First-preference votes

Transcription

- 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.