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Apportionment Schemes

Apportionment Schemes. Dan Villarreal MATH 490-02 Tuesday, Sept. 15, 2009. But first…a quick PSA. What is Apportionment?.

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Apportionment Schemes

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  1. Apportionment Schemes Dan Villarreal MATH 490-02 Tuesday, Sept. 15, 2009

  2. But first…a quick PSA Apportionment Schemes

  3. What is Apportionment? • The apportionment problem is to round a set of fractions so that their sum is maintained at original value. The rounding procedure must not be an arbitrary one, but one that can be applied constantly. Any such rounding procedure is called an apportionment method. Apportionment Schemes

  4. Example In the 1974-75 NHL season, the Stanley Cup Champion Philadelphia Flyers won 51 games, lost 18 games, and tied 11 games. Won: 51  63.75%  64% Lost: 18  22.5%  23% Tied: 11  13.75%  14% But this adds up to 101%, an impossibility! Apportionment Schemes

  5. Dramatis Personae • George Washington • Alexander Hamilton • Thomas Jefferson • Daniel Webster • Delaware • Virginia Apportionment Schemes

  6. The Constitution • Amendment 14, Section 2: “Representatives shall be apportioned among the several States according to their respective numbers, counting the whole number of persons in each State” • Article I, Section 2: • “The actual Enumeration shall be made within three Years after the first Meeting of the Congress of the United States, and within every subsequent Term of ten Years, in such Manner as they shall by Law direct. The Number of Representatives shall not exceed one for every thirty Thousand, but each State shall have at Least one Representative” Apportionment Schemes

  7. The First Apportionment • For the third session of Congress (1793-1795) • House of Representatives set at 105 • 15 states • U.S. Population: 3,615,920 • 3,615,920 105 = 34,437 people/district Apportionment Schemes

  8. Standard Divisors • 34,437 is our standard divisor for 1790. • More generally, SDt = Where HSt is the size of the House of Representatives(or whatever overall body) for year t. Poptotal HSt Apportionment Schemes

  9. Popi SDt Quotas • The number of Congressional districts a state should get is its quota: Qi = • Take Delaware, for example… Apportionment Schemes

  10. If only it were that easy… THERE’S NO SUCH THING AS .613 CONGRESSPERSONS. Hence the need for apportionment schemes, a way to map the quotas in R onto apportionments in Z. Apportionment Schemes

  11. If only it were that easy… Apportionment Schemes

  12. More on quotas • The lower quota is the quota rounded down (or the integer part of the quota): LQi = ⌊Qi⌋ • The upper quota is the quota rounded up: UQi = ⌈Qi⌉ = ⌊Qi⌋ + 1 Apportionment Schemes

  13. If only it were that easy… Apportionment Schemes

  14. Alexander Hamilton • 1755-1804 • One author of Federalist Papers • First Secretary of the Treasury • Most importantly for our purposes, devised the Hamilton Method for apportioning Congressional districts to states Apportionment Schemes

  15. The Hamilton Method • State i receives either its lower quota or upper quota in districts; those states that receive their upper quota are those with the greatest fractional parts Apportionment Schemes

  16. Back to 1790 Apportionment Schemes

  17. If only it were that easy… Apportionment Schemes

  18. If only it were that easy… Apportionment Schemes

  19. Apportionment Schemes

  20. Back to Square One • President Washington vetoed the Apportionment Bill because he believed, following the counsel of Edmund Randolph and Thomas Jefferson, that it was unconstitutional: ADE 2 1 PopDE 55,540 30,000 = < Apportionment Schemes

  21. The Alabama Paradox • The Hamilton Method was Congress’s preferred method of apportionment from 1850 to 1900. • In 1881, the Alabama Paradox was first discovered. • The Census Bureau, as a matter of course, calculated apportionments for a range of House sizes; in this case, 275-350 • Something interesting and very weird happened between the tables for HS = 299 and 300… Apportionment Schemes

  22. The Alabama Paradox • US population in 1880 was 49,369,595 Apportionment Schemes

  23. The Alabama Paradox • US population in 1880 was 49,369,595 • For HS = 299, SD = 165,116 Apportionment Schemes

  24. The Alabama Paradox • US population in 1880 was 49,369,595 • For HS = 299, SD = 165,116 • For HS = 300, SD = 164,565 Apportionment Schemes

  25. The Alabama Paradox • US population in 1880 was 49,369,595 • For HS = 299, SD = 165,116 • For HS = 300, SD = 164,565 Apportionment Schemes

  26. The Alabama Paradox • US population in 1880 was 49,369,595 • For HS = 299, SD = 165,116 • For HS = 300, SD = 164,565 Apportionment Schemes

  27. The Alabama Paradox • US population in 1880 was 49,369,595 • For HS = 299, SD = 165,116 • For HS = 300, SD = 164,565 Apportionment Schemes

  28. The Alabama Paradox • US population in 1880 was 49,369,595 • For HS = 299, SD = 165,116 • For HS = 300, SD = 164,565 Apportionment Schemes

  29. The Alabama Paradox • US population in 1880 was 49,369,595 • For HS = 299, SD = 165,116 • For HS = 300, SD = 164,565 Apportionment Schemes

  30. The Alabama Paradox • US population in 1880 was 49,369,595 • For HS = 299, SD = 165,116 • For HS = 300, SD = 164,565 Apportionment Schemes

  31. The Alabama Paradox • US population in 1880 was 49,369,595 • For HS = 299, SD = 165,116 • For HS = 300, SD = 164,565 Apportionment Schemes

  32. The Alabama Paradox • US population in 1880 was 49,369,595 • For HS = 299, SD = 165,116 • For HS = 300, SD = 164,565 Apportionment Schemes

  33. The Alabama Paradox • US population in 1880 was 49,369,595 • For HS = 299, SD = 165,116 • For HS = 300, SD = 164,565 Apportionment Schemes

  34. Back to 1793… • This particular issue with the Hamilton Method was not discovered until 1881, but the Constitutional constraints meant that it could not be used in 1793. • A new method was proposed by Thomas Jefferson: the Jefferson Method. Apportionment Schemes

  35. Thomas Jefferson • Biographical Information: You know this all already… • Had the good fortune never to take a class in Morton Hall Apportionment Schemes

  36. Popi d The Jefferson Method • Rather than use the standard divisor SD, the Jefferson Method uses the population of the smallest district, d. • Each state receives an adjusted quota; this will need to be rounded down the actual apportionment: Ai = ⌊⌋ Apportionment Schemes

  37. The Jefferson Method • In 1793, Jefferson used d = 33,000, so AVA = ⌊630,560 / 33,000⌋ = ⌊19.108⌋ = 19 ADE = ⌊55,540 / 33,000⌋ = ⌊1.683⌋ = 1 • But how do we determine d in the first place? Apportionment Schemes

  38. Finding the Critical Divisor • Start with the lower quota of each state; this is its tentative apportionment, ni. • Next, find the critical divisor for each state: Popi ni + 1 • For example, dVA= 630,560 / (18 + 1) = 33,187 dDE = 55,540 / (1 + 1) = 27,770 di = Apportionment Schemes

  39. The Critical Divisor • The critical divisor for each state is the divisor for which the state will be entitled to ni + 1 seats. • For example, if d > 27,770, Delaware gets only 1 seat, but for d ≤ 27,770, Delaware gets 2. • But then Virginia gets ⌊630,560 / 27,770⌋ = ⌊22.707⌋ = 22 seats. This will surely result in an overfull House • Thus, d will need to be greater than 27,770 Apportionment Schemes

  40. The Jefferson Method • Step 1: Assume a tentative apportionment of the lower quota for each state: ni = LQi • Step 2: Determine the critical divisor di for each state and rank by di • Step 3: If any seats remain to be filled, grant one to the state with the highest di; recompute di for this state since its ni has now increased by 1. • Step 4: Iterate Step 3 until the House is filled. Apportionment Schemes

  41. The Jefferson Method • This method actually was used for the 1793 apportionment, and it resulted in Virginia receiving 19 seats to Delaware’s one. • Used until about 1840 • Not subject to the Alabama paradox • But fails to satisfy the quota condition… Apportionment Schemes

  42. The Quota Condition • The quota condition is twofold: • 1. No state may receive fewer seats than its lower quota • 2. No state may receive more seats than its upper quota • The Jefferson Method does just fine with 1, but not 2 Apportionment Schemes

  43. Example • U.S. population in 1820 was 8,969,878, with a House size of 213, so SD = 8,969,878 / 213 = 42,112 • New York had a population of 1,368,775: QNY = 1,368,775 / 42,112 = 32.503 • So if the quota condition was satisfied, New York’s delegation should be either 32 or 33 • Using the Jefferson Method and d = 39,900, we actually get 34 seats for New York Apportionment Schemes

  44. What’s the Problem? • The Jefferson Method always skews in favor of the large states. • Let ui = pi / d be the state’s adjusted quota. Then Ai = ⌊ui⌋. Now compare ui with the state’s quota: M = = / = × = • Then ui = M * Qi => ai = ⌊M *Qi⌋ • The rich only get richer… ui Popi Popi Popi SD SD Qi d SD d Popi d Apportionment Schemes

  45. The Webster Method • Daniel Webster devised an apportionment method that was similar in nature to Jefferson’s, but that did not unconditionally favor large states. • Used for 1840-1850 reapportionments, then 1900-1930 Apportionment Schemes

  46. The Webster Method • Step 1: Determine SD, and find the quota Qi for each state i. • Step 2: Round each quota up or down and let this be the tentative apportionment ni for each state. • Step 3: Determine the total apportionment at this point. 3 cases: • 1. The total apportionment equals HS • 2. The total apportionment is greater than HS • 3. The total apportionment is less than HS Apportionment Schemes

  47. Adjusting the Apportionment • If we have an overfill, at least one or more seats needs to be pared off. Let the critical divisor be di- = pi / (ni-1/2). The state with the smallest di- will be the next to lose a seat. • Conversely, if we have an underfill, we need to add more seats. Let the critical divisor be di+ = pi / (ni + 1/2). The state with the smallest di+ will be the next to gain a seat. • Iterate either process until done. Apportionment Schemes

  48. Large State Bias • How does the Webster Method avoid susceptibility to the large-state bias exhibited by the Jefferson Method? • We get a similar expression for M: M = SD/d Apportionment Schemes

  49. Large State Bias • M > 1 when there is an underfill, thus in this circumstance, the larger states are more likely to receive another seat • But when there is an overfill and we must subtract, M < 1, and the larger states are more likely to get a seat subtracted • Equally likely to get an overfill or underfill • Thus, equally likely that the Webster Method will favor neither large nor small Apportionment Schemes

  50. Webster Method Timeline Jefferson Method Hamilton Method 1790 1840 1850 1900 Webster Method Hill-Huntington Method 1940 Present 1900 Apportionment Schemes

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