- WELCOME TO A LESSON ON THE BORDA COUNT METHOD OF VOTING.
IN THIS LESSON WE'LL DEFINE THE BORDA COUNT METHOD,
THEN DETERMINE WINNERS OF ELECTIONS
USING THE BORDA COUNT METHOD.
BORDA COUNT IS A VOTING METHOD NAMED AFTER
JEAN CHARLES DE BORDA, WHO DEVELOPED THE SYSTEM IN 1770.
IN THIS METHOD POINTS ARE ASSIGNED TO CANDIDATES
BASED UPON THEIR RANKING, WHERE CANDIDATES RECEIVE 1 POINT
FOR LAST, 2 POINTS FOR SECOND TO THE LAST, AND SO ON.
SO FOR EXAMPLE, IF THERE WERE FOUR CHOICES,
FIRST CHOICE VOTES WOULD BE WORTH 4 POINTS,
SECOND CHOICE VOTES WORTH 3 POINTS,
THIRD CHOICE VOTES WORTH 2 POINTS,
AND FINALLY FOURTH CHOICE VOTES WOULD BE WORTH 1 POINT.
THEN THE POINTS OF THE BALLOTS ARE TOTALED,
AND THE CANDIDATE WITH THE LARGEST POINT TOTAL
IS THE WINNER.
LET'S LOOK AT AN EXAMPLE.
HERE'S THE PREFERENCE TABLE FOR A CAMPING CLUB
THAT IS DECIDING WHERE TO GO ON THEIR NEXT TRIP,
WHERE "A" REPRESENTS THE ARCHES, G REPRESENTS THE GRAND CANYON,
Y REPRESENTS YOSEMITE, AND Z = ZION.
NOTICE HOW WE HAVE FOUR CHOICES HERE, AND THEREFORE FIRST PLACE,
OR FIRST CHOICE, WOULD RECEIVE 4 POINTS,
SECOND PLACE WOULD RECEIVE 3 POINTS, THIRD PLACE 2 POINTS,
AND FOURTH PLACE 1 POINT.
WHICH MEANS EVERY VOTE IN THIS FIRST ROW
WOULD BE WORTH 4 POINTS,
EVERY VOTE IN THE SECOND CHOICE ROW IS WORTH 3 POINTS,
AND SO ON.
SO TO FIND THE TOTAL POINTS,
WE'LL SET THIS UP USING A LARGER TABLE, AS WE SEE HERE.
AGAIN, ALL OF THESE FIRST PLACE VOTES HERE ARE WORTH 4 POINTS
EACH FOR EACH CANDIDATE, OR IN THIS CASE EACH LOCATION.
SO LOOKING AT THIS FIRST ROW FOR A MOMENT,
NOTICE HOW 12 VOTERS VOTED YOSEMITE FIRST.
SO YOSEMITE RECEIVES 12 x 4 OR 48 POINTS.
LOOKING AT THE SECOND COLUMN HERE,
NOTICE HOW FIVE VOTERS VOTED THE GRAND CANYON FIRST,
AND THEREFORE THE GRAND CANYON RECEIVED 5 x 4 OR 20 POINTS.
THE SECOND CHOICE ROW HERE IS WORTH 3 POINTS FOR EVERY VOTE.
THE THIRD CHOICE ROW IS WORTH 2 POINTS,
AND THE FOURTH CHOICE ROW IS WORTH 1 POINT.
SO YOU MAY WANT TO PAUSE THE VIDEO NOW
JUST TO VERIFY THAT I DID CALCULATE
THESE POINTS CORRECTLY.
LET'S GO AHEAD AND FIND THE TOTAL FOR EACH CHOICE.
LET'S FIRST FIND THE POINTS FOR THE ARCHES,
SO WE'LL LOOK FOR ALL THE "A"s.
SO IN THE FIRST ROW WE HAVE AN "A" HERE AND HERE,
THE SECOND ROW WE HAVE AN "A" HERE,
THE THIRD ROW WE HAVE AN "A" HERE AND HERE,
AND THE FOURTH ROW WE HAVE AN "A" HERE.
SO "A" RECEIVES 36 + 24 + 12 + 10 + 20 + 12 POINTS,
OR 114 POINTS.
NEXT WE'LL FIND THE POINTS FOR THE GRAND CANYON,
SO WE'LL FIND ALL OF THE Gs.
FIRST ROW WE HAVE A G HERE, SECOND ROW WE HAVE A G HERE,
THIRD ROW WE HAVE A G HERE, HERE, AND HERE,
AND THE FOURTH ROW WE HAVE A G HERE.
SO THE GRAND CANYON RECEIVES
20 + 30 + 24 + 8 + 18 + 6, OR 106 POINTS.
NEXT, FOR YOSEMITE WE HAVE A Y HERE IN THE FIRST ROW,
SECOND ROW WE HAVE A Y HERE AND HERE,
THIRD ROW WE HAVE A Y HERE,
AND FOURTH ROW WE HAVE A Y HERE AND HERE.
SO YOSEMITE RECEIVES 48 + 15 + 27 + 12 + 4 + 10, OR 116 POINTS.
AND THEN FINALLY ZION IS HERE IN THE FIRST ROW AND HERE,
SECOND ROW WE HAVE A Z HERE AND HERE.
THERE ARE NO Zs IN THE THIRD ROW,
BUT THE FOURTH ROW WE HAVE A Z HERE AND HERE.
SO ZION RECEIVES 16 + 40 + 36 + 18 + 5 + 9 POINTS,
WHICH IS 124 POINTS.
AND SINCE ZION HAS THE MOST POINTS,
UNDER THE BORDA COUNT METHOD ZION IS THE WINNER.
NOW LET'S TALK ABOUT THE FLAW,
OR WHAT'S WRONG WITH THE BORDA COUNT METHOD.
ONE POTENTIAL FLAW OF BORDA COUNT
IS A CANDIDATE COULD RECEIVE
A MAJORITY OF THE FIRST CHOICE VOTES,
AND STILL LOSE THE ELECTION.
TO ILLUSTRATE THIS, LET'S TAKE A LOOK AT ANOTHER EXAMPLE.
LET'S CONSIDER THE FOLLOWING PREFERENCE SCHEDULE.
FIRST, NOTICE IN THIS ELECTION THERE ARE A TOTAL OF 20 VOTES,
AND NOTICE HOW CANDIDATE "A" RECEIVES 9 + 2,
OR 11 OF THE 20 VOTES, WHICH WOULD BE 55%,
OR A MAJORITY WIN.
NOTICE CANDIDATE B RECEIVES 8 FIRST CHOICE VOTES,
OR 40% OF THE FIRST CHOICE VOTES,
AND CANDIDATE C ONLY RECEIVES 1, OR 5% OF THE FIRST CHOICE VOTES.
SO "A" WOULD WIN BY MAJORITY.
BUT NOW LET'S FIND THE WINNER OF THIS ELECTION
BY USING THE BORDA COUNT METHOD.
FIRST, SINCE THERE ARE THREE CANDIDATES,
FIRST CHOICE VOTES ARE WORTH 3 POINTS,
SECOND CHOICE VOTES ARE WORTH 2 POINTS,
AND THIRD CHOICE VOTES ARE WORTH 1 POINT.
I DIDN'T SHOW AS MUCH WORK HERE,
BUT NOTICE HOW LOOKING AT THIS FIRST COLUMN
9 VOTERS VOTED "A" FIRST,
SO "A" RECEIVES 9 x 3, OR 27 POINTS.
IN THIS COLUMN 2 VOTERS VOTED "A" FIRST,
SO "A" RECEIVES ANOTHER 2 x 3, OR 6 POINTS.
HERE B IS VOTED FIRST BY 8 VOTERS,
SO B RECEIVES 8 x 3, OR 24 POINTS, AND SO ON.
EVERY VOTE IN THE SECOND ROW IS WORTH 2 POINTS,
SO B RECEIVES 18 POINTS HERE, SINCE 9 x 2 = 18, AND SO ON.
SO NOW LET'S FIND THE TOTAL NUMBER OF POINTS
FOR EACH CANDIDATE.
"A" RECEIVES 27 + 6 + 8 + 1 OR 42 POINTS.
REMEMBER "A" WAS THE MAJORITY WINNER.
BUT NOW IF WE TAKE A LOOK AT CANDIDATE B,
B RECEIVES 24 + 18 + 2 + 2,
WHICH NOTICE HOW IT COMES TO 46 POINTS.
NOTICE HOW CANDIDATE B HAS MORE POINTS THAN "A",
EVEN THOUGH "A" IS THE MAJORITY WINNER.
AND THEN FINALLY CANDIDATE C RECEIVES
3 + 4 + 16 + 9, OR 32 POINTS.
SO NOTICE HOW, AGAIN, B WOULD WIN USING BORDA COUNT,
WHICH BRINGS UP ANOTHER FAIRNESS CRITERION,
THE MAJORITY CRITERION.
IF A CHOICE HAS A MAJORITY OF FIRST PLACE VOTES,
THAT CHOICE SHOULD BE THE WINNER.
BORDA COUNT IS SOMETIMES DESCRIBED
AS A CONSENSUS BASED VOTING SYSTEM,
SINCE IT CAN SOMETIMES CHOOSE A MORE BROADLY ACCEPTABLE OPTION
OVER ONE WITH MAJORITY SUPPORT.
THIS IS A DIFFERENT APPROACH THAN PLURALITY
AND INSTANT RUNOFF VOTING THAT FOCUS ON FIRST CHOICE VOTES.
BORDA COUNT CONSIDERS VOTERS ENTIRE RANKING
TO DETERMINE THE OUTCOME.
BECAUSE OF THIS CONSENSUS BEHAVIOR, BORDA COUNT,
OR SOME VARIATION OF IT,
IS COMMONLY USED IN SELECTING SPORTS AWARDS.
VARIATIONS ARE USED TO DETERMINE
THE MOST VALUABLE PLAYER IN BASEBALL,
TO RANK TEAMS IN THE NCAA,
AND ALSO TO AWARD THE HEISMAN TROPHY.
I HOPE YOU FOUND THIS HELPFUL.