11 Weighted Voting Systems

Review Vocabulary

Banzhaf power index In a voting system, the number of winning coalitions in which voter V is a critical voter. (p. 475)

Coalition A set of participants in a voting system that might vote in favor of a measure. The empty coalition is allowed, and it represents a situation when the voters unanimously oppose a motion. (p. 474)

Combinations and Combinations symbol The combinations symbol $_{n} C_{k}$ stands for the number of ways to choose $k$ members from a set of $n$ elements—that is, the number of combinations of $n$ objects, taken $k$ at a time. (p. 480)

Critical voter Let W be a winning coalition, and let V be a voter who belongs to W. Then V is a critical voter in W if the coalition consisting of all the voters in Wexcept V is a losing coalition. (p. 475)

Dictator A voter $D$ is a dictator if a motion will pass if and only if $D$ is in favor and the votes of the other members make no difference. (p. 463)

Dummy voter A participant in a voting system who never has an opportunity to cast a deciding vote is called a dummy voter. (p. 464)

Equivalent voting systems Two voting systems are equivalent if there is a way for all the voters of the first system to exchange places with the voters of the second system and preserve all winning coalitions. (p. 489)

Extra votes The number of votes by which a winning coalition’s total weight exceeds the quota. (p. 476)

Factorial The number of permutations of a set of $n$ voters is called the factorial of $n$ and is denoted $n!$ . (p. 470)

Formulas involving combinations

Basic combinations formula: $_{n} C_{k} = \frac{n!}{(n - k)! \times k!}$ (p. 481)

Symmetry formula: $_{n} C_{k} =_{n} C_{n - k}$ (p. 482)

Sum formula: $_{n} C_{(k - 1)} +_{n} C_{k} =_{(n + 1)} C_{k}$ (p. 486)

Approximation formula: $_{2 n} C_{n} = \frac{2^{2 n}}{R n π}$ , approximately (p. 486)

Grand coalition The coalition that includes every voter. (p. 477)

Losing coalition A set of participants in a weighted voting system whose combined voting weight is less than the quota; or, in an unweighted voting system, a set of voters that does not contain a minimal winning coalition. (p. 474)

Minimal winning coalition A winning coalition is minimal if every voter who belongs to the coalition is critical in that coalition. (p. 478)

Pivotal voter The first voter in a voting permutation who, when joined by those coming before him or her, would have enough voting weight to win. Each voting permutation has exactly one pivotal voter. (p. 467)

Quota In a weighted voting system, the required number of votes, or total voting weight, necessary to pass a measure. (p. 462)

Shapley–Shubik power index Among $n$ voters, the number of voting permutations in which a voter is pivotal, divided by $n!$ (the factorial of $n$ ). (p. 469)

Veto power A voter whose vote is necessary to pass any motion is said to have veto power. (p. 464)

Voting permutation An ordered list of all the voters in a voting system. (p. 466)

Page 493

Weighted voting system Specifies the voting weights $w_{1}, w_{2}, \dots, w_{n}$ of the participants, and the quota, $q$ . A motion will pass if the sum of the weights of the voters in favor is at least equal to the quota. The notation $[q : w_{1}, w_{2}, \dots, w_{n}]$ is used to denote a system in which there are $n$ voters, with voting weights $w_{1}, w_{2}, \dots, w_{n}$ ; and the quota is $q$ . (p. 462)

Winning coalition A set of participants in a weighted voting system whose combined voting weight is greater than or equal to the quota; or, in an unweighted voting system, a set of voters that contains a minimal winning coalition. (p. 474)