- WELCOME TO A LESSON ON THE ALABAMA PARADOX
OF APPORTIONMENT.
THE ALABAMA PARADOX
WAS THE FIRST OF THE APPORTIONMENT PARADOXES
TO BE DISCOVERED.
THE U.S. HOUSE OF REPRESENTATIVES
IS CONSTITUTIONALLY REQUIRED TO ALLOCATE SEATS
BASED UPON POPULATION COUNTS
WHICH ARE ACQUIRED EVERY 10 YEARS AFTER EVERY CENSUS.
THE SIZE OF THE HOUSE IS SET BY STATUTE.
AFTER THE 1880 CENSUS, C.W. SEATON,
CHIEF CLERK OF THE UNITED STATES CENSUS BUREAU,
COMPUTED APPORTIONMENTS FOR ALL HOUSE SIZES BETWEEN 275 AND 350.
IT WAS DISCOVERED THAT ALABAMA WOULD GET 8 SEATS
WITH A HOUSE SIZE OF 299,
BUT ONLY 7 SEATS WITH A HOUSE SIZE OF 300.
IN GENERAL, THE TERM ALABAMA PARADOX REFERS
TO ANY APPORTIONMENT SITUATION
WHERE INCREASING THE TOTAL NUMBER OF ITEMS
WOULD DECREASE ONE OF THE SHARES, AS WE SEE HERE,
WHICH HAPPENED TO ALABAMA.
NOTICE HOW WHEN THE HOUSE SIZE INCREASED BY 1,
FROM 299 TO 300,
ALABAMA ACTUALLY LOST A SEAT.
THE NUMBER OF SEATS WENT FROM 8 TO 7.
A SIMILAR EXERCISE BY THE CENSUS BUREAU AFTER THE 1900 CENSUS
COMPUTED APPORTIONMENT FOR ALL HOUSE SIZES BETWEEN 350 AND 400.
COLORADO WOULD HAVE RECEIVED 3 SEATS IN ALL CASES,
EXCEPT WITH A HOUSE SIZE OF 357,
IN WHICH CASE IT WOULD HAVE ONLY RECEIVED 2.
LET'S TAKE A LOOK AT A MORE DETAILED EXAMPLE.
HERE WE'LL USE HAMILTON'S METHOD TO APPORTION 10 SEATS
AND THEN 11 SEATS AMONG THREE STATES.
LOOKING AT THE TABLE BELOW,
NOTICE HOW THE TOTAL POPULATION WOULD BE 6 + 6 + 2, OR 14,
AND THEREFORE WHEN THERE ARE 10 SEATS,
THE STANDARD DIVISOR WOULD BE 14 DIVIDED BY 10, OR 1.4.
WHEN THERE ARE 11 SEATS THE STANDARD DIVISOR WOULD BE
14 DIVIDED BY 11, OR APPROXIMATELY 1.2727.
AND NOW TO REVIEW, REMEMBER TO FIND THE QUOTAS
WE TAKE THE POPULATION OF EACH STATE
AND DIVIDE BY THE STANDARD DIVISOR.
SO FOR EXAMPLE, WHEN THERE ARE 10 SEATS AND THE DIVISOR IS 1.4,
THE QUOTAS FOR STATE "A" AND B WOULD BE THE SAME,
BECAUSE THE POPULATION IS THE SAME.
THEY WOULD BE 6 DIVIDED BY 1.4, WHICH IS APPROXIMATELY 4.286,
AS WE SEE HERE.
AND THEN FOR STATE C WE WOULD HAVE 2 DIVIDED BY 1.4,
GIVING A QUOTA OF APPROXIMATELY 1.429.
AND NOW TO FIND THE INITIAL ALLOCATION,
ALSO CALLED THE LOWER QUOTA, WE REMOVE THE DECIMAL PART,
GIVING US AN ALLOCATION OF 4, 4, AND 1.
BUT BECAUSE THIS SUM IS 9, AND THERE ARE 10 SEATS,
THE EXTRA SEAT GOES TO THE STATE
THAT HAS THE LARGEST DECIMAL PART OF THE QUOTA,
WHICH WOULD BE STATE C.
SO STATE C GETS ONE MORE SEAT,
GIVING A FINAL ALLOCATION AS WE SEE HERE AS 4, 4, AND 2,
GIVING US A TOTAL OF 10 SEATS.
NOW LET'S COMPARE THIS TO WHEN THERE ARE 11 SEATS.
WE ALREADY KNOW OUR STANDARD DIVISOR
IS GOING TO BE APPROXIMATELY 1.2727.
LET'S GO AHEAD AND CHECK OUR QUOTAS.
SO AGAIN, FOR BOTH STATE "A" AND B WE'D HAVE 6 DIVIDED BY 1.2727,
SO THE QUOTA IS APPROXIMATELY 4.714.
AND THEN FOR STATE C WE WOULD HAVE 2 DIVIDED BY 1.2727,
SO THE QUOTA IS APPROXIMATELY 1.571.
AND NOW TO FIND THE INITIAL APPORTIONMENT OR LOWER QUOTA,
WE REMOVE THE DECIMAL PART, GIVING US AN ALLOCATION OF 4,
4, AND 1.
NOTICE HOW THIS SUM IS 9,
BUT NOW BECAUSE WE HAVE 11 SEATS WE HAVE 2 EXTRA SEATS,
WHICH WOULD BE AWARDED TO THE TWO STATES
WITH THE LARGEST DECIMAL PART OF THE QUOTA.
NOTICE HERE THAT WOULD BE STATE "A" AND B,
SO "A" GETS AN EXTRA SEAT AND SO DOES STATE B,
GIVING A FINAL ALLOCATION OR FINAL APPORTIONMENT
OF 5, 5, AND 1.
SO NOTICE HOW WHEN THE SEATS WENT FROM 10 TO 11,
STATES "A" AND B PICKED UP AN EXTRA SEAT,
BUT STATE C ACTUALLY LOST A SEAT.
THEY WENT FROM TWO SEATS TO ONE SEAT.
THIS IS AN EXAMPLE OF THE ALABAMA PARADOX.
THIS OCCURS BECAUSE INCREASING THE NUMBER OF SEATS
INCREASES THE FAIR SHARE OR QUOTA FASTER
FOR THE LARGER STATES THAN FOR THE SMALLER STATES.
IN PARTICULAR, THE LARGER STATES "A" AND B
HAD THEIR FAIR SHARE INCREASED FASTER THAN SMALLER STATE C.
THEREFORE THE FRACTIONAL OR DECIMAL PARTS OF THE QUOTA,
OR FAIR SHARE,
FOR "A" AND B INCREASED FASTER THAN THOSE FOR C.
IN FACT, THEY OVERTOOK C'S FRACTION,
CAUSING C TO LOSE ITS SEAT,
SINCE THE HAMILTON METHOD EXAMINES
WHICH STATES HAVE THE LARGEST FRACTION
OR DECIMAL PART OF THE QUOTA.
THE ALABAMA POPULATION AND NEW STATES PARADOXES
CAN OCCUR WITH THE HAMILTON METHOD.
THEY WILL NOT OCCUR WITH ANY OTHER APPORTIONMENT METHODS.
HOWEVER, THE DIVISOR METHOD MAY VIOLATE THE QUOTA RULE.
THE QUOTA RULE SAYS THAT THE NUMBER OF REPRESENTATIVES
A STATE RECEIVES
SHOULD BE WITHIN ONE OF THE STATE'S QUOTA OR FAIR SHARE.
I HOPE YOU FOUND THIS LESSON HELPFUL.