- WELCOME TO AN INTRODUCTION TO FAIR DIVISION.
IN THIS LESSON WE'LL DEFINE A FAIR DIVISION METHOD,
DEFINE FAIR SHARE, AND ALSO, GIVEN A SITUATION,
DETERMINE A PERSON'S FAIR SHARE.
FAIR DIVISION IS THE PROBLEM OF DIVIDING A SET OF GOODS
OR ASSETS BETWEEN SEVERAL PEOPLE SUCH THAT EACH PERSON RECEIVES
HIS OR HER DUE SHARE, OR FAIR SHARE.
THIS PROBLEM ARISES IN VARIOUS REAL-WORLD SETTINGS,
SUCH AS DIVIDING FOOD AMONG FRIENDS,
DIVIDING BUSINESS PROFITS OR ASSETS,
DIVIDING LOTTERY WINNINGS WHEN A GROUP OF PLAYERS
POOL THEIR TICKETS, DIVORCE SETTLEMENT, AIR TRAFFIC
MANAGEMENT, NATURAL RESOURCES, AND SEVERAL OTHER SETTINGS.
THIS IS AN ACTIVE RESEARCH AREA OF MATH, ECONOMICS,
SOCIAL CHOICE THEORY, GAME THEORY, AND ESTATE RESOLUTION.
A FAIR DIVISION METHOD IS A PROCEDURE
THAT CAN BE FOLLOWED AND RESULT IN A DIVISION OF ITEMS
IN A WAY SUCH THAT EACH PARTY FEELS
THEY HAVE RECEIVED THEIR FAIR SHARE.
FOR THESE METHODS TO WORK, WE DO HAVE TO MAKE A FEW ASSUMPTIONS.
NUMBER ONE, THE PARTIES ARE NON-COOPERATIVE,
SO THE METHOD MUST OPERATE
WITHOUT COMMUNICATION BETWEEN THE PARTIES.
TWO, THE PARTIES HAVE NO KNOWLEDGE
OF WHAT THE OTHER PLAYERS LIKE, OR DO NOT KNOW THEIR VALUATIONS.
THREE, THE PARTIES ACT RATIONALLY,
MEANING THEY ACT IN THEIR BEST INTERESTS
AND DO NOT MAKE EMOTIONAL DECISIONS.
AND THEN FINALLY, THE METHOD SHOULD ALLOW
THE PARTIES TO MAKE A FAIR DIVISION
WITHOUT REQUIRING AN OUTSIDE ARBITRATOR
OR OTHER INTERVENTION.
A FAIR DIVISION METHOD NEEDS TO GUARANTEE THAT EACH PARTY
WILL RECEIVE A SHARE THEY VIEW AS FAIR.
HOWEVER, FAIR DIVISION DOES NOT NEED TO BE ENVY FREE,
WHERE AN ENVY FREE DIVISION IS ONE IN WHICH NO PARTY
WOULD PREFER ANOTHER PARTY'S SHARE OVER THEIR OWN.
AGAIN, FAIR DIVISION DOES NOT NEED TO BE ENVY FREE.
FAIR DIVISION ALSO DOES NOT NEED TO BE PARETO OPTIMAL,
WHERE A PARETO OPTIMAL DIVISION IS ONE
IN WHICH NO OTHER DIVISION WOULD MAKE A PARTICIPANT BETTER OFF
WITHOUT MAKING SOMEONE ELSE WORSE OFF.
AGAIN, FAIR DIVISION DOES NOT NEED TO BE PARETO OPTIMAL.
FAIR DIVISION ALSO DOES NOT NEED TO BE EQUITABLE,
WHERE AN EQUITABLE DIVISION IS ONE
IN WHICH THE PROPORTION OF THE WHOLE EACH PARTY RECEIVES,
JUDGED BY THEIR OWN VALUATION, IS THE SAME.
AGAIN, IT DOES NOT HAVE TO BE EQUITABLE.
SO BASICALLY A FAIR DIVISION METHOD
DOESN'T HAVE TO BE THE BEST POSSIBLE DIVISION.
IT JUST HAVE TO GIVE EACH PARTY THEIR FAIR SHARE.
SO WHEN N PARTIES DIVIDE SOMETHING EQUALLY,
EACH PARTY'S FAIR SHARE IS THE AMOUNT THEY ARE ENTITLED TO.
AS A FRACTION IT WOULD BE 1/Nth OF THE TOTAL VALUE.
SO FOR EXAMPLE, IF THERE ARE FIVE PEOPLE,
EACH PERSON'S FAIR SHARE WOULD BE 1/5 OF THE VALUE.
LET'S CONSIDER AN EXAMPLE WHERE FOUR FRIENDS
PURCHASE AN APPLE PIE AND A BERRY PIE TO SHARE.
EACH PIE COSTS $6, AND SINCE THE TOTAL COST WOULD BE $6 x 2,
OR $12, AND AGAIN, THERE ARE FOUR FRIENDS,
EACH PERSON'S FAIR SHARE WOULD BE $12 DIVIDED BY 4,
OR $3 FOR EACH FRIEND.
SO LET'S SAY THE FOUR FRIENDS WANT TO SPLIT THE PIE EQUALLY.
IF MANNY LIKES BERRY PIE THREE TIMES
AS MUCH AS HE LIKES APPLE PIE, DESCRIBE A FAIR SHARE FOR MANNY.
SO EVEN THOUGH EACH PIE COSTS $6 EACH,
BECAUSE MANNY LIKES BERRY PIE THREE TIMES AS MUCH,
IT WILL CHANGE MANNY'S VALUATION FOR EACH PIECE OF PIE.
SO IF WE LET X DOLLARS BE MANNY'S VALUE OF THE APPLE PIE,
THEN 3X DOLLARS WOULD BE MANNY'S VALUE OF THE BERRY PIE.
AND BECAUSE THE TWO PIES COST $12,
WE CAN SAY THAT X + 3X = 12, OR 4X = 12.
SO IF WE SOLVE FOR X BY DIVIDING BOTH SIDES BY 4,
NOTICE HERE X = 3.
WHICH MEANS FOR MANNY THE APPLE PIE IS WORTH $3
AND THE BERRY PIE IS WORTH $3 x 3, OR $9.
NOTICE HOW THE BERRY PIE IS WORTH 3 TIMES
THAT OF THE APPLE PIE FOR MANNY.
NOW, IF WE CUT THE PIES INTO 4 EQUAL PIECES, AS WE SEE HERE,
MANNY'S VALUE FOR EACH PIECE OF APPLE PIE WOULD BE $3
DIVIDED BY 4, AND HIS VALUE FOR EACH PIECE OF THE BERRY PIE
WOULD BE $9 DIVIDED BY 4.
SO WE CAN SAY THAT FOR MANNY EACH PIECE OF APPLE PIE
IS WORTH 75 CENTS,
AND EACH PIECE OF BERRY PIE IS WORTH $2.25.
AND AGAIN, BECAUSE EACH PERSON'S FAIR SHARE IS $3.00,
SINCE 75 CENTS + $2.25 = $3.00,
THIS WOULD BE ONE WAY THAT MANNY CAN RECEIVE A FAIR SHARE.
SO AGAIN, WE COULD SAY A FAIR SHARE FOR MANNY
WOULD BE ONE PIECE OF BERRY PIE AND ONCE PIECE OF APPLE PIE.
BUT NOTICE HOW THIS IS NOT THE ONLY WAY THAT MANNY
CAN RECEIVE A FAIR SHARE.
NOTICE HOW IF HE RECEIVED 4 PIECES OF APPLE PIE,
SINCE 4 x 75 CENTS = $3,
ANOTHER POSSIBILITY WOULD BE FOR MANNY
TO RECEIVE 4 PIECES OF APPLE PIE.
NOW LET'S CONTINUE THE SAME SITUATION,
BUT IF RITA IS ALLERGIC TO BERRIES,
DESCRIBE A FAIR SHARE FOR RITA.
WELL, IF RITA IS ALLERGIC TO BERRIES,
AND SHE CAN'T EAT THE BERRY PIE,
RITA'S VALUE OF THE BERRY PIE WOULD BE $0.
WHICH MEANS FOR RITA THE APPLE PIE CONTAINS ALL OF THE VALUE,
WHICH REMEMBER WOULD BE $12.
SO AGAIN, IF WE DIVIDE EACH PIE INTO 4 EQUAL PIECES,
AS WE SEE HERE, FOR RITA EACH PIECE OF APPLE PIE IS WORTH $3,
OR $12 DIVIDED BY 4,
AND EACH PIECE OF BERRY PIE IS STILL WORTH $0.
THEREFORE, IN THIS CASE RITA WOULD RECEIVE HER FAIR SHARE
OF $3 WORTH BY RECEIVING 1 PIECE OF APPLE PIE.
SO NOTICE HOW IN THIS CASE EVEN THOUGH 1 PIECE OF APPLE PIE
WOULD BE A FAIR SHARE FOR RITA,
IF WE GO BACK TO MANNY JUST FOR A MOMENT,
1 PIECE OF APPLE PIE FOR MANNY WOULD ONLY BE WORTH 75 CENTS,
NOT HIS FAIR SHARE.
SO RECEIVING A FAIR SHARE REALLY DEPENDS
ON EACH PERSON'S VALUATIONS, IN THIS CASE OF EACH PIECE OF PIE.
NOW, THERE ARE TWO BROAD CLASSIFICATIONS
OF FAIR DIVISION METHODS,
THOSE THAT APPLY TO CONTINUOUSLY DIVISIBLE ITEMS,
AND THOSE THAT APPLY TO DISCREETLY DIVISIBLE ITEMS.
CONTINUOUSLY DIVISIBLE ITEMS ARE THINGS THAT CAN BE DIVIDED
INTO PIECES OF ANY SIZE,
LIKE CUTTING OUR PIE INTO PIECES OF DIFFERENT SIZE,
OR LIKE DRAWING BORDERS TO SPLIT A PIECE OF LAND
INTO SMALLER PLOTS.
DISCREETLY DIVISIBLE ITEMS ARE WHEN YOU ARE DIVIDING
SEVERAL ITEMS THAT CANNOT BE BROKEN APART EASILY,
SUCH AS ASSETS IN A DIVORCE,
FOR EXAMPLE, A HOUSE, CAR, FURNITURE, AND ETCETERA.
I HOPE YOU FOUND THIS INTRODUCTION HELPFUL.
WE'LL NOW BEGIN TO TAKE A LOOK AT METHODS OF FAIR DIVISION.