Largest remainder method

Transcription

- WELCOME TO A LESSON ON HAMILTON'S METHOD OF APPORTIONMENT. ALEXANDER HAMILTON PROPOSED THE METHOD THAT NOW BEARS HIS NAME. HIS METHOD WAS APPROVED BY CONGRESS IN 1791, BUT WAS VETOED BY PRESIDENT WASHINGTON. IT WAS LATER ADOPTED IN 1852, AND USED THROUGH 1911. SINCE HE WAS INTERESTED IN THE QUESTION OF CONGRESSIONAL REPRESENTATION, WE'LL USE THE LANGUAGE OF STATES AND REPRESENTATIVES. HAMILTON'S METHOD PROVIDES A PROCEDURE TO DETERMINE HOW MANY REPRESENTATIVES EACH STATE SHOULD RECEIVE. SO THE FIRST STEP IN HAMILTON'S METHOD IS TO DETERMINE HOW MANY PEOPLE EACH REPRESENTATIVE SHOULD REPRESENT. WE DO THIS BY DIVIDING THE TOTAL POPULATION OF ALL THE STATES BY THE TOTAL NUMBER OF REPRESENTATIVES. THIS ANSWER IS CALLED THE STANDARD DIVISOR, OR DIVISOR. STEP TWO, WE DIVIDE EACH STATE'S POPULATION BY THE DIVISOR TO DETERMINE HOW MANY REPRESENTATIVES IT SHOULD HAVE. WE WOULD RECORD THIS ANSWER TO SEVERAL DECIMAL PLACES, AND THIS ANSWER IS CALLED THE QUOTA. SINCE WE CAN ONLY ALLOCATE WHOLE REPRESENTATIVES, HAMILTON RESOLVES THE WHOLE NUMBER PROBLEM AS FOLLOWS: STEP THREE, WE CUT OFF THE DECIMAL PARTS OF ALL THE QUOTAS, AND THESE ARE CALLED THE LOWER QUOTAS. THEN WE ADD THE LOWER QUOTAS. THIS SUM WILL ALWAYS BE LESS THAN OR EQUAL TO THE TOTAL NUMBER OF REPRESENTATIVES. STEP FOUR, ASSUMING THAT THE TOTAL FROM STEP THREE WAS LESS THAN THE TOTAL NUMBER OF REPRESENTATIVES, ASSIGN THE REMAINING REPRESENTATIVES ONE EACH TO THE STATES WHOSE DECIMAL PART OF THE QUOTA WERE LARGEST UNTIL THE DESIRED TOTAL IS REACHED. WE DO WANT TO MAKE SURE THAT EACH STATE ENDS UP WITH AT LEAST ONE REPRESENTATIVE. LET'S TAKE A LOOK AT OUR FIRST EXAMPLE. AGAIN, THE FIRST STEP, WE WANT TO FIND THE DIVISOR, OR STANDARD DIVISOR. SO WE TAKE THE SUM OF THE POPULATION FROM ALL THE STATES, WHICH IS 189,000, AND DIVIDE BY THE TOTAL OF SEATS IN CONGRESS, WHICH IS 30. NOTICE HOW THIS GIVES US A DIVISOR OF 6,300. AND NOW, TO FIND THE QUOTAS WE TAKE EACH STATE POPULATION AND DIVIDE BY 6,300. SO FOR STATE "A" THE QUOTA WOULD BE 27,500 DIVIDED BY 6,300. IF WE ROUND TO FOUR DECIMAL PLACES, THE QUOTA FOR STATE "A" WOULD BE APPROXIMATELY 4.3651. FOR STATE B WE WOULD HAVE 38,300 DIVIDED BY 6,300, SO THE QUOTA FOR STATE B WOULD BE APPROXIMATELY 6.0794, AND SO ON. TO SAVE SOME TIME WE WON'T SHOW ALL THIS DIVISION. SO HERE ARE OUR QUOTAS TO FOUR DECIMAL PLACES, AND NOW FOR THE INITIAL APPORTIONMENT OR INITIAL ASSIGNMENT OF THE SEATS IN CONGRESS, WE REMOVE THE DECIMAL PARTS OF THE QUOTA. WHICH MEANS STATE "A" WOULD RECEIVE 4, STATE B WOULD RECEIVE 6, STATE C WOULD RECEIVE 7, AND STATE D WOULD RECEIVE 12. WE NOTICE HOW THE SUM HERE IS 29, AND WE HAVE A TOTAL OF 30 SEATS. WE NOW ASSIGN THE REMAINING SEAT TO THE STATE WHOSE QUOTA HAS THE LARGEST DECIMAL PART. NOTICE STATE "A" HAS THE LARGEST DECIMAL PART AT .3651, AND THEREFORE STATE "A" RECEIVES ONE MORE SEAT. WHICH MEANS FOR THE FINAL APPORTIONMENT STATE "A" RECEIVES 5 SEATS, STATE B RECEIVES 6, STATE C RECEIVES 7, AND STATE D RECEIVES 12. NOTICE HERE THE TOTAL IS 30. WE'VE USED ALL THE SEATS IN CONGRESS. AND HERE'S THE RESULT IN A NICE TABLE. LET'S TAKE A LOOK AT A SECOND EXAMPLE. HERE A TEACHER WISHES TO DISTRIBUTE 10 IDENTICAL PIECES OF CANDY AMONG 4 STUDENTS, BASED UPON HOW MANY PAGES OF A BOOK THEY READ LAST MONTH, USING HAMILTON'S METHOD. THE TABLE BELOW LISTS THE TOTAL NUMBER OF PAGES READ BY EACH STUDENT. SO IN THIS PARTICULAR QUESTION WE'RE ASKED TO FIND THE DIVISOR, THE QUOTA FOR ANTONIO, AND THE INITIAL APPORTIONMENT FOR ANTONIO. BUT WE'LL ACTUALLY GO AHEAD AND GO THROUGH THIS ENTIRE PROCESS FOR THIS PROBLEM. SO THE FIRST STEP IS TO FIND THE DIVISOR, SO WE'LL SET THIS UP AS A TABLE AS WE SEE HERE. NOTICE HOW WE ALREADY FOUND THE SUM OF THE TOTAL NUMBER OF PAGES, WHICH IS 1,120, AND THERE ARE 10 PIECES OF CANDY TO APPORTION. SO OUR DIVISOR IS 1,120 DIVIDED BY 10, OR 112. SO TO FIND THE QUOTA WE'LL TAKE THE NUMBER OF PAGES EACH STUDENT READ AND DIVIDE BY 112. SO THE QUOTE FOR ALAN WOULD BE 580 DIVIDED BY 112, WHICH WOULD BE APPROXIMATELY 5.1786. FOR ANTONIO THE QUOTA WOULD BE 230 DIVIDED BY 112, GIVING A QUOTA OF APPROXIMATELY 2.0536, AND SO ON. SO AGAIN, HERE ARE OUR QUOTAS. WE FOUND FROM THE PREVIOUS SLIDE THE DIVISOR IS 112, AND THE QUOTA FOR ANTONIO WE NOW KNOW IS 2.0536. AND NOW FOR THE INITIAL APPORTIONMENT WE REMOVE THE DECIMALS FROM THE QUOTA. SO ALAN WOULD RECEIVE 5 PIECES, ANTONIO WOULD RECEIVE 2, ALEX WOULD RECEIVE 1, AND LUCAS WOULD ALSO RECEIVE 1. SO THE INITIAL APPORTIONMENT FOR ANTONIO, WHICH IS WHAT OUR QUESTION ASKED FOR, IS 2. BUT NOTICE HOW THERE ARE 10 PIECES OF CANDY, AND THIS ONLY ADDS TO 9. SO GOING BACK TO THE ORIGINAL QUOTAS, SINCE ALEX HAS THE LARGEST DECIMAL PART IN HIS QUOTA, HE WOULD RECEIVE THE EXTRA PIECE OF CANDY. AND THEREFORE THE FINAL APPORTIONMENT WOULD BE 5, 2, 2, AND 1. NOTICE HOW HERE THE TOTAL IS 10 PIECES, SO WE'VE USED ALL THE CANDY. AND AGAIN, HERE IS THE RESULT IN A NICE TABLE. HAMILTON'S METHOD DOES SATISFY WHAT IS CALLED THE QUOTA RULE. THE QUOTA RULE SAYS THAT THE FINAL NUMBER OF REPRESENTATIVES A STATE GETS SHOULD BE WITHIN ONE OF THAT STATE'S QUOTA. SINCE WE'RE DEALING WITH WHOLE NUMBERS FOR OUR FINAL ANSWERS, THAT MEANS THAT EACH STATE SHOULD EITHER GO UP TO THE NEXT WHOLE NUMBER ABOVE ITS QUOTA OR GO DOWN TO THE NEXT WHOLE NUMBER BELOW ITS QUOTA. AND THEN FINALLY, THERE IS SOME CONTROVERSY WHEN USING HAMILTON'S METHOD. AFTER SEEING HAMILTON'S METHOD, MANY PEOPLE FIND THAT IT MAKES SENSE, AND IT'S NOT THAT DIFFICULT TO USE. SO WHY WOULD ANYONE WANT TO USE ANOTHER METHOD? WELL, THE PROBLEM IS THAT HAMILTON'S METHOD IS SUBJECT TO SEVERAL WHAT WE CALL PARADOXES. THREE OF THEM HAPPENED ON SEPARATE OCCASIONS WHEN HAMILTON'S METHOD WAS USED TO APPORTION THE UNITED STATES HOUSE OF REPRESENTATIVES. AND THOSE THREE PARADOXES ARE NUMBER ONE, THE ALABAMA PARADOX, TWO, THE NEW STATES PARADOX, AND THREE, THE POPULATION PARADOX. WE'LL TALK ABOUT EACH OF THESE IN FUTURE LESSONS. I HOPE YOU FOUND THIS HELPFUL.