- WELCOME TO A LESSON ON THE PLURALITY VOTING METHOD.
IN THIS LESSON WE'LL DEFINE THE PLURALITY VOTING METHOD,
DETERMINE WINNERS OF ELECTIONS USING THE PLURALITY METHOD,
DEFINE THE CONDORCET FAIRNESS CRITERION
AND ALSO FIND A CONDORCET WINNER.
THE PLURALITY VOTING METHOD IS PROBABLY THE METHOD
YOU'RE MOST FAMILIAR WITH,
WHERE THE CHOICE WITH THE MOST FIRST PREFERENCE
VOTES IS DECLARED THE WINNER.
TIES ARE POSSIBLE AND WOULD HAVE TO BE SETTLED THROUGH
SOME SORT OF RUN OFF.
THIS METHOD IS SOMETIMES MISTAKENLY
CALLED THE MAJORITY METHOD, OR MAJORITY RULES,
BUT IT IS NOT NECESSARY FOR A CHOICE
TO HAVE GAINED A MAJORITY OF VOTES TO WIN,
WHERE A MAJORITY IS OVER 50% OF THE VOTES.
SO IT IS POSSIBLE FOR A WINNER TO HAVE A PLURALITY
WITHOUT HAVING A MAJORITY.
LET'S TAKE A LOOK AT A COUPLE OF EXAMPLES.
THE SURVEY ASKED TO RANK WHICH WEST COAST STATES
PEOPLE PREFER TO LIVE.
THE RESULTS ARE BELOW.
USE THE PLURALITY METHOD TO SELECT THE WINNER.
WE'RE LOOKING AT THE PREFERENCE TABLE HERE,
C = CALIFORNIA, O = OREGON, AND W = WASHINGTON.
NOTICE THAT WE FIND THE SUM OF THESE VALUES HERE,
WE CAN DETERMINE THE TOTAL VOTES IS 300.
TO DETERMINE THE PLURALITY WINNER WE'LL DETERMINE
HOW MANY FIRST CHOICE VOTES CALIFORNIA RECEIVED,
THEN HOW MANY FIRST CHOICE VOTES OREGON RECEIVED,
AND THEN HOW MANY FIRST CHOICE VOTES WASHINGTON RECEIVED.
WELL, CALIFORNIA RECEIVED 75 + 94 FIRST CHOICE VOTES,
WHERE 75 + 94 = 169.
OREGON RECEIVED 51 + 12 FIRST CHOICE VOTES,
WHICH IS 63 FIRST CHOICE VOTES.
AND FINALLY, WASHINGTON RECEIVED
43 + 25 OR 68 FIRST CHOICE VOTES.
SO IN THIS CASE, NOTICE THAT CALIFORNIA RECEIVED
THE MOST FIRST CHOICE VOTES.
AND THEREFORE, CALIFORNIA IS THE PLURALITY WINNER.
NOTICE HOW IN THIS CASE
CALIFORNIA RECEIVED 169 FIRST CHOICE VOTES OUT OF 300,
WHICH IS APPROXIMATELY 56.3%, WHICH IS MORE THAN 50%.
AND THEREFORE, CALIFORNIA WOULD ALSO BE THE MAJORITY WINNER.
REMEMBER, A WINNER DOES NOT HAVE TO BE A MAJORITY WINNER
TO BE THE PLURALITY WINNER.
LET'S TAKE A LOOK AT A SECOND EXAMPLE,
WHERE HERE A SMALL GROUP OF COLLEGE STUDENTS
RANK THE BEST DESTINATION FOR SPRING BREAK
WHERE S = SAN DIEGO, L = LAKE HAVASU, AND R = ROCKY POINT.
AGAIN, BY FINDING THE SUM OF THESE VALUES
HERE WE CAN DETERMINE THERE ARE A TOTAL OF 17 VOTES.
NOTICE, SAN DIEGO RECEIVED A TOTAL OF 4 + 4, OR 8,
FIRST PLACE VOTES.
LAKE HAVASU RECEIVED A TOTAL OF TWO FIRST PLACE VOTES.
AND ROCKY POINT RECEIVED A TOTAL OF 5 + 2, OR 7,
FIRST PLACE VOTES.
AND SINCE SAN DIEGO RECEIVED THE MOST FIRST CHOICE VOTES,
OR FIRST PLACE VOTES, SAN DIEGO IS THE WINNER.
NOTICE IN THIS CASE, SAN DIEGO RECEIVED A TOTAL
OF 8 FIRST PLACE VOTES OUT OF 17,
WHICH IS APPROXIMATELY 47.1%.
SO NOTICE HOW HERE EVEN THOUGH
SAN DIEGO IS NOT THE MAJORITY WINNER,
IT STILL IS THE WINNER USING THE PLURALITY METHOD.
THIS LEADS US TO A DISCUSSION
ABOUT WHAT CAN BE WRONG ABOUT THE PLURALITY VOTING METHOD.
IF THERE ARE THREE OR MORE CHOICES IT IS POSSIBLE
THAT A CHOICE COULD LOSE,
BUT WHEN COMPARED IN A ONE TO ONE COMPARISON
IT COULD BE PREFERRED OVER THE PLURALITY WINNER.
AND THIS VIOLATES WHAT'S CALLED A FAIRNESS CRITERION
WHERE THE FAIRNESS CRITERIA ARE STATEMENTS
THAT SEEM LIKE THEY SHOULD BE TRUE IN A FAIR ELECTION.
THE FIRST FAIRNESS CRITERION WE'LL CONSIDER
IS CALLED THE CONDORCET CRITERION
WHERE IF THERE IS A CHOICE,
IT IS PREFERRED IN EVERY ONE TO ONE COMPARISON
WITH THE OTHER CHOICES.
THAT CHOICE SHOULD BE THE WINNER
AND WE CALL THIS WINNER THE CONDORCET WINNER
OR CONDORCET CANDIDATE.
LET'S LOOK AT TWO MORE EXAMPLES.
THIS IS THE EXAMPLE THAT WE SAW BEFORE
WHERE WE KNOW THE PLURALITY WINNER WAS SAN DIEGO
WITH A TOTAL OF 8 VOTES,
BUT NOW WE WANT TO FIND THE CONDORCET WINNER.
SO TO FIND THE CONDORCET WINNER WE'LL DO A ONE TO ONE COMPARISON
WITH OUR THREE OPTIONS.
SO WE'LL COMPARE SAN DIEGO VERSUS LAKE HAVASU.
WE'LL COMPARE SAN DIEGO VERSUS ROCKY POINT.
AND WE'LL COMPARE LAKE HAVASU VERSUS ROCKY POINT.
TO DO THE ONE TO ONE COMPARISON WITH SAN DIEGO AND LAKE HAVASU
WE WOULD IGNORE ROCKY POINT.
SO WE'LL IGNORE ROCKY POINT
HERE, HERE, HERE, HERE, AND HERE.
REMEMBER, WE HAVE A TOTAL OF 17 VOTES.
SO OF THE 17, SAN DIEGO IS PREFERRED OVER LAKE HAVASU
4 + 4 + 5 TIMES, SO THAT WOULD BE 8 + 5 = 13.
SO SAN DIEGO WINS OVER LAKE HAVASU 13 TO 4.
NOW WE'LL COMPARE SAN DIEGO TO ROCKY POINT
SO WE'LL IGNORE LAKE HAVASU.
SO NOTICE SAN DIEGO BEATS ROCKY POINT HERE AND HERE,
BUT NOTICE HOW ROCKY POINT WINS HERE, HERE, AND HERE.
AND THEREFORE, FOR SAN DIEGO VERSUS ROCKY POINT THE VOTE
IS 8 TO 9.
NOTICE IN THIS ONE TO ONE COMPARISON ROCKY POINT WINS.
AND THEN FINALLY, WE WANT TO CONSIDER
LAKE HAVASU VERSUS ROCKY POINT.
SO NOW WE'LL IGNORE SAN DIEGO.
SO LAKE HAVASU'S PREFERRED OVER ROCKY POINT HERE AND HERE
AND THEREFORE, LAKE HAVASU VERSUS ROCKY POINT
WOULD BE 6 TO 11.
NOW, LOOKING AT THESE ONE TO ONE COMPARISONS
NOTICE HOW ROCKY POINT BEATS LAKE HAVASU HERE
AND ROCKY POINT ALSO BEATS SAN DIEGO HERE.
THEREFORE ROCKY POINT ALWAYS WINS IN A ONE TO ONE COMPARISON.
AND THEREFORE, ROCKY POINT IS THE CONDORCET WINNER.
SO EVEN THOUGH SAN DIEGO WAS THE PLURALITY WINNER,
UNDER THE CONDORCET FAIRNESS CRITERION
ROCKY POINT SHOULD BE THE WINNER.
LET'S TAKE A LOOK AT ONE MORE EXAMPLE.
WE WANT TO FIND THE CONDORCET WINNER, OR CONDORCET CANDIDATE,
IF THERE IS ONE.
SO THE CANDIDATES ARE "A," B, AND C
SO WE'LL DO A ONE TO ONE COMPARISON.
WE'LL HAVE "A" VERSUS B, "A" VERSUS C, AND B VERSUS C.
NOTICE THE PLURALITY WINNER
WOULD BE C WITH A TOTAL OF 16 FIRST CHOICE VOTES.
SO FOR "A" VERSUS B WE'LL IGNORE C.
SO "A" WOULD WIN OVER B HERE AND HERE.
SO "A" VERSUS B WOULD BE 31 TO 10.
NEXT, FOR "A" VERSUS C WE'LL IGNORE B.
NOTICE, "A" WINS ONLY HERE SO "A" VERSUS C WOULD BE 15 TO 26.
AND THEN FOR B VERSUS C WE'LL IGNORE "A".
NOTICE HOW B WINS HERE AND C WINS HERE AND HERE.
SO B VERSUS C WOULD BE 10 TO 31.
SO AGAIN, LOOKING AT THESE TWO HERE NOTICE C WINS OVER B
AND HERE C ALSO WINS OVER "A"
AND THEREFORE CANDIDATE C IS THE CONDORCET WINNER,
BUT NOTICE HOW C IS ALSO THE PLURALITY WINNER HERE.
I HOPE YOU FOUND THIS HELPFUL.