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