] [15]

H. P. Young. Monotonic solutions of cooperative games.International Journal of Game Theory14(1985), 65–72. 34

work page 1985