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Hamilton’s Method and the Quota Rule & The Alabama and Other Paradoxes

Hamilton’s Method and the Quota Rule & The Alabama and Other Paradoxes. Notes 18 – Sections 4.2 & 4.3. Essential Learnings. Students will understand and be able to use the Hamilton Method of apportionment and understand the Quota Rule.

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Hamilton’s Method and the Quota Rule & The Alabama and Other Paradoxes

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  1. Hamilton’s Method and the Quota Rule&The Alabama and Other Paradoxes Notes 18 – Sections 4.2 & 4.3

  2. Essential Learnings • Students will understand and be able to use the Hamilton Method of apportionment and understand the Quota Rule. • Students will understand the Alabama Paradox and other paradoxes that exist with apportionment.

  3. Hamilton’s Method Hamilton’s method can be described quite briefly: Every state gets at least itslower quota.As many states as possible get their upper quota, with the one withhighest residue (i.e.,fractional part) having first priority, theone with second highest residue second priority, and so on.

  4. HAMILTON’S METHOD Step 1 Calculate each state’s standard quota. Step 2 Give to each state its lower quota. Step 3 Give the surplus seats (one at a time) to thestates with the largest residues(fractional parts)until there are no more surplus seats.

  5. Example 4.4 Parador’s Congress(Hamilton’s Method) Recalling the Parador Congress, we will use Hamilton’s Method to determine how to divide the seats. Table 4-6 shows all of the details. (Reminder: The standard quotas in the second row of the table were computed in Example 4.3.)

  6. Example 4.4 Parador’s Congress(Hamilton’s Method) Hamilton’s method (also known as Vinton’s method or the method of largest remainders) was used in the United States only between 1850 and 1900.

  7. Hamilton’s Method Hamilton’s method is still used today to apportion the legislatures of Costa Rica, Namibia, and Sweden. At first glance, Hamilton’s method appears to be quite fair. It could be reasonably argued that Hamilton’s method has a major flaw inthe way it relies entirely on the size of the residues without consideration of whatthose residues represent as a percent of the state’s population. In so doing,Hamilton’s method creates a systematic bias in favor of larger states over smallerones.

  8. Hamilton’s Method This is bad–a good apportionment method should be population neutral,meaning that it should not be biased in favor of large states over small ones orvice versa. Hamilton’smethod has two important things going for it:(1) It is very easy to understand, and (2) it satisfies an extremely importantrequirement for fairness called the quota rule.

  9. QUOTA RULE No state should be apportioned a number of seats smaller than its lowerquota or larger than its upper quota. (When a state is apportioned a numbersmaller than its lower quota, we call it a lower-quota violation; when a state isapportioned a number larger than its upper quota, we call it an upper-quotaviolation.)

  10. Hamilton’s Method & the Quota Rule An apportionment method that guarantees that every state will be apportioned either its lower quota or its upper quota is said to satisfy the quota rule. Itis not hard to see that Hamilton’s method satisfies the quota rule: Step 2 ofHamilton’s method hands out to each state its lower quota. Right off the bat thisguarantees that there will be no lower-quota violations. In Step 3 some states getone extra seat, some get none; no state can get more than one. This guaranteesthat there will be no upper-quota violations.

  11. Alabama Paradox - Hamilton’s Method The most serious (in fact, the fatal) flaw of Hamilton’s method is commonlyknown as the Alabama paradox. The Alabama paradox occurs whenan increase in the total number of seats being apportioned, in and of itself, forces astate to lose one of its seats.

  12. Example 4.5 More Seats Means Fewer Seats The small country of Calavos consists of three states: Bama, Tecos, and Ilnos. With a total population of 20,000 and 200 seats in the House of Representatives,the apportionment of the 200 seats under Hamilton’s method is shown in Table 4-7.

  13. Example 4.5 More Seats Means Fewer Seats Now imagine that overnight the number of seats is increased to 201, butnothing else changes. Since there is one more seat to give out, the apportionmenthas to be recomputed.

  14. Example 4.5 More Seats Means Fewer Seats Here’s the new apportionment (Hamilton’s method) for a House with 201 seats. (Notice that for M = 200,the SD is 100; for M = 201,theSD drops to 99.5.)

  15. Example 4.5 More Seats Means Fewer Seats The shocking part of this story is the fate of Bama, the “little guy.” When theHouse of Representatives had 200 seats, Bama got 10 seats, but when the numberof seats to be divided increased to 201, Bama’s apportionment went down to9 seats.

  16. Example 4.5 More Seats Means Fewer Seats How did this paradox occur? Notice the effect of the increase in M on thesize of the residues: In a House with 200 seats, Bama is at the head of the priorityline for surplus seats, but when the number of seats goes up to 201, Bama getsshuffled to the back of the line.

  17. Arithmetic of the Alabama Paradox Example 4.5 illustrates the quirk of arithmetic behind the Alabama paradox:When we increase the number of seats to be apportioned, each state’s standardquota goes up, but not by the same amount. As the residues change, some statescan move ahead of others in the priority order for the surplus seats. This can result in some state or states losing seats they already had.

  18. The Population Paradox Sometime in the early 1900s it was discovered that under Hamilton’s method astate could potentially lose some seats because its population got too big! Thisphenomenon is known as the population paradox. Thepopulation paradox occurs when state Aloses a seat to state B even though thepopulation of A grew at a higher rate than the population of B.

  19. Example 4.6 A Tale of Two Planets In the year 2525 the five planets in the Utopia galaxy finally signed a peace treatyand agreed to form an Intergalactic Federation governed by an Intergalactic Congress. This is the story of the two apportionments that broke up the Federation.

  20. Example 4.6 A Tale of Two Planets Part I. The Apportionment of 2525. The first Intergalactic Congress was apportionedusing Hamilton’s method, based on the population figures (in billions) shown inthe second column of Table 4-10. There were 50 seats apportioned.

  21. Example 4.6 A Tale of Two Planets Part I. The Apportionment of 2525.

  22. Example 4.6 A Tale of Two Planets Let’s go over the calculations. Since the total population of the galaxy is 900billion, the standard divisor is SD = 900/50 = 18 billion. Dividing the planetpopulations by this standard divisor gives the standard quotas shown in the thirdcolumn of Table 4-10. After the lower quotas are handed out (column 4), thereare two surplus seats. The first surplus seat goes to Conii and the other one toEllisium. The last column shows the apportionments. (Keep an eye on the apportionments of Betta and Ellisium.)

  23. Example 4.6 A Tale of Two Planets Part II. The Apportionment of 2535. After 10 years of peace, all was well in theIntergalactic Federation. The Intergalactic Census of 2535 showed only a fewchanges in the planets’ populations–an 8 billion increase in the population ofConii, and a 1 billion increase in the population of Ellisium. The populations ofthe other planets remained unchanged from 2525. Nonetheless, a new apportionment was required.

  24. Example 4.6 A Tale of Two Planets Part II. The Apportionment of 2535. Here are the details of the 2535 apportionmentunder Hamilton’s method.

  25. Example 4.6 A Tale of Two Planets Notice that the total population increased to 909billion, so the standard divisor for this apportionment was SD = 909/50 = 18.18. The one remarkable thing about the 2535 apportionment is that Ellisium losta seat while its population went up and Betta gained that seat while its populationremained unchanged!

  26. Hamilton’s Method-Population Paradox UnderHamilton’s method, it is possible for a state with a positive population growth rate tolose one (or more) of its seats to another state with a smaller (or zero) populationgrowth rate.

  27. The New-States Paradox In 1907 Oklahoma joined the Union. Prior to Oklahoma becoming a state, therewere 386 seats in the House of Representatives. At the time the fair apportionmentto Oklahoma was five seats, so the size of the House of Representatives waschanged from 386 to 391. The point of adding these five seats was to give Oklahomaits fair share of seats and leave the apportionments of the other states unchanged.

  28. The New-States Paradox However, when the new apportionments were calculated, another paradox surfaced: Maine’s apportionment went up from three to four seats and New York’s wentdown from 38 to 37 seats. The fact that the addition of a new state with itsfair share of seats can, in and of itself, affect the apportionments of other states iscalled the new-states paradox.

  29. Example 4.7 Garbage Time The Metro Garbage Company has a contract to provide garbage collection andrecycling services in two districts of Metropolis, Northtown (with 10,450 homes)and the much larger Southtown (89,550 homes). The company runs 100 garbagetrucks, which are apportioned under Hamilton’s method according to the numberof homes in the district. A quick calculation shows that the standard divisor isSD = 1000homes, a nice, round number which makes the rest of the calculations(shown in Table 4-12) easy.

  30. Example 4.7 Garbage Time As a result of the apportionment, 10 garbage trucks areassigned to service Northtown and 90 garbage trucks to service Southtown.

  31. Example 4.7 Garbage Time Now imagine that the Metro Garbage Company is bidding to expand itsterritory by adding the district of Newtown (5250 homes) to its service area. In itsbid to the City Council the company promises to buy five additional garbagetrucks for the Newtown run so that its service to the other two districts is not affected. But when the new calculations (shown in Table 4-13) are carried out, thereis a surprise: One of the garbage trucks assigned to Southtown has to be reassigned to Northtown!

  32. Example 4.7 Garbage Time Noticethat the standard divisor has gone up a little and is now approximately 1002.38.

  33. Hamilton’s Method There are two key lessons we should take from this section: In terms offairness, Hamilton’s method leaves a lot to be desired; and the critical flaw inHamilton’s method is the way it handles the surplus seats. Clearly, there must bea better apportionment method.

  34. Assignment 4.2 – p. 145: 11, 13, 14, 15, 19, 20 4.3 – p. 145: 6, 10, 21

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