A vote worth nothing until it was outnumbered

The Treaty of Rome gave the six founding members of the European Economic Community weighted votes in its Council. France, Germany and Italy held four each, Belgium and the Netherlands two each, and Luxembourg one, seventeen in all; a qualified majority needed twelve. The weights were negotiated with some care, and Luxembourg’s single vote was a recognition that it was very much smaller than the others without being nothing.

It was nothing. In every one of the $2^5 = 32$ ways the other five members could line up for or against a proposal, adding Luxembourg’s vote never changed the result. A coalition of the others that reached twelve passed without it, and one that fell short fell short by at least two. For fifteen years, from 1958 until the first enlargement in 1973, Luxembourg’s vote on qualified-majority questions was decorative — and the figure below shows what happened when the Council grew.

In 1973 Britain, Denmark and Ireland joined, the weights were rescaled, and Luxembourg held two votes of fifty-eight — a smaller share than one of seventeen. Its power went from nothing to just under one per cent. In 1981 Greece joined and Luxembourg’s power tripled, to three per cent, while its share of the votes fell again. The larger members’ shares of power fell at each step, as dilution should make them. Luxembourg’s rose, and the only thing that had happened to Luxembourg was that it had been outnumbered.

This is the paradox of new members, named by Steven Brams and Paul Affuso in 1976, and it is one of several ways in which a share of the votes differs from a share of the power that the essay on the subject did not reach. That essay measured one assembly at a time and found that power is monotone within an assembly: more votes never mean less power than somebody else’s fewer. Here the assembly changes, and the reassurance does not survive the change.

How power is being counted #

The measure throughout is the one Lloyd Shapley and Martin Shubik proposed in 1954, which is the average over orders of arrival applied to a vote. Imagine the members declaring their support for a motion one at a time, in some order. At some point the declared total first reaches the quota, and the member whose declaration did it is pivotal for that order. A member’s power is the fraction of all possible orders in which it is pivotal.

With six members there are $6! = 720$ orders, and the 1958 numbers come from checking all of them: France, Germany and Italy are pivotal in 168 orders each, Belgium and the Netherlands in 108 each, Luxembourg in none. The fractions add to one, because every order has exactly one pivotal member, and that is checked in every count in this essay. The 1995 Council, with fifteen members, has more than a trillion orders, and they are counted rather than sampled — not by listing them, which would take a long time, but by grouping them by which members precede the pivotal one, which reduces the work to the $2^{14}$ coalitions of the other members.

Power counted this way is not a forecast of how often a member gets its way, and it is not a model of how real ministers behave. It is a measure of what the voting rule itself makes possible: how often, in an assembly where every alignment is equally likely, a member’s vote is the one that decides. A member with no power by this measure is a member the rule has made irrelevant, whatever happens in the room.

Why one vote of seventeen bought nothing #

Luxembourg’s zero is not a coincidence of the arithmetic, and the reason fits in one line: every other member’s weight was even.

A coalition of France, Germany, Italy, Belgium and the Netherlands in any combination holds an even number of votes: four plus four plus two, or four plus two plus two, always even. Luxembourg, holding one, can be pivotal only by joining a coalition that holds exactly one vote less than the quota, which at a quota of twelve means exactly eleven. Eleven is odd. No coalition of the others can hold it, so Luxembourg is never pivotal, in any order, by any count.

The histogram shows the consequence and the escape. Had the quota been thirteen, Luxembourg would have needed a coalition of exactly twelve, and four coalitions of the others hold twelve — France, Germany and Italy; or two of the large members with both middle ones. Luxembourg would then have been pivotal in a definite fraction of orders. The single vote’s worth depended entirely on the parity of the quota, a property nobody negotiating the treaty is likely to have considered.

That is worth stating as a general caution, because it is the shape of the defect rather than an accident of one treaty. When most weights share a common factor and a few do not, the few are powerless at every quota that the common factor divides. The weights $4, 4, 4, 2, 2$ share a factor of two; at any even quota the odd member is a dummy.

The quota as a switch #

The vote-power essay observed that power is a step function of the quota. On the 1958 weights the steps have a pattern that is visible at once.

Luxembourg’s power switches off and on as the quota steps by one: ten per cent at nine, nothing at ten, ten per cent at eleven, nothing at twelve, nearly twelve per cent at thirteen. The large members’ power moves in the opposite direction at most steps, absorbing what Luxembourg loses. The middle members’ line does something more erratic, and at a quota of fourteen falls to a twentieth while Luxembourg’s would have been a tenth — had fourteen not been even.

A designer reading this chart would draw two conclusions. First, that the choice between twelve and thirteen was not a choice about how demanding the majority should be; the difference in demandingness is one vote in seventeen. It was a choice about whether Luxembourg had any power at all. Second, that the right way to choose a quota is to look at this chart, and the wrong way is to pick a round fraction of the total. Seventy per cent of seventeen is 11.9, and rounding it up gave the founders a quota that silenced one of their six members.

Newcomers who make old members stronger #

The Council’s history shows one direction of the paradox. A census shows how common it is.

More than half of the enlargements — 396 of 756 — leave at least one original member with more power than it had, even though every original member’s share of the votes has fallen. The smallest example the census found is almost a cartoon. Weights $3, 1, 1, 1$ with a quota of four: the large member needs exactly one of the three small ones, and each small member is pivotal only in orders where the large member is already in and no other small member has arrived — one twelfth of the time. Add a fourth small member, raise the quota to five, and each small member’s power rises to a tenth, because the large member now needs two small members and there are more orders in which any given small member supplies the second.

Nothing about the newcomer’s weight predicts the effect cleanly. The bars rise and fall without a trend, because whether an old member gains depends on how the new total interacts with the rounded quota, which is the same parity-like arithmetic that silenced Luxembourg. The phenomenon is closely related to the Alabama paradox of apportionment, where enlarging a legislature costs a state a seat: in both, a rule that looks proportional is a rule about rounding, and rounding does not respect the comparisons that proportion seems to promise.

Two members merge #

The opposite of enlarging is merging. Two members combine into a bloc holding both their votes, and the natural expectation is that the bloc is at least as strong as the two were separately — they can always vote together, and now they must.

The census says the natural expectation is wrong slightly more often than it is right: 3,694 mergers lose power, 3,245 gain it, and 3,441 break even. The first two worked cases show why both directions are possible, using the simplest assembly there is, four equal members.

Under a majority of three out of four, each member has a quarter of the power. Two who merge become a bloc of two votes with two singletons, and now the bloc is pivotal in every order except those in which it arrives first — two thirds of the time, more than the half the pair held together. Merging helped.

Under unanimity, four out of four, each member is pivotal exactly when it arrives last: a quarter each. Two who merge become one member among three, pivotal when it arrives last among three — a third. The pair held a half and the bloc holds a third. Merging hurt, because under unanimity power is the ability to block, and two members who can each block separately have more chances to be the decisive blocker than one who blocks for both.

The third case shows the effect without unanimity. Weights $2, 2, 2, 1$ with a quota of six: the only winning coalitions contain all three members holding two, so the member holding one is a dummy and each of the others holds a third of the power — a veto, shared three ways. Two of them merge into a bloc of four. Now the bloc and the remaining member holding two are both needed and nobody else is, so each holds a half. The two members held two of the three vetoes, two thirds of the power, and the bloc holds one of two, a half. They had more say as two separate obstacles than as one large one.

The lesson for anybody forming a bloc is the one the Luxembourg case taught about quotas. Combining votes is not combining power, because power is not a quantity that sits in the members; it is a property of how their votes can combine to reach the quota, and merging two members changes the combinations available to everybody.

A vote given away #

The strangest of these paradoxes depends on which measure of power is used, and the census separates the two.

The vote-power essay introduced a second measure, John Banzhaf’s: count the coalitions in which a member is a swing, turning a losing coalition into a winning one by joining it, and normalise so that the counts add to one. It is a count over coalitions rather than orders, and the two measures usually agree about the direction of every change.

In the drawn case they do not. A member holding four of nine votes gives one to the member holding two, and its share of the pivotal orders falls, as anyone would expect. Its normalised share of the swings rises, from forty per cent to forty-one. Giving away a vote has made it stronger, by that measure. The mechanism is that the gift destroys several of the other members’ swings — the coalitions in which the small members were decisive — faster than it destroys the donor’s, and a normalised share rises when everybody else’s count falls further.

The census puts a number on how rare this is, and on the difference between the measures. In 39,440 one-vote gifts across every five-member assembly with weights up to six, the donor’s share of pivotal orders never once rose. Its normalised Banzhaf share rose eighteen times. Dan Felsenthal and Moshé Machover, who catalogued paradoxes of this kind in the 1990s, called it the donation paradox, and their work made it a test of a measure rather than a curiosity: a power index that can reward a member for giving its votes away is measuring something other than what designers want from it. The average over orders passed the test everywhere the census looked. That is evidence, not proof, and in larger assemblies the census says nothing.

The surprising thing, stated once #

The four paradoxes in this essay are one fact seen four ways. A member’s power is not a function of its own weight. It is a function of the whole list of weights and the quota, and it depends on them through the arithmetic of which totals can be reached — a question about sums of subsets, with all the irregularity that subset sums have. Luxembourg’s vote was worthless because the others’ weights were all even. It gained power when new members brought odd totals into reach. Mergers lose when they destroy a way of reaching the quota that two separate members could each supply. A gift raises a normalised share when it destroys reachable totals for everybody else.

The original average over orders is forced by four conditions, and none of the four is spare. None of them says anything about what happens when the game itself changes — they compare the members of one game. The paradoxes live entirely in that gap, which is why a rule with an impeccable axiomatic pedigree behaves this way, and why no rule that answers only to those four conditions could avoid it.

Power counted as if every order of arrival were equally likely #

The power counted here assumes every order of arrival is equally likely. That is the model behind the Shapley–Shubik index, and it is a model of ignorance rather than of politics: real ministers form coalitions by affinity, and France and Germany were far more likely to vote together than at random. Under a model of real alignments Luxembourg might have mattered, or mattered less; the figures measure what the rule allows, which is a fact about the treaty and not about the Council’s history.

Qualified majority was not the only rule. Many Council decisions required unanimity, under which every member including Luxembourg could block alone, and the treaty assigned qualified majority only to certain questions. Luxembourg’s zero is a statement about those questions. Nothing in the figures says Luxembourg was without influence in the Community, only that one of its voting rules gave it none.

The censuses are complete for the ranges they state, and silent beyond them. Four or five members with weights up to six or seven is a small world. The donation census found no gift that raised a donor’s share of orders; whether one exists in larger assemblies is not something a census of small ones can say, and the claim here is only the count.

Still open: designing for the paradoxes #

Every paradox here can be found by computation for a given assembly, and that is the practical answer: anybody designing a weighted vote can compute the powers at every quota and every plausible enlargement and look. The Jagiellonian compromise for the enlarged European Union was designed exactly that way, with the quota solved for numerically, and it was for exactly that reason that its proposed quota, 61.5 per cent, was a number nobody would have chosen by rule of thumb.

What is not known is how to design against the paradoxes in general. A rule that never lets a newcomer strengthen an old member, never rewards a merger or a split, and never pays a donor would be valuable, and results going back to Brams and Felsenthal suggest it is impossible for weighted voting with a fixed quota: some paradox can always be produced by some change. Which combinations of paradoxes can be ruled out together, for which classes of weights, and whether a family of quota rules exists that avoids the worst of them in large assemblies, are questions answered for particular cases and not in general. The Council itself moved in 2014 to a double majority — fifty-five per cent of members representing sixty-five per cent of the population — which removes the weights altogether and replaces them with populations; its own power distribution, and its own paradoxes, have been computed member by member and argued over ever since. Counting too many orders to list is now routine; deciding which counts a fair rule should satisfy is not.