Ron’s connection between veto power and Banzhaf power
The weight of a player is not a good predictor of a player’s Banzhaf power. As Ron
pointed out in class, the key is not the weight, but the number of times a player is a
critical player in a winning coalition.
Ron reminded us that since a player P with
veto power is critical in every winning coalition, then player P will be critical the
maximum number of times possible, giving player P the largest amount of Banzhaf
power.
In summary, if a player has veto power, then this player will have the largest
amount of
Banzhaf power.
We have:
Ron’s Theorem
:
If a player P in a weighted voting system has veto power,
then player P has the largest amount of Banzhaf power in
this weighted voting system.
Proof:
Player P has veto power. Hence player P is in every winning
coalition. Moreover, player P is critical in every winning coalition. So
the numerator of the Banzhaf power distribution for player P will be the
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 Spring '07
 Berkowitz
 Math, 50%, 20%, 0%, Ron, 60%, AirTrain Newark

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