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Section 2.4

Section 2.4. The Shapley-Shubik Power Index. A different way to measure power which centers around the concept of sequential coalition Sequential coalition - every coalition starts with a first player who may then be joined by a second player, then a third, and so on

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Section 2.4

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  1. Section 2.4 The Shapley-Shubik Power Index

  2. A different way to measure power which centers around the concept of sequential coalition • Sequential coalition- every coalition starts with a first player who may then be joined by a second player, then a third, and so on • The order of the players does matter!

  3. Example 1 • In Banzhaf, {P1,P2,P3} is a coalition and all vote together. We do not care who joined the coalition 1st. With Shapley-Shubik, the same three players form six different sequential coalitions: <P1,P2,P3> <P1,P3,P2> <P2,P1,P3> <P2,P3,P1> <P3,P1,P2> <P3,P2,P1> Notice < > is used to represent sequential coalitions

  4. Number of Sequential Coalitions • To determine the number of sequential coalitions, we use factorials • If there are four players in a coalition, then there are 4! Sequential coalitions. • 4! = 4 x 3 x 2 x 1 = 24 Again, the order of the players is important because it is used to determine the Shapley-Shubik power distribution

  5. Pivotal player- the one player who tips the scales or changes the coalition from a losing to a winning coalition the moment it joins • Shapley-Shubik focuses on this player

  6. Shapley-Shubik Power Index- a player’s power depends on the total number of times that it is pivotal in relation to all of the other players • Shapley-Shubik Power Distribution- a listing of the power index for each player

  7. Finding the Shapley-Shubik Power Index of Player P: • Make a list of all sequential coalitions containing all the players (N players) • In each sequential coalition, determine the pivotal player (each coalition should have one) • Count the number of times P is the pivotal player and call this S • Calculate S/N! which is the SSPI

  8. Example: [12: 8, 5, 3] • Find the Shapley-Shubik power distribution. • List all sequential coalitions: • ABC • ACB • BAC • BCA • CAB • CBA

  9. Determine the pivotal players in each coalition: • ABC B is pivotal • ACB B is pivotal • BAC A is pivotal • BCA A is pivotal • CAB B is pivotal • CBA A is pivotal

  10. Power Distribution • A: 3/6 = ½ = 50% • B: 3/6 = ½ = 50% • C: 0/6 = 0%

  11. Example 2 • Find the Shapley-Shubik power distribution for the following: [10: 6, 5, 4]

  12. List of coalitions and pivotal players • ABC B is pivotal • ACB C is pivotal • BAC A is pivotal • BCA A is pivotal • CAB A is pivotal • CBA A is pivotal A: 4/6 = 66.7% B: 1/6 = 16.7 % C: 1/6 = 16.7%

  13. Example 3 • Find the Shapley-Shubik power distribution for the following: [10: 7, 5, 4, 3]

  14. List of Coalitions • Should be 4! Or 24 total ABCD BACD CABD DABC ABDC BADC CADB DACB ACBD BCAD CBAD DBAC ACDB BCDA CBDA DBCA ADBC BDAC CDAB DCAB ADCB BDCA CDBA DCBA

  15. Determine the critical players ABCD BACD CABD DABC ABDC BADC CADB DACB ACBD BCAD CBAD DBAC ACDB BCDA CBDA DBCA ADBC BDAC CDAB DCAB ADCB BDCA CDBA DCBA

  16. Determine Distribution • A: 12/24 or 50% • B: 4/24 or 16.7% • C: 4/24 or 16.7% • D: 4/24 or 16.7%

  17. Section 2.5 Applications of the Shapley-Shubik Power Index

  18. Electoral college • 51 States (including District of Columbia) means 51! sequential coalitions. This number has 67 digits • In other words, this is all too large to work with, so we’d need to look for mathematical shortcuts and use computers to help.

  19. The United Nations Security Council has 5 permanent members and 10 non-permanent members for a total of 15 members • 15! means about 1.3 trillion sequential coalitions

  20. Assignment • Page 75-76 # 23-27, 28 a, e, 29, 30 b, c

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