### V: The Mathematics of Voting Quick Summary

5 VOTING METHODS (4 Preferential*, 1 Nonpreferential)
(*Preferential Voting Methods are those which require the use of a Preference Schedule...a list of candidates in order of preference).

PREFERENTIAL VOTING METHODS

• Plurality (candidate with the most first place votes wins)
• Plurality with Elimination (winner is selected through a series of Plurality elections in which the candidate with the fewest votes is eliminated in each round)
• Borda Count (1 pt. for each last place vote, 2 pts. for each next-to-last place vote, etc.; candidate with thelargest point total wins)
• Pairwise Comparisons (look at all one-on-one matchups; 1 pt. for each one-on-one win (½ pt. for a tie); candidate with the largest point total wins)

A NONPREFERENTIAL VOTING METHOD
(Arrow's Impossibility Theorem does not apply to nonpreferential voting methods.)

• Approval Voting (voters ‘approve’ as many of the candidates as they wish; candidate with the most approval votes wins)

2 RANKING PROCEDURES*
(*Ranking Methods provide a complete ranking from first place to last place)

• Extended Rankings (run the election; rank candidates based on how they did relative to the other candidates)
• Recursive Rankings (run the election & find winner—gets 1st Place overall; eliminate winner, run a new election & find winner—gets 2nd Place overall; continue until all candidates are ranked)

Note: Each of these Ranking Methods can be applied to each of the Voting Methods.
Note: For Approval Voting, Extended and Recursive Ranking Methods always produce the same results.

4 FAIRNESS CRITERIA*
(*Fairness Criteria represent some ‘common sense’ ideas on which one might judge the ‘fairness’ of various Voting Methods)

• Majority Criterion (Any candidate receiving a majority of first place votes should be the winner.)
• Condorcet Criterion (Any candidate who wins all one-on-one matchups with the remaining candidates should be the winner.)
• Monotonicity Criterion (An election is run and produces a winner. This original winner should remain the winner in any revote in which all ballot changes are in favor of the original winner.)
• Irrelevant Alternatives Criterion (An election is run and produces a winner. If any of the losing candidates are dropped & election results recalculated, the winner should be the same.)

Note: Arrow’s Impossibility Theorem tells us that no preferential voting method can satisfy all of these Fairness Criteria.

MISCELLANEOUS TIDBITS

• The total number of points in a Borda Count election (with 1 pt. for the last place candidate, etc.) is given by:
(SUM OF THE POINT VALUE FOR EACH PLACE)(NUMBER OF VOTERS).

Ex. With 5 candidates & 100 voters, there are a total of (1+2+3+4+5)(100)=(15)(100)=1500 points in a Borda Count election.

• With N candidates, there are N(N-1)/2 possible one-on-one matchups.

Ex. With 4 candidates, there are 4(3)/2=12/2=6 possible one-on-one matchups.

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