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V OTING W EIGHTS SYSTEM Chapter 12 sec 3.

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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

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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)

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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”.

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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.

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

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

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

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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)

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[ 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

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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.

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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.

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

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

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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.

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

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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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