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Applications of the Banzhaf Power Index

Applications of the Banzhaf Power Index. Notes 8 – Section 2.3. Essential Learnings. Students will understand and be able to apply the Banzhaf Power Index. Nassau County (N.Y.) Board of Supervisors. Weighted voting in New York dates back to the 1800’s.

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Applications of the Banzhaf Power Index

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  1. Applications of the Banzhaf Power Index Notes 8 – Section 2.3

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

  3. Nassau County (N.Y.) Board of Supervisors • Weighted voting in NewYork dates back to the 1800’s. • Weighted voting systems usedin some New York counties were seriously flawed (and unconstitutional).

  4. Nassau County (N.Y.) Board of Supervisors • John Banzhaf introduced the Banzhaf Power Index in an article entitledWeighted Voting Doesn’t Work. • Key point wasthat in weighted voting votes do not necessarily imply power.

  5. Example – Nassau County • In the 1960’s, Nassau County was divided into 6 uneven districts, 4 withhigh populations and 2 rural districts with low populations.

  6. Example – Nassau County • Nassau County Board (1964) NameWeight Hempstead #1 31 Hempstead #3 31 Oyster Bay 28 North Hempstead 21 Long Beach 2 Glen Cove 2 Simple majority The idea is that power should be proportional to its population.

  7. Example – Nassau County • Use the Banzhaf Power Index to determine the power each county had.

  8. Example – Nassau County • Banzhaf’s mathematical analysis of the unfair power distribution in the Nassau County Board se the stage for a series of lawsuits against Nassau and other New York state counties based on the argument that weighted voting violated the “equal protection” guarantees of the Fourteenth Amendment. • The final result was a federal court decision in 1993 abolishing weighted voting in N.Y. counties.

  9. Example – The United Nations Security Council • The U.N. Security Council currently consist of 15 voting nations with 5 permanent members (Britain, China, France, Russia, and the U.S.). • The other 10 nations are nonpermanent members appointed for a two-year period on a rotating basis.

  10. Example – The United Nations Security Council • To pass a motion in the Security Council requires a yes vote from each of the five permanent members (essentially giving each one veto power) plus at least 4 additional yes votes from the 10 nonpermanent members. • There are 848 possible winning coalitions in the Security Council – too many to list one by one.

  11. Example – The United Nations Security Council • Winning coalitions with 9 members, (5 permanent members plus 4 additional nonpermanent members) every member is a critical player. • In all the other winning coalitions, only the permanent members are critical players.

  12. Example – The United Nations Security Council • Out of the 848 possible winning coalitions, there are 210 with 9 members and the rest have 10 or more members. • Total number of coalitions (winning and losing): 5080 • Banzhaf Power Index: Permanent member: Nonpermanent member: 848/5080 = 16.7% 84/5080 = 1.65%

  13. Example – The European Union • The European Union (EU) is a political and an economic confederation of European nations. • As of the writing of the book, the EU consists of 27 member nations with three more countries (Turkey, Croatia, and Macedonia) expected to join the EU in the near future.

  14. Example – The European Union • The legislative body for the EU (called the EU Council of Ministers) operates as a weighted voting system where different member nations have weights that are roughly proportional to their respective populations.

  15. Example – The European Union • The European Union (EU) is a political and an economic confederation of European nations. • As of the writing of the book, the EU consists of 27 member nations with three more countries (Turkey, Croatia, and Macedonia) expected to join the EU in the near future.

  16. Example – The European Union • The total number of votes is V = 345, with a quota q = 255 votes. • There is a very close match between Banzhaf power and weights. • This weighted voting system works!

  17. Assignments Banzhaf Power Practice Problems p. 68: 21, 23, 24

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