Extension for 3/23

Weighted voting systems are voting systems based on the idea that not all voters are equal. Instead, it can be desirable to recognize differences by giving voters different amounts of say (weights) concerning the outcome of an election. This is in contrast to normal parliamentary procedure, which assumes that each member's vote carries equal weight. This type of voting system is used in everything from shareholder meetings, where votes are weighted by the number of shares that each shareholder owns, to the United States Electoral College. This typer of voting has been used elsewhere in the world throughout history.

The Roman assemblies provided for weighted voting after the person's tribal affiliation and social class (i.e. wealth). Rather than counting one vote per citizen, the assemblies convened in blocs, with the plurality of voters in each bloc deciding the vote of the bloc as an entity. Men of certain tribes and a higher social standing convened in smaller blocs, thus giving their individual vote the effect of many poor citizens' votes. In the Plebeian Council, where only the plebs could participate, these effects were somewhat relaxed, thus making the decision to grant its decisions the full force of law controversial.

In several Western democracies, such as Sweden weighted voting preceded equal and universal suffrage, as well as women's suffrage, to different extents. In Sweden, universal and equal male suffrage to the lower house was introduced by Arvid Lindman's first cabinet, while voting for city and county councils, which indirectly decided the composition of the upper house, was graded along a 40-degree scale. Certain corporations also had votes of their own, thus multiplying the political strength of its owners. Weighted voting was abolished in Nils Eden's reforms of 1918-19, when female suffrage was also introduced.

Extension 3/20

             Just a little history lesson on our voting techniques  that we currently use today.  The Australian ballot also called secret ballot, is the system of voting in which voters mark their choices in privacy on uniform ballots printed and distributed by government or designate their choices by other secret means.  Victoria and South Australia were the first states that introduced secrecy of the ballot in 1856, and for that reason the secret ballot is referred to as the Australian ballot.

            The system had then spread to Europe and the United States to meet the growing public and parliamentary demand for protection of voters.  In the United States the Australian ballot system was widely adopted after the presidential election of 1884.  This was the election that came very close in votes and the deciding state was New York.  The winner was democrat Grover Cleveland.  He was the first democrat to win an election since 1856, before the American Civil War.

                Voting by the Australian ballot usually takes place in a prescribed manner.  The voting boxes are required to be a certain dimension and closed.  The containers are examined before the poll begins and obviously cannot be opened until the count begins.  The voter then marks a ballot, in a special booth, and (after the voter’s right to vote has been verified) the vote is placed inside the container.  If you have voted before than this is not something new to you.  I found it very interesting how the secrecy of voting became, and we the U.S. did not come up with this idea.  Like most things in the United States we adopt it from another country.

An Extension of Different Voting Methods

Posted for Whitney:

Today in class we learned about three different types of ways to evaluate votes. Weighted voting is a system where different voter’s votes are worth different amounts. A dictator is a voter who has the only vote that counts. A dummy is a voter who’s vote does not count. A weighted voting system has a quota. That is the number of votes needed to win or pass a motion and the weights of all the votes in the system. Dictators have all the power and dummies have none. There is also a power index.  A power index assigns a number to each voter that measures the amount of power that voter has in that system.

In 1961, In Southern Rhodesia they used the weighted voting system called cross voting. This system used two types of voters;  A rolls and B rolls. The A rolls were made up of individuals with European decent that lived in the African country.  The B rolls were made up of Native Africans. There were sixty-five total parliamentary seats.  At the conclusion of the election despite there being more Native Africans in Rhodesia the A rolls elected fifty out of sixty-six parliamentary seats. There were also a special group of voters. They were one thousand tribal leaders. They elected eight seats. In despite of the huge population of Native Rhodesians, they were left with the remainder of the few parliamentary seats.

This is an example of weighted voting because the A rolls votes disproportionately weighed more than the B rolls. The tribal leader’s votes also outweighed the B rolls. Do you think this is a fair system of voting?  

3/13 Summary

Posted on behalf of Hannah:

In class we learned about weighted voting. Weighted voting is a voting system in which not all voters’ votes count the same. A weighted voting system is expressed in numbers as a list. There is a quota. A quota is how many votes are needed for a motion to pass. The weights of the voters in the system are how many votes each has. For example [3: 3, 1, 1] 3 is the quota and 3, 1, 1 are the votes. 3 is the dictator because that’s the only vote that counts. The other numbers are dummy voters because they don’t count. In this example [8: 5, 3, 1] there is no dictator because no voter has enough votes to decide the election. Voter C (1) is a dummy because no matter how A and B vote, Cs vote will never matter. In this example [9: 5, 3, 1] there is no dictator or dummy voter. Everyone has the same power which is called an unanimous consent system. It will only pass if all voters have the same vote. The second topic that we learned about is Shapely-Shubik Power Index. A Power Index assigns a number to each voter in a system. The number indicates how much power that voter has in that system. This Power Index uses the idea of         roll-call vote (a vote in which each voter is called on to vote in some order). With this power index we consider every possible order in which the voters in the system could vote, and then we see which other voter casts the pivotal vote (the vote that decides the election). 

Extension_3-13-13

The Banzhaf Power Index

Tallying votes when the stakes are high is always going to be a complicated process.  Weighted voting is one of the better methods for obtaining the fairest outcome.  We’ve explored the current method, Shapley-Shubik index, which measures power by the number of different permutations in which each voter has the pivotal vote. 
 
A previous voting index used was the Banzhaf power index.  Instead of looking at the number of permutations, the Banzhaf method looks at the number of coalitions which a voter is critical.  A coalition is defined as a group of voters who all agree to vote a certain way to have a motion blocked or passed.  The Banzhaf method counted the critical voters – a voter who, if he/she changed their vote from yes to no, would cause the coalition to lose. The voter power is measured as the fraction of all swing votes that voter could cast. To calculate the power of a voter using the Banzhaf index, you would list all the winning coalitions and then count the critical voters in each.

An example of the Banzhaf method was the 2000 election when Bush-Cheney ran against Gore-Lieberman.   The Bush-Cheney team won with 271 electoral votes – only one vote over quota.  This means that almost every voter in the Bush-Cheney coalition was a critical voter. This is important because even the state with the smallest weight would still have a high amount of power in the 2000 election because they were a critical voter in that coalition. 

Summary from class Feb 13 2013

Today in class we covered another method of Apportionment called The Huntington-Hill Method.  This method was adopted until 1941 after several methods by several different politicians(in some case their own) failed to fix the problem.  I mean I can see how this confusion was going on with the addition of new states joining the Union after the Civil War and in fact at each census the population would change resulting in changes + or - for some states.  But the states would end up trying to sue or declare it unconstitutional but nothing ever really got anywhere.  So there are 435 seats in the house 200 held by Democrats, 232 by Republicans, and 3 vacant. 
The Huntington-Hill Method has the same first three steps as the Hamilton method.  
1.Find the total population 
2.Calculate the ideal district size, also known as the Standard Divisor: total population ÷ house size
3.Calculate the Quotas: state population ÷ standard divisor
But after that things start to get a little complicated and steps cant be skipped.
4.Then we find each states upper quota in which we round up, then each states lower quota in which we round down.
5.Then we take the geometric mean of each states upper and lower quota.
6.After that we then create a tentative apportionment as follows.  If the quota is less then Q* then we use the lower quota.  If the quota is greater than or equal to Q* use the upper quota.
7.Sometimes we might still have a seat to give out.  So lets find out who would get the seat.  Calculate the critical divisor for each state.  Remember if there are too many seats divide the state population by the geometric mean of the states tentative apportionment N and N-1.  If there are too few seats, do the same thing but use N and N+1.
8.If the tentative apportionment is too large the state with the smallest critical divisor loses a seat.  If the tentative apportionment were too small the seat with the largest critical divisor would gain a seat.
9.You may still need to do some work if the correct house size still hasnt been achieved.  We would recalculate the critical divisors of the states whose apportionment has changed and repeat the process.

Extension-- Huntington-hill method of apportionment

We first learned the simple method of Hamilton for apportionment, well at least simple compared to the die hard method created by Huntington and Hill. However they are both similar due to the first three steps. They are; find the total population, calculate the ideal district size, and calculate the quotas. Simple right? For that part yes, it gets much more difficult. From there you must find each states upper quota and lower quota then find the geometric mean. Then you create a tentative apportionment and calculate the critical divisor for each state. So on and so on.

Edward Vermilye Huntington was a great mathematician of his time. He spent most of his career at Harvard. During world war one he would actually create important statistics for generals of the army. It was among that time when he learned of the apportionment ideas of Joseph A. Hill. He began to study and revise Joseph's thoughts. He then came up with a method he referred to as "the method of equal proportions" otherwise known as the Huntington-Hill method. This is the current method used in congress today to decide how many seats each state receives. It was signed into law the year he retired from Harvard, November 15th, 1941.

After researching methods such as Webster's, Dean's, and Jefferson's, I believe this method shall be used for a very long time. Although great, not perfect. There were many extensive debates about which method to use. This went on for a long time at the beginning of the 20th century until the speaker of the house, Nicholas Longworth, suggested a team of mathematicians be sent out to find the right method. They then found Huntington's to work the best, thus the current method being used.