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Social Choice_Lecture 8_SSI

Social Choice_Lecture 8_SSI - Power in Voting...

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Power in Voting: Shapley-Shubik Index 1 Matt Van Essen University of Alabama 1 These slides are based on Chapter 3 of Taylor and Pacelli (2008) Van Essen (U of A) Power 1 / 22
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Shapley-Shubik Index of Power Suppose we have a yes no voting system with three votes: Ann, Bob, and Cate Ann has 3 votes Bob has 3 votes Cate has 1 vote The quota for passage is 4 votes. We are interested in the °power± of the individual voters ²i.e., the degree to which they control outcomes. For instance, in the above example is Ann or Cate more powerful? In this set of notes we introduce a measure of power due to Shapley and Shubik. Van Essen (U of A) Power 2 / 22
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Shapley-Shubik Index of Power: Pivotal Player Central to the notion of Shapley and Shubik³s index is the idea of a pivotal player. Suppose we have a yes-no voting system with 7 players: p 1 , p 2 , p 3 , p 4 , p 5 , p 6 , and p 7 . There are 7! orderings of these players. Suppose we take one of them, say, p 3 p 5 p 1 p 6 p 7 p 4 p 2 We want to identify one of the players as being pivotal for this ordering. So we start with a coalition with nobody in it (i.e., the empty coalition) start adding players. The voter who makes a coalition go from losing to winning as the pivotal voter . Van Essen (U of A) Power 3 / 22
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Shapley-Shubik Index of Power De´nition Suppose p is a voter in a yes no voting system and let X be the set of all voters. Then the Shapley-Shubik index of p , denoted by SSI ( p ) is the number given by SSI ( p ) = the total number of orderings of X for which p is pivotal the total number of orderings of X Van Essen (U of A) Power 4 / 22
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Shapley-Shubik Index of Power 1 The denominator of the SSI is just n ! if there are n voters 2 For every voter p , it must be that 0 ° SSI ( p ) ° 1 3 If the voters are p 1 , ..., p n , then n i = 1 SSI ( p i ) = 1 Think of the SSI has the fraction of power that each individual has.
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