Hi,
There's been some confusion about the following homework problem in
Chapter 3, so here's a start.
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Find the Banzhaf Power Distribution for the weighted voting
system [10:5,4,3,2,1]. Only list the winning coalitions.
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There are no winning coalitions with just one or two players. (Even
Players 1 and 2, who have the most votes, only total to 9.)
There are a number of ways to build a Winning Coalition with three
players, but they all need player 1 (with her 5 votes):
{P1, P2, P3} -- 12 votes total -- every player is critical
{P1, P2, P4} -- 11 votes total -- every player is critical
{P1, P2, P5} -- 10 votes total -- every player is critical
{P1, P3, P4} -- 10 votes total -- every player is critical
(So the total number of times all the players are critical
in three-player coalitions is 4 times 3 = 12.)
Any coalition with four players is a Winning Coalition. (Even if Player 1
votes against a motion with her five votes, the other players still have
exactly 10 votes, which are enough to pass a motion.)
{P1, P2, P3, P4} -- 14 votes total -- only P1 is critical
{P1, P2, P3, P5} -- 13 votes total -- both P1 and P2 are critical
{P1, P2, P4, P5} -- 12 votes total -- both P1 and P2 are critical
{P1, P3, P4, P5} -- 11 votes total -- P1, P3, and P4 are critical
{P2, P3, P4, P5} -- 10 votes total -- every player is critical
(So the total number of times all the players are critical
in four-player coalitions is 1+2+2+3+4=12.)
There are five players, so there is only one coalition -- the "Grand
Coalition" -- with five players.
{P1, P2, P3, P4, P5} -- 15 votes total -- no players are critical
So we only have critical players in coalitions with three or four players,
and by counting we see that the total number of times all players are
critical is 12+12=24.
That means "24" is the number in the denominator of any Banzhaf Power
Index. (But note that the back of the book might reduce the fractions in
some answers, for example 20/24 = 10/12.)
Hope this helps,
Jon