Research Investigations: Weighted Voting

Prepared by:

Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk/

The weighted voting game [ 6; 4, 3, 2, 1 ] has 4 players with different weights, however, the Banzhaf Power of these players is (5/12, 3/12,3/12, 1/12).
If one looks at the game [ 6; 5, 3, 3, 1 ] the Banzhaf power vector is (5/12, 3,/12, 3/12, 1/12). Thus, we have been able to represent the original game with a new set of weights with the property that the Banzhaf power of both games is the same and the weights are proportional to Banzhaf power. Note, that the game [ 7; 5, 3, 3, 1 } will have exactly the same Banzhaf power as [ 6; 5, 3, 3, 1 ] as well as the same minimal winning coalitions at that game.

Here is another example. Given the game [ 5; 4, 2, 1, 1 ] the Banzhaf power vector for the players is ( 7/10, 1/10, 1/10, 1/10). [ 8; 7, 1, 1, 1 ] has the same power vector as the original.

Question:

1. Is it true that every weighted voting game (where the quota is at least one half the sum of the weights plus one) can be represented by a set of weights where the weights are proportional to the Banzhaf power?

2. If the answer to Question 1 is no, give an example where such a representation is not possible.

3. If the answer to Question 1 is no, can you describe those games where such a representation is possible?

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