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Weighted Voting: The Shapley-Shubik Power Index - YouTube
Channel: Mathispower4u
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- WELCOME TO A LESSON ON THE
SHAPLEY-SHUBIK POWER INDEX.
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THE SHAPLEY-SHUBIK POWER INDEX
WAS INTRODUCED IN 1954
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BY ECONOMISTS LLOYD SHAPLEY
AND MARTIN SHUBIK.
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IT PROVIDES A DIFFERENT APPROACH
FOR CALCULATING POWER
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IN A WEIGHTED VOTING SYSTEM
THAT IS DIFFERENT
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THEN THE BANZHAF POWER INDEX.
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IN SITUATIONS LIKE POLITICAL
ALLIANCES,
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THE ORDER IN WHICH PLAYERS JOIN
AN ALLIANCE COULD BE CONSIDERED
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THE MOST IMPORTANT
CONSIDERATION.
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IN PARTICULAR, IF A PROPOSAL
IS INTRODUCED,
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THE PLAYER THAT JOINS
THE COALITION
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AND ALLOWS IT TO REACH QUOTA
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MIGHT BE CONSIDERED
THE MOST ESSENTIAL.
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THE SHAPLEY-SHUBIK POWER INDEX
COUNTS
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HOW LIKELY A PLAYER IS TO BE
PIVOTAL,
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BUT WHAT DOES IT MEAN
FOR A PLAYER TO BE PIVOTAL?
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FIRST, WE NEED TO CHANGE
OUR APPROACH TO COALITIONS.
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PREVIOUSLY, THE COALITIONS
CONTAINING PLAYERS ONE AND TWO
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AND PLAYERS TWO AND ONE WOULD BE
CONSIDERED EQUIVALENT
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SINCE THEY CONTAIN THE SAME
PLAYERS.
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WE NOW NEED TO CONSIDER
THE ORDER IN WHICH PLAYERS
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JOIN THE COALITION.
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FOR THAT WE WILL CONSIDER WHAT'S
CALLED SEQUENTIAL COALITIONS,
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WHERE SEQUENTIAL COALITIONS
THAT ARE COALITIONS
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THAT CONTAIN ALL PLAYERS
IN WHICH THE ORDER OF PLAYERS
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ARE LISTED TO REFLECT THE ORDER
THEY JOINED THE COALITION.
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FOR EXAMPLE, THE SEQUENTIAL
COALITION HERE
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CONTAINING PLAYER TWO,
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PLAYER ONE AND PLAYER THREE
WOULD MEAN PLAYER TWO
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JOINED THE COALITION FIRST,
PLAYER ONE JOINED SECOND,
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AND FINALLY, PLAYER
THREE JOINED THIRD.
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THE ANGLE BRACKETS ARE USED
INSTEAD OF CURLY BRACKETS
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TO DISTINGUISH SEQUENTIAL
COALITIONS.
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AND NOW LET'S DEFINE A PIVOTAL
PLAYER.
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A PIVOTAL PLAYER IS A PLAYER
IN A SEQUENTIAL COALITION
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THAT CHANGES THE COALITION
FROM A LOSING COALITION
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TO A WINNING ONE.
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NOTICE THERE CAN ONLY BE ONE
PIVOTAL PLAYER
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IN ANY SEQUENTIAL COALITION.
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LET'S DETERMINE
THE PIVOTAL PLAYER
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IN THE WEIGHTED VOTING SYSTEM
FOR EACH SEQUENTIAL COALITION.
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SO WE'LL ADD THE WEIGHT OF EACH
PLAYER
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IN EACH PARTICULAR ORDER
TO SEE WHICH PLAYER
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ALLOWS THE COALITION TO REACH
THE QUOTA, IN THIS CASE, OF 16.
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SO PLAYER THREE HAS A WEIGHT
OF SEVEN
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SO WE HAVE SEVEN + THE WEIGHT OF
PLAYER TWO.
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THAT WOULD BE 8.
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7 + 8 = 15,
WE STILL DON'T HAVE QUOTA.
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AND THEN PLAYER FOUR
HAS A WEIGHT OF THREE SO + 3.
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NOTICE THAT PLAYER FOUR BRINGS
THE TOTAL WEIGHT TO 18
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AND NOW THE COALITION REACHES
QUOTA.
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AND THEREFORE, PLAYER FOUR
IS THE PIVOTAL PLAYER.
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LOOKING AT THE NEXT
SEQUENTIAL COALITION,
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WE FIRST HAVE PLAYER ONE
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THAT HAS A WEIGHT OF 12 + NEXT
PLAYER TWO JOINS A COALITION
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WITH A WEIGHT OF 8.
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AND SINCE 12 + 8 = 20,
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NOTICE AFTER PLAYER TWO JOINS
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THE COALITION THE COALITION
REACHES QUOTA.
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AND THEREFORE, WE SAY PLAYER
TWO IS PIVOTAL.
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NOW LETS GO OVER THE STEPS
ON HOW TO CALCULATE
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THE SHAPLEY-SHUBIK POWER INDEX.
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NUMBER ONE, WE LIST ALL
SEQUENTIAL COALITIONS.
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REMEMBER THESE COALITIONS
CONTAIN ALL THE PLAYERS.
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TWO, IN EACH SEQUENTIAL
COALITION
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WE'LL DETERMINE THE PIVOTAL
PLAYER.
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THREE, WE'LL COUNT HOW MANY
TIMES EACH PLAYER IS PIVOTAL.
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AND THEN FOUR, WE'LL CONVERT
THESE COUNTS TO FRACTIONS,
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DECIMALS, OR PERCENTAGES
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BY DIVIDING BY THE TOTAL NUMBER
OF SEQUENTIAL COALITIONS.
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BUT BEFORE WE DO THIS, WE NEED
TO MENTION ONE MORE THING.
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IF A WEIGHTED VOTING SYSTEM
HAS N PLAYERS,
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THEN THERE ARE N FACTORIAL
SEQUENTIAL COALITIONS.
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SO FOR EXAMPLE, IF A VOTING
SYSTEM HAS FIVE PLAYERS,
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THERE WOULD BE FIVE FACTORIAL
SEQUENTIAL COALITIONS,
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WHERE FIVE FACTORIAL IS 5 x 4 x
3 x 2 x 1 = 120.
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SO NOW LET'S TAKE A LOOK
AT OUR EXAMPLE.
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WE WANT TO FIND
THE SHAPLEY-SHUBIK POWER INDEX
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FOR THE GIVEN WEIGHTED VOTING
SYSTEM.
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NOTICE HOW THERE ARE THREE
PLAYERS AND THEREFORE,
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THERE ARE THREE FACTORIAL
OR SIX SEQUENTIAL COALITIONS
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GIVEN HERE.
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REMEMBER, FOR A SEQUENTIAL
COALITION THE ORDER MATTERS.
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SO WE HAVE THE COALITION
IN THE ORDER OF PLAYERS
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ONE, TWO, AND THREE.
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THEN THE COALITION OF PLAYERS
ONE, THREE, AND TWO,
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THEN THE COALITION WITH PLAYERS
TWO, ONE, AND THREE,
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THEN THE COALITION OF PLAYERS
TWO, THREE, AND ONE,
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AND THEN FINALLY, WE HAVE THE
TWO SEQUENTIAL COALITIONS
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WHERE WE HAVE PLAYERS
THREE, ONE AND TWO
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AND THEN PLAYERS
THREE, TWO, AND ONE.
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SO WE'VE DONE STEP ONE.
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WE'VE LISTED ALL THE SEQUENTIAL
COALITIONS
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AND NOW FOR EACH SEQUENTIAL
COALITION
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WE'LL DETERMINE THE PIVOTAL
PLAYER OR THE PLAYER
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THAT CHANGES THE COALITION
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FROM NOT MEETING
QUOTA TO MEETING QUOTA.
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SO FOR THIS FIRST COALITION
WE HAVE PLAYER ONE FIRST,
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WHICH HAS A WEIGHT OF 17 + THE
WEIGHT OF PLAYER TWO,
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WHICH IS 13.
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NOTICE HOW ONCE PLAYER TWO
JOINS THE COALITION
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THE COALITION HAS A WEIGHT
OF 30, WHICH MEETS QUOTA.
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AND THEREFORE,
PLAYER TWO IS PIVOTAL.
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NEXT ORDER IS PLAYER ONE, PLAYER
THREE, AND THEN PLAYER TWO.
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SO PLAYER ONE HAS A WEIGHT OF 17
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+ THE WEIGHT OF PLAYER THREE
IS 10.
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WELL, 17 + 10 = 27.
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SO PLAYER THREE TURNS
THE COALITION
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FROM NOT MAKING QUOTA TO MAKING
QUOTA SINCE THE QUOTA'S 25.
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SO PLAYER THREE IS PIVOTAL.
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FOR THE NEXT COALITION
WE HAVE PLAYER TWO FIRST,
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WHICH HAS A WEIGHT OF 13 +
PLAYER ONE SO WEIGHT OF 17.
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NOTICE PLAYER ONE MAKES
THE SEQUENTIAL COALITION
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MEET THE QUOTA AND THEREFORE,
PLAYER ONE IS PIVOTAL.
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NEXT, WE HAVE THE ORDER PLAYER
TWO, PLAYER THREE,
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AND PLAYER ONE.
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PLAYER TWO IS 13 + PLAYER THREE
IS 10.
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NOTICE HOW HERE THE WEIGHT
IS 23,
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WE STILL HAVE NOT MET QUOTA.
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IT TAKES PLAYER ONE TO JOIN
WITH A WEIGHT OF 17
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FOR THE COALITION TO MAKE QUOTA.
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THIS WOULD BE 40, WHICH ONCE
AGAIN, MAKES QUOTA.
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SO PLAYER ONE IS PIVOTAL
FOR THIS ORDER.
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NEXT,
PLAYER THREE HAS A WEIGHT OF 10
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+ PLAYER ONE HAS A WEIGHT OF 17.
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NOTICE HOW WHEN PLAYER
ONE JOINS,
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THE COALITION MEETS QUOTA AT 27.
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SO PLAYER ONE IS PIVOTAL.
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AND THEN FINALLY,
WE HAVE THE ORDER OF PLAYER
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THREE, TWO, AND THEN ONE.
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PLAYER THREE HAS A WEIGHT OF 10.
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PLAYER TWO HAS A WEIGHT OF 13,
STILL HAVE NOT MADE QUOTA.
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IT TAKES PLAYER ONE TO JOIN WITH
A WEIGHT OF 17 TO MAKE QUOTA.
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SO PLAYER ONE IS PIVOTAL.
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NOW WE'LL COUNT HOW MANY TIMES
EACH PLAYER IS PIVOTAL.
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SO PLAYER ONE IS PIVOTAL HERE,
HERE, HERE, AND HERE,
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SO FOUR TIMES.
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PLAYER TWO IS PIVOTAL HERE
ONLY ONCE.
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AND PLAYER THREE IS ALSO ONLY
PIVOTAL ONLY ONCE.
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NOTICE HOW THE SUM OF THE NUMBER
OF PIVOTS IS SIX,
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SO THE SHAPLEY-SHUBIK POWER
INDEX FOR PLAYER ONE
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WOULD BE 4 DIVIDED BY 6, OR 2/3,
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WHICH AS A PERCENTAGE
WOULD BE APPROXIMATELY 66.7%.
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THE POWER INDEX FOR PLAYER TWO
WOULD BE 1 DIVIDED BY 6, OR 1/6,
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WHICH AS A PERCENTAGE
WOULD BE APPROXIMATELY 16.7%
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AND THE SAME THING FOR PLAYER
THREE.
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REMEMBER, THE POWER OF INDEX
CAN BE EXPRESSED AS A FRACTION,
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DECIMAL, OR PERCENTAGE.
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I DO WANT TO TAKE A MOMENT
AND MENTION
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THAT THE BANZHAF POWER INDEX AND
THE SHAPLEY-SHUBIK POWER INDEX
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ARE USUALLY SIMILAR.
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HOWEVER, SINCE THE TWO
APPROACHES ARE DIFFERENT,
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THE VALUES ARE OFTEN SLIGHTLY
DIFFERENT.
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FOR EXAMPLE,
IF WE WERE TO CALCULATE
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THE BANZHAF POWER INDEX
FOR THE PREVIOUS EXAMPLE,
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NOTICE HOW THE POWER INDEX
VALUES ARE SLIGHTLY DIFFERENT
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USING THIS METHOD.
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WE WON'T TAKE THE TIME
TO CALCULATE
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THE BANZHAF POWER INDEX IN THIS
LESSON
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SINCE WE ALREADY LEARNED
THIS IN THE LAST LESSON,
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BUT NOTICE HOW THESE VALUES
ARE SLIGHTLY DIFFERENT
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THEN THE VALUES THAT WE FOUND
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USING THE SHAPLEY-SHUBIK POWER
INDEX.
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HERE ARE THE VALUES FOR THE
BANZHAF POWER INDEX
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AND HERE ARE THE VALUES
THAT WE FOUND
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USING
THE SHAPLEY-SHUBIK POWER INDEX.
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I HOPE YOU FOUND
THIS LESSON HELPFUL.
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