The Pairwise-Comparison Method
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1 The Pairwise-Comparison Method Lecture 12 Section 1.5 Robb T. Koether Hampden-Sydney College Mon, Sep 19, 2016 Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
2 1 The Method of Pairwise Comparisons 2 Example 3 The Number of Comparisons 4 A Shortcoming of the Method 5 Assignment Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
3 Outline 1 The Method of Pairwise Comparisons 2 Example 3 The Number of Comparisons 4 A Shortcoming of the Method 5 Assignment Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
4 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 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. Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
5 Outline 1 The Method of Pairwise Comparisons 2 Example 3 The Number of Comparisons 4 A Shortcoming of the Method 5 Assignment Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
6 Example Example Suppose that there are 3 candidates: A, B, C. The following table summarizes the voters preferences. How many pairings are there? List the pairings. Preferences No. of voters st A B B C 2nd B A C B 3rd C C A A Count the votes for each pairing and determine the winner. Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
7 Example Example Suppose that there are 5 candidates: A, B, C, D, E. The following table summarizes the voters preferences. Preferences No. of voters st B B D E A C C 2nd A A A C D B D 3rd C D E D C A A 4th D E C B B D B 5th E C B A E E E How many pairings are there? List the pairings. Count the votes for each pairing and determine the winner. Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
8 Outline 1 The Method of Pairwise Comparisons 2 Example 3 The Number of Comparisons 4 A Shortcoming of the Method 5 Assignment Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
9 The Number of Comparisons How many comparisons are there? With 3 candidates, there are 3 comparisons. With 5 candidates, there are 10 comparisons. Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
10 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? Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
11 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? How many with 7 candidates? Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
12 Outline 1 The Method of Pairwise Comparisons 2 Example 3 The Number of Comparisons 4 A Shortcoming of the Method 5 Assignment Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
13 A Shortcoming This method seems to take pretty much everything into account. So what could go wrong? Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
14 A Shortcoming Example (A Shortcoming) Preferences No. of voters st B B D E A C C 2nd A A A C D B D 3rd C D E D C A A 4th D E C B B D B 5th E C B A E E E Reconsider the previous example. Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
15 A Shortcoming Example (A Shortcoming) Preferences No. of voters st B B D E A C C 2nd A A A C D B D 3rd C D E D C A A 4th D E C B B D B 5th E C B A E E E At the last minute, candidate C drops out. Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
16 A Shortcoming Example (A Shortcoming) Preferences No. of voters st B B D E A B D 2nd A A A D D A A 3rd D D E B B D B 4th E E B A E E E Now who is the winner? Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
17 A Shortcoming Example (A Shortcoming) Preferences No. of voters st B B D E A B D 2nd A A A D D A A 3rd D D E B B D B 4th E E B A E E E Now who is the winner? Is that surprising? Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
18 Shortcomings The Perfect Voting Method Is there a voting method that has no shortcoming? Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
19 Outline 1 The Method of Pairwise Comparisons 2 Example 3 The Number of Comparisons 4 A Shortcoming of the Method 5 Assignment Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
20 Assignment Assignment Chapter 1: Exercises 41, 42, 43, 44, 45, 47, 49, 50. Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, / 17
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