Handful Apps / Articles / Voting weight is not voting power
Give one bloc twice the votes of another and you have not necessarily given it any more influence. Sometimes you have given it exactly none. The gap between how many votes a party holds and how much its votes are worth is measurable, and the arithmetic is simple enough to do on paper.
Four blocs share ten votes. Decisions need more than half, so six votes carry a motion.
| Bloc | Votes | Share of votes |
|---|---|---|
| S | 5 | 50% |
| D | 2 | 20% |
| E | 2 | 20% |
| M | 1 | 10% |
Looking at that table, D and E each appear twice as strong as M. Hold onto that thought.
John Banzhaf's idea, published in 1965 and originally aimed at a county board in New York whose weighted voting he argued was unconstitutional, is to stop counting votes and start counting moments when a bloc's votes decide the outcome.
Take every group of blocs that could form. Keep the ones that win, meaning they hold at least six votes between them. Then, within each winning group, ask of each member: if this bloc walked out, would the group still win? If the answer is no, that bloc is critical to that group. Count those moments.
Here are the winning groups, all seven of them:
| Group | Votes | Critical members |
|---|---|---|
| S + D | 7 | both |
| S + E | 7 | both |
| S + M | 6 | both |
| S + D + E | 9 | S only |
| S + D + M | 8 | S only |
| S + E + M | 8 | S only |
| S + D + E + M | 10 | S only |
Two things jump out of that table before any arithmetic. First, S appears in every winning group, because the other three together hold only five votes and five is not enough. Second, in the larger groups S is the only member who matters, since removing D, E or M still leaves six or more.
Now count the critical moments. S is critical seven times, once in each winning group. D is critical once, in S + D. E is critical once, in S + E. M is critical once, in S + M. That is ten critical moments in total, so the shares are:
| Bloc | Share of votes | Share of power |
|---|---|---|
| S | 50% | 70% |
| D | 20% | 10% |
| E | 20% | 10% |
| M | 10% | 10% |
D and E hold twice as many votes as M and exactly as much power. The second vote each of them was given buys nothing whatsoever. Whatever D can do, M can do, and vice versa: each of them can complete a majority with S, and neither can do anything without S.
Meanwhile S, holding half the votes, holds seventy percent of the power. It is what the literature calls a veto player. No motion passes without it, and that fact is worth more than the twenty percentage points of weight it appears to be missing.
Notice what would happen if you were D, negotiating your share of a coalition. Arguing for one more vote, taking you from two to three, would change nothing at all: with three votes you would still need S, and S would still need only one partner. To gain anything, you would need to reach six, or reduce S below six. The useful question in that negotiation is never how many votes you hold, it is which groups you complete.
Anywhere votes are weighted rather than one per head. Company boards with shareholdings. Coalition governments allocating ministries by seat count. Federations weighting members by population. Committees where a chair holds a casting vote. In all of them, the seat table is published and the power table is not, and it is the second one that predicts who has to be in the room.
Two related traps are worth naming. A dummy is a bloc that is never critical to any winning group. It holds votes and no power at all, and can be safely ignored by everyone else, which is a strange thing to discover about yourself after an election. The opposite case is the one above, where a single bloc sits in every winning combination.
Both are properties of the arithmetic between the weights and the threshold, not of anyone's politics. Change the threshold from six to seven and every number in this article changes.
The Banzhaf index counts combinations, and in doing so it treats every group of blocs as equally likely to form. Real coalitions are not like that: parties have positions, and some partnerships are unthinkable regardless of the arithmetic. The index measures what the voting rule permits, not what politics will deliver.
It also says nothing about what gets proposed in the first place. A bloc with a large power share and no say over the agenda may matter less in practice than a smaller one that decides what is voted on, which is a separate mechanism and worth its own article.
Every figure above is reproducible. Agenda Simulator computes Banzhaf shares for whatever you put on the map, and it produces exactly the numbers in this article for these four blocs: 0.700, 0.100, 0.100 and 0.100. It also flags veto players and dummies as they appear, so you can watch a bloc become powerless by changing the threshold rather than its votes.
The app is free on the Mac App Store. The user guide covers the power table alongside everything else on screen.