A share of the votes is not a share of the power

A weighted vote looks like it needs no analysis. Members hold votes, a motion needs a quota, and a member’s influence is the share of the votes they hold — which is a division and takes a second.

It is not, and the failure is not a subtlety at the margin. There are assemblies in which a member holding a ninth of the votes has exactly as much power as one holding four ninths, and assemblies in which a member with a fifth of the votes has none at all.

What a weighted vote actually is #

Write down the game rather than the weights. A coalition is winning if its total weight reaches the quota, and losing otherwise; give a winning coalition the value one and a losing one zero.

That is a cooperative game — a number for every subset — and the rule from the bottom of this ladder applies to it unchanged. A member’s share is their average marginal contribution over every order of arrival, and a marginal contribution here is zero or one: it is one exactly when the coalition was losing before the member arrived and winning after.

So a member’s power is the fraction of orders in which they are the one who carries the coalition over the quota. That is the Shapley–Shubik index, published by Shapley and Shubik in 1954, and it is the general rule specialised to a game whose values are all zero or one.

The specialisation is what makes it computable and interpretable. There is no arithmetic about how much anybody adds; there is a count of the orders in which somebody is pivotal, divided by $n!$.

Four, four, one #

Take the first row of the figure. Weights $4, 4, 1$ and a quota of $5$.

Which coalitions win? Not any single member — the largest holds four, below five. Every pair: $4 + 4 = 8$, $4 + 1 = 5$, $4 + 1 = 5$. And all three. So the winning coalitions are exactly the pairs and the whole, which is the same game as three members with one vote each and a quota of two.

Every member is pivotal in exactly the orders where they arrive second, which is a third of the orders for each. The power is $1/3, 1/3, 1/3$ against weights of $4/9, 4/9, 1/9$.

The member with a ninth of the votes has a third of the power. Nothing has gone wrong; the weights simply do not determine the game, and two very different weight assignments give the same game.

A weighted vote is a representation of a game, and many weightings represent one game. Once that is seen, the surprise dissolves: the power is a property of the game, and the weights are one description of it among many.

The dummy #

The sharpest case is a member who is never pivotal at all. Weights $3, 1, 1$ with a quota of $3$: the first member wins alone, and no coalition of the other two reaches the quota. So the first is a dictator and the other two are dummies — their votes make no difference to any outcome, ever.

A dummy holds a fifth of the votes and none of the power, and the null-player condition is exactly the statement that the rule gives such a member zero. Both indices in the figure do, and the figure asserts it: a member who can never swing a vote gets nothing under both counts, and the assertion is checked rather than assumed.

Dummies are not an artefact of small examples. They occur in real weighted systems whenever a quota sits above the total of the small members, and they are one of the standard things a power analysis is done to find — a member whose weight was negotiated at length and buys nothing.

Moving the quota #

The weights are only half the specification and the quota is the other half, and changing it alone changes everything.

Take weights $4, 3, 2$ and move the quota. At $5$, every pair wins and no single member does, so all three are symmetric and each has a third of the power. At $6$, the pair $3+2$ falls short, so the largest member is in every winning coalition and holds two thirds. At $9$, only the whole assembly wins, so every member is pivotal exactly when they arrive last — a third each again, by unanimity rather than by symmetry of the weights.

Power is not a continuous function of the quota. It is a step function, jumping wherever some coalition’s total is crossed, and between jumps nothing changes at all. So a quota chosen to be “about two thirds” may sit anywhere in a wide plateau, and moving it by one vote may reorganise the whole assembly.

That has a practical consequence worth stating. Anybody negotiating a weighted system negotiates weights, because weights are what look like the thing being divided — and the quota, usually settled afterwards by a rule of thumb, does at least as much work.

More votes never means less power #

One reassurance the counting does give: the indices are monotone in the weight. A member holding at least as many votes as another is pivotal in at least as many orders, because any coalition the smaller member swings is one the larger swings too.

So the failures above are all of one kind. A member can hold more votes and the same power — the four-four-one assembly, where a ninth buys as much as four ninths — and a member can hold votes and no power at all. What cannot happen is a reversal.

That is worth having because it bounds how badly the intuition fails. The naive reading of a weight is not merely wrong; it is an over-reading of something that does carry a correct ordering. Weight determines the order of the power shares and says nothing whatever about their sizes, and every example in this essay is a case where the sizes are the surprise.

Counting coalitions instead of orders #

There is a second index, and comparing it with the first is the most useful thing this rung does.

The Banzhaf count asks: in how many coalitions is this member a swing — how many subsets of the others turn from losing to winning when this member joins? That is a count over $2^{n-1}$ coalitions rather than over $n!$ orders, and the raw counts are normalised to add to one.

The two indices are different quantities computed over different objects, and they usually disagree in the second decimal place while agreeing about the qualitative structure. The figure computes them separately and prints both.

Where they always agree is on dummies: a member who swings no coalition is pivotal in no order, and conversely, so both give zero to exactly the same members. Where they can disagree is on how much power the non-dummies have relative to one another.

The difference has an interpretation. Shapley–Shubik weights each coalition by the number of orders producing it, which favours coalitions of middling size; Banzhaf weights every coalition equally. Which is right depends on a model of how coalitions form, and there is no fact of the matter — which is why the two coexist rather than one having replaced the other.

A different question from the one voting rules ask #

This site’s applied field already contains a run of essays about voting, and they are asking something else. Keeping the two apart is worth a section because the vocabulary overlaps completely.

The rules essays take a profile of ranked ballots and ask which candidate a rule selects — a question about preferences and about aggregating them. The famous results there are impossibilities: no rule has all four of Arrow’s conditions, and no rule is safe from a strategic lie.

This essay takes no preferences at all. There are no candidates, no rankings, and no rule to select a winner. What there is is a quota and a set of weights, and the question is how much each member’s vote matters to whether a motion passes — which is a question about the structure of the winning coalitions and about nothing else.

The two connect at exactly one point, and it is worth naming. A member’s power index is the probability of being pivotal under the assumption that every order of arrival is equally likely — which is a model of how coalitions form, and the only place any assumption about preferences enters. Change the model and the index changes; the Banzhaf count is the same calculation under a different one.

The measure is only as good as the model of coalition formation behind it, and both standard indices use models chosen for symmetry rather than for realism.

What the analysis is used for #

Three uses, and the third is the one that changed practice.

Detecting dummies before agreeing to a rule. A member negotiating a weight wants to know whether the weight buys anything, and the answer is not readable from the number.

Designing a quota. For a fixed set of weights, the power distribution changes as the quota moves, sometimes discontinuously — a small change in the quota can turn a dummy into a pivotal member or collapse three members into one bloc. Choosing a quota with a target power distribution in mind is a search problem, and the search is over the same finite set of coalitions.

Checking that a system does what it was designed to. The best-known instance is the Nassau County board of supervisors in the United States, where weights were assigned in proportion to population and a Banzhaf analysis in 1965 showed that three of the six districts had zero power — their votes could never change an outcome. The analysis was used in litigation and the system was changed. It is the reason the index is often called the Banzhaf index rather than the Penrose index, though Penrose had it in 1946.

Why the square root, and where it comes from #

There is a design principle that follows from the same counting and is worth stating because it is unintuitive and correct.

If members represent groups of different sizes and the aim is that every individual in every group has equal influence on the final outcome, the members’ powers should be proportional to the square roots of the group sizes — not to the sizes.

The reason is that a single individual’s chance of being decisive within a group of size $m$ falls like $1/\sqrt m$, by the same square-root law that governs how far from the average a sum of many small contributions strays — the same exponent a bell curve assembled out of coin flips produces. Multiplying that by the member’s power in the assembly and equalising gives the square-root rule.

It is Penrose’s, from 1946, and it is the clearest case of this analysis producing a design recommendation rather than a diagnosis. And it is about powers rather than weights, so applying it means solving the inverse problem — finding weights whose power indices are proportional to given numbers — which has no closed form and is done by search.

A member who is worth nothing, drawn #

The dummy is the case worth seeing twice, because it is the one whose existence is genuinely hard to believe from the weights alone.

The second row is the interesting one, because there is no dictator. Weights $5, 3, 1$ with a quota of $6$: the winning coalitions are $\{A,B\}$ at eight, $\{A,C\}$ at six, and the whole. The smallest member is pivotal only when it completes $\{A,C\}$ — but that coalition is winning at exactly the quota, so the member is a swing there and is not a dummy after all.

Change the quota to $8$ and it becomes one: only $\{A,B\}$ and the whole win, so the smallest member is never the one that carries anything. A single vote of quota, and a member’s power goes from a positive share to nothing.

That is the sharpest form of the essay’s point. The member’s weight has not changed, the other members’ weights have not changed, and what the member is worth has gone from something to nothing — because power is a property of the winning coalitions and the quota is what decides which coalitions win.

What the picture cannot show #

The figure draws bars for three-member assemblies, where every coalition can be listed and the indices are exact. Real assemblies have dozens of members, $n!$ is astronomical, and both indices have to be estimated — which is the next rung and is a different problem.

It cannot show the quota’s effect, which is one of the most useful things the analysis says. Power as a function of the quota is a step function with jumps at the points where some coalition’s total is crossed, and drawing it would need an axis the figure does not have.

And it cannot show what a power index leaves out. Both indices assume every coalition is equally likely to form, which is false in every real assembly — members have affinities, and a coalition of natural opponents is not as available as one of natural allies. Indices weighted by coalition likelihood exist and need data, which puts them outside what this field measures.

Where the ladder goes next #

Above: what to do when the orders cannot be listed, which is where every real application of this analysis starts. And then the rule dividing a cost rather than a gain, where the same arithmetic answers a question with the sign reversed.

One debt. The inverse problem — find weights realising a target power distribution — is named here and not treated. It is the question anybody designing a voting system actually has, it has no closed-form answer, and the search over integer weights is a small and genuinely interesting exhaustion.

What the weights were not #

A weighted vote is a description of a game, and the power belongs to the game.

Two weightings with the same winning coalitions have the same power distribution however different the numbers look, and a member’s weight tells nobody anything until the winning coalitions are worked out. The counting that does work it out is the same average over orders that answers the very different question of what a partner is owed — one rule, two questions, and in this one the values being averaged are only ever nought or one.