Transcript Weighted Voting Systems

Weighted Voting
• When we try to make collective decisions, it is
only natural to consider how things are done in
society.
• We are familiar with voting for class president –
one for per person, winner is one with most votes.
• The electoral college seeks to give more power to
states having more population
• We want to know who has the most power – as
that could influence whose mind we try to change
or whether the system is fair.
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Weighted Voting
• We are trying to decide whether or not a
measure passes.
• The coalition is the group of people that can
get the measure passed.
• Not everyone has the same “clout”.
Modeled as some having multiple votes.
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Notation
• A weighted voting system is characterized
by three things — the players, the weights
and the quota. The voters are the players (P1
, P2, . . ., PN). N denotes the total number of
players. A player's weight (w) is the number
of votes he controls. The quota (q) is the
minimum number of votes required to pass a
motion. [q:w1,w2…wn]. Normally we
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w  w  ...  w 
q

require that
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as we can say more interesting things that
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way.
1
2
n
Power
• A player's power is defined as that player's ability to influence
decisions.
• The power of a coalition is not simply determined by its size.
• Consider the voting system [6: 5, 3, 2].
• Notice that a motion can only be passed with the support of
P1. In this situation, P1 has veto power. A player is said to
have veto power if a motion cannot pass without the support
of that player. This does not mean a motion is guaranteed to
pass with the support of that player – as player 1 doesn’t have
enough votes by himself.
• Who has the most power? How is power divided between the
players with 3 and 2?
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Weighted Voting Systems – Terms
i. Coalition: any subset of a group of voters that bands
together to either support a measure.
ii. Winning/Losing Coalition: a coalition that has
enough votes to pass a measure is a winning
coalition, otherwise it is a losing coalition.
iii. Dummy: a voter in a winning coalition whose vote
isn’t needed to pass the measure.
iv. Voters Weight: the number of votes each voter has.
v. Quota: the number of votes, q, necessary to pass
a measure.
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Weighted Voting Systems - Terms
vi. Notation for voting system: [q : w1 , w2 ,..., wn ] where q is
the quota, wi are the individual weights of the voters,
and n is the number of voters.
vii. Requirements:
as otherwise definition of
1.
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q  w1  w2  ...  wn  dictator is problematic
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2. q  w  w  ...  w  as no point is having q larger
1
2
n
viii. Changing q affects the way power is distributed.
ix. Blocking Coalition: subset of voters opposing a
motion with enough votes to defeat it. Any coalition
with weight  w  q . w  w1  w2  ... wn 
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Weighted Voting Systems – Terms
x. Dictator: voter whose voting weight meets or exceeds
the quota for passing a measure. All other voters
are dummies.
xi. Veto Power: a voter who has enough votes to block a
measure is said to have veto power. A voter with
weight  w  q . A dictator automatically has veto
power.
xii. Critical Voter: in any winning coalition, he is the
voter whose votes are essential to win.
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Power
• Now let us look at the weighted voting system [10: 11, 6,
3]. With 11 votes, P1 is called a dictator. A player is typically
considered a dictator if his weight is equal to or greater than the quota.
The difference between a dictator and a player with veto power is that a
motion is guaranteed to pass if the dictator votes in favor of it.
• The dictator has veto power. The measure passes if and only if he votes
for it. Since the quota must be more than half the total, a dictator
always has veto power.
• A dummy is any player, regardless of his weight, who has
no say in the outcome of the election. A player without any
say in the outcome is a player without power. Dummies
always appear in weighted voting systems that have a
dictator (provided the quota is more than half total) but also
occur in other weighted voting systems
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Power
• Consider the voting system [8: 5, 3, 2]. Which are
dictators? have veto power? are dummies?
5 and 3 have veto power. 2 is a dummy
• Consider the voting system [8: 9, 3, 2]. Which are
dictators? have veto power? are dummies?
• Consider the voting system [20:10,10,9]. Which are
dictators? have veto power? are dummies?
• Consider the voting system [7:4,2,1]. Which are dictators?
have veto power? are dummies?
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Banzhaf power index
(sometimes called Penrose-Banzhaf index)
• Designed to quantify the power a voter has
• defined by the probability of changing an
outcome of a vote
• To calculate the power of a voter using the Banzhaf index,
list all the winning coalitions, then count the critical voters.
A critical voter is a voter who, if he changed his vote from
yes to no, would cause the measure to fail. A voter's power
is measured as the fraction of all swing votes that he could
cast
• Warning: in our electoral college of 50 states, there are
51,476,301,254,318 winning coalitions!
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An example Game Theory and Strategy by Phillip
D. Straffin:
• [6; 4, 3, 2, 1]
• The winning groups, with underlined critical voters,
are as follows:
• AB, AC, ABC, ABD, ACD, BCD, ABCD
• Notice we assume that we only worry about what
ONE player does in each case.
• There are 12 total critical votes, so by the Banzhaf
index, power is divided thus.
• A = 5/12 B = 3/12 C = 3/12 D = 1/12
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Consider the U.S. Electoral College.
• There are a total of 538 electoral votes. A majority vote is considered
270 votes. The Banzhaf Power Index would be a mathematical
representation of how likely a single state would be able to swing the
vote. For a state such as California, which is allocated 55 electoral
votes, they would be more likely to swing the vote than a state such as
Montana, which only has 3 electoral votes.
• Example: The United States is having a presidential election between a
Republican and a Democrat. For simplicity, suppose that only three
states are participating: California (55 electoral votes), Texas (34
electoral votes), and New York (31 electoral votes).
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Consider having republicans win. The democrats winning is
similar. Need 61 votes to win.
Calif Texas
(55) (34)
R R
R R
R D
D R
NY
(31)
R
D
R
R
R
D
States that could
votes votes swing the vote
120
0none
89 31California, Texas
86 34California , New York
65 55Texas , New York
Power, each state has 1/3
Consider a different set of states
Need 55 to win
California
(55)
R
R
R
R
Texas
(34)
R
R
D
D
Ohio
(20)
R
D
R
D
States that could
R D swing
109 0California
89 20California
75 34California
55 54California
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factor twenty
difference
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Shapley-Shubik Power Index:
i. Shapley-Shubik Power Index:
a. Permutation: total number of ways n things can
n!
be taken r at a time . P n 
.
r
(n  r )!
Order is important in a permutation.
n
b. Pn
 n!
is used to find the number of ways to
order n elements in a set.
2. 1st voter in a permutation whose vote would make
the coalition a winning coalition is called a pivotal
voter.
3. Shapley-Shubik Power Index is fraction of
permutations in which a voter is pivotal.
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Shapley-Shubik Power Index:
Given a voting system [q : w1 , w2 ,..., wn ] create a ShapleyShubik table: For this example use {A,B,C} with the voting system
[3:2,1,1]
P 3  3!  6 Pivotal voter is underlined.
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Banzhaf
AB
AC
ABC
4 1 1
 , , 
6 6 6
1. Count number of times A,B, and C are pivotal voters. Divide each
value by 6 to get the Shapley-Shubik Power Index:
4 1
2. Voter A is 6  6  4
times more powerful than B or C. Voter A
has 4/6 or 66.67% of the power in this voting system.
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factor twenty
difference.
Quite similar.
Factor of 4.1-4.3
difference.
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Try this one
• Suppose decisions are made by majority
rule in a body consisting of A, B, C, D, who
have 3, 2, 1 and 1 votes, respectively. The
majority vote threshold is 4. There are 24
possible orders for these members to vote:
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A=1/2, B=C=D = 1/6
[4:3,2,1,1]
ABCD
ABDC
ACBD
ACDB
ADBC
ADCB
BACD
BADC
BCAD
BCDA
BDAC
BDCA
CABD
CADB
CBAD
CBDA
CDAB
CDBA
DABC
DACB
DBAC
DBCA
DCAB
DCBA
B has no more power than C or D
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Shapley-Shubik Power Index:
1. Sometimes permutations are too large to list all of them
so we do it by grouping.
Consider the voting system [5:3,1,1,1,1,1,1]. 7! = 5040
GSSSSSS 3456789
SSSSGSS 1234789
SGSSSSS 1456789
SSSSSGS 1234589
SSGSSSS 1256789
SSSSSSG 1234569
SSSGSSS 1236789
G is pivotal 3/7 of the time. S is pivotal in (4/7)/6=2/21 of the
time. Therefore, the Shapley-Shubik Power Index is
3 2 2 2 2 2 2 
 , , , , , , 
 7 21 21 21 21 21 21
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