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class+February+4 - Math 103, Section 11, Friday, February...

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Math 103, Section 11, Friday, February 4, 2011 1. Submitted Homework for Chapter 2 is at Sakai. It is in the Assignments section and is called: Assignment Chapter 2A Please do Exercises 18b,18c,20 and 24 on page 68 and upload to Sakai. Homework is due by Friday, February 11 at midnight The Exercises are at Sakai, Resources, Chapter 2 Exercises from text.doc Homework to be discussed on Tuesday, February 8: Page 68 in the text, Exercises 17a,17b,17c, 19,23 Setup and terminology: A weighted voting system has some number of players. These are, essentially, the voters, i.e. the entities which cast votes. They may be individual people, or states in the electoral college, or nations sitting on the UN Security Council, etc. We will call them "player 1, player 2, player 3. .." and use the shorthand notation P 1 , P 2 , P 3 ,... The players vote on a motion , which can be thought of as a proposal, or a legislative bill, or a UN resolution, etc. Each player votes either "yes" or "no". Each player casts a certain number of votes. This number is the player's " weight " in the system. The minimum number of "yes" votes required to pass a motion is called the quota, q of the weighted voting system. This could be a simple majority of the total number of votes cast (more than 50%), or a 60% majority, or a two thirds majority, etc. The quota could even equal 100% of the total number of votes cast. Notation: [quota: weight of P 1 , weight of P 2 , weight of P 3 , . ..]. Thus [12: 7, 5, 3, 2] describes a weighted voting systems with 4 players whose weights are 7, 5, 3, and 2 respectively, and with a quota of 12 votes needed to pass a motion. A coalition is a set of players who all vote the same way on a specific motion. We usually think of a coalition as a set of players who are all voting "yes", since the quota is defined in terms of how may "yes" votes are needed to pass a motion. The coalition consisting of all the players is called the grand coalition . When all players unanimously vote "yes" for the motion, they are forming the grand coalition. A coalition with enough votes to win is called a winning coalition . A coalition without enough votes to win is called a losing coalition. A player is called a critical player for a specific winning coalition if that player's departure from the coalition would change it from a winning to a losing coalition. In other words, the decision of the player to stay or leave is critical to whether the motion will pass. "Critical" makes sense only in relation to a specific winning coalition.
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A player who, by voting "No", can singlehandedly prevent a motion from passing, is said to have veto power . A player with veto power cannot force a motion to be passed, but the motion cannot pass unless the player with veto power votes “Yes” for the motion. A player with veto power cannot force a motion to pass, but can force a motion to fail.
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This note was uploaded on 01/20/2012 for the course MATH 103 taught by Professor Berkowitz during the Spring '07 term at Rutgers.

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class+February+4 - Math 103, Section 11, Friday, February...

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