01:640:103 Chap.3 Weighted Voting and the Banzhaf Power Index
Weighted Voting and the Banzhaf Power Index
Weight is not the same as power
In a weighted voting system each player casts a number of votes, its weight, all yes or all no, and a motion passes when the yes votes reach the quota. The system is written as the quota followed by the weights in decreasing order. The quota has to exceed half the total weight, or two separate groups could pass contradictory motions.
Handing out votes in proportion to population feels fair, but a committee weighted 5, 4 and 3 with a quota of 7 gives all three members identical power: any two of them can pass anything.
Counting critical players
A coalition is a group voting yes; it is winning when its weight reaches the quota. A player is critical in a winning coalition when leaving it would drop the total below the quota.
The Banzhaf power index of a player is that player's critical count as a fraction of every critical appearance across all winning coalitions.
The indices must add to 1. In the guide's funding board with quota 9 and weights 6, 4, 3 and 2, the players are critical 5, 3, 3 and 1 times, so the two middle members hold equal power despite unequal votes.
Three special roles
A dictator can pass a motion alone, which happens exactly when its weight reaches the quota.
A player with veto power can block any motion by voting no, because all the others together fall short of the quota. A dummy is never critical and has no power at all.
Raising or lowering the quota while keeping the weights fixed can create or remove each of these roles, which is why the chapter treats the quota as seriously as the weights.
How the quota reshapes power
Keeping the weights at 6, 4, 3 and 2 and moving the quota from 8 to 15 changes the Banzhaf picture repeatedly. At quota 8, a simple majority of the 15 votes, the largest player holds half of the power.
At quota 10 the other three together fall one vote short, so the largest player gains a veto. At quota 13 the smallest player becomes a dummy, and at quotas 14 and 15 every player has a veto and all four share power equally. The lesson for exams is to recheck every role whenever a question changes the quota, even by one vote.
What this chapter covers
- 01
Players, weights and the quota
- 02
Coalitions and winning coalitions
- 03
Critical players and the Banzhaf index
- 04
Dictators, veto power and dummies
- 05
Power from a list of winning coalitions
Worked example · free
A small system with a dummy
- 2Winning coalitions: {A, B} with weight 6 and {A, B, C} with weight 7; no other coalition reaches 6.
- 3In {A, B} both players are critical. In {A, B, C}, removing A leaves 3 and removing B leaves 5, so both are critical; removing C leaves 6, so C is not.
- 1A and B are each critical twice and C never, so T = 4.
Key terms
- Quota
- The quota is the minimum total weight of yes votes needed for a motion to pass.
- Critical player
- A critical player is a member of a winning coalition whose departure would make the coalition losing.
- Banzhaf power index
- The Banzhaf power index is a player's count of critical appearances divided by the total count for all players.
- Veto power
- Veto power is the ability of one player to block every motion by voting no.
- Dummy
- A dummy has no critical appearance in any winning coalition and so holds no power.
- Winning coalition
- A winning coalition is a group of players whose combined weight reaches the quota.
Weighted Voting and the Banzhaf Power Index FAQ
Why must the quota be more than half of the total weight?
If it were not, two groups with no members in common could each reach the quota, so contradictory motions could both pass on the same day. Setting the quota above half the total rules that out.
Can I find Banzhaf power without knowing the weights?
Yes, provided the list of winning coalitions is complete. For each coalition, remove one player at a time and see whether the smaller group is still on the list; if it is not, that player was critical.
What is the difference between a dictator and a player with veto power?
A dictator can pass a motion alone because its weight reaches the quota. A veto player can only block motions, because the others cannot reach the quota without it. Every dictator has veto power, but most veto players are not dictators.
Exam move
List coalitions systematically by size, from the grand coalition down, and mark critical players as you go. Finish every problem by checking that the indices sum to one.
Working through Weighted Voting and the Banzhaf Power Index in 01:640:103? Sia is AskSia’s AI Mathematics tutor — ask any 01:640:103 Weighted Voting and the Banzhaf Power Index question and get a clear, step-by-step explanation grounded in how 01:640:103 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
