• Electoral College
• United Nations Security Council
• Dictatorships
• MOM

To make things simple we are only going to look at a vote which only involves two choices. Referred to in your books as a motion.

Some Basic Definitions:

• Weighted Voting - described as one voter --> x votes, opposite of the idea of one voter - one vote.

• Players - the voters; denoted P₁ , P₂ , P₃ , . . . . , P_n

  
  

• Weight - the number of votes each player controls; denoted w₁ , w₂ , w₃ , . . . . , w_n

  
  

• Quota - the minimum number of votes needed to pass a motion denoted q
  

  • NOTE: The quota does not have to be a majority of votes, any number may be chosen as the quota as long as it is more than half the number of total number of votes, but not more than the total number of votes

• Dictator - a player whose weight is bigger than or equal to the quota

  
  

  • NOTE: Whenever there is a dictator, all the other players, regardless of their weights, have absolutely no power; such a player is called a dummy.

Standard notation: [ q; w₁ , w₂ , w₃ , . . . . , w_n ]
The quota is given first, followed by the respective weights of the individual players

EXAMPLE

Consider: [ 25; 8, 6, 5, 3, 3, 3, 2, 2, 1, 1, 1, 1 ]

• This is a weighted voted system with 12 players ( P₁ , P₂ , P₃ , . . . . , P₁₂ )
• The quota is 25
• P₁ has 8 votes, P₂ has 6 votes , P₃ has 5 votes, etc.

Some Basic Definitions:

• Coalition - any set of players that might join forces and vote together
  

• Weight of the Coalition - the total number of votes controlled by a coalition
  • NOTE : Some coalitions have enough votes to win and some don't; We call the former winning coalitions, the latter losing coalitions

• Notation: { P₁ , P₂ } is the coalition consisting of the players P₁ and P₂
  
  • _{We will be looking for players whose desertion turns a winning coalition into a losing coalition; these are called critical players}
  • _{Banzhaf's key idea is that a player's power is proportional to the number of coalitions for which a player is critical}

• _{Step 1. Make a list of all possible coalitions}
• _{Step 2. Determine which of them are winning coalitions}
• _{Step 3. In each winning coalition, determine which of the players are critical players}
• _{Step 4. Count the total number of times player P is critical (B)}
• _{Step 5. Count the total number of times all players are critical (T)}

Here is an example:

We have a weighted voting system [ 6; 4, 3, 2 ]

Step 1.

Coalition	Coalition Weight	Win or Lose
{P1}	4	Lose
{P2}	3	Lose
{P3}	2	Lose
{P1, P2}	7	Win
{P1, P3}	6	Win
{P2, P3}	5	Lose
{P1, P2, P3}	9	Win

The winning coalitions are {P1, P2}, {P1, P3}, {P1, P2, P3}

Step 3.

Winning Coalitions	Critical Players
{P1, P2}	P1 and P2
{P1, P3}	P1 and P3
{P1, P2, P3}	P1 only

Step 4.

• P1 is critical three times ( B = 3)
• P2 is critical one time ( B= 1)
• P3 is critical one time ( B = 1)

Step 5.
T = 5

Step 6. Power index for each player is given by B/T

P1; 3/5

P2; 1/5

P3; 1/5

For our example than:

P1; 60%

P2; 20%

P3; 20%

You can check your work by adding up the percentages. If everything is done correctly, they should add up to 100% The same is true for the fractions, they should add up to 1.