1 5 Weighted voting and power

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					      1.5 Weightd voting and voting power

Goal: For given holdings, find the power index
of each stockholder.

Is a person’s influence or power proportional to
the fraction of votes he casts?
    Weighted Voting Systems
    Outline/learning Objectives
• Represent a weighted voting system using a
  mathematical model.
• Use the Banzhaf index to calculate the
  distribution of power in a weighted voting
     Weighted Voting Systems
• The Players
  The voters in a weighted voting system.
• The Weights
  That each player controls a certain number
  of votes.
• The Quota
  The minimum number of votes needed to
  pass a motion (yes-no votes)
• Dictator
  The player’s weight is bigger than or equal
  to the quota.

Consider [11:12, 5, 4]
    owns enough votes to carry a motion
 single handedly.
• Dummy
  A player with no power.

Consider [30: 10, 10, 10, 9]
    turns out to be a dummy! There is never
 going to be a time when      is going to
 make a difference in the outcome of the
• Veto Power
  If a motion cannot pass unless player votes
  in favor of the motion.

Consider [12: 9, 5, 4, 2]
    has the power to obstruct by preventing
 any motion from passing.
• Coalitions
  Any set of players that might join forces and vote
  the same way. The coalition consisting of all the
  players is called a grand coalition.
• Winning Coalitions
  Some coalitions have enough votes to win
  and some don’t. We call the former
  winning coalitions and the latter losing
• Critical players
  In a winning coalition, a player is said to be a
  critical player for the coalition if the coalition
  must have that player’s votes to win. If and
  only if
  – Computing a Banzhaf Power Distribution
• Step 1. Make a list of all possible winning
Step 2. Within each winning coalition
  determine which are the critical players. (To
  determine if a given player is critical or not
  in a given winning coalition, we subtract the
  player’s weight from the total number of
  votes in the coalition- if the difference drops
  below the quota q, then that player is
  critical. Otherwise, that player is not
• Step 3.Count the number of times that
P 1is critical. Call this numberB1 Repeat for
  each of the other players to find B 2, B3, ...,BN
Step 4. Find the total number of times
 all players are critical. This total is
 given by

       T =B + B
             1     2   + ... + BN
  – Computing a Banzhaf Power Distribution
• Step 5. Find the ratio 1       .
                            =B T
This gives the Banzhaf power index of        .
Repeat for each of the other players to find
2, 3, …, N . The complete list of ’s gives
  the Banzhaf power distribution of the
  weighted voting system.
Applications of Banzhaf Power
• The Nassau County Board of
  John Banzhaf first introduced the concept
• The United Nations Security Council
  Classic example of a weighted voting
• The European Union (EU)
  Relative Weight vs Banzhaf Power Index

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