Apport ionment Q uot a M et hods T erminology: Seat s It ems t o - PDF document

Apport ionment Q uot a M et hods T erminology: Seat s – It ems t o be apport ioned House Size – Number of Seat s St at es – Part ies t o whom seat s are apport ioned Populat ion – M easurement of st at e’s size T ot al Populat ion – Sum of populat ions of all st at es Nat ural D ivisor = T ot al Populat ion House Size St at e’s Nat ural Q uot a = St at e’s Populat ion Nat ural D ivisor

Alt ernat ive: St at e’s Nat ural Q uot a = St at e’s Populat ion T ot al Populat ion · ( House Size) O bvious Solut ion: Simple Rounding – Round each st at e’s nat ural quot a t o nearest int eger. Problem: House Size may not come out cor- rect ly.

Hamilt on’s M et hod – M et hod of L argest Fract ional Part s • Calculat e t he nat ural quot a for each st at e. • Round each st at e’s nat ural quot a down t o t he nearest int eger t hat does not exceed t he quot a. • Assign ext ra seat s t o t he st at es whose quo- t as have t he largest fract ional part s.

L owndes’ M et hod – M et hod of L argest Relat ive Fract ions • Calculat e t he nat ural quot a for each st at e. • Round each st at e’s nat ural quot a down t o t he nearest int eger t hat does not exce ed t he quot a. • Assign ext ra seat s t o t he st at es whose quo- t as have t he largest relat ive fract ional part s. T he relat ive fract ional part is t he fract ional part of t he nat ural quot a divided by it s int e- gral part . Definition 1 (Quota Property). An appor- t ionment is said t o sat isfy t he Q uot a Property if each st at e’s allocat ion diff ers from it s nat ural quot a by less t han one.

Early D ivisor M et hods Definition 2 (Jefferson’s Method). Find a divisor such t hat t he correct number of seat s is allot t ed when all t he result ing modifi ed quot as are rounded down t o t he nearest whole num- ber. T o fi nd a divisor t hat would make a st at e’s modifi ed quot a a cert ain value, divide it s pop- ulat ion by it s desired quot a. In ot her words, t hreshold divisor for n seat s = populat ion . n Possible Problem: Violat ion of t he Q uot a Prop- erty. All divisor met hods violat e t he Q uot a Property. Smaller divisors result in larger modifi ed quot as and larger divisors result in smaller modifi ed quot as.

Jeff erson’s M et hod has a bias in favor of t he larger st at es. Definition 3 (Webster’s Method). Find a di- visor such t hat t he correct number of seat s is allocat ed when t he result ing modifi ed quot as are rounded in t he nat ural way. t hreshold divisor for n seat s = populat ion . n − 0 . 5

� � Apport ionment in T oday’s House of Represent at ives Definition 4 (Hill-Huntington Method). Find a divisor such t hat t he correct number of seat s is allocat ed when t he result ing modifi ed quot as are rounded using t he geomet ric mean ( n − 1) n for rounding between n − 1 and n . t hreshold divisor for n seat s = populat ion ( n − 1) n Alt ernat ives • Adam’s M et hod – Q uot a’s are rounded up from t he modifi ed quot a. B ias in favor of smaller st at es. • D ean’s M et hod – Cut off for rounding is ( n − 1) n n − 0 . 5 .

n − 0 . 5 = ( n − 1) n n Not e: n − 1 < ( n − 1) · n − 0 . 5 = n · n − 1 n − 0 . 5 < n . • Condorcet ’s M et hod – Q uot as are rounded up at 0 . 4.

Search for an Ideal Apport ion- ment M et hod Definition 5 (Quota Property). An appor- t ionment is said t o sat isfy t he Q uot a Property if each st at e’s allocat ion diff ers from it s nat ural quot a by less t han one. Theorem 1. No divisor met hod sat isfi es t he Q uot a Property. Definition 6 (House Size Property). No st at e ever loses a seat when t he size of t he house in- creases. Theorem 2. Every divisor met hod sat isfi es t he House Size Property. Definition 7 (Population Property – Popu- lation Monotonicity). For a fi xed house size, no st at e whose populat ion increases ever loses a seat t o anot her st at e whose populat ion de- creases. Theorem 3. An apport ionment met hod sat is- fi es t he Populat ion Property if and only if it is a “ generalized” divisor met hod, where t he rounding met hod may depend on t he number of st at es and t he house size. Consequent ly, no apport ionment met hod sat isfi es bot h t he Q uot a Property and t he Populat ion Property.