Banzhaf Power Index, explained.
The Banzhaf index measures voting power by counting coalitions whose losing-to-winning status changes when a particular voter joins.
Why it happens
Three voters have weights 4,3,2 and an adjustable quota from one to nine. Enumerate all coalitions excluding each voter, count critical joins and normalise the swing counts. A separate coalition selector shows whether its selected weight passes the current quota.
The Banzhaf index measures voting power by counting coalitions whose losing-to-winning status changes when a particular voter joins.
Read the result
Compare vote-weight share with swing-power share. The index counts decisive joins across all coalitions, not how frequently real members will negotiate together.
A worked example
A small board with unequal vote weights
Set quota six for weights four, three and two, then select coalitions to inspect which pass.
Count critical joins: the four-vote member has three swings, while each other member has one.
Normalise to sixty, twenty and twenty percent. Those shares differ from four-ninths, three-ninths and two-ninths of vote weight.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
Three voters have weights 4,3,2 and an adjustable quota from one to nine. Enumerate all coalitions excluding each voter, count critical joins and normalise the swing counts. A separate coalition selector shows whether its selected weight passes the current quota.
Where this idea is useful
A practical use
A board member's share of votes need not equal their share of decisive opportunities.
A common misconception
“Twice as many votes must mean twice as much influence.”
The quota and possible coalitions determine pivotal opportunities. Weight alone does not specify the index, and the index does not describe every practical source of influence.
What this explanation leaves out
- Every coalition is treated equally in this index. Real alliances, likelihoods, agenda control and strategic preferences can change practical influence. Normalised swing power is not a prediction of actual political outcomes.
What happens if the quota changes?
The set of winning coalitions changes, so critical joins and normalised power can change even with identical vote weights.
Which quota or coalition assumption matters more than the headline share of votes in a decision you follow?
Associated thinkers
Associations marked provisional are awaiting source review.
Further reading
Explore the original research or the teaching reference behind this experiment.