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arxiv: 2606.18559 · v1 · pith:47HNVFBUnew · submitted 2026-06-17 · 💻 cs.GT

Principal Component Analysis and Power Indices

Xavier Molinero , Enric Mons\'o , Daniel Samaniego This is my paper

Pith reviewed 2026-06-26 19:20 UTC · model grok-4.3

classification 💻 cs.GT
keywords power indexsimple gamesprincipal component analysiseigenvalueswinning coalitionscharacterizationvoting power
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The pith

A new power index based on winning coalitions coincides with the eigenvalues from principal component analysis and is uniquely characterized by four properties.

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

The paper proposes a fresh power index for measuring player influence in simple games, built directly from the set of winning coalitions. It proves that the values this index assigns to players are exactly the eigenvalues that principal component analysis produces when the game is represented as data. The authors further establish that four specific properties fully determine this index. A sympathetic reader would care because the result ties a classical concept from cooperative game theory to a standard tool in data science for ranking feature importance. If the coincidence holds, computing power in voting systems could borrow efficient algorithms already used for large datasets.

Core claim

The paper defines a power index in terms of winning coalitions in a simple game. It shows that this index coincides with the eigenvalues obtained by the Principal Component Analysis method. It also provides a characterization of the index by means of four properties.

What carries the argument

The proposed power index defined directly from winning coalitions, shown to equal PCA eigenvalues.

If this is right

  • The index can be obtained by running standard PCA on the matrix whose rows encode winning coalitions.
  • The four properties together single out this index among all possible power measures.
  • Power measurement in simple games becomes interchangeable with an established multivariate statistics procedure.
  • The result supplies an explicit link between coalition structure and the principal directions of variation in the game data.

Where Pith is reading between the lines

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

  • The connection may let analysts apply existing PCA software packages to compute power indices for games too large for direct enumeration.
  • Similar eigenvalue interpretations could be explored for other solution concepts in cooperative games that also depend on coalition data.
  • If the four properties turn out to be the minimal set needed, they could serve as an axiomatic starting point for designing new indices tuned to specific voting contexts.

Load-bearing premise

The new index is defined using the structure of winning coalitions in the simple game.

What would settle it

For any concrete simple game, compute the proposed index values for each player and separately extract the eigenvalues of the principal component analysis applied to the game's winning-coalition incidence data; systematic mismatch between the two sets of numbers would disprove the claimed coincidence.

Figures

Figures reproduced from arXiv: 2606.18559 by Daniel Samaniego, Enric Mons\'o, Xavier Molinero.

Figure 1
Figure 1. Figure 1: Matrix representation of the SVD. W.l.o.g., we assume [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
read the original abstract

Measuring the influence of a player in a simple game is a widely studied topic. Shapley-Shubik power index is perhaps the maximum exponent in terms of relevance. Furthermore, other power indexes have been proposed over time. In this paper, we propose yet another power index, defined in terms of winning coalitions. We show that this index coincides with eigenvalues obtained with the Principal Component Analysis method, a broadly used technique in data science to determine the influence of different features given a dataset. Furthermore, we provide a characterization of this proposed index in terms of four properties.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes a new power index for players in simple games, defined in terms of winning coalitions. It claims that this index coincides with the eigenvalues obtained via Principal Component Analysis and provides a characterization of the index in terms of four properties.

Significance. If the claimed coincidence and characterization hold with explicit derivations, the result would connect cooperative game theory power indices to a standard data-science technique, potentially enabling new computational approaches or interpretations of influence in voting systems. The axiomatic characterization would add rigor to the proposal.

major comments (1)
  1. [Abstract] Abstract: the central claims—that the proposed index coincides with PCA eigenvalues and admits a four-property characterization—are asserted without any definition of the index, mapping to the PCA input matrix, derivation of the coincidence, proof sketch, or statement of the four properties. This absence makes it impossible to evaluate whether the mathematics supports the claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the comments. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claims—that the proposed index coincides with PCA eigenvalues and admits a four-property characterization—are asserted without any definition of the index, mapping to the PCA input matrix, derivation of the coincidence, proof sketch, or statement of the four properties. This absence makes it impossible to evaluate whether the mathematics supports the claims.

    Authors: The abstract is deliberately concise to summarize the contributions within typical length limits. The definition of the proposed power index (in terms of winning coalitions), the explicit mapping to the PCA input matrix, the derivation establishing coincidence with the eigenvalues, the proof, and the statement plus verification of the four characterizing properties are all contained in the body of the manuscript. We will revise the abstract to include a brief indication of the four properties and the coincidence result. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines its proposed power index directly from winning coalitions in simple games (a conventional, non-circular starting point in cooperative game theory). It then proves coincidence with PCA eigenvalues and supplies an independent four-property axiomatic characterization. No step reduces a claimed result to a fitted parameter, self-citation chain, or definitional tautology; the equivalence and characterization are presented as derived consequences rather than inputs renamed as outputs. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Insufficient information available from abstract alone to identify free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5613 in / 787 out tokens · 20991 ms · 2026-06-26T19:20:23.893034+00:00 · methodology

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Reference graph

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