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IntroductionNotation and TerminologyApportionment MethodsFairnessU.S. ConstitutionApportionment TimelineNotesApportionmentJWRJuly 27, 2006Contents1 Introduction 12 Notation and Terminology 23 Apportionment Methods 54 Fairness 7A U.S. Constitution 8B Apportionment Timeline 9C Notes 111 IntroductionThe constitution says that apportionment of representatives to a s tate shouldbe proportional to its population. If taken literally, this would mean giving frac-tional representatives, so some method of converting fractions to whole numbersmust be used. Various methods of doing this have been used over the years.Often the result seem s unfair to many people and bitter disputes have resulted.Here is a list of surprising things that happened as a result of using one methodor another.1.1. The Alabama Paradox. An increase in the total number of seats to beapportioned can cause a state to lose a seat. The Alabama Paradox first surfacedafter the 1870 census. With 270 members in the House of Representatives,Rhode Island got 2 representatives but when the House size was increased to 280,Rhode Island lost a seat. After the 1880 census, C. W. Seaton (chief clerk of U.S. Census Office) computed apportionments for all House sizes between 275 and350 members. He then wrote a letter to Congress pointing out that if the Houseof Representatives had 299 seats, Alabama would get 8 seats but if the Houseof Representatives had 300 seats, Alabama would only get 7 seats. The methodused at the time was Hamilton’s method explained below in paragraph ??.11.2. The Population Paradox. An increase in a state’s population can causeit to lose a seat. The Population Paradox was discovered around 1900, when itwas shown that a state could lose seats in the House of Representatives as a resultof an increase in its population. (Virginia was growing much faster than Maine,but Virginia lost a seat in the House while Maine gained a seat.) The methodused at the time was Hamilton’s method explained below in paragraph ??.1.3. The New States Paradox. Adding a new state with its fair share of seatscan affect the number of seats due other states. The New States Paradox wasdiscovered in 1907 when Oklahoma became a state. Before Oklahoma becamea state, the House of Re presentatives had 386 seats. Comparing Oklahoma’spopulation to other states, it was clear that Oklahoma should have 5 seats so theHouse size was increased by five to 391 seats. The intent was to leave the numberof seats unchanged for the other states. However, when the apportionment wasmathematically recalculated, Maine gained a seat (4 instead of 3) and New Yorklost a seat (from 38 to 37). The method used at the time was Hamilton’s methodexplained below in paragraph ??.1.4. The Quota Paradox. The percentage of a state’s population timesthe house size is called the quota for that state; the lower quota is the quotarounded down and the upper quota is the quota rounded up. It can happenthat a state receives a number of seats which is smaller than its lower quota orlarger than its upper quota. For example, in the apportionment based on the1820 census New York had a population of 1,368,775, the total U.S. populationwas 8,969,878, and the size of the house was 213. New York’s quota was thusq =1,368,7758,969,878× 213 = 32.503. However the apportionment method used at thetime (Jefferson’s method explained below in paragraph ??) awarded New York34 seats.2 Notation and Terminology2.1. Assume that there are n states numbered i = 1, 2, . . . , n. We call a vectorp = (p1, p2, . . . , pn) ∈ Rn+of positive numbers a population vector; pirepresents the population of theith state. We suppose that h denotes the total number of seats to be allocatedand call it the house size. An apportionment of house size h among n statesis a vectora = (a1, a2, . . . , an) ∈ Nnof nonnegative integers such thath = a1+ a2+ ··· + an.The number aiis the number of seats assigned to the ith state by the appor-tionment. LetNnh:= {a ∈ Nn: a = (a1, a2, . . . , an), a1+ a2+ ··· + an= h}2denote the set of all apportionments of house size h among n states. Roughlyspeaking an apportionment method is a function assigns an apportionment toeach population vector. However, both to allow for ties and to ease the ex-position, we allow multivalued functions. Thus an apportionment functionfor n states and house size h is a function M which assigns to each populationvector p a nonempty set M(p) ⊂ Nnhof apportionments of house size h amongn states. An apportionment system is a function which assigns apportion-ment function for n states and house size h to each positive integer n and eachnonnegative integer h.2.2. The quantity¯pi=pipis the ith states percentage of the total population. (In order to allow forrescaling, we do not impose the condition that the population piof the ith stateis an integer.) The vector¯p = (¯p1, ¯p2, . . . , ¯pn)is called the normalized population vector. For each nonnegative integer hthe quantityqi= ¯pi× his called the quota of the ith state for house size h: it represents the number ofseats that would get if fractional numbers were allowed. The numbers bqic anddqie are c alled the lower and upper quotes respectively. Thus bqic and dqie arenonnegative integers andbqic ≤ qi≤ dqie.3 Apportionment Methods3.1. Hamilton’s Method. For each i calculate the quota qi= pi/h of the ithstate. The sum h0= bq1c + bq2c + ··· + bqnc will satisfy h − n ≤ h0< h. (Theequality h = h0occurs only in the unlikely event that all the quotas qiare wholenumbers.) Award the ith state either its lower quota bqic or its upper quotadqie. The states which receive their upp er quota are the h − h0states with thelarger fractional parts qi− bqic.3.2. Divisor Methods. These methods all involve a notion of “rounding”as explained below. The idea is that the (exact) quota for each state can bedetermined by dividing the population of each state by the number of personsrepresented (ideally) by each representative, i.e.qi=pip× h =piλ0, where λ0=ph.For a divisor λ which is near λ0the numbers pi/λ will be close to quotasqi= pi/λ0but rounding pi/λ might give a different whole number than rounding3pi/λ0. The various divisor methods differ in how they define “rounding”. Thevarious notions of rounding involve the choice, for each nonnegative integer aof a number µ(a) between a and a + 1. Then the result of rounding a numberq is bqc if bqc ≤ q < µ(bqc) and dq e if


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UW-Madison MATH 141 - Apportionment

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