Research Investigations: Weighted Voting
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451
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.
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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