Sect 13.2 Flaws of Voting Methods

Transcription

1 218 Sect 13.2 Flaws of Voting Methods From an example the previous section, we had 48 sports writers rank the top four Spurs players of all time. Below is the preference table. Number of votes st choice TD GG DR GG 2 nd choice DR DR GG TD 3 rd choice GG TD TD DR 4 th choice TP TP TP TP As we saw in the last section, determining who is the top Spurs player of all time depends upon which voting method is employed: 1) The Plurality Method: Tim Duncan 2) The Borda Count Method: David Robinson 3) The Plurality-with-Elimination Method: George Gervin 4) The Pairwise Comparison Method: David Robinson The question arises as to which method is the most democratic and fair. In this section, we will look at the four criteria that a voting method should meet for the voting system to be fair. This is known as the fairness criteria: Fairness Criteria 1) The Majority 2) The Head-to-Head 3) The Monotonicity, and 4) The Irrelevant Alternatives Objective #1 Understanding the majority criterion. Most people believe that if a candidate receives more than half of the first place votes, the candidate should win the election. This is known as the majority criterion. This is different from the plurality method where the candidate that received the most first place votes is the winner. Just because a candidate receives the most first place votes does not mean he or she received more than half of the first place votes. Tim Duncan is a example of that for the last section. He had the most first place votes (20) so the plurality method said he was the winner, but he did not receive more than half of the first place votes (25 or higher).

2 The Majority If the candidate receives a majority of first-place votes in an election, then that candidate should win the election. Find the following: Ex. 1 Fifty-two tech reviewers were asked to rank the quality of cell phones by the brands Apple (A), Google (G), Moto (M), and Samsung (S). The results are displayed in the preference table below. Number of votes st choice A G S M G 2 nd choice G S G S S 3 rd choice S M M G A 4 th choice M A A A M a) Which brand has the majority of the vote? b) Which brand is the best brand using the Borda count method? c) Does the Borda count method for this problem the majority criterion? Solution: a) Apple has 27 first-place votes which is more than half of 52 votes, so it has received the majority of the first-place votes. b) Since there are four choices, each first-place vote is worth four points, each second-place vote is worth 3 points, each third-place vote is worth 2 points, and each last place vote is worth 1 point. 219 Number of votes st choice A: 4 27 = 108 G: 4 18 = 72 S: 4 4 = 16 M: 4 2 = 8 G: 4 1 = 4 2 nd choice G: 3 27 = 81 S: 3 18 = 54 G: 3 4 = 12 S: 3 2 = 6 S: 3 1 = 3 3 rd choice S: 2 27 = 54 M: 2 18 = 36 M: 2 4 = 8 G: 2 2 = 4 A: 2 1 = 2 4 th choice M: 1 27 = 27 A: 1 18 = 18 A: 1 4 = 4 A: 1 2 = 2 M: 1 1 = 1 Now total the points for each brand separately. Apple: = 134 Google: = 173 Moto: = 80 Samsung: = 133

3 So, by the Borda count method, Google is the winner since it has the most points. c) No, the Borda count method does not the majority criterion since Apple had the majority of the first-place votes. The Borda count method is the only one that can potentially violate the majority criterion. The plurality method, plurality-with-elimination method, and the pairwise comparison method will never violate the majority criterion. 220 Objective #2: Understanding the Head-to-Head. If one candidate is favored by the voters when compared to each of the other candidates, then most people feel that the candidate should win the election. This is known as the head-to-head criterion. Head-to-Head If a candidate is favored when compared separately (head-to-head) with every other candidate, then that candidate should win the election. Find the following: Ex. 2 Sixty frequent flyers were asked to rank in order the airlines they like the most among American Airline (A), Delta Airlines (D), and Southwest Airlines (S). The results are displayed below. Number of votes st choice A S D S D 2 nd choice D D A A S 3 rd choice S A S D A a) Which airline is preferred over the others using the head-to-head comparison? b) Which airline wins using the plurality method? c) Does the plurality method for this problem the head-to-head criterion? Solution: a) Since we have four candidates, we will need to do 3 C 2 = 3 comparisons. The comparisons are A vs. D, A vs. S, and D vs. S

4 A vs. D Number of votes st choice A D D 2 nd choice D D A A 3 rd choice A D A = 27 prefer American while = 33 prefer Delta, so Delta gets 1 point. 221 A vs. S Number of votes st choice A S S 2 nd choice A A S 3 rd choice S A S A = 28 prefer American while = 32 prefer Southwest, so Southwest gets 1 point. D vs. S Number of votes st choice S D S D 2 nd choice D D S 3 rd choice S S D = 35 prefer Delta while = 25 prefer Southwest, so Delta gets 1 point. Notice that Delta is preferred over the other two airlines using the head-to head comparison. b) We only need to examine the first choice row of the table: Number of votes st choice A S D S D 1 st place votes for American: 18 1 st place votes for Delta: = 17 1 st place votes for Southwest: = 25 Since Southwest Airlines has the most first place votes, then Southwest is the winner. c) No, the plurality method does not the head-to-head criterion since Southwest was not preferred over the other two in the head-to-head comparison.

5 The pairwise comparison method is the only voting method that does not have the potential to violate the head-to-head criterion. The other three methods the plurality method, the Borda count method, and the pluralitywith-elimination method have the potential to violate the head-to-head criterion. 222 Objective #3: Understanding the Monotonicity. In the first round of an election, sometimes a preliminary election or straw vote is taken where the votes do not count. This is done to measure the voter's intentions. Generally, it is believe that if a candidate wins the first election and then gains additional support without losing any of the original support, then the candidate should win the second election. This criterion is called the monotonicity criterion. The Monotonicity If a candidate wins an election and, in reelection, the only changes are changes that favor the candidate, then the candidate should win the reelection. Find the following: Ex. 3 Eighty-seven participants in a focus group are asked to rank their preferences among the networks ABC, CBS, and NBC at the beginning of the focus group and at the end of the focus group. The results are displayed below. Beginning of the focus group: Number of votes st choice NBC CBS ABC ABC 2 nd choice ABC NBC CBS NBC 3 rd choice CBS ABC NBC CBS End of the focus group: Number of votes st choice NBC CBS ABC ABC 2 nd choice ABC NBC CBS NBC 3 rd choice CBS ABC NBC CBS

6 223 a) Use the plurality-with-elimination to determine which network won the voting at the beginning of the focus group. b) Use the plurality-with-elimination to determine which network won the voting at the end of the focus group. c) Does this violate the montonicity criterion. Solution: a) Since there were a total of 87 ballots, a network will need 44 first place votes in order to have a majority. Number of votes st choice NBC CBS ABC ABC 2 nd choice ABC NBC CBS NBC 3 rd choice CBS ABC NBC CBS The number of first place votes are: 1 st place votes for ABC: = 33 1 st place votes for CBS: 24 1 st place votes for NBC: 30 Since none of the networks have a majority, eliminate the network with the fewest first place votes which would be CBS. Number of votes st choice NBC NBC ABC ABC 2 nd choice ABC ABC NBC NBC The number of first place votes are: 1 st place votes for ABC: = 33 1 st place votes for NBC: = 54 Since NBC has the majority of the first place votes, then NBC is the winner. b) Since there were a total of 87 ballots, a network will need 44 first place votes in order to have a majority. Number of votes st choice NBC CBS ABC ABC 2 nd choice ABC NBC CBS NBC 3 rd choice CBS ABC NBC CBS The number of first place votes are:

7 1 st place votes for ABC: = 22 1 st place votes for CBS: 24 1 st place votes for NBC: 41 Since none of the networks have a majority, eliminate the network with the fewest first place votes which would be ABC. Number of votes st choice NBC CBS CBS NBC 2 nd choice CBS NBC NBC CBS The number of first place votes are: 1 st place votes for CBS: = 45 1 st place votes for NBC: = 42 Since CBS has the majority of the first place votes, then CBS is the winner. 224 c) All that changed between the vote at the beginning of the focus group and the vote the end of the focus group was that 11 of the 12 voters in the last column switched their ballots to the first column. NBC won the first election and then picked additional support from the eleven voters that switched to the first column. Since NBC lost the second election, this violates the monotonicity criterion. The plurality method is the only voting method that does not have the potential to violate the monotonicity criterion. The other three methods the Borda count method, the plurality-with-elimination method, and pairwise comparison method have the potential to violate the head-to-head criterion. Objective #4: Understanding the Irrelevant Alternatives. Suppose a candidate wins the election and then one or more of the other candidates drops out and a recount is done. Most people would expect that the previous winner should still win the election. This is know as the irrelevant alternatives criterion. Irrelevant Alternatives If a candidate wins an election and, in a recount, the only changes are that one or more of the other candidates are removed from the ballot, then the previous winning candidate should still win the election.

8 Find the following: Ex. 4 Four candidates, Cruz (C), Jones (J), Ortega (O), and Wentworth (W), are running for mayor of Tuna, Texas. The election results are displayed below. 1 st choice C W O O 2 nd choice J J C C 3 rd choice W O W J 4 th choice O C J W 225 a) Use the pairwise comparison method to determine the winner. b) Prior to the announcement of the results, both Jones and Wentworth are forced to withdraw due to a major scandal. Use the pairwise comparison method to determine the winner. c) Does this violate the irrelevant alternatives criterion? Solution: a) Since we have four candidates, we will need to do 4 C 2 = 6 comparisons. The comparisons are C vs. J, C vs. O, C vs. W, J vs. O, J vs. W, and O vs. W. C vs. J 1 st choice C 2 nd choice J J C C 3 rd choice J 4 th choice C J = 858 people prefer Cruz and 330 people prefer Jones, so Cruz gets 1 point. C vs. O 1 st choice C O O 2 nd choice C C 3 rd choice O 4 th choice O C 528 people prefer Cruz and = 660 people prefer Ortega, so Ortega gets 1 point.

9 C vs. W 1 st choice C W 2 nd choice C C 3 rd choice W W 4 th choice C W = 858 people prefer Cruz and 330 people prefer Wentworth, so Cruz gets 1 point. J vs. O 1 st choice O O 2 nd choice J J 3 rd choice O J 4 th choice O J = 858 people prefer Jones and = 330 people prefer Ortega, so Jones gets 1 point. J vs. W 1 st choice W 2 nd choice J J 3 rd choice W W J 4 th choice J W = 594 people prefer Jones and = 594 prefer Wentworth so each gets ½ of a point. O vs. W 1 st choice W O O 2 nd choice 3 rd choice W O W 4 th choice O W = 330 people prefer Ortega and = 858 people prefer Wentworth, so Wentworth gets 1 point. Total points: Points for Cruz: = 2 Points for Jones: 1 + ½ = 1½ Points for Ortega: 1 Points for Wentworth: 1 + ½ = 1½ Since Cruz has the most points, Cruz is the winner. 226

10 227 b) Since both Jones and Wentworth are forced to withdraw due to a major scandal, then Cruz and Ortega are the only two candidates left: 1 st choice C O O O 2 nd choice O C C C Since we have two candidates, we will need to do 2 C 2 = 1 comparison. The comparison is C vs. O. C vs. O 1 st choice C O O O 2 nd choice O C C C 528 people prefer Cruz and = 660 people prefer Ortega, so Ortega gets 1 point. Total points: Points for Cruz: 0 Points for Ortega: 1 Since Ortega has the most points, Ortega is the winner. c) Cruz was the winner in the first election result. Since both Jones and Wentworth were not winners, their removal made Ortega the winner. This violates the irrevelant alternatives criterion. All four of the voting methods the plurality method, the Borda count method, the plurality-with-elimination method, and pairwise comparison method may violate the irrevelant alternatives criterion. Objective #5: Understanding Arrow's Impossibility Theorem. For an election to truly be democratic and fair, it must each of the four fairness criteria majority criterion, head-to-head criterion, monotonicity criterion, and irrevelant alternatives criterion. Below is a summary of each of the four fairness criteria:

11 228 Majority Head-to-Head Monotonicity Irrelevant Alternatives Description If the candidate receives a majority of first-place votes in an election, then that candidate should win the election. If a candidate is favored when compared separately (head-to-head) with every other candidate, then that candidate should win the election. If a candidate wins an election and, in reelection, the only changes are changes that favor the candidate, then the candidate should win the reelection. If a candidate wins an election and, in a recount, the only changes are that one or more of the other candidates are removed from the ballot, then the previous winning candidate should still win the election. The next table shows which voting methods the criteria. Voting Method Plurality-withelimination Borda count Method Method Always satisfies Fairness Criteria Majority Head-to- Head Monotonicity Irrelevant Alternatives Plurality Method Always satisfies Always satisfies Pairwise Comparison Method Always satisfies Always satisfies None of the voting methods discussed satisfies all of the fairness criteria. In fact, for an election involving more than two candidates, there is no perfectly democratic and fair voting method. This is known as Arrow's Impossibility Theorem. Arrow's Impossibility Theorem It is mathematically impossible for any democratic voting system to each of the four fairness criteria.