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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 2N 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?

 
{P1,P2,P3}
 
 
{P1, P2}
 
 
{P1, P3}
 
 
{P2,P3}
 
 
{P1}
 
 
{P2}
 
 
{P3}
 
 
{}
 
Now which of these are winning coalitions??
{P1,P2,P3}
 
 
{P1, P2}
 
 
{P1, P3}
 
In each of these winning coalitions, which player(s) are critical?
P1 is critical 3 times.
P2 is critical 1 time.
P3 is critical 1 time.

Now P1 is given a Banzhaf power index of 3/5 or 60%. P2 and P3 each have a Banzhaf power index of 1/5 or 20% each.

Another Example

A committee consists of four players, P1, P2, P3, and P4. Each committee member has one vote, and a motion is carried by majority vote except in the case of a 2-2 tie. In this case, if P1 voted for the motion, then it carries. (P1 plays the tie-breaker here.) Determine the Banzhaf power index of each of these four players.

How many coalitions are there?

How many winning coalitions are there?

 
{P1,P2,P3, P4}
 
 
{P1, P2, P3}
 
 
{P1, P2, P4}
 
 
{P1,P3,P4}
 
 
{P2,P3,P4}
 
 
{P1, P2}
 
 
{P1, P3}
 
 
{P1, P4}
 
How many times are each of the players critical?
P1 is critical 6 times.
P2 is critical 2 times.
P3 is critical 2 times.
P4 is critical 2 times.

Then the Banzhaf power index of each is given by

P1: [6/12] or 50 percent
P2: [2/12] or 16.67 percent (approximately)
P3: [2/12] or 16.67 percent (approximately)
P4: [2/12] or 16.67 percent (approximately)
File translated from TEX by TTH, version 0.9.

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