# V OTING W EIGHTS SYSTEM Chapter 12 sec 3.. V OTING P OWER Def. The probability that a single vote is decisive Is affected by the rule for aggregating.

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V OTING W EIGHTS SYSTEM Chapter 12 sec 3.

V OTING P OWER Def. The probability that a single vote is decisive Is affected by the rule for aggregating votes into a single outcome

F ACTS Not all members of the UN Security Council have the same voting power. The present council consists of 5 permanent members and 10 nonpermanent members.(Do you know the 5 are?) (Great Britain, France, U.S., China, Russia)

F ACTS According to the rules, the council cannot pass a resolution unless all the permanent members vote “yes” and, in addition, four of the nonpermanent members also vote “yes”.

W EIGHTED VOTING SYSTEM Def. A weighted voting system with n voters is described by a set of numbers. The quota is the number of votes necessary in this system to get a resolution passed. The number that follow, called the weights, are the amount of votes controlled by voter 1, voter 2, etc.

F ORMAT [quota: weight of voter 1, weight of voter 2, …, weight of voter n ]

D ICTATOR Def. If a single voter has the equal or greater weight then the quota.

V ETO P OWER Def. A voter who can, by him- or herself, prevent a motion from passing.

P ROBLEMS Explain each of the following weighted voting system, identify: a) quota, b) dictator, c) who has veto power, and d) the weights of each voter. A) [4:1, 1, 1, 1, 1, 1, 1] A) quota is 4, and a simple majority suffices to pass a resolution. (one person, one vote)

[ 14: 15, 2, 5, 4] Quota is 14, voter 1 is a dictator [ 10: 4, 3, 2, 1] Quota is 10, every voter is needed to pass every resolution-all have the same power

COALITION Def. Any set of voters who vote the same way is called a coalition. The sum of the weights of the voters in a coalition is called the weight of the coalition. If a coalition has a weight that is greater than or equal to the quota, then that coalition is called the winning coalition.

E XAMPLE PROBLEM A town has two large political parties, (R)epublican and (D)emocrat, and on small party, (I)ndependent. Membership is proportional to the size of the parties. We will assume that R has 9 members on the council, D has 8, and I only have 3. Traditionally, each party votes as a single bloc, and resolutions are passed by a simple majority.

List all possible coalitions and their weights, and identify the winning coalitions.

S OLUTION CoalitionWeight {R}9 {D}8 {I}3 {R, D}17Winning {R, I}12Winning {D, I}11Winning {R, D, I}20Winning

C RITICAL Def. A voter in a winning coalition is called critical if it is the case that if he or she were to leave the coalition then the coalition would no longer be winning.

Identify the critical voters in the winning coalitions in the town council.

S OLUTION CoalitionWeightCritical Voters {R}9 {D}8 {I}3 {R, D}17WinningR,D {R, I}12WinningR, I {D, I}11WinningD, I {R, D, I}20Winningnone

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