- WELCOME TO A LESSON ON COPELAND'S METHOD.
IN THIS LESSON WE WILL DEFINE COPELAND'S METHOD,
AND ALSO DETERMINE WINNERS OF AN ELECTION
USING COPELAND'S METHOD.
COPELAND'S METHOD TRIES TO SATISFY THE CONDORCET CRITERION
BY LOOKING AT PAIRWISE OR ONE-TO-ONE COMPARISONS.
WHEN USING COPELAND'S METHOD,
EACH PAIR OF CANDIDATES IS COMPARED USING ALL PREFERENCES,
TO DETERMINE WHICH OF THE TWO IS MORE PREFERRED.
THE MORE PREFERRED CANDIDATE IS AWARDED 1 POINT.
IF THERE IS A TIE, EACH CANDIDATE IS AWARDED
HALF A POINT.
AFTER ALL THE PAIRWISE COMPARISONS ARE MADE,
THE CANDIDATE WITH THE MOST POINTS,
AND THEREFORE THE ONE THAT HAS THE MOST PAIRWISE WINS,
IS DECLARED THE WINNER.
COPELAND'S METHOD IS A CONDORCET METHOD.
PROPONENTS ARGUE THAT THIS METHOD
IS EASILY UNDERSTOOD BY THE GENERAL POPULATION,
WHICH IS GENERALLY FAMILIAR WITH THE SPORTING EQUIVALENT.
FOR EXAMPLE, IN MANY ROUND ROBIN TOURNAMENTS
THE WINNER IS THE PARTICIPANT WITH THE MOST VICTORIES.
IT IS ALSO EASY TO CALCULATE.
HOWEVER, COPELAND'S METHOD CAN LEAD TO TIES,
AND SOME SAY THAT IT PUTS TOO MUCH EMPHASIS ON THE QUANTITY
OF PAIRWISE VICTORIES AND DEFEATS,
RATHER THAN THEIR MAGNITUDES.
LET'S TAKE A LOOK AT AN EXAMPLE.
USING THE GIVEN PREFERENCE TABLE,
WE WANT TO FIND THE WINNER OF THIS ELECTION
USING PAIRWISE COMPARISON,
OR COPELAND'S METHOD.
NOTICE IF WE SUM THIS FIRST ROW THERE ARE A TOTAL OF 42 VOTES,
AND BECAUSE THERE ARE FOUR CANDIDATES,
WE'LL HAVE SIX COMPARISONS: "A" VERSUS B, "A" VERSUS C,
AND "A" VERSUS D, AS WELL AS B VERSUS C, B VERSUS D,
AND C VERSUS D.
SO TO BEGIN WE'LL COMPARE "A" AND B.
TO MAKE THIS EASIER TO DO, WE'LL DELETE C AND D FROM THE TABLE.
NOTICE "A" IS PREFERRED HERE AND HERE,
SO "A" WOULD HAVE 21 VOTES.
AND B IS PREFERRED HERE AND HERE.
NOTICE B ALSO RECEIVES 21 VOTES, AND SINCE IT'S A TIE,
"A" AND B BOTH GET HALF A POINT.
NEXT, FOR "A" VERSUS C WE'LL REMOVE B AND D FROM THE TABLE.
NOTICE "A" IS PREFERRED HERE, HERE AND HERE,
SO "A" RECEIVES 10 + 11 + 16, OR 36 VOTES,
AND C RECEIVES 6 VOTES.
"A" WINS.
"A" GETS 1 POINT.
FOR "A" VERSUS D WE'LL REMOVE B AND C FROM THE TABLE.
NOTICE "A" IS PREFERRED HERE, SO "A" GETS 11 VOTES.
BUT D RECEIVES 10 + 6 + 15, OR 31 VOTES,
SO D WINS, AND D RECEIVES 1 POINT.
FOR B VERSUS C WE'LL REMOVE "A" AND D FROM THE TABLE.
B RECEIVES 15 VOTES, AND C RECEIVES 10 + 6 + 11,
OR 27 VOTES.
SO C WINS, AND C GETS 1 POINT.
FOR B VERSUS D WE'LL REMOVE "A" AND C FROM THE TABLE.
SO B RECEIVES 11 + 15, OR 26 VOTES, AND D RECEIVES 10 + 6,
OR 16 VOTES.
B WINS, AND B GETS 1 POINT.
AND NOW FOR THE LAST COMPARISON OF C AND C,
WE'LL REMOVE "A" AND B FROM THE TABLE.
SO C IS PREFERRED HERE AND HERE, SO C RECEIVES 17 POINTS.
AND D IS PREFERRED HERE AND HERE, SO D GETS 25 VOTES.
D WINS. D GETS 1 POINT.
AND NOW TO FIND THE WINNER WE'LL SUM THE POINTS
FOR EACH CANDIDATE.
SO NOTICE CANDIDATE "A" RECEIVES 1/2 + 1, OR 1 1/2 POINTS,
CANDIDATE B RECEIVES 1/2 + 1, OR 1 1/2 POINTS,
CANDIDATE C ONLY HAS 1 POINT,
AND CANDIDATE D RECEIVES 1 + 1, OR 2 POINTS.
THEREFORE CANDIDATE D WINS.
NOTICE THAT D IS NOT A CONDORCET WINNER.
COPELAND'S METHOD WOULD PICK THE CONDORCET WINNER
IF THERE WAS ONE.
NOW, LET'S TALK ABOUT WHAT CAN GO WRONG
WHEN USING COPELAND'S METHOD.
COPELAND'S METHOD DOES SATISFY THE CONDORCET CRITERION,
THE MAJORITY CRITERION, AND THE MONOTONICITY CRITERION.
HOWEVER, COPELAND'S METHOD IS NOT PERFECT.
SOMETIMES THE COPELAND METHOD
CAN VIOLATE THE FOLLOWING FAIRNESS CRITERION,
OR THE FAIRNESS CRITERION CALLED
THE INDEPENDENCE OF IRRELEVANT ALTERNATIVES CRITERION.
WHICH IS IF A NON-WINNING CHOICE IS REMOVED FROM THE BALLOT,
IT SHOULD NOT CHANGE THE WINNER OF THE ELECTION.
SO FOR EXAMPLE, IF CHOICE "A" IS PREFERRED OVER B,
INTRODUCING OR REMOVING C SHOULD NOT CAUSE B
TO BE PREFERRED OVER "A".
AND WE'LL TALK MORE ABOUT THIS IN THE NEXT LESSON.
I HOPE YOU FOUND THIS HELPFUL.