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| {{Other uses|Power index (disambiguation){{!}}Power index}}
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| The '''Shapley–Shubik power index''' was formulated by [[Lloyd Shapley]] and [[Martin Shubik]] in 1954<ref>Shapley, L.S. and M. Shubik, A Method for Evaluating the Distribution of Power in a Committee System, ''American Political Science Review'', 48, 787–792, 1954.</ref> to measure the powers of players in a voting game. The index often reveals surprising power distribution that is not obvious on the surface.
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| The constituents of a voting system, such as legislative bodies, executives, shareholders, individual legislators, and so forth, can be viewed as players in an [[n-player game|''n''-player game]]. Players with the same preferences form coalitions. Any coalition that has enough votes to pass a bill or elect a candidate is called winning, and the others are called losing. Based on [[Shapley value]], Shapley and Shubik concluded that the power of a coalition was not simply proportional to its size.
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| The power of a coalition (or a player) is measured by the fraction of the possible voting sequences in which that coalition casts the deciding vote, that is, the vote that first guarantees passage or failure.<ref>Hu, X., An asymmetric Shaplay–Shubik power index, ''International Journal of Game Theory,'' 34, 229–240, 2006.</ref>
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| The power index is normalized between 0 and 1. A power of 0 means that a coalition has no effect at all on the outcome of the game; and a power of 1 means a coalition determines the outcome by its vote. Also the sum of the powers of all the players is always equal to 1.
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| == Examples ==
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| 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 [[Factorial|4!]] = 24 possible orders for these members to vote:
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| {| class="wikitable"
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| |-
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| | A'''B'''CD
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| | A'''B'''DC
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| | A'''C'''BD
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| | A'''C'''DB
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| | A'''D'''BC
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| | A'''D'''CB
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| |-
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| | B'''A'''CD
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| | B'''A'''DC
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| | BC'''A'''D
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| | BC'''D'''A
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| | BD'''A'''C
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| | BD'''C'''A
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| |-
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| | C'''A'''BD
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| | C'''A'''DB
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| | CB'''A'''D
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| | CB'''D'''A
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| | CD'''A'''B
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| | CD'''B'''A
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| |-
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| | D'''A'''BC
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| | D'''A'''CB
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| | DB'''A'''C
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| | DB'''C'''A
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| | DC'''A'''B
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| | DC'''B'''A
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| |}
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| For each voting sequence the pivot voter – that voter who first raises the cumulative sum to 4 or more – is bolded. Here, A is pivotal in 12 of the 24 sequences. Therefore, A has an index of power 1/2. The others have an index of power 1/6. Curiously, B has no more power than C and D. When you consider that A's vote determines the outcome unless the others unite against A, it becomes clear that B, C, D play identical roles. This reflects in the power indices.
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| Suppose that in another majority-rule voting body with <math>2n+1</math> members, in which a single strong member has <math>k</math> votes and the remaining <math>2n</math> members have one vote each. It then turns out that the power of the strong member is <math>\dfrac{k}{2n+2-k}</math>. As <math>k</math> increases, his power increases disproportionately until it approaches half the total vote and he gains virtually all the power. This phenomenon often happens to large shareholders and business takeovers.
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| ==See also==
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| * [[Shapley value]]
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| * [[Arrow theorem]]
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| * [[Banzhaf power index]]
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| == References ==
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| <references/>
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| == External links ==
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| *[http://www.warwick.ac.uk/~ecaae/ Computer Algorithms for Voting Power Analysis] Web-based algorithms for voting power analysis
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| *[http://korsika.informatik.uni-kiel.de/~stb/power_indices/index.php Power Index Calculator] Computes various indices for (multiple) weighted voting games online. Includes some examples.
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| {{DEFAULTSORT:Shapley-Shubik power index}}
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| [[Category:Game theory]]
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| [[Category:Cooperative games]]
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| [[Category:Voting systems]]
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