2.1 Weighted Voting Systems
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Transcript 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 , . . . , wN .
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 + . . . + w N
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