arxiv: 2605.00962 · v1 · submitted 2026-05-01 · 💻 cs.GT · math.CO· math.GR· math.RT

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Induced Representations in Cooperative Games with Homogeneous Groups of Players

Windsor Kiang

Authors on Pith no claims yet

Pith reviewed 2026-05-09 18:21 UTC · model grok-4.3

classification 💻 cs.GT math.COmath.GRmath.RT

keywords cooperative gamesShapley valueinduced representationsYoung subgroupsLittlewood-Richardson rulehomogeneous groupssymmetric linear valuesgame theory

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The pith

Homogeneous group cooperative games are supported solely by irreducible representations whose interaction depth is bounded by minority group size.

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

The paper establishes that cooperative games with homogeneous groups of players admit a simplified algebraic structure that filters out much of the complexity in the general kernel of symmetric values. By modeling the games via induced representations from Young subgroups, it shows that only irreducible representations appear, with their depth strictly limited by the size of the smallest group according to the Littlewood-Richardson rule. This filtration restricts any symmetric linear value to a specific subspace of the full space of games. For the case of exactly two such groups the method recovers the Shapley value as the unique solution satisfying the usual axioms. The same reduction is then applied to concrete large-scale examples such as voting bodies and markets to demonstrate computational feasibility.

Core claim

We identify the spectrum of homogeneous group games by using an induced representation from a Young subgroup. We then prove that such games are supported solely by irreducible representations, via the Littlewood-Richardson rule, where the depth of interactions is strictly bounded by the size of the minority group. Therefore, the algebraic structure of the game filters out the complexities of the general kernel W. We then show that this filtration constrains any symmetric linear value to a specific subspace. This recovers the Shapley value uniquely for m=2 under standard axioms.

What carries the argument

Induced representation from a Young subgroup, which via the Littlewood-Richardson rule selects only irreducible components whose interaction depth is bounded by minority-group size and thereby filters the kernel for symmetric linear values.

If this is right

The spectrum of the game is supported solely by irreducible representations whose depth is bounded by minority-group size.
The algebraic filtration removes the complexities of the general kernel W.
Any symmetric linear value is constrained to a specific subspace of the space of games.
For exactly two homogeneous groups the Shapley value is recovered uniquely under standard axioms.
The same reduction applies directly to voting structures such as the UN Security Council and to complementary-goods markets.

Where Pith is reading between the lines

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

The bounded-depth property may extend to other classes of symmetric cooperative games that admit a similar Young-subgroup decomposition.
Empirical checks of interaction depth could be performed on real market or voting data to test whether the minority-size bound holds in observed behavior.
The subspace constraint might be combined with existing polynomial-time algorithms for symmetric games to yield further speed-ups for multi-group settings.

Load-bearing premise

The games consist of homogeneous groups of players, which permits the use of Young subgroups and their induced representations to produce the claimed filtration of the kernel.

What would settle it

Exhibit a concrete homogeneous-group game with two groups in which a symmetric linear value distinct from the Shapley value satisfies the standard axioms, or in which the representation decomposition contains an irreducible summand whose depth exceeds the minority-group size.

read the original abstract

Oftentimes, the Shapley value becomes infeasible for games with many players. However, establishing symmetry allows for polynomial-time computation. To examine this reduction, we identify the spectrum of homogeneous group games by using an induced representation from a Young subgroup. We then prove that such games are supported solely by irreducible representations, via the Littlewood-Richardson rule, where the depth of interactions is strictly bounded by the size of the minority group. Therefore, the algebraic structure of the game filters out the complexities of the general kernel $W$. We then show that this filtration constrains any symmetric linear value to a specific subspace. This recovers the Shapley value uniquely for $m=2$ under standard axioms. Finally, we explore applications to the UN Security Council and complementary goods markets to illustrate the practical power of this approach.

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 tries to simplify Shapley values in homogeneous-group games via induced representations from Young subgroups, but invokes the Littlewood-Richardson rule where Young's rule and Kostka numbers apply, which undercuts the claimed filtration and uniqueness result.

read the letter

The paper's main move is to treat cooperative games with homogeneous player groups as living inside an induced representation from the corresponding Young subgroup, then use that to bound interaction depth and filter the space of symmetric linear values down to something that recovers the Shapley value when there are only two groups. The abstract presents this as a way to make computation feasible for large n in restricted but relevant cases like voting and markets.

Referee Report

1 major / 2 minor

Summary. The paper develops a representation-theoretic approach to cooperative games with homogeneous groups of players. It identifies the spectrum of such games via induced representations from Young subgroups, claims these games are supported solely on specific irreducible representations (via the Littlewood-Richardson rule) with interaction depth bounded by the minority group size, shows this filters the general kernel W and constrains symmetric linear values to a subspace, and recovers the Shapley value uniquely for m=2 under standard axioms. Applications to the UN Security Council and complementary goods markets are explored.

Significance. If the central claims hold after correction, the work provides an algebraic method to exploit symmetry for efficient value computation in large homogeneous cooperative games, reducing complexity via representation filtration. The explicit recovery of the Shapley value for m=2 offers a valuable consistency check, and the applications demonstrate relevance to political and economic settings. This integrates tools from symmetric group representation theory into cooperative game theory in a potentially fruitful way.

major comments (1)

[Abstract and main proof of spectrum support] The central claim identifying the spectrum and filtration (stated in the abstract and developed in the proof that games are supported solely by irreps with depth bounded by minority size): the paper invokes the Littlewood-Richardson rule for the decomposition of Ind_Y^{S_n} 1 from the Young subgroup Y of homogeneous block sizes. However, this decomposition is governed by Young's rule, with multiplicities of Specht modules S^μ given by Kostka numbers K_{μλ}, not LR coefficients (which govern tensor products or inductions from two subgroups). This is load-bearing, as the incorrect rule underpins the claimed filtration of W, the subspace constraint, and the uniqueness of the Shapley value for m=2.

minor comments (2)

[Section introducing the kernel W] The kernel W is referenced without an explicit equation or prior definition in the main text; adding a numbered equation or section reference at first use would clarify the filtration argument.
[Applications section] In the applications, the concrete computation of the induced representation or the resulting bounded subspace for the UN Security Council example is not shown explicitly; including a small table or calculation would strengthen the illustration.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need for greater precision regarding the representation-theoretic decomposition used in the paper. We address the major comment below.

read point-by-point responses

Referee: The central claim identifying the spectrum and filtration (stated in the abstract and developed in the proof that games are supported solely by irreps with depth bounded by minority size): the paper invokes the Littlewood-Richardson rule for the decomposition of Ind_Y^{S_n} 1 from the Young subgroup Y of homogeneous block sizes. However, this decomposition is governed by Young's rule, with multiplicities of Specht modules S^μ given by Kostka numbers K_{μλ}, not LR coefficients (which govern tensor products or inductions from two subgroups). This is load-bearing, as the incorrect rule underpins the claimed filtration of W, the subspace constraint, and the uniqueness of the Shapley value for m=2.

Authors: We appreciate the referee pointing out this terminological inaccuracy. We agree that the decomposition of Ind_Y^{S_n} 1 for a general Young subgroup Y corresponding to arbitrary block sizes is governed by Young's rule, with multiplicities given by the Kostka numbers K_{μλ}. The reference to the Littlewood-Richardson rule was imprecise in the abstract and main proof sections. However, for the case m=2 that is central to the uniqueness result, Y = S_k × S_{n-k} (k = minority size), and the induction of the trivial representation coincides exactly with the Littlewood-Richardson decomposition of the tensor product of two trivial representations. In this case the LR coefficients are nonzero precisely for the irreps S^{(n-j,j)} with j ≤ k, which establishes the claimed bound on interaction depth by the minority group size. For general m we will revise the manuscript to cite Young's rule and Kostka numbers explicitly, while confirming that the support property on irreps with depth constrained by the group sizes continues to hold. This ensures the filtration of the kernel W, the resulting subspace constraint on symmetric linear values, and the recovery of the Shapley value for m=2 remain valid. We will update the abstract and relevant proofs accordingly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external representation theory

full rationale

The paper identifies the spectrum of homogeneous games via induced representations from Young subgroups, then invokes the Littlewood-Richardson rule to claim support solely on specific irreps with interaction depth bounded by minority group size. This filters the kernel W and constrains symmetric linear values, recovering the Shapley value uniquely for m=2 under standard axioms. All steps rely on external standard results in the representation theory of symmetric groups rather than self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations. The central argument does not reduce to its own inputs by construction and remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper draws on standard representation theory without introducing new free parameters or invented entities in the abstract; axioms are the usual properties of Young subgroups and the Littlewood-Richardson rule.

axioms (2)

standard math Littlewood-Richardson rule governs the decomposition of tensor products of representations
Invoked to bound the depth of interactions by minority group size
domain assumption Induced representations from Young subgroups capture the symmetry of homogeneous groups
Core modeling choice for the game structure

pith-pipeline@v0.9.0 · 5435 in / 1431 out tokens · 38851 ms · 2026-05-09T18:21:14.395464+00:00 · methodology

discussion (0)

Reference graph

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