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Understanding Weighted Voting Power
A weighted voting rule gives each voter a number of votes. A proposal passes when the coalition reaches the quota. The Banzhaf power index asks a sharper question. It asks how often one voter can turn a losing coalition into a winning coalition. That voter is called critical, decisive, or a swing voter. The score is not just the voter weight. A small voter can matter if many close coalitions need that vote.
Why Efficient Counting Matters
The direct method lists every coalition. That is clear, but it grows fast. Ten voters create one thousand twenty four coalitions. Twenty voters create more than one million. This calculator therefore offers exact enumeration for clear reports and dynamic counting for larger integer games. Dynamic counting checks subset totals instead of printing every coalition. It is faster when coalition lists are not needed.
Formula Used
For voter i, count each coalition S that does not include i. If weight(S) is below the quota, but weight(S) plus weight(i) reaches the quota, voter i has one swing. Raw Banzhaf power is beta_i. The normalized index is beta_i divided by the sum of all beta values. The absolute index is beta_i divided by two raised to n minus one, because each voter has that many possible outside coalitions.
How to Use This Calculator
Enter voter names in one field and matching weights in the next field. Choose a quota rule or enter a custom quota. Select exact enumeration when you want coalition names, minimal winners, and detailed checking. Select dynamic programming when you have many integer weights and want efficient swing counts. Set the precision for percentages. Then press calculate.
Reading the Results
The table shows raw swings, normalized power, absolute power, and each voter’s swing range. A zero score marks a dummy voter. A dictator can pass alone and can block every other group. A veto voter appears in every winning coalition. Minimal winning coalitions are useful because every listed member is critical. Compare these values with weight share. Large gaps can reveal hidden influence, weak representation, or unfair voting design.
Useful Efficient Methods
Enumeration is best for teaching and audits. It shows every winning coalition and makes each swing easy to inspect. It also helps spot ties, veto power, and dummy voters. Dynamic programming is better for routine conversion work. It builds counts by possible vote totals. The method reuses partial totals, so it avoids repeated coalition scanning. For very large councils, sampling can estimate power, but estimates should show error ranges. This page focuses on exact results. Use smaller coalition limits when reports become long. Use integer weights when you want the dynamic option. Keep the quota realistic. If the quota is too high, nobody can win. If it is too low, almost everyone may look powerful. Always compare index share with actual vote share carefully.
Frequently Asked Questions
What does this calculator measure?
It measures voting influence in a weighted voting system. It counts when each voter can change a losing coalition into a winning one.
Is weight share the same as power share?
No. Weight share is the vote weight percentage. Power share depends on swing positions near the quota, so it can be very different.
What is a critical voter?
A critical voter is a member of a winning coalition whose removal would make that coalition lose. Each such case adds one raw swing.
What is normalized Banzhaf power?
It is one voter’s raw swing count divided by all raw swing counts. The normalized values add to one, or one hundred percent.
What is absolute Banzhaf power?
It divides a voter’s raw swings by all possible coalitions formed by the other voters. It shows the direct swing probability under equal coalition likelihood.
When should I use exact enumeration?
Use exact enumeration when the voter count is small and you want listed winning coalitions, critical voters, and minimal winners.
When should I use dynamic programming?
Use dynamic programming for larger games with integer weights. It calculates swing counts by subset totals instead of listing every coalition.
What is a dummy voter?
A dummy voter has zero raw swings. The voter may have weight, but that weight never changes a losing coalition into a winning one.
What is a veto voter?
A veto voter must appear in every winning coalition. Without that voter, the remaining total weight cannot reach the quota.
Can decimal weights be used?
Yes, decimal weights can be used with exact enumeration. The dynamic method needs integer weights and an integer quota.
Why does the quota matter so much?
The quota defines which coalitions win. A changed quota can create new swing positions, remove old ones, and reshape every voter’s power.