Methods of allotment are mathematical techniques used to apportion resources such as constabulary officers in a certain metropolis or congressional seats. These techniques are rather complicated and are based on several variables depending on which method one is taking to utilize. Two of the most celebrated methods for work outing apportionment jobs are known as The Hamilton Method and The Huntington-Hill Principle. In this paper we will get down by treatment the Hamilton Method by feigning that 10 different provinces are to be assigned 100 congressional seats by utilizing allotment. The Hamilton Method of Apportionment

The Hamilton Method is a “common sense” method that Alexander Hamilton used to allocate the really first United States Congress. With that being said. one could feign that they have to split or allocate 100 congressional seats among 10 provinces of the Union. To make this utilizing The Hamilton Method the population for each of the 10 provinces would hold to be known. Then the population for all 10 provinces would necessitate to be totaled. Once this sum is received. so the entire population will necessitate to be divided into each single provinces population. For illustration. province 1 has a population of 1500 and province 2 has a population of 2000 for a population sum of 3500 ( Pirnot. n. d. ) . 1500/3500 = 0. 42857143 ( province 1 )

2000/3500 = 0. 57142857 ( province 2 )
Next the denary topographic points in the Numberss above will necessitate to be moved two topographic points to the right and unit of ammunition to the nearest 100 if necessary. This should give the replies 42. 86 for province 1 and 57. 14 for province 2. These Numberss are known as your Hamilton Numberss. Now in The Hamilton Method the Numberss before the decimal are known as the Integers and they represent how many seats each province gets. and the denary Numberss are known as the fractional Numberss determine who will acquire the staying seats. if there are any. The staying seats are given to the provinces that have the largest fractional Numberss foremost and work their manner down. Therefore. presuming there are a 100 seats to be apportioned. so 42 seats will travel to province one and 57 seats will travel to province 2. However. we must retrieve that there are 100 seats to apportion. 42+57 = 99. hence there is 1 staying place to be apportioned. Since province 1 has a fractional portion of. 86 and province 2 has a fractional portion of 14. province 1 receives the excess place because it has the larger fractional figure ( Pirnot. n. d. ) .

Now allow us acquire back to the original job of 10 provinces allocating 100 seats. Sing how this is a instead big job with big Numberss one might desire to utilize a reckoner or dispersed sheet to find how many seats are assigned to each start. By utilizing a dispersed sheet one can see that the seats are assigned as followed:

Population

Hamilton

Assign Additional
State
Insert Below
% Representation
Numbers
Integer Part
Fractional Part
Members Manually

The inquiry now becomes. are these seats all apportioned reasonably? To happen out we need to cognize the “Average Constituency” of each province. ” The Average Constituency measures the equity of an allotment ( Pirnot. n. d. pg. 534 ) . ” To happen the Average Constituency one would take the population of a province and split it by the assigned seats. and the comparison them to find equity. Giving an illustration from the computations above. one can see that province 1 has a population of 15475 and province 2 has a population of 35644. State 1 has 3 assigned seats and province 2 has 7 ( Pirnot. n. d. ) . 15457/3 = 5158

Components

35644/7 = 5092 Components
In comparing. merely by looking at the figure of constitutional poetries the figure of seats ; one would presume that the provinces are non truly represented reasonably. because province one has more components and fewer representatives than province 2. Below is the mean constituency of all 10 provinces in the given job above ( Pirnot. n. d. ) .

Having these Numberss to compare helps us acquire a better apprehension of how ailing some province can be represented. One would wish to believe that holding the same sum of components in each province would be the sure-fire reply to work outing that job. but harmonizing to ( Pirnot. n. d. . pg. 535 ) . “it is normally non possible to accomplish this ideal when devising and existent allotment. ” Therefore we should at least attempt to do mean constituencies every bit equal as possible. One can really mensurate this by utilizing what is called “Absolute Unfairness” ( Pirnot. n. d. ) .

Absolute Unfairness

Absolute Unfairness is defined as being “the difference in mean constituencies” ( Pirnot. n. vitamin D ) . To happen the absolute unfairness of two of the provinces given supra. we should utilize this simple expression. ( mean
constituencies of province A ) – ( mean constituencies of province B ) =

Now to utilize this expression to see if any of the provinces in our job has any absolute unfairness. we will pick provinces 3 and 2 to utilize as a comparing. ( province 3 ) 5486 – ( province 2 ) 5092 = 394 Absolute Unfairness

One can now see that the absolute unfairness of constituencies between provinces 3 & A ; 2 is 394. Therefore. harmonizing to absolute unfairness these two provinces are non every bit represented. The constituencies would hold to hold been the same in both provinces in order for the provinces to be every bit represented. and this is seldom the instance. With that being said. absolute unfairness is non what one would desire to utilize to mensurate the unfairness of two allotments. because it truly demo the instability of an allotment of two provinces. In other words. absolute unfairness might give some people the incorrect decision about the instability. Meaning. merely because there is a big absolute unfairness Department of Energy non foretell a greater instability. In all actuality. the sized of the province demands to be taken into consideration as good. when mensurating unfairness. For illustration. in a province with a larger sum of electors like Texas. if a politician loses by 100. 000 to 1. 500. 000 ballots. it is considered a close race. in a little town election where the ballots tally as 100 to 30 so the difference is considered to be rather big. This is why it is of import to mensurate the “relative unfairness” ( Pirnot. n. vitamin D ) .

Relative Unfairness

“Relative unfairness considers the size of constituencies in a ciphering absolute unfairness ( Pirnot. n. d. pg. 356 ) . ” To cipher the comparative unfairness of dealt out seats between two provinces one would utilize this expression. absolute unfairness of allotment / smaller mean constituency of the two provinces =

So. utilizing the two provinces were given to calculate out the absolute unfairness we can state that 0. 08 is the comparative unfairness of the two provinces. 394 ( absolute unfairness ) / 5092 ( province 2 ) = 0. 07737628 ( rounded to the nearest 100 ) = 0. 08 comparative unfairness
To acquire a comparing we will utilize two other provinces. State 1 has 5158 norm constituencies. and province 4 has 5196 for a sum of 38 absolute unfairness. Remember to deduct the province with the smallest sum of constituencies from the larger state’s constituencies to acquire the absolute unfairness. To happen the comparative unfairness. take the absolute unfairness and split it by the province with the lowest constituency figure which was province 1. 38/5158 = 0. 007367197 ( rounded to the nearest 100 ) = 0. 007 comparative unfairness
The comparative unfairness of provinces 1 and 4 is 0. 007. Therefore in comparing with provinces 2 and 3’s larger comparative unfairness of 0. 08. it tells us that there is more of an unjust allotment for provinces 2 and 3 than the provinces of 1 and 4. In other words. when comparing comparative unfairness the larger figure in comparing agencies it’s apportioned more below the belt. However. due to the fact that all of these computations were based on The Hamilton Method all of the information could perchance alter if there were a sudden population alteration due to growing. This is called a population paradox ( Pirnot. n. d. ) .

Population Paradox

A population paradox occurs when one province grows in population faster than the other. and the province with the faster growing loses a place or representative to the other province ( Pirnot. n. d. ) . For illustration. province 6 has a population of 85663 and province 8 has a population of 84311 for a entire population of 169974. Now we want to delegate these two provinces 100 seats of Congress utilizing The Hamilton Method. First take the entire population and divide by 100 seats to acquire our criterion factor ( Pirnot. n. d. ) . 169976/100 = 1699. 74 ( standard factor )

Now divide each province by 1699. 74 to acquire your Hamilton Number. 85663/1699. 74 = 50. 4 ( province 6 )
84311/1699. 74 = 49. 6 ( province 8 )
Hamilton Numbers Lower Quota ( Integer ) Fractional Part Assigned Seats province 6: 50. 6 50 0. 4 50 province 8: 49. 6 49 0. 6 50 = 100
seats ( Notice that the sum for the whole number or lower quota is 99. so hence there was one excess place to delegate and it went to the province with the highest fractional portion which was province 8. )

Now if we increase province 6’s population by 1000 and province 8’s population by 100 you will acquire a population paradox. To happen out how this happens you will necessitate to do the same computations by utilizing The Hamilton Methods. except you will necessitate to increase the population of both provinces to acquire the new sums. whole numbers. fractional parts. and assigned seats ( Pirnot. n. d. ) . ( province 6 ) 85663 + 1000 = 86663 ( new population )

( province 8 ) 84311 + 100 = 84411 ( new population )
86663 + 84411 = 171074 ( entire population )
171074/100 = 1710. 74 ( standard factor )
86663/ 1710. 74 = 50. 66 ( Hamilton figure )
84411 / 1710. 74 = 49. 34 ( Hamilton figure )
Notice that the fractional portion has changed for the two provinces Hamilton Numberss. Therefore since province 6 now has the larger fractional portion due to the population alteration it will take the excess place from province 8 for a sum of 100 seats. State 6 will hold 51 and province 8 will hold 49. To happen out which province received the greatest sum of growing we merely split the growing by the original population ( Pirnot. n. d. ) . 1000/85663 = 1. 16 % ( province 6 ) and 100/84311 ( province 8 ) = 1. 19 % One can now see that this is a population paradox that occurs when utilizing The Hamilton Method. because the province that had the most growing in population lost a place to the province with the least of sum of growing due to how the fractional portion of the Hamilton Numberss changed. However. a population paradox is non the lone paradox associated with The Hamilton Method. The Alabama Paradox has besides shown its ugly face when utilizing The Hamilton Method of allotment ( Pirnot. n. d. ) .

Alabama Paradox

In 1870. after the nose count. the Alabama paradox surfaced. This occurred when a house of 270 members increased to 280 members of the House of Representatives doing Rhode Island to lose one of its 2 seats. Later on after the nose count a adult male by the name of C. W. Seaton calculated the allotments for all House sizes that ranged from 275 to 350 members. Harmonizing to ( ua. edu. n. d. ) . “He so wrote a missive to Congress indicating out that if the House of Representatives had 299 seats. Alabama would acquire 8 seats but if the House of Representatives had 300 seats. Alabama would merely acquire 7 seats. ” This became known as the Alabama paradox. It is merely when the entire figure of seats to be apportioned additions. and in bend causes a province to lose a place. There is a method called the Huntington-Hill Principle that helps avoid the Alabama paradox. This method merely apportions the new seats when the House of Representatives additions in size. This is what avoids the Alabama paradox. To use the Huntington-Hill Principle we would utilize this simple algebraic expression below for each of the provinces for comparing that are in inquiry of deriving the excess place ( Pirnot. n. d. ) . ( population of Y ) ^2 / Y * ( y + 1 )

Let us state that Y has a population of 400 and allow Y equal 5. and let’s say that X has a population of 300 and allow X equal 2. Now let us see which one of these gets the excess place. ( 400 ) ^2 / 5 * ( 5 + 1 ) and ( 300 ) ^2 / 2 * ( 2 + 1 )

160. 000 / 5 * 6 = 90. 000 / 2 * 3 =
= 160. 000 / 30 = 90. 000 / 6
= 5333. 33 = 15. 000
By utilizing the Huntington-Hill Principle method of allotment we can now compare the two provinces to see which one will acquire the excess place. Notice that province Ten with the Huntington -Hill figure of 15. 000 is great than that of province Y. therefore province X should acquire the excess place. With this being said. if I were to utilize allotment as my manner of delegating seats to the House of Representatives. I would decidedly take to utilize The Huntington-Hill Principle method of allotment ( Pirnot. n. d. ) .

Apportionment is a great manner to accomplish just representation every bit long as we are non utilizing the Hamilton Method. The Hamilton Method has the possibility of cause three types of paradoxes: the Alabama paradox. the population paradox. and the new provinces paradox. Even though the Hamilton Method does non go against the quota regulation. avoiding these paradoxes are more of import when seeking to give equal representation to each province of the Union. There are other apportionment methods that are every bit every bit great as The Huntington-Hill Principle. such as Webster’s method ( Pirnot. n. d. ) .

Webster’s Method of allotment

What truly sets Webster’s method apart from Huntington-Hill is that Webster uses modified divisor alternatively of a criterion factor to cipher what is called a modified quota or Integer. A modified factor is a factor that is smaller than the standard factor. A modified quota is a quota that is larger than the standard quota. One would fundamentally pick a figure smaller than the criterion factor and work their manner down until they end up with one that will give them and modified quota. Once that quota or Integer is found so it will necessitate to be rounded either up or down depending on the figure ( the criterion manner of rounding ) to find who will acquire the allotted seats. Webster’s method is really precisely like Huntington-Hill except for the rounding portion. and it was the apportionment method used until it was replaced by Huntington-Hill ( Pirnot. n. d. )

Decision

Apportionment methods are a great manner to every bit split certain Numberss of substances among changing Numberss. every bit long as one stays off from the Hamilton Method. Sure the Hamilton Method is rather simple to utilize. but causes many jobs such as paradoxes. The Alabama paradox. the population paradox. and the new province paradox are among the 1s that the Hamilton Method can do. This causes provinces to lose seats due to new Representatives. new population growing and even a new boundary line or province fall ining the Union. Thankfully there were some people out at that place that were smart plenty to come up with new methods of allotment that eliminated the issues of the paradoxes. such as the Huntington-Hill method and Webster’s method. Both of these methods are the best allotment methods out at that place to assist do certain that provinces are represented every bit by Congress. . and sing the fact that I live in a really hapless. poorness afflicted province. I want to do certain that our province gets the best representation possible. so that possibly our representatives will be able to listen to all of their components and do something to assist hike our economic system. increase employment rates. and convey people out of poorness.

Mentions
Apportionment Paradoxes. Alabama Paradox. Retreived from hypertext transfer protocol: //www. ctl. ua. edu/math103/apportionment/paradoxs. htm # Exemplifying the Alabama Paradox Pirnot. T. Mathematics All Around. Fourth Addition. Apportionment. Retrieved from hypertext transfer protocol: //media. pearsoncmg. com/aw/aw_pirnot_mathallaround_4/ebook/pma04_flash_main. hypertext markup language? chapter=null & A ; page=531 & A ; anchory=null & A ; pstart=null & A ; pend=null