A coalition ledger, EEC Council, 1958 to 1972
The Vote That Never Counted Under the Commission-Proposal Rule
In one branch of the original EEC Council's qualified-majority rule, Luxembourg held one of 17 weighted votes, yet that vote could not change a single outcome: every other weight was even and the threshold was 12. Enumerate all 64 coalitions, inspect Luxembourg's 32 cases, then rebuild the voting body and watch power refuse to follow vote share. The page also exposes the treaty condition that gives Luxembourg one pivotal coalition, and keeps formal power separate from the Council's consensus-seeking practice.
One vote, no swing
A vote has formal power here only when it changes a losing coalition into a winning one. The machine below enumerates the entire voting body. Luxembourg starts in the Commission-proposal branch of Article 148. Open its ledger and inspect every coalition it could join.
Computing the Commission-proposal branch.
Enumerating coalitions.
| Member | Votes | Critical coalitions | Banzhaf | Shapley-Shubik |
|---|
Computing.
| Other members | Their weight | Without LUX | With LUX | Changes? |
|---|
The quota-only arithmetic has a one-cell trap. Luxembourg can matter only if the other members reach exactly ..., one short of the threshold. But their weights are all even. Their coalition totals are even too. The missing total is unreachable.
The title needs a boundary. Original Article 148 had two qualified-majority branches. Acts on a Commission proposal needed the weighted quota. Other qualified-majority acts needed that quota and votes from at least four members. Press the second rule above. Luxembourg then has one critical coalition because France, Germany, and Italy already supply the weight, while Luxembourg supplies the fourth state.
This is why the page says the vote never counted in one branch of the rule. Calling Luxembourg formally powerless across every EEC Council vote would be false.
Weight is not power
The objection is sensible: surely this is just a tiny-system oddity. Build a different weighted body. Drag a weight, move the quota, add or remove a member, and the machine will enumerate every coalition again. Vote share and power can move in different directions.
| Player | Votes | Vote share | Swings | Banzhaf | Shapley-Shubik |
|---|
Computing.
Player A's power as A's weight changes
Same active members, same other weights, same quota. Each column is recomputed. Press a column to set A's weight.
computing
Load the new-member preset, then admit D. C's share of all votes falls, while both displayed power indices rise.
computing
In the named preset, giving A one more vote raises A's raw swing count but lowers A's normalized Banzhaf share. Shapley-Shubik moves the other way.
Two measures, not two rankings
Banzhaf asks which players are critical across equally weighted coalitions. The normalized value divides each player's critical count by the total critical count. Shapley-Shubik asks who becomes pivotal as players arrive in every possible order. Those are different probability stories, and the builder shows different magnitudes.
There is also a useful limit. In a single weighted-threshold game, these two measures cannot reverse the player order. If player A has at least player B's weight, every same-sized coalition that makes B critical also makes A critical. Banzhaf adds those cases equally. Shapley-Shubik adds them with a positive weight based on coalition size. The ordering survives both sums.
The brief for this piece proposed showing the two indices rank the same weighted players differently. The computation does not support that claim, and the dominance argument above rules it out for this class of games. Different magnitudes are real. A reversed ranking here is not.
Formal power is not a record of practice
These indices describe a rule before anyone's preferences, bargaining, alliances, agenda control, or reluctance to call a vote are known. They do not count how often Luxembourg won, lost, spoke, negotiated, or changed a proposal.
That distinction matters especially in this period. The 1965 empty-chair crisis ended with the January 1966 Luxembourg Compromise. When very important interests were invoked, Council members would seek a solution acceptable to all. This strengthened consensus-seeking in practice. It did not rewrite the arithmetic in Article 148, and it does not license the claim that every Council matter from 1958 through 1972 legally required unanimity.
The check
Every power number on this page is produced by enumerating the displayed body in the browser. No result row contains a prewritten index.
- Verified source values: the original treaty weights, both Article 148 qualified-majority conditions, the 1 January 1958 effective date, the 1966 compromise, and the first enlargement on 1 January 1973.
- Derived here: all coalition outcomes, critical counts, normalized Banzhaf shares, Shapley-Shubik shares, the parity obstruction, and both paradox examples.
- Free modelling choices: binary yes or no votes, the winning rule, coalition symmetry for Banzhaf, ordering symmetry for Shapley-Shubik, and normalization of Banzhaf counts to a total of one.
- Not measured: member preferences, issue frequency, proposal power, bargaining, stable alliances, abstention strategy, consensus pressure, and diplomatic influence. The indices are a priori models, not observations.
Offline reproduction: node research/the-vote-that-never-counted/verify-the-vote-that-never-counted.mjs. It independently rebuilds every featured body and fails on any drift.
Definitions and normalization
- Critical or swing: a player belongs to a winning coalition, and removing that player makes the coalition lose.
- Normalized Banzhaf: a player's swing count divided by the swing counts of all players. Some literature instead reports the absolute Penrose-Banzhaf probability, swing count divided by the possible coalitions of the other players. The page labels its normalized choice.
- Shapley-Shubik: the fraction of all player orderings in which that player is first to turn the accumulated predecessors into a winning coalition. The code obtains the same value without listing every permutation, weighting each critical coalition by its factorial count.
- Rule boundary: the main historic table defaults to the Commission-proposal quota branch. The adjacent control separately computes Article 148's additional four-member condition for other qualified-majority acts.
Sources
- Treaty establishing the European Economic Community, original Article 148, authentic French text on EUR-Lex, signed 25 March 1957. This is the source for the weights and both qualified-majority branches.
- European Parliament, Developments up to the Single European Act, for the empty-chair crisis and Luxembourg Compromise.
- Council treaty record for the 1972 Accession Treaty, effective 1 January 1973.
- L. S. Shapley and Martin Shubik, "A Method for Evaluating the Distribution of Power in a Committee System," American Political Science Review 48, no. 3 (1954), 787-792, DOI 10.2307/1951053.
- John F. Banzhaf III, "Weighted Voting Doesn't Work: A Mathematical Analysis," Rutgers Law Review 19, no. 2 (1965), 317-343.