Shapley-Shubik Index Calculator
Enter voter weights and the quota to compute each voter's Shapley-Shubik power.
Result
How it is calculated
The tool lists every possible order in which the voters could join a coalition: for n voters that is n! orders. In each order it adds up the weights one voter at a time until the running total first reaches the quota (total ≥ quota). The voter whose weight tips the total over is the pivot for that order. A voter's index is pivot count ÷ n!, shown as a percentage, and all indices add up to 100% when the quota can be reached. Weights are entered separated by commas, spaces or semicolons, the quota is a single number above zero. The tool allows up to 9 voters (362,880 orders). If the quota exceeds the total weight nobody is ever pivotal and every voter gets 0.00%. Voters with equal weights always get equal indices.
Worked example
Four voters have weights 4, 3, 2 and 1, and the quota is 6 of 10 total votes. With 4 voters there are 24 orders. Counting who tips the total past 6 gives pivot counts of 10, 6, 6 and 2. Dividing by 24: 41.67%, 25.00%, 25.00% and 8.33%. The weight-1 voter is pivotal only in the two orders where 3 and 2 come first, then 1, then 4, so 2 ÷ 24 = 8.33%. Voter 1 holds 40% of the votes but 41.67% of the power.
Common mistakes
- Entering weights as percentages and the quota as a fraction (0.5). The quota must be in the same units as the weights, for example 6 of 10 votes, not 0.6.
- Setting the quota above the total weight, which makes every voter powerless (0.00% each) instead of giving an error.
- Assuming power is proportional to weight. In the example the voter with weight 1 has 8.33% power, not the 10% share of votes.
- Expecting more than 9 voters. The number of orders explodes factorially, so larger bodies need a different method.
- Confusing this index with the Banzhaf index. Banzhaf counts swing votes in coalitions, Shapley-Shubik counts pivots in orders, and the two give different numbers.
About this calculator
The Shapley-Shubik power index measures how much real influence each voter has in a weighted voting game, not just their nominal vote share. It considers every possible ordering of voters (all n! permutations), finds the pivotal voter in each ordering — the one whose vote pushes a growing coalition from below the quota to meeting it — and gives each voter the fraction of orderings in which they were pivotal.
Political scientists and legal scholars use this index to analyze bodies with weighted votes, such as the EU Council of Ministers or corporate shareholder meetings, showing that a voter's actual power often differs sharply from their raw vote weight — a large bloc can sometimes have zero power if it can never be pivotal, while a small one can be decisive.
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