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- WELCOME TO A LESSON ON THE LAST DIMINISHER METHOD.
TO APPLY THE LAST DIMINISHER METHOD, A PARTY OF THREE OR MORE
ARE RANDOMLY ASSIGNED AN ORDER.
STEP ONE, THE FIRST PERSON CUTS A SLICE
THEY VALUE AS A FAIR SHARE,
BASED UPON THEIR VALUE SYSTEM.
TWO, THE SECOND PERSON EXAMINES THE PIECE.
IF THEY THINK IT IS WORTH LESS THAN A FAIR SHARE,
THEN THEY PASS ON THE PIECE UNCHANGED.
IF THEY THINK THE PIECE IS WORTH MORE THAN A FAIR SHARE,
THEY TRIM OFF THE EXCESS AND LAY CLAIM TO THE PIECE.
THE TRIMMINGS ARE ADDED BACK TO THE TO-BE-DIVIDED PILE.
STEP THREE, EACH REMAINING PERSON IN TURN
CAN EITHER PASS OR TRIM THE PIECE.
FOUR, AFTER THE LAST PERSON HAS MADE THEIR DECISION,
THE LAST PERSON TO TRIM THE SLICE RECEIVES IT.
IF NO ONE HAS MODIFIED THE SLICE,
THEN THE PERSON WHO CUT IT RECEIVES IT.
FIVE, WHOEVER RECEIVES THE PIECE LEAVES WITH THE PIECE,
AND THE PROCESS REPEATS WITH THE REMAINING PEOPLE.
WE CONTINUE UNTIL ONLY TWO PEOPLE REMAIN,
AND THEY CAN DIVIDE WHAT IS LEFT WITH THE DIVIDER-CHOOSER METHOD.
LET'S WALK THROUGH AN EXAMPLE ON HOW THIS WORKS.
WE'LL LOOK AT AN EXAMPLE OF DIVIDING UP
A SMALL TROPICAL ISLAND AMONG FOUR PLAYERS.
SO WE HAVE PLAYER ONE THROUGH PLAYER FOUR,
AND WE'LL GO IN THIS ORDER.
SO FOR ROUND ONE, PLAYER ONE CLAIMS PART OF THE ISLAND
WORTH 25% ACCORDING TO THEIR VALUE SYSTEM,
WHICH WE'LL CALL C.
NOTICE EACH PERSON'S FAIR SHARE WOULD BE 25%,
BECAUSE WE HAVE FOUR PLAYERS.
SO THIS IS C TO BEGIN WITH.
NOW PLAYER TWO EXAMINES THE REGION.
PLAYER TWO DECIDES THE REGION IS WORTH LESS THAN 25% AND PASSES,
SO NOW IT GOES TO PLAYER THREE.
PLAYER THREE EXAMINES THE REGION.
PLAYER THREE DECIDES THE REGION IS WORTH MORE THAN 25%,
SO PLAYER THREE TRIMS THE REGION, WHICH BECOMES THE NEW C,
AND NOW PLAYER THREE IS THE CLAIMANT.
SO IF PLAYER THREE TRIMS THE REGION,
IT MAY LOOK SOMETHING LIKE THIS.
SO THIS IS THE NEW C, AND PLAYER THREE IS NOW THE CLAIMANT.
STILL GOES TO PLAYER FOUR.
PLAYER FOUR EXAMINES THE REGION.
PLAYER FOUR DECIDES THE REGION IS WORTH LESS THAN 25%
AND PASSES, THEREFORE PLAYER THREE CLAIMS C AND IS OUT.
SO NOW PLAYERS ONE, TWO, AND FOUR REMAIN,
AND BECAUSE THERE ARE ONLY THREE PLAYERS,
THEIR FAIR SHARE OF THE REMAINING PIECE OF THE ISLAND
WOULD BE 33.3% APPROXIMATELY.
SO FOR ROUND TWO IT GOES BACK TO PLAYER ONE.
PLAYER ONE CLAIMS PART OF THE ISLAND WORTH APPROXIMATELY 33.3%
ACCORDING TO P SUB 1, WHICH AGAIN WE NOW CALL C.
SO THIS WOULD BE THE NEW C.
PLAYER TWO NOW EXAMINES THE REGION.
PLAYER TWO DECIDES THE REGION
IS WORTH LESS THAN 33.3% AND PASSES.
SO IT GOES TO PLAYER FOUR, AND PLAYER FOUR EXAMINES THE REGION.
AND THERE IS AN ADVANTAGE
TO BEING LAST IN THE LAST DIMINISHER METHOD.
PLAYER FOUR DECIDES THE REGION IS WORTH MORE THAN 33.3%,
AND SINCE PLAYER FOUR IS LAST,
P SUB 4 TRIMS A PIECE THAT IS ESSENTIALLY 0%
AND CLAIMS THE REGION.
SO P SUB 4 IS REQUIRED TO TRIM THE REGION BEFORE CLAIMING IT,
BUT BECAUSE PLAYER FOUR IS LAST, THEY CAN TRIM ALMOST NOTHING.
SO IT MAY LOOK SOMETHING LIKE THIS,
WHERE IT'S JUST A VERY MINOR TRIM.
AGAIN, P SUB 3 HAS THIS PIECE.
THIS IS THE NEW C.
SO AGAIN, GOING BACK TO THE PREVIOUS SLIDE,
PLAYER FOUR IS LAST,
AND NOTICE HOW PLAYER FOUR JUST SLIGHTLY TRIMS IT,
AND NOW CAN CLAIM THIS PIECE
WHICH THEY DO VALUE AS MORE THAN 33.3% OF THE VALUE.
SO PLAYER FOUR CLAIMS THIS PIECE, AND IS OUT.
FOR ROUND THREE, SINCE ONLY TWO PLAYERS REMAIN,
PLAYER ONE AND PLAYER TWO, WE USE THE DIVIDER-CHOOSER METHOD.
SO A COIN IS FLIPPED, AND LET'S JUST SAY P SUB 2 IS THE DIVIDER.
P SUB 2 DIVIDES THE REGION INTO TWO REGIONS
WORTH 50% OF THE REMAINING VALUE,
WHICH WE SEE HERE DIVIDED BY THIS BLUE SEGMENT.
PLAYER ONE CHOOSES A REGION,
AND PLAYER TWO RECEIVES THE REMAINING REGION.
SO JUST TO FINISH THE EXAMPLE,
LET'S SAY PLAYER ONE SELECTS THIS REGION.
THEN PLAYER TWO WOULD BE LEFT WITH THIS REGION HERE.
EVERYONE RECEIVES AT LEAST A FAIR SHARE,
BUT NOTICE HOW PLAYER FOUR DID RECEIVE MORE THAN A FAIR SHARE.
NOW LET'S TAKE A LOOK AT TWO MORE EXAMPLES.
A SIX-FOOT SUB VALUED AT $25 IS DIVIDED AMONG FIVE PLAYERS,
P SUB 1 THROUGH P SUB 5.
USING THE LAST DIMINISHER METHOD THE PLAYERS PLAY
IN A FIXED ORDER.
P SUB 1 IS FIRST, FOLLOWED BY P SUB 2, P SUB 3, AND SO ON.
AND IN ROUND ONE P SUB 1 MAKES THE FIRST CUT,
AND MAKES A CLAIM ON A C PIECE.
FOR EACH OF THE REMAINING PLAYERS,
THE VALUE OF THE CURRENT C PIECE AT THE TIME
IT IS THEIR TURN IS GIVEN IN THE FOLLOWING TABLE.
LET'S BEGIN BY DETERMINING EACH PLAYER'S FAIR SHARE,
WHICH WOULD BE THE TOTAL VALUE OF THE SUB
DIVIDED BY THE NUMBER OF PLAYERS.
SO EACH PLAYER'S FAIR SHARE IS $5.
SO PLAYER ONE CUTS A PIECE WHICH THEY VALUE AT $5.
THEN, NOTICE THAT PLAYER TWO VALUES THAT PIECE AT $3.50,
WHICH IS LESS THAN THE FAIR SHARE.
SO PLAYER TWO WOULD PASS, SO IT GOES TO PLAYER THREE.
PLAYER THREE VALUES THE PIECE AT $7,
SO PLAYER THREE IS GOING TO TRIM THE PIECE
SO ACCORDING TO THEM IT'S WORTH $5, AND TAKE CLAIM TO THE PIECE.
AND NOW IT GOES TO PLAYER FOUR.
PLAYER FOUR VALUES THE CURRENT PIECE AT $6.50,
AGAIN MORE THAN THE FAIR SHARE.
SO ONCE AGAIN, PLAYER FOUR IS NOW GOING TO TRIM THE PIECE
AND TAKE CLAIM TO THE PIECE. PLAYER FOUR IS GOING TO TRIM
THE PIECE SO THAT IN THEIR MIND
IT'S WORTH THEIR FAIR SHARE OF $5,
AND NOW IT GOES TO PLAYER FIVE.
PLAYER FIVE FEELS THE CURRENT PIECE IS ONLY WORTH $4,
LESS THAN THEIR FAIR SHARE, AND PASSES.
AND THEREFORE PLAYER FOUR RECEIVES THE PIECE,
WHICH IS VALUED AT THEIR FAIR SHARE OF $5.
SO WHICH PLAYER GETS HIS OR HER SHARE AT THE END OF ROUND ONE?
THIS WOULD BE PLAYER FOUR.
AND THE VALUE OF THE SHARE, OR VALUE OF THE PIECE,
WOULD BE THEIR FAIR SHARE AT $5.
THE ONLY TIME IT WOULD BE MORE THAN THE FAIR SHARE
WOULD BE IF THE LAST PERSON VALUED THE CURRENT PIECE
AT MORE THAN $5.
THEN THEY COULD ESSENTIALLY TRIM NOTHING OFF THE PIECE,
TAKE CLAIM OF THAT PIECE, AND THEN RECEIVE IT.
LET'S LOOK AT AN EXAMPLE LIKE THAT.
AGAIN, IT'S THE SAME QUESTION,
BUT NOW THE SUB IS VALUED AT $35,
AND AGAIN THERE ARE FIVE PLAYERS.
SO EVERYONE'S FAIR SHARE WOULD BE $35 DIVIDED BY 5, OR $7.
SO PLAYER ONE CUTS A PIECE WHICH THEY VALUE AT $7.
THEN IT GOES TO PLAYER TWO.
NOTICE PLAYER TWO'S VALUE IS LESS THAN $7, SO THEY PASS.
PLAYER THREE VALUES THE CURRENT PIECE AT $7.75,
SO THEY WOULD TRIM THE PIECE SO THAT THE VALUE IS $7
IN THEIR MIND, AND TAKE CLAIM TO THE PIECE.
NEXT IT WOULD GO TO PLAYER FOUR,
WHO VALUES THE CURRENT PIECE AT LESS THAN THE FAIR VALUE,
SO PASSES.
AND NOW IT GOES TO PLAYER FIVE,
AND AGAIN, NOTICE HOW PLAYER FIVE IS LAST,
AND THEY CAN TAKE ADVANTAGE OF THIS.
PLAYER FIVE VALUES THE CURRENT PIECE AT $8,
AND PLAYER FIVE KNOWS IF THEY TRIM IT,
THEY'RE GUARANTEED TO RECEIVE THE PIECE.
SO IN THIS CASE PLAYER FIVE IS GOING TO TRIM ESSENTIALLY 0%
OFF THE PIECE, KEEPING THE VALUE AT $8, AND RECEIVING THAT PIECE.
WHICH MEANS AT THE END OF ROUND ONE PLAYER FIVE
RECEIVES HIS OR HER SHARE,
AND THE VALUE OF THIS SHARE WOULD BE $8.
REMEMBER IN THE LAST EXAMPLE
THE PLAYER ONLY RECEIVED THEIR FAIR SHARE,
BECAUSE THEY WERE NOT LAST.
LET'S FINISH BY LOOKING AT WHY THE LAST DIMINISHER METHOD
IS FAIR.
IF PLAYER ONE IS LEFT WITH THE INITIAL CLAIM
BECAUSE EVERYONE ELSE PASSES,
THEN THE PLAYER GOT WHAT WAS IDENTIFIED AS A FAIR SHARE.
NO ONE ELSE CAN TAKE IT AWAY WITHOUT DIMINISHING IT.
IF PLAYER TWO DIMINISHES IT,
IT SHOULD APPEAR LESS THAN A FAIR SHARE TO PLAYER ONE,
AS LONG AS PLAYER ONE IS FAIR AND NOT GREEDY.
THIS WOULD MAKE WHATEVER IS LEFT APPEAR POTENTIALLY
MORE THAN A FAIR SHARE.
THE SAME ARGUMENT APPLIES TO EVERYONE ELSE.
NOTE THAT THE PART REMOVED BY A DIMINISHER
CAN BE ARBITRARILY SMALL,
AND BECAUSE OF THIS IT DOES PENALIZE POTENTIAL GREED.
EVEN WHEN THE LAST PLAYER RECEIVED MORE THAN A FAIR SHARE,
IT'S STILL FAIR,
BECAUSE EVERYONE ELSE VALUED THAT PIECE AT LESS THAN
OR POSSIBLY EQUAL TO A FAIR SHARE.
I HOPE YOU FOUND THIS HELPFUL.