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Applications of the Shapley- Shubik Power Index

Applications of the Shapley- Shubik Power Index. Notes 10 – Section 2.5. Essential Learnings. Students will understand and be able to apply the Shapley- Shubik Power Index. Example - The Electoral College.

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Applications of the Shapley- Shubik Power Index

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  1. Applications of the Shapley-Shubik Power Index Notes 10 – Section 2.5

  2. Essential Learnings • Students will understand and be able to apply the Shapley-Shubik Power Index.

  3. Example - The Electoral College • Calculating the Shapley-Shubik power index of the states inthe Electoral College is no easy task. • There are 51! sequential coalitions. A number 67 digits long. • Computers can calculate the power indexes quickly.

  4. Example – The Electoral College • The handout shows the Shapley-Shubik and Banzhaf power indexes for each state and the District of Columbia. • What do you notice? • There is a very small difference between the two power indexes.

  5. Example – The United Nations Security Council • 15 members: 5 permanent & 10 nonpermanent • Fora motion to pass it must have a Yes vote from each of the 5 permanent membersplus at least 4 of the 10 nonpermanent members.

  6. Example – The United Nations Security Council • It can be shown that the permanent members have 7 votes each, thenonpermanent members have 1 vote each, and the quota is 39 votes. • The simplified version of finding the Shapley-Shubik power indexes follows.

  7. Example – The United Nations Security Council • Step 1: There are 15! sequential coalitions of 15 players (roughly about 1.3trillion).

  8. Example – The United Nations Security Council • Step 2: A nonpermanent member can be pivotal only if it is the 9th player in the coalition, preceded by all five of the permanent members and three nonpermanent members. There are approximately 2.44 billion sequential coalitions of this type.

  9. Example – The United Nations Security Council • Step 3: From steps 1 and 2 we can conclude that the Shapley-Shubik power index of anonpermanent member is approximately 2.44 billion/1.3 trillion ≈ 0.0019 = 0.19%. • The correspondingBanzhaf power index is 1.65% (big difference!).

  10. Example – The United Nations Security Council • Step 4: The 10 nonpermanent members have together 1.9% (0.19% times 10). • The remaining 98.2% is divided equally among the 5 permanent members. • Thus, the Shapley-Shubikpower index of each permanent member is approximately 98.2/5 =19.64%.

  11. Example – The United Nations Security Council • Permanent members have roughly 100 times the Shapley-Shubik power of non-permanent members!

  12. Summary • Depending on the situation, the Banzhaf power distribution and the Shapley-Shubik power distributions can be very similar or vastly different.

  13. Assignment p. 69: 28b, 29, 36 a-c, 37, 43 Power Practice WS 1

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