## Thinking Mathematically (6th Edition)

Published by Pearson

# Chapter 13 - Voting and Apportionment - 13.2 Flaws of Voting Methods - Exercise Set 13.2: 19

#### Answer

(a) Candidate A is selected as the winner using the plurality method. (b) The majority criterion is satisfied. (c) The head-to-head criterion is satisfied. (d) After two voters change their votes, we can see the new preference table below. Candidate A is selected as the winner using the plurality method. (e) The irrelevant alternatives criterion is satisfied. (f) The results do not contradict Arrow's Impossibility Theorem.

#### Work Step by Step

(a) Candidate A has a majority of first-place votes, so Candidate A is selected as the winner using the plurality method. (b) The majority criterion is satisfied. Candidate A has a majority of first-place votes, and candidate A is selected as the winner. (c) Since Candidate A has a majority of first-place votes, Candidate A is favored over all other candidates in a head-to-head comparison. Since Candidate A is selected as the winner, the head-to-head criterion is satisfied. (d) After two voters change their votes, we can see the new preference table below. Since Candidate A has a majority of first-place votes, Candidate A is selected as the winner using the plurality method. (e) Even if Candidate C drops out, Candidate A still has a majority of first-place votes, so Candidate A is selected as the winner using the plurality method. Therefore, the irrelevant alternatives criterion is satisfied. (f) The results do not contradict Arrow's Impossibility Theorem. Specific election results may satisfy all four fairness criteria because of the way the votes were cast in that specific election. However, this does not mean that the particular voting method used in that election will always satisfy the four fairness criteria in all elections.

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