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arxiv: 1907.01323 · v1 · pith:I7FN2XQSnew · submitted 2019-07-02 · 💻 cs.GT

An Axiomatization of the Shapley-Shubik Index for Interval Decisions

Sascha Kurz , Issofa Moyouwou , Hilaire Touyem This is my paper

Pith reviewed 2026-05-25 10:43 UTC · model grok-4.3

classification 💻 cs.GT
keywords Shapley-Shubik indexpower indexinterval decisionsaxiomatizationsimple gamescooperative game theoryvoting power
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The pith

The Shapley-Shubik index extends to games with a continuum of decision options via an axiomatization whose discretizations recover the classical versions.

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

This paper establishes an axiomatization for a power measure in cooperative games where each player selects an approval level from a continuous interval. It demonstrates that the standard Shapley-Shubik index for simple games and the version for (j,k) simple games both arise exactly as special cases when the continuous model is discretized. A sympathetic reader would care because the construction unifies power measurement across binary, finite-level, and gradual decision settings while preserving the original axioms in the limit. The result suggests the new measure can be viewed as the Shapley-Shubik index for interval decisions and can further be extended to a value.

Core claim

For these games with interval decisions we prove an axiomatization of a power measure and show that the Shapley-Shubik index for simple games, as well as for (j,k) simple games, occurs as a special discretization. This relation and the closeness of the stated axiomatization to the classical case suggests to speak of the Shapley-Shubik index for games with interval decisions, that can also be generalized to a value.

What carries the argument

The set of axioms (efficiency, symmetry, dummy player, and marginal contribution) extended directly to the continuous interval-decision model.

If this is right

  • The classical Shapley-Shubik index for simple games is recovered exactly by discretizing the continuous version to two levels.
  • The (j,k) simple-game version of the index likewise appears as a finite discretization of the interval version.
  • The power measure satisfies the same four axioms as the discrete Shapley-Shubik index when restricted to the appropriate subdomain.
  • The construction admits a direct extension from an index to a full value for interval games.

Where Pith is reading between the lines

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

  • The continuous formulation could be used to assign power in settings where approval levels vary smoothly, such as resource allocation with fractional participation.
  • Similar continuous generalizations might exist for other established power indices that rely on marginal contributions.
  • The axiomatization supplies a template for deriving limit-consistent extensions of discrete solution concepts in cooperative game theory.

Load-bearing premise

The chosen axioms for the interval case are the natural continuous generalization of the discrete Shapley-Shubik axioms, with the discretization limit preserving the index exactly.

What would settle it

A calculation of the continuous power measure on a binary simple game that yields a value different from the classical Shapley-Shubik index for that game.

Figures

Figures reproduced from arXiv: 1907.01323 by Hilaire Touyem, Issofa Moyouwou, Sascha Kurz.

Figure 1
Figure 1. Figure 1: Moves at phase 1 Phase 1 We start with the interval simple game given by u 0 (x) = 0 for all x ∈ [0, 1]2\ {(1, 1)}, where we clearly have Ψ(u 0 ) = 1 2 , 1 2  . Next we consider the local increment u 0 ∅,0.1,[0,1]2 ,→ u 1 , so that Ψ(u 1 ) = 1 2 , 1 2  ; see [PITH_FULL_IMAGE:figures/full_fig_p026_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Moves at phase 2 consists in u 4 {1},0.2,[0, 1 4 ] ,→ u 5 gives Ψ(u 5 ) = 9 20 + 2 80 , 11 20 − 2 80  . Finally, u 5 {2},−0.2,[ 1 4 ,1] ,→ u 6 implies Ψ(u 6 ) = 19 40 + 6 80 , 21 40 − 6 80  [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Moves at phase 3 28 [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
read the original abstract

The Shapley-Shubik index was designed to evaluate the power distribution in committee systems drawing binary decisions and is one of the most established power indices. It was generalized to decisions with more than two levels of approval in the input and output. In the limit we have a continuum of options. For these games with interval decisions we prove an axiomatization of a power measure and show that the Shapley-Shubik index for simple games, as well as for $(j,k)$ simple games, occurs as a special discretization. This relation and the closeness of the stated axiomatization to the classical case suggests to speak of the Shapley-Shubik index for games with interval decisions, that can also be generalized to a value.

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 paper claims to provide an axiomatization of a power measure for cooperative games with interval decisions (a continuum of approval levels), using axioms that are continuous analogues of efficiency, symmetry, dummy player, and additivity/marginal contribution. It further asserts that the classical Shapley-Shubik index for simple games, as well as its generalization to (j,k) simple games, arise exactly as special cases via discretization of the interval model.

Significance. If the axiomatization and exact-recovery claims hold, the work supplies a natural continuous extension of the Shapley-Shubik index that recovers the discrete cases without approximation error. This unifies discrete and continuous power-index theory under a single axiomatic framework and supports generalization to a value, which would be a substantive contribution to cooperative game theory.

major comments (1)
  1. [Abstract] Abstract: the central claim rests on the existence of a proof that the stated axioms characterize the interval power measure and that the discretization limit recovers the classical indices exactly. No derivation, independence verification, or explicit limit argument is visible in the provided text, so the support for the claim cannot be assessed beyond the assertion itself.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and the positive evaluation of the potential contribution of the work. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim rests on the existence of a proof that the stated axioms characterize the interval power measure and that the discretization limit recovers the classical indices exactly. No derivation, independence verification, or explicit limit argument is visible in the provided text, so the support for the claim cannot be assessed beyond the assertion itself.

    Authors: The full manuscript contains the required material beyond the abstract. Section 3 states and proves the axiomatization (Theorem 1) for the interval power index using the continuous versions of efficiency, symmetry, dummy, and additivity, including a complete derivation and explicit verification that the axioms are independent. Section 4 defines the discretization map and proves (Theorems 2 and 3) that the interval index restricts exactly to the classical Shapley-Shubik index on simple games and on (j,k)-games, with no approximation error; the proofs are direct rather than limit arguments. These sections supply the derivations, independence checks, and exact-recovery arguments. If only the abstract was visible in the review copy, we are happy to highlight the relevant pages. revision: no

Circularity Check

0 steps flagged

No significant circularity; axiomatization is self-contained

full rationale

The paper presents an axiomatization result for a continuous power index on interval games, with the discrete Shapley-Shubik indices recovered exactly via discretization as a special case. The abstract and reader's summary indicate the axioms are stated as direct continuous analogues of the classical efficiency, symmetry, dummy-player, and additivity properties, with no parameter fitting, self-referential definitions, or load-bearing self-citations invoked to justify the central claim. The derivation chain therefore rests on independent axiom verification rather than reducing to its own inputs by construction. This is the expected outcome for a pure axiomatization paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, invented entities, or non-standard axioms are mentioned. The work relies on standard cooperative-game axioms generalized to the interval setting.

axioms (1)
  • domain assumption The power measure satisfies efficiency, symmetry, dummy-player, and marginal-contribution axioms generalized from the discrete Shapley-Shubik case.
    Invoked when the paper claims the discrete indices arise as special discretizations of the interval version.

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Reference graph

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