Question 1171056: A weighted voting is an electoral system in which the voters do not have the same amount of influence over the result of an election. Each voting system can be described using the form {q: w1, w2, w3, w4,…,wn}
where q is the quota, w are the weights and n is the number of voters. The weights are listed in
numerical order, starting with the highest weight.
Banzhaf Power Index, named after John F. Banzhaf III, is a power index
defined by the probability of changing an outcome of a vote where voting rights
are not necessarily equally divided among the voters or shareholders. 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.
Flaws of Voting Systems
In 1948, Kenneth J. Arrow outlined various criteria for a fair voting
system and came up with the Arrow’s Impossibility theorem which states that
”It is mathematically impossible to create any system of voting with three or
more candidates that satisfy all four fairness criteria.”
These criteria are as follows:
Majority Criterion - If the candidate receives a majority of the votes, then that
candidate should be declared the winner.
Monotonicity Criterion - If candidate A wins an election and, in a subsequent
election, the only changes are changes in favor of
candidate A, then-candidate A should be declared the winner.
Head-to-Head Criterion - This is sometimes referred to as the Condorcet criterion. If candidate A wins when compared head-to-head with each of the other candidates, then-candidate A should be declared the winner.
Irrelevant Alternatives Criterion - If candidate A wins an election and, in a recount, the
only changes are that one or more of the losing
candidates withdraw the race, then candidate A should still be declared the winner.
Weighted Voting System
A weighted voting is an electoral system in which the voters do not
have the same amount of influence over the result of an election. Some voters
have more weight than the others. This type of voting is usually used in
shareholder meetings where the votes of the shareholders are weighted
according to the number of shares they owned. Another characteristic of
weighted voting system is the quota. It is the required minimum number of
votes to pass a resolution or a measure. Each voting system can be described
using the form
{q: w1, w2, w3, w4,…,wn}
where q is the quota, w are the weights and n is the number of voters. The
weights are listed in numerical order, starting with the highest weight. Some
examples of the voting systems are as follows:
Voting System Example
One person, one
vote
{ 4: 1, 1, 1, 1, 1, 1, 1}
In this system, the quota is 4 and there are 7
voters where every voter has one vote. To pass
a resolution, it requires 4 votes which are a
majority.
Dictatorship
{ 10: 11, 4, 3, 1, 1}
In this system, one of the voters has 11 votes
which is greater than the quota; hence, he
alone can pass a resolution. The remaining
voters, even if they combine their votes cannot
reach a quota.
Null System
{ 15: 5, 4, 2, 2, 1}
Even if all voters in this system combine their
votes, it cannot reach the quota, thereby, no
resolution can be passed.
Veto Power System
{ 30: 10, 8, 6, 4, 2}
The sum of all the votes in this system is equal
to the quota; hence, if anyone withdraws his
vote, then no resolution can be passed. This is
the case where everyone in the system has a
veto power.
A coalition in a weighted voting system is a set of voters in which they
vote the same way. A winning coalition is a set of voters in which the combined
votes are greater than or equal to the quota. A losing coalition is a set of voters
in which the combined votes are less than the quota. When a voter leaves a
winning coalition and turns it into a losing coalition, then that voter is called a
critical voter. The number of coalitions that can possibly form in a system with
n voters is 2
n - 1.
Example.
A corporation has four shareholders, A, B, C, and D, with 49, 48, 2 and
1 share, respectively. It uses the weighted voting system
{51 : 49, 48, 2, 1}
a. Determine the winning coalitions.
b. For each winning coalition, determine the critical voters.
Solution.
a. The winning coalitions are voters with total votes equal to or greater
than the quota.
Winning coalition Number of votes
{A, B} 97
{A, C} 51
{A, B, C} 99
{A, B, D} 98
{B, C, D} 51
{A, B, C, D} 100
b. The critical voter of a winning coalition is a voter that will turn it a loser
after leaving the coalition. The table below shows the critical voter for each winning
coalition.
Winning coalition Number of votes Critical voters
{A, B} 97 A, B
{A, C} 51 A, C
{A, B, C} 99 A
{A, B, D} 98 A, B
{B, C, D} 51 B, C, D
{A, B, C, D} 100 none
Problem:
1. Calculate, if possible, the Banzhaf power index for each voter. Round to the nearest hundredth.
a. {8: 4, 3, 2}
b. {15: 8, 6, 4}
c. {20: 7, 3, 2, 1}
d. {10: 7, 5, 1, 1}
e. {12: 14, 12, 4, 3, 1}
f. {5: 1, 1, 1, 1, 1, 1}
g. {21: 21, 7, 3, 3, 1, 1}
h. {12: 6, 6, 4, 3, 1}
i. {80: 60, 40, 30, 25, 5}
j. {85: 50, 40, 25, 5}
Which, if any, of the voting systems is a dictatorship?
Which, if any, of the voting systems is a veto power system?
Which, if any, of the voting systems is a null system?
Which, if any, of the voting systems is a one-person, one-vote system?
Answer by ikleyn(52898)   (Show Source):
You can put this solution on YOUR website! .
Hello,
please accept my congratulations (!)
Finally, under my guidance, you get the logically complete presentation of your problem.
Now, the rest step for you, is to read it attentively from the beginning to the end.
May be, several times, to better assimilate/adopt/adsorb the major conceptions and procedures.
As soon as you really adopt them, the rest is simple and elementary.
So simple and elementary that any help from outside is UNNECESSARY.
I am more than confident that everyone who is able to read this text in whole,
will be able to complete the assignment on his or her own (if not lose his mind :).
One more good news for you is THIS:
I found in the Internet another source which treats similar problems.
It is this pdf-file http://www.opentextbookstore.com/mathinsociety/current/WeightedVoting.pdf
It treats and teaches the subject from the very beginning conceptions to more complicated issues
step by step, in small logic paces.
So, boldly go forward and solve this elementary assignment.
You may keep me informed about your progress.
And one question, please, from my side to satisfy my curiosity:
at which high school / college / university / research institution / thinking tank
did you get this problem ?
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