Weighted voting systems:
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WEIGHTED VOTING SYSTEMS
• sets of voters (slide #3--17)
• the power of coalitions (#18--23)
• power of individual voters(#24--26)
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WEIGHTED VOTING SYSTEMS
Example:
• A certain small business has 4 shareholders.
ALICE owns 45 shares of stock
BRADLEY owns 30 shares
CHARLES owns 15 shares
DOROTHY owns 10 shares
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WEIGHTED VOTING SYSTEMS
• ALICE owns 45 shares Each stockholder
of stock
attends monthly
• BRADLEY owns 30
shares meetings, at which they
• CHARLES owns 15 are able to vote on
shares matters that affect the
• DOROTHY owns 10
shares
business.
We will look at 3 ways
that the voting might be
accomplished, and
discuss the “fairness” of
each of the voting
systems considered.
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WEIGHTED VOTING SYSTEMS
WVS NOTATION: A wvs with “n” voters...
[q: w1, w2, w3, … wn]
where q = quota (number of votes needed to
pass a motion)
and wn = the weight of voter n (number of
votes that particular voter has)
• NOTE: We will say that the wvs is valid as long
as the QUOTA is at least a *simple majority and
the QUOTA is not more than the total number of
votes possible.
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WEIGHTED VOTING SYSTEMS
WVS NOTATION: A wvs with “n” voters...
[q: w1, w2, w3, … wn] where q = quota & wn = the weight of voter n
•NOTE: We will say that the wvs is valid as long as the QUOTA is at
least a simple majority and the QUOTA is not more than the total
number of votes possible.
NOTE also…
• The voting system is made up of the actual
voters…
• (ex) {Alice, Brad, Chuck, Dorothy}
• We will not allow the voters to simply be “named”
by their # of votes.
• i.e. If Alice has 45 votes, we won’t call her “voter
45” … we’ll still call her Alice!!!
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WEIGHTED VOTING SYSTEMS
•ALICE owns 45 shares of • Wvs#1
stock
The number of votes
•BRADLEY owns 30 shares each person has
•CHARLES owns 15 shares (weight of each
•DOROTHY owns 10 shares voter) is equal to
the number of
We will look at 3 ways that the shares of stock
voting might be (s)he owns.
accomplished, and discuss
the “fairness” of each of the And the number of
voting systems considered. votes needed to
pass a measure
(quota) is the
simple majority.
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WEIGHTED VOTING SYSTEMS
•ALICE owns 45 shares of stock Wvs#1
•BRADLEY owns 30 shares The number of votes each
•CHARLES owns 15 shares person has (weight of each
•DOROTHY owns 10 shares voter) is equal to the number of
shares of stock (s)he owns.
And the number of votes
needed to pass a measure
• Wvs#1 (quota) is the simple majority.
FIND the simple majority: 45+30+15+10 = 100 total
votes.
The simple majority is the first integer that is more
than half of 100 (the total # of votes possible).
That’s 51 votes!
(note: a vote of 50 to 50 would be a tie…
a vote of 51 to 49 would produce a winner & loser)
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WEIGHTED VOTING SYSTEMS
ALICE owns 45 shares of stock
BRADLEY owns 30 shares WVS #1
CHARLES owns 15 shares
DOROTHY owns 10 shares shorthand notation:
[51: 45, 30, 15, 10]
• Let’s use some shorthand notation for the
voters’ too… so that we can take a look at
the whole system, and how the voters might
vote on certain issues.
• We’ll look at the set of voters as follows:
{A, B, C, D}
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WEIGHTED VOTING SYSTEMS
How the voting could work...
• The next set of slides will deal with looking at all
of the possibilities for a vote on a motion.
• We will assume that no voter is allowed to abstain
from voting.
• When voters join together on one side of an issue
(FOR or AGAINST), they will form a subset of
voters called a COALITION.
• After you have done this look at the workings of a
weighted voting system, you will NOT have to do
this for any other problem.
• This is the BIG PICTURE… so you get a feel for
the possibilities.
10
Let’s do an exhaustive look at all of the possible ways
voting could work here. (**handout**)
Coalition voting FOR Coalition voting Motion PASS
weight status weight status or FAIL
the motion AGAINST the motion
none 0 lose ABCD 100 block FAIL
A 45 lose BCD 55 block FAIL
B 30 lose ACD 70 block FAIL
C 15 lose ABD 85 block FAIL
D 10 lose ABC 90 block FAIL
AB 75 win CD 25 lose PASS
AC __ __ __ __ __ __
AD __ __ __ __ __ __
BC __ __ __ __ __ __
BD __ __ __ __ __ __
CD __ __ __ __ __ __
ABC __ __ __ __ __ __
ABD __ __ __ __ __ __
ACD __ __ __ __ __ __
BCD __ __ __ __ __ __
ABCD __ __ __ __ __ __ 11
Coalition voting FOR Coalition voting Motion PASS
weight status weight status or FAIL
the motion AGAINST the motion
none 0 lose ABCD 100 block FAIL
A 45 lose BCD 55 block FAIL
B 30 lose ACD 70 block FAIL
C 15 lose ABD 85 block FAIL
D 10 lose ABC 90 block FAIL
AB 75 win CD 25 lose PASS
AC __ __ __ __ __ __
AD __ __ __ __ __ __
BC __ __ __ __ __ __
BD __ __ __ __ __ __
CD __ __ __ __ __ __
ABC __ __ __ __ __ __
ABD __ __ __ __ __ __
ACD __ __ __ __ __ __
BCD __ __ __ __ __ __
ABCD __ __ __ __ __ __ 12
[51: 45,30,15,10] WEIGHTED VOTING SYSTEMS
Coalition voting FOR Coalition voting Motion PASS
weight status weight status or FAIL
the motion AGAINST the motion
none 0 lose ABCD 100 block FAIL
A 45 lose BCD 55 block FAIL
B 30 lose ACD 70 block FAIL
C 15 lose ABD 85 block FAIL
D 10 lose ABC 90 block FAIL
AB 75 win CD 25 lose PASS
AC 60 win BD 40 lose PASS
AD 55 win BC 45 lose PASS
BC 45 lose AD 55 block FAIL
BD 40 lose AC 60 block FAIL
CD 25 lose AB 75 block FAIL
ABC 90 win D 10 lose PASS
ABD 85 win C 15 lose PASS
ACD 70 win B 30 lose PASS
BCD 55 win A 45 lose PASS
ABCD 100 win none 0 lose PASS
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WEIGHTED VOTING SYSTEMS
• Notice that there are 8
winning coalitions;
• And that ALICE is in 7 out • So, for WVS#2,
of the 8 winning coalitions, let’s make the
• While the other three QUOTA higher
voters are each in 5 out of
8 winning coalitions. than just a
• Brad might not be too simple majority
happy about that! of the votes…
• He’s got a lot more votes
than Chuck and Dot… • And for WVS#3,
• Maybe if the QUOTA was we’ll try
higher, he would gain something
some “power?”
different...
14
ALICE owns 45 shares
BRADLEY owns 30 shares
CHARLES owns 15 shares
DOROTHY owns 10 shares
•WVS#2: • WVS#2:
The number of votes [70: 45, 30, 15, 10]
will be the same as in
WVS#1,
but the QUOTA will be
70 votes.
(70 votes needed to pass a
motion).
15
ALICE owns 45 shares
BRADLEY owns 30 shares • WVS#3:
CHARLES owns 15 shares
DOROTHY owns 10 shares [?: 4, 3, 1, 1]
• the simple majority
•WVS#3: will be the first integer
that is more than half
The number votes will of:
be the tens digit of the
4+3+1+1 = 9
# of shares of stock,
• 9/2 = 4.5
and the QUOTA will be • 5 votes will be the
the simple majority. simple majority.
(notice that a vote of 5 to 4
would have a winner and
loser.)
[5: 4, 3, 1, 1]
16
Complete wvs charts
• If available, use the • Notice the similarities
handout(s) and and differences in the
complete a chart STATUS and PASS/FAIL
that looks at the way columns
each of these wvs • You should see that in
work. Method #1 and
• If a handout is not Method #3, the
available, create the STATUS columns are
charts on your own. the same for
corresponding
coalitions.
• These are called
EQUIVALENT wvs
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WVS power • Winning Coalitions
have a weight that is
at least the quota.
• Notice that a
They pass a motion.
coalition of voters
can either: • Losing Coalitions
have a weight that is
• WIN
less than the quota.
• LOSE or
• *Blocking Coalitions
• *BLOCK have enough votes
when voting on a to keep their
motion opponents from
passing a motion.
18
WVS power How many votes are needed to
BLOCK a measure??
• If you know the TOTAL # of
• Winning Coalitions
have a weight that
votes in the system
is at least the • and you know the QUOTA
quota. • It is always true that a
• Losing Coalitions LOSING coalition will have a
have a weight that
is less than the weight that is less than or
quota. equal to the
• *Blocking TOTAL - QUOTA
Coalitions have
• So, in order to BLOCK, a
enough votes to
keep their coalition must have MORE
opponents from THAN the (TOTAL - QUOTA)
winning.
19
WVS power (ex) In WVS#1,
How many votes are • the total # votes = 100
needed to BLOCK
a measure?? • the quota = 51
• In order to A coalition with (100 - 51) =
BLOCK, a
coalition must 49 votes is a losing
have coalition
MORE THAN the
(in fact, any coalition with less
(TOTAL - QUOTA)
than 49 votes loses, too!)
Any coalition (voting
AGAINST a motion) that
has MORE THAN 49
votes can BLOCK a
motion from being
passed. 20
WVS power REMEMBER…
(ex) In WVS#1,
A BLOCKING COALITION
Any coalition with
MORE THAN is voting AGAINST the
49 votes can motion under consideration.
BLOCK a
motion from
being passed. This coalition might have a
weight equal to the quota,
but it might not!
It just depends on how
many votes the coalition
voting FOR the motion has!
21
WVS power •The quota is 70
In WVS#2, how
•The total # of votes is 100
many votes are
needed to To BLOCK a motion, a
BLOCK a coalition must have
measure from MORE THAN (100 - 70)
being passed? that is MORE THAN 30
votes.
(Look at the coalitions you’ve
marked BLOCK to confirm this)
22
WVS power
• Your text does not mention BLOCKING
coalitions
• However, in order to thoroughly discuss
POWER in a wvs, it is important to
remember that whatever coalition has
enough votes to attain their goal, has some
type of power.
• A WINNING COALITION: has the power to PASS
a measure.
• A BLOCKING COALITION: has the power to
prohibit a measure from being passed (blocking
power.)
23
WVS power
• Winning or Blocking Power is related to the
status of the entire coalition.
• We must also discuss the POWER OF
INDIVIDUAL VOTERS in the system and
within their coalitions.
• A voter whose weight is greater than or equal
to the quota is called a DICTATOR.
• A DICTATOR is a one-person winning
coalition.
24
WVS power
• A voter whose weight is greater than or equal to the
quota is called a DICTATOR.
• A DICTATOR is a one-person winning coalition.
• A voter has VETO POWER if (s)he can
prevent a measure from being passed even
if all others vote for the measure.
• A voter with VETO POWER is a one-person
blocking coalition.
• A voter whose vote is NEVER necessary in
any winning coalition is called a DUMMY
VOTER (if that voter leaves, the remaining
members will still make up a winning coalition) .
25
WVS: The power of a voter...
• In any wvs, there will • We will look at TWO
always be the measures of power.
argument about • The first is called the
“fairness.” BANZHAF POWER
• Do some voters have INDEX.
too much power? • This index measures
• Are there any voters the power of a voter
who have no power by describing how
at all? many winning
• How can we describe coalitions the voter is
the “power of a voter” critical for the success
in a system? of.
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End of show #1
Going on?...
Weighted voting system:
Show #2: Banzhaf Power
Distribution
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