§ 2.1 Weighted Voting Systems

							§ 2.1 Weighted Voting Systems
         Weighted Voting
 So far we have discussed voting
 methods in which every individual’s vote
 is considered equal--these methods
 were based on the concept of “one
 voter--one vote.”

 Weighted voting, is based on the idea
 of “one voter--x votes”--in other words,
 some voters ‘count more’ than others.
   Weighted Voting Systems
 More rigorously stated, any formal
 voting system arrangement in which the
 voters are not necessarily equal in
 terms of the number of votes they
 control is called a weighted voting
 system.
 For the sake of simplicity we shall only
 examine motions--votes involving only
 two choices/candidates.
   Weighted Voting Systems
 Weighted voting systems are
 comprised of:

 1. Players - The groups, or individuals
 that can cast votes.
 2. Weights of the players - The number
 of votes each player controls.
 3. Quota - The smallest number of votes
 needed to pass a motion.
   Weighted Voting Systems
 Notation:
 We will use N to refer to the number of
 players in our system.
 The players will be denoted P1 , P2 , P3 ,
 . . . , PN .
 Their corresponding weights are w1 , w2
 , w3 , . . . , w N .
 The letter q will be used to represent the
 quota.
   Weighted Voting Systems
 Using this notation we can represent
 the entire weighted voting system as:

          [ q : w1 , w2 , w 3 , . . . , wN ]

 Here the quota is listed first and the
 weights are given in decreasing order.
   Weighted Voting Systems
 The quota, q, must always be larger
 than half the number of votes and not
 more than the total number of votes.
 Stated mathematically,

 w 1 + w2 + w 3 + . . . + w N < q ≤ w 1 + w 2 +
 w3 + . . . + wN
            2
Example 1: Suppose that the board of
a small corporation has four shareholders,
P1 , P2 and P3 . P1 has 8 votes, P2 has 4
votes, P3 has 2 votes and P4 has 1. If at
least two-thirds of the votes are needed to
pass a motion then describe this system
using the ‘bracketed’ notation.
Example 2: Consider weighted voting
system with four players, P1 , P2 , P3 and
P4 . P1 has three times as many votes as
P2 . P2 has twice as many votes as P3
and P4 (which have the same number of
votes). If a simple majority is all that is
necessary to pass a motion then describe
this weighted voting system.
   Weighted Voting Systems
 Notice in the last example that P1 could
 pass or block any motion. In such a
 situation, P1 would be called a dictator.
 In general, a player is a dictator if the
 player’s weight is bigger than or equal
 to the quota.
 Whenever there is a dictator, all of the
 other players are irrelevant--such a
 player with no power is called a dummy.
   Weighted Voting Systems
 Now look back at example 1. You
 might notice that no motion could pass
 in that weighted system without the
 support of P1 , but that P1 would still
 need the support of at least one other
 voter in order to pass a motion.
 Any player who is not a dictator, but
 can block the passing of any motion has
 what is referred to as veto-power.
Example 3: (Exercise #10 pg 73) In each
of the following weighted voting systems,
determine which players, if any, (i) are dictators;
(ii) have veto power; (iii) are dummies.

(a) [ 27 : 12, 10, 4, 2 ]
(b) [ 22 : 10, 8, 7, 2, 1 ]
(c) [ 21 : 23, 10, 5, 2 ]
(d) [ 15 : 11, 5, 2, 1 ]
Example 4: The US Senate is currently
composed of 55 Republicans, 44 Democrats
and 1 Independent (who votes with the
Democrats). Suppose 6 Republican senators
decided to form their own “Consensus Party”
(yes, I know this is even sillier than voting
muppets). Further suppose that following such
defections each party keeps its members strictly
in line.
Example 4: The US Senate is currently
composed of 55 Republicans, 44 Democrats
and 1 Independent (who votes with the
Democrats). Suppose 6 Republican senators
decided to form their own “Consensus Party”
(yes, I know this is even sillier than voting
muppets). Further suppose that following such
defections each party keeps its members strictly
in line.

We might describe this weighted voting system
as:
                [ 51 : 49, 45, 6 ]
Example 4: The US Senate is currently
composed of 55 Republicans, 44 Democrats
and 1 Independent (who votes with the
Democrats). Suppose 6 Republican senators
decided to form their own “Consensus Party”
(yes, I know this is even sillier than voting
muppets). Further suppose that following such
defections each party keeps its members strictly
in line.

We might describe this weighted voting system
as:
                  [ 51 : 49, 45, 6 ]
While it might seem that both the Democrats
and Republicans hold more power than the