An upper bound on the dimension of the voting system of the European Union Council under the Lisbon rules
Pith reviewed 2026-05-24 17:12 UTC · model grok-4.3
The pith
The dimension of the European Union Council's voting system under Lisbon rules is at most 25.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
By proving that the union of two weighted games equals the intersection of certain weighted games, the authors reduce the dimension of the Lisbon voting system from an upper bound of 13,368 to an upper bound of 25.
What carries the argument
Theorem 1, a representation that expresses the union of two weighted games as the intersection of weighted games.
If this is right
- The Lisbon voting system equals the intersection of 25 weighted games.
- The same representation technique yields an upper bound after the United Kingdom exits the Council.
- The gap between the known lower bound of 7 and the new upper bound of 25 narrows the possible range for the actual dimension.
- Real-world voting bodies can have dimensions that are small enough for explicit representation.
Where Pith is reading between the lines
- The same union-to-intersection conversion might apply to other voting systems whose winning sets are built from multiple weighted rules.
- If the lower bound of 7 can be raised, the actual dimension might lie between 7 and 25 and become computable.
- The technique could be tested on smaller artificial voting systems to measure how often the bound is tight.
Load-bearing premise
The specific winning coalitions of the Lisbon rules can be decomposed into unions of two weighted games to which the new representation applies directly without extra non-weighted factors.
What would settle it
A proof or explicit construction showing that any representation of the Lisbon winning coalitions as an intersection of weighted games requires at least 26 components.
read the original abstract
The voting rules of the European Council (EU) under the Treaty of Lisbon became effective on 1 November 2014. Kurz \& Napel (2015) showed that the dimension of this voting system is between $7$ and $13,368$. The lower bound $7$ actually sets a new world record for the dimension of the real-world voting bodies. In this article, by finding a new way to represent the union of two weighted games as an intersection of certain weighted games (Theorem 1), we greatly reduce the upper bound $13,368$ to just $25$. We also consider what will happen to our upper bound and Kurz \& Napel's lower bound if the United Kingdom is no longer a member of the European Union Council.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims that a new representation theorem (Theorem 1) allows the union of two weighted voting games to be expressed as the intersection of 25 weighted games; applying this to the EU Council voting system under the Lisbon Treaty reduces the known upper bound on its dimension from 13,368 to 25 (while the lower bound remains 7). It also recomputes the bounds after the United Kingdom's departure.
Significance. If the theorem and its application are correct, the result substantially tightens the dimension bounds for a major real-world voting body and introduces a potentially reusable technique for decomposing unions of weighted games. The lower bound of 7 already sets a record; the improved upper bound makes the dimension far more tractable for further study.
major comments (2)
- [Theorem 1 and its application (presumably §3)] The reduction to an upper bound of 25 rests entirely on expressing the Lisbon rules (population threshold + member-state threshold + blocking minority) as the union of exactly two weighted games and then invoking Theorem 1. The manuscript must supply the explicit weight vectors and quotas for these two games so that readers can verify that their union reproduces the Lisbon winning coalitions without requiring additional non-weighted factors or deeper nesting.
- [Proof of Theorem 1] The derivation of the concrete number 25 in Theorem 1 must be shown to be tight for the specific parameters of the two Lisbon games; if the construction introduces extra factors or replaces weighted games by non-weighted ones for these thresholds, the claimed bound does not follow.
minor comments (2)
- [Section on UK exit] The post-Brexit recalculation should state whether the same two-game decomposition still holds after removing the UK weights and thresholds, or whether a new decomposition is required.
- [Introduction] Ensure the citation to Kurz & Napel (2015) is complete and that any notation for weighted games is defined before first use.
Simulated Author's Rebuttal
We thank the referee for the careful review and for identifying points where greater explicitness will strengthen the manuscript. We address each major comment below and will revise the paper to incorporate the requested details.
read point-by-point responses
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Referee: [Theorem 1 and its application (presumably §3)] The reduction to an upper bound of 25 rests entirely on expressing the Lisbon rules (population threshold + member-state threshold + blocking minority) as the union of exactly two weighted games and then invoking Theorem 1. The manuscript must supply the explicit weight vectors and quotas for these two games so that readers can verify that their union reproduces the Lisbon winning coalitions without requiring additional non-weighted factors or deeper nesting.
Authors: We agree that the explicit weight vectors and quotas for the two weighted games whose union yields the Lisbon winning coalitions should be stated in the manuscript. In the revised version we will add these vectors and quotas (derived from the population and member-state thresholds together with the blocking-minority condition) so that readers can directly verify that the union reproduces the Lisbon rules without additional non-weighted components. revision: yes
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Referee: [Proof of Theorem 1] The derivation of the concrete number 25 in Theorem 1 must be shown to be tight for the specific parameters of the two Lisbon games; if the construction introduces extra factors or replaces weighted games by non-weighted ones for these thresholds, the claimed bound does not follow.
Authors: Theorem 1 supplies a general construction whose dimension bound depends on the number of players and the concrete weights of the two input games. For the specific weights and quotas of the two Lisbon games the construction yields an intersection of exactly 25 weighted games; no non-weighted games are introduced. The revised manuscript will include the short, explicit calculation that substitutes the Lisbon parameters into the general bound of Theorem 1, confirming that the factor 25 is obtained directly and without extraneous terms. revision: yes
Circularity Check
No circularity; new theorem provides independent bound reduction
full rationale
The paper's central step is the introduction and proof of Theorem 1, a new representation converting the union of two weighted games into an intersection of weighted games. This is applied to the Lisbon rules (previously shown by external citation Kurz & Napel 2015 to be the union of two weighted games) to obtain the dimension-25 upper bound. No step reduces by construction to fitted inputs, self-definitions, or load-bearing self-citations; the prior lower bound 7 and upper bound 13368 are external, and the new theorem is self-contained mathematical content. The derivation chain is therefore independent of its target result.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Standard axioms of simple games and weighted voting games, including closure under intersection and union operations.
discussion (0)
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