Review of Common Test #1 Mistakes

Plurality versus Majority Plurality winner = candidate with more first place votes than any other Majority winner = candidate with more than 50% of the first place votes. Plurality = the most votes Majority = most of the votes 50% is not a majority and more than 50% does not mean 51% or more

Majority criterion versus Majority rules method Majority criterion = If a candidate has a majority of first places votes, then that candidate should win (by whatever method is used) Majority rules method = the candidate with more than 50% of the votes is the winner.

Difference between the voting methods and the fairness criteria Majority criterion = If a candidate has a majority of votes, that candidate should win (regardless of the method used to determine the winner) Majority method = The candidate with a majority of votes is the winner. Condorcet criterion = If a candidate beats all others in one-on-one contests, then that candidate should win (regardless of the method used to determine the winner). Condorcet method = The candidate that beats all other candidates one-on-one is the winner. Incidentally, neither the majority method nor the Condorcet method guarantee a winner (when voting with 3 or more candidates) and therefore are not “legitimate” methods of voting (with 3 or more candidates).

Showing the failure of fairness criteria All of the fairness criteria can be written in the form of a conditional statement. To show the failure of the fairness criteria requires demonstrating the negation of the corresponding conditional. For example: Given a condition of fairness in the form: “If P then Q” The negation is “P and not Q”. This shows the condition of fairness was not met.

Negating the majority criterion Majority criterion = If a candidate has a majority of first place votes, then that candidate should win. Negation: A candidate has a majority of first place votes and doesn’t win. Notice that saying a candidate wins and doesn’t have a majority is not a negation of the majority criterion (it is actually negating the converse which is not logically the same)

Majority criterion and its converse Majority criterion = If a candidate has a majority of first place votes, then that candidate should win. The converse = If a candidate wins, then they have a majority of first place votes. To show failure of the majority criterion, you must demonstrate a negation of the criterion not its converse. Demonstrating that a candidate wins and doesn’t have a majority of votes does not demonstrate a failure of the majority criterion, it only shows failure of the converse.

Negating the Condorcet Winner Criterion CWC = If a candidate beats all others in one-on- one contests, then that candidate should win the election. Negation = A candidate beats all others in one- on-one contests and doesn’t win the election (by whatever method). Notice that winning and not beating all other candidates one-on-one is negating the converse of the CWC and is not logically the same thing.

CWC and its converse CWC = If a candidate beats all others one- on-one, then that candidate should win. The converse = If a candidate wins, then the candidate beats all others one-on-one. To show a candidate wins and does not beat all others one-on-one shows failure of the converse and does not demonstrate a failure of the CWC.