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- WELCOME TO A LESSON ON THE NEW STATES PARADOX OF APPORTIONMENT.
IF A NEW STATE IS ADDED,
AS WELL AS A PROPORTIONAL NUMBER OF NEW SEATS,
IT SEEMS LOGICAL THAT THE NUMBER OF SEATS OF EXISTING STATES
WOULD NOT CHANGE.
HOWEVER, THIS IS NOT ALWAYS THE CASE.
ADDING A NEW STATE WITH ITS FAIR SHARE OF SEATS
CAN AFFECT THE NUMBER OF SEATS APPORTIONED TO OTHER STATES.
WHEN A STATE GAINS OR LOSES A SEAT IN THIS SITUATION,
THIS IS CALLED THE NEW STATES PARADOX.
THE NEW STATES PARADOX WAS DISCOVERED IN 1907
WHEN OKLAHOMA BECAME A STATE.
BEFORE OKLAHOMA BECAME A STATE,
THE HOUSE OF REPRESENTATIVES HAD 386 SEATS.
COMPARING OKLAHOMA'S POPULATION TO OTHER STATES,
IT WAS CLEAR THAT OKLAHOMA SHOULD RECEIVE 5 SEATS.
AS A RESULT, THE HOUSE SIZE WAS INCREASED BY 5 TO 391 SEATS.
THE INTENT WAS TO LEAVE THE NUMBER OF SEATS UNCHANGED
FOR THE OTHER STATES.
HOWEVER, WHEN THE 391 SEATS WERE APPORTIONED,
MAINE GAINED A SEAT, AND NEW YORK LOST A SEAT.
MAINE WENT FROM 3 SEATS TO 4 SEATS,
AND NEW YORK WENT FROM 38 SEATS TO 37 SEATS.
THIS IS THE NEW STATES PARADOX.
LET'S TAKE A LOOK AT OUR OWN EXAMPLE.
LET'S SUPPOSE A CERTAIN CITY HAS TWO DISTRICTS "A" AND B.
THE POPULATIONS OF THE DISTRICTS WERE GIVEN IN THE TABLE BELOW.
THERE ARE 100 POLICE OFFICERS THAT ARE EMPLOYED
TO PATROL THESE TWO DISTRICTS.
APPORTION THE OFFICERS USING HAMILTON'S METHOD.
FIRST, NOTICE HOW THE TOTAL POPULATION IS 100,000,
AND THE NUMBER OF OFFICERS IS 100.
SO THE STANDARD DIVISOR, AS WE SEE HERE,
IS EQUAL TO 1,000.
SO TO FIND THESE QUOTAS HERE,
WE DIVIDE THE DISTRICT POPULATION BY 1,000,
WHICH IS EQUIVALENT TO MOVING A DECIMAL POINT TO THE LEFT
THREE PLACES.
FOR THE NEXT STEP, WE REMOVE THE DECIMAL PART OF THE QUOTA
FOR THE INITIAL ALLOCATION, OR LOWER QUOTA.
SO DISTRICT "A" RECEIVES 10, DISTRICT B RECEIVES 89.
BUT NOTICE HOW THIS SUM IS 99,
AND WE HAVE 100 OFFICERS TO APPORTION.
SO WE GIVE THE EXTRA OFFICER TO THE DISTRICT
THAT HAS THE LARGEST DECIMAL PART TO ITS QUOTA.
SO COMPARING .45 TO .55, BECAUSE .55 IS LARGER,
THE EXTRA OFFICER IS ASSIGNED TO DISTRICT B.
THEREFORE THE FINAL APPORTIONMENT IS 10 OFFICERS
FOR DISTRICT "A" AND 90 OFFICERS FOR DISTRICT B,
GIVING US A TOTAL OF 100 OFFICERS.
NOW LET'S SUPPOSE THE CITY EXPANDS TO COVER A NEW DISTRICT,
CALLED DISTRICT C.
NOTICE DISTRICT C HAS A POPULATION
THAT'S ROUGHLY 5,000,
WHICH IS A 5% INCREASE FROM THE PREVIOUS POPULATION.
AS A RESULT, THE CITY HIRES FIVE NEW POLICE OFFICERS
TO COVER THE NEW DISTRICT,
BRINGING THE TOTAL TO 105 OFFICERS.
SO THE POPULATION INCREASED BY APPROXIMATELY 5%,
AND THE NUMBER OF OFFICERS ALSO INCREASED BY 5%.
NOW WE'LL APPORTION THE 105 OFFICERS
USING HAMILTON'S METHOD.
TO FIND THE STANDARD DIVISOR
WE TAKE THE TOTAL POPULATION OF ALL THE DISTRICTS,
AND DIVIDE BY THE NUMBER OF OFFICERS, WHICH IS NOW 105,
GIVING US A DIVISOR OF APPROXIMATELY 1,002.381.
LET'S GO AHEAD AND CHECK THESE QUOTAS HERE.
WE TAKE THE POPULATION OF EACH DISTRICT
AND DIVIDE BY THE NEW DIVISOR.
SO 10,450 DIVIDED BY 1,002.381 GIVES THE QUOTA
OF APPROXIMATELY 10.43.
FOR DISTRICT B--
WE HAVE A QUOTA OF APPROXIMATELY 89.34.
AND FINALLY, FOR DISTRICT C, THE NEW DISTRICT--
WE HAVE A QUOTA OF APPROXIMATELY 5.24.
AND NOW FOR THE INITIAL ALLOCATION, OR THE LOWER QUOTA,
WE REMOVE THE DECIMAL PART.
SO WE HAVE 10, 89, AND 5.
NOTICE HOW THIS GIVES A TOTAL OF 104 OFFICERS,
BUT BECAUSE WE HAVE 105 OFFICERS TO APPORTION,
WE APPORTION THE EXTRA OFFICER TO THE DISTRICT
THAT HAS THE LARGEST DECIMAL VALUE OF ITS QUOTA.
SO COMPARING .43, .34, AND .24,
NOTICE DISTRICT "A" HAS THE LARGEST DECIMAL VALUE.
SO DISTRICT "A" RECEIVES THE EXTRA OFFICER,
SO THE FINAL APPORTIONMENT IS 11, 89, AND 5,
BRINGING THE TOTAL TO 105.
NOTICE IF WE COMPARE THIS FINAL APPORTIONMENT HERE
TO THE FINAL APPORTIONMENT WHEN THERE WERE ONLY TWO DISTRICTS,
NOTICE DISTRICT "A" WENT FROM HAVING 10 OFFICERS
TO 11 OFFICERS,
AND DISTRICT B WENT FROM HAVING 90 OFFICERS TO 89 OFFICERS.
THIS IS AN EXAMPLE OF THE NEW STATES PARADOX.
I HOPE YOU FOUND THIS HELPFUL.