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Weighted Voting When we try to make collective decisions, it is only natural to consider how things are done in society. We are familiar with voting for class president – one for per person, winner is one with most votes. The electoral college seeks to give more power to states having more population We want to know who has the most power – as that could influence whose mind we try to change or whether the system is fair. 1

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Weighted Voting We are trying to decide whether or not a measure passes. The coalition is the group of people that can get the measure passed. Not everyone has the same “clout”. Modeled as some having multiple votes. 2

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Notation A weighted voting system is characterized by three things — the players, the weights and the quota. The voters are the players (P 1, P 2,..., P N ). N denotes the total number of players. A player's weight (w) is the number of votes he controls. The quota (q) is the minimum number of votes required to pass a motion. [q:w 1,w 2 …w n ]. Normally we require that as we can say more interesting things that way. 3

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Power A player's power is defined as that player's ability to influence decisions. The power of a coalition is not simply determined by its size. Consider the voting system [6: 5, 3, 2]. Notice that a motion can only be passed with the support of P 1. In this situation, P 1 has veto power. A player is said to have veto power if a motion cannot pass without the support of that player. This does not mean a motion is guaranteed to pass with the support of that player – as player 1 doesn’t have enough votes by himself. Who has the most power? How is power divided between the players with 3 and 2? 4

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5 Weighted Voting Systems – Terms i. Coalition: any subset of a group of voters that bands together to either support a measure. ii. Winning/Losing Coalition: a coalition that has enough votes to pass a measure is a winning coalition, otherwise it is a losing coalition. iii. Dummy: a voter in a winning coalition whose vote isn’t needed to pass the measure. iv. Voters Weight: the number of votes each voter has. v. Quota: the number of votes, q, necessary to pass a measure.

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6 Weighted Voting Systems - Terms vi. Notation for voting system: where q is the quota, are the individual weights of the voters, and n is the number of voters. vii. Requirements: 1. 2. as no point is having q larger viii. Changing q affects the way power is distributed. ix. Blocking Coalition: subset of voters opposing a motion with enough votes to defeat it. Any coalition with weight. as otherwise definition of dictator is problematic

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7 Weighted Voting Systems – Terms x. Dictator: voter whose voting weight meets or exceeds the quota for passing a measure. All other voters are dummies. xi. Veto Power: a voter who has enough votes to block a measure is said to have veto power. A voter with weight. A dictator automatically has veto power. xii.Critical Voter: in any winning coalition, he is the voter whose votes are essential to win.

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Power Now let us look at the weighted voting system [10: 11, 6, 3]. With 11 votes, P 1 is called a dictator. A player is typically considered a dictator if his weight is equal to or greater than the quota. The difference between a dictator and a player with veto power is that a motion is guaranteed to pass if the dictator votes in favor of it. The dictator has veto power. The measure passes if and only if he votes for it. Since the quota must be more than half the total, a dictator always has veto power. A dummy is any player, regardless of his weight, who has no say in the outcome of the election. A player without any say in the outcome is a player without power. Dummies always appear in weighted voting systems that have a dictator (provided the quota is more than half total) but also occur in other weighted voting systems 8

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Power Consider the voting system [8: 5, 3, 2]. Which are dictators? have veto power? are dummies? 5 and 3 have veto power. 2 is a dummy Consider the voting system [8: 9, 3, 2]. Which are dictators? have veto power? are dummies? Consider the voting system [20:10,10,9]. Which are dictators? have veto power? are dummies? Consider the voting system [7:4,2,1]. Which are dictators? have veto power? are dummies? 9

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Banzhaf power index (sometimes called Penrose-Banzhaf index) Designed to quantify the power a voter has defined by the probability of changing an outcome of a vote To calculate the power of a voter using the Banzhaf index, list all the winning coalitions, then count the critical voters. A critical voter is a voter who, if he changed his vote from yes to no, would cause the measure to fail. A voter's power is measured as the fraction of all swing votes that he could cast Warning: in our electoral college of 50 states, there are 51,476,301,254,318 winning coalitions! 10

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An example Game Theory and Strategy by Phillip D. Straffin: [6; 4, 3, 2, 1] The winning groups, with underlined critical voters, are as follows: AB, AC, ABC, ABD, ACD, BCD, ABCD Notice we assume that we only worry about what ONE player does in each case. There are 12 total critical votes, so by the Banzhaf index, power is divided thus. A = 5/12 B = 3/12 C = 3/12 D = 1/12 11

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Consider the U.S. Electoral College. There are a total of 538 electoral votes. A majority vote is considered 270 votes. The Banzhaf Power Index would be a mathematical representation of how likely a single state would be able to swing the vote. For a state such as California, which is allocated 55 electoral votes, they would be more likely to swing the vote than a state such as Montana, which only has 3 electoral votes. Example: The United States is having a presidential election between a Republican and a Democrat. For simplicity, suppose that only three states are participating: California (55 electoral votes), Texas (34 electoral votes), and New York (31 electoral votes). 12

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Calif (55) Texas (34) NY (31) R votes D votes States that could swing the vote RRR1200none RRD8931California, Texas RDR8634California, New York DRR6555Texas, New York Power, each state has 1/3 Consider having republicans win. The democrats winning is similar. Need 61 votes to win.

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Consider a different set of states Need 55 to win 14 California (55) Texas (34) Ohio (20)RD States that could swing RRR1090California RRD8920California RDR7534California RDD5554California

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15 factor twenty difference

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16 Shapley-Shubik Power Index: i. Shapley-Shubik Power Index: a. Permutation: total number of ways n things can be taken r at a time.. Order is important in a permutation. b. is used to find the number of ways to order n elements in a set. 2. 1 st voter in a permutation whose vote would make the coalition a winning coalition is called a pivotal voter. 3. Shapley-Shubik Power Index is fraction of permutations in which a voter is pivotal.

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17 Shapley-Shubik Power Index: Given a voting system create a Shapley- Shubik table: For this example use {A,B,C} with the voting system [3:2,1,1] Pivotal voter is underlined. Banzhaf AB AC ABC 1. Count number of times A,B, and C are pivotal voters. Divide each value by 6 to get the Shapley-Shubik Power Index: 2. Voter A is times more powerful than B or C. Voter A has 4/6 or 66.67% of the power in this voting system.

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18 factor twenty difference. Quite similar. Factor of 4.1-4.3 difference.

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Try this one Suppose decisions are made by majority rule in a body consisting of A, B, C, D, who have 3, 2, 1 and 1 votes, respectively. The majority vote threshold is 4. There are 24 possible orders for these members to vote: 19

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A=1/2, B=C=D = 1/6 20 [4:3,2,1,1] ABCDABDCACBDACDBADBCADCB BACDBADCBCADBCDABDACBDCA CABDCADBCBADCBDACDABCDBA DABCDACBDBACDBCADCABDCBA B has no more power than C or D

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21 Shapley-Shubik Power Index: 1. Sometimes permutations are too large to list all of them so we do it by grouping. Consider the voting system [5:3,1,1,1,1,1,1]. 7! = 5040 GSSSSSS 3456789SSSSGSS 1234789 SGSSSSS 1456789 SSSSSGS 1234589 SSGSSSS 1256789SSSSSSG 1234569 SSSGSSS 1236789 G is pivotal 3/7 of the time. S is pivotal in (4/7)/6=2/21 of the time. Therefore, the Shapley-Shubik Power Index is

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