Chapter 2  The Power Game
The Banzhaf Power Index
Terminology

·

coalition  Any set of players that join forces to vote together
(this may be only one player)

·

weight of a coalition  The total number of votes controlled by
a coalition

·

winning coalitions  Coalitions that have enough votes to win

·

losing coalitions  Coalitions who can't win due to a lack of sufficient
votes

·

critical player  A player whose desertion turns a winning coalition
into a losing coalition
Banzhaf's Key
A player's power is proportional to the number of times that player
is a critical player.
Important Note
To determine the Banzhaf power index, we will have to count all the
possible coalitions and then only keep the winning ones.
One might ask the following: How many coalitions are there if we have
3 players in the ``game.'' Let's find out with three volunteers from the
audience.
Some Set Theory
Given N players, there are a total of 2^{N} possible coalitions,
including the ``empty'' coalition which has no one in it.
Note the connection with Pascal's triangle.
Focus on Banzhaf's Power Index
Let's now look at an example to motivate Banzhaf's power index. Consider
the example voting system [6:5, 3, 1]. What are the coalitions?
Now which of these are winning coalitions??
In each of these winning coalitions, which player(s) are critical?

P_{1} is critical 3 times.

P_{2} is critical 1 time.

P_{3} is critical 1 time.
Now P_{1} is given a Banzhaf power index of 3/5 or 60%. P_{2}
and P_{3} each have a Banzhaf power index of 1/5 or 20% each.
Another Example
A committee consists of four players, P_{1}, P_{2},
P_{3}, and P_{4}. Each committee member has one vote, and
a motion is carried by majority vote except in the case of a 22 tie. In
this case, if P_{1} voted for the motion, then it carries. (P_{1}
plays the tiebreaker here.) Determine the Banzhaf power index of each
of these four players.
How many coalitions are there?
How many winning coalitions are there?
{P_{1},P_{2},P_{3}, P_{4}} 

How many times are each of the players critical?

P_{1} is critical 6 times.

P_{2} is critical 2 times.

P_{3} is critical 2 times.

P_{4} is critical 2 times.
Then the Banzhaf power index of each is given by

P_{1}: [6/12] or 50 percent

P_{2}: [2/12] or 16.67 percent (approximately)

P_{3}: [2/12] or 16.67 percent (approximately)

P_{4}: [2/12] or 16.67 percent (approximately)
File translated from T_{E}X by T_{T}H,
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