The Pairwise-Comparison Method: A Voting System Examined, Study Guides, Projects, Research of Discrete Mathematics

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The Pairwise-Comparison Method

Lecture 11 Section 1.

Robb T. Koether

Hampden-Sydney College

Wed, Feb 7, 2018

(^1) The Method of Pairwise Comparisons

(^2) Examples

(^3) The Number of Comparisons

(^4) A Shortcoming of the Method

(^5) Assignment

The Method of Pairwise Comparisons

Definition (The Method of Pairwise Comparisons)

By the method of pairwise comparisons, each voter ranks the candidates. Then, for every pair (for every possible two-way race) of candidates, Determine which one was preferred more often. That candidate gets 1 point. If there is a tie, each candidate gets 1/2 point. The candidate who gets the greatest number of points is the winner.

Then rank the candidates according to the number of points received.

Outline

(^1) The Method of Pairwise Comparisons

(^2) Examples

(^3) The Number of Comparisons

(^4) A Shortcoming of the Method

(^5) Assignment

Example

Example

Suppose that there are 4 candidates: A, B, C, D. The following table summarizes the voters’ preferences.

Preferences No. of voters 11 8 7 4 1st A B D C 2nd B D A A 3rd C C B D 4th D A C B

How many pairings are there? List the pairings. Count the votes for each pairing and determine the winner.

Example

Example

Suppose that there are 5 candidates: A, B, C, D, E. The following table summarizes the voters’ preferences.

Preferences No. of voters 6 4 4 4 2 1 1 1st B B D C A E E 2nd A A A E D B D 3rd E D C D E A A 4th D C E B B D B 5th C E B A C C C

How many pairings are there? List the pairings. Count the votes for each pairing and determine the winner.

The Number of Comparisons

How many comparisons are there? With 3 candidates, there are 3 comparisons. With 5 candidates, there are 10 comparisons.

The Number of Comparisons

How many comparisons are there? With 3 candidates, there are 3 comparisons. With 5 candidates, there are 10 comparisons. With 6 candidates, how many comparisons would there be?

Outline

(^1) The Method of Pairwise Comparisons

(^2) Examples

(^3) The Number of Comparisons

(^4) A Shortcoming of the Method

(^5) Assignment

A Shortcoming

This method seems to take pretty much everything into account. So what could go wrong?

A Shortcoming

Example (A Shortcoming)

Preferences No. of voters 6 4 4 4 2 1 1 1st B B D C A E E 2nd A A A E D B D 3rd E D C D E A A 4th D C E B B D B 5th C E B A C C C

At the last minute, candidate E drops out.

A Shortcoming

Example (A Shortcoming)

Preferences No. of voters 6 4 4 4 2 1 1 1st B B D C A B D 2nd A A A D D A A 3rd D D C B B D B 4th C C B A C C C

Now who is the winner?

Shortcomings

The Perfect Voting Method

Is there a voting method that has no shortcoming?

Outline

(^1) The Method of Pairwise Comparisons

(^2) Examples

(^3) The Number of Comparisons

(^4) A Shortcoming of the Method

(^5) Assignment