Section 14
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MGF1107: Section 14.2 Flaws of Voting In the previous section, we learned about four voting methods. We also discovered that the four methods can produce the same winner or different winners. So, although the methods appear to provide reasonable ways to determine winners, there are flaws with each method. Mathematicians and political scientists have agreed that a voting method should meet the following four fairness criteria
in order for the voting method to be considered fair. The Head-to-Head criterion is also known as the Condorcet criterion
. This criterion is named after the Marquis de Condorcet. The Marquis de Condorcet (1743-1794) was one of the most influential mathematicians during the American and French revolutions. He discovered, much to his disappointment that occasionally there can be no clear and fair way to declare the winner of an election. (
https://en.wikipedia.org/wiki/Marquis_de_Condorcet ) All from etext pages 900-905 Example: Applying the majority criterion
The members of the Fayetteville Town Board are holding an election to select a company to maintain the property at the town park. The choices are Green World Landscaping (G), Lawn and Garden Haven (L), and Jubilant Gardens (J). The 13 board members rank the three choices use the Borda count method to make their selection. The preference table is shown below.
from etext page 900
The Borda count method resulted in Lawn and Garden Haven being selected. Does the winner have a majority of first-place votes? Is the majority criterion satisfied? Solution:
Lawn and Garden has 6 out of 13 first place votes, which is less than 50%. So the winner of the election, Lawn and Garden Have does not have a majority of the first-place votes. In fact, a majority of the town board members, 7 out of 13, selected Green World Landscaping. Therefore, the Majority criterion is not satisfied. This example demonstrates that the Borda count method has the potential to violate the Majority criterion. Example:
The preference table below shows the outcomes of 33 votes. Decide which candidate would be chosen with each of the following voting methods. Also discuss whether or not the method violates the majority criterion. from etext page 900
a.
Plurality method The plurality method awards the election to A since it received the most votes for first place: 17 out of 33 total votes cast. This is also a majority (51.5%). The majority criterion states the candidate with the majority of first place votes must win. In general, a candidate who holds a majority of the first-place votes also holds a plurality of first-place votes. Therefore
, the plurality method never violates the majority criterion. However, this does not mean that the plurality method always produces a winner with a majority of the votes.
b.
Borda count method Each first place vote is worth 3 points, 2
nd
place vote 2 points and 3
rd
place 1 point. Candidate A: (17 x 3) + (16 x 1) = 67 points Candidate B: (9 x 3) + (24 x 2) = 75 points Candidate C: (7 x 3) + (9 x 2) + (17 x 1) = 56 points Candidate B wins using the Borda count method. Since candidate A holds a majority of the first place votes, this method violates the majority criterion. The Borda count method has the potential for violating the majority criterion.
It does not violate in every case but the potential is there. c.
Plurality with elimination This method awards the election to Candidate A since this candidate received the majority of the first place votes. No elimination of candidates is needed. In general, a candidate who holds a majority of first-place votes is awarded the election without having to hold a second election, or without having to consider eliminating a candidate and re-
aligning voters’ choices.
Therefore, the plurality with elimination method never violates the majority criterion. d.
Pairwise comparison method There are 3 candidates so there are 3(2)/2 = 3 comparisons Candidate A vs. Candidate B
Candidate A: 17 votes Candidate B: 9 + 7 = 16 votes Candidate A gets one point Candidate A vs. Candidate C Candidate A: 17 points Candidate B: 9+7 = 16 points Candidate A gets one point Candidate B vs. Candidate C Candidate B: 17 + 9 = 26 points Candidate C: 7 points Candidate B gets one point Totals: Candidate A: 2 points Candidate B: 1 points Candidate C: 0 points
This method awards the election to candidate A. In general, if a candidate holds a majority of the first place votes, this candidate always wins every pairwise comparison. Thus, the pairwise comparison method never violates the majority criterion. Of the four methods, only the Borda count method can violate the majority criterion. Example: Applying the head to head criterion Suppose that four candidates [Alvarez (A), Buchannan (B), Czechanski (C), and Davis (D)] are running for mayor of Springwater. The election involves ranking the candidates with the results shown below: from etext page 902
a.
Is there one candidate who is favored over all others using a head-to-head comparison? 4 candidates so 4(3)/2 = 6 comparisons Alvarez vs. Buchannan: Alvarez: 129 + 90 = 219 votes Buchanan: 87 + 78 + 42 = 207 votes Alvarez wins Alvarez vs. Czechanski: Alvarez: 129 + 90 = 219 points Czechanski: 87 + 78 + 42 = 207 votes Alvarez wins Alvarez vs. Davis: Alvarez: 90 + 87 + 42 = 219 votes Davis: 129 + 78 = 207 votes Alvarez wins Buchannan vs. Czechanski: Buchannan: 129 + 87 = 216 votes Czechanski:
90+ 78 + 42 = 210 votes Buchannan wins Buchannan vs. Davis: Buchannan: 90 + 87 + 78 + 42 = 297 votes Davis: 129 votes Buchannan wins Czechanski vs. Davis: Czechanski:
90+ 87 + 78 + 42 = 297 votes Davis: 129 votes Czechanski wins
Summary: Alvarez is favored over all others. b.
Who wins the election if the plurality method is used? Does this result violate the head-to-head criterion? Using the plurality method, Davis wins because he received 129 first-place votes. Therefore, this result violates the head-to-head comparison. This example demonstrates that the plurality method has the potential to violate the head-to-head criterion. c.
Who wins if the Borda count method is used? Does this result violate the head-to-
head criterion? There are four candidates so first-place is 4 points, second place 3 points, etc. Alvarez: (3 x 129) + (4 x 90) + (2 x 87) + (1 x 78) + (2 x 42) = 1083 Buchannan: (2 x 129) + (2 x 90) + (4 x 87) + (3 x 78) + (3 x 42) = 1146 Czechanski: (1 x 129) + (3 x 90) + (3 x 87) + (4 x 78) + (4 x 42) = 1140 Davis: (4 x 129) + (1 x 90) + (1 x 87) + (2 x 78) + (1 x 42) = 891 Buchannan is elected. Therefore, this result violates the head-to-head criterion. This example demonstrates that the Borda count method has the potential to violate the head-to-head criterion. d.
Who wins if the plurality with elimination method is used? Does this result violate the head-to-head criterion? Using the plurality with elimination method, Buchannan is eliminated in the first round. The preference table is updated below: # of votes 129 90 87 78 42 1
st
D A C C C 2
ND
A C A D A 3
RD
C D D A D Now Alvarez has 90 first place votes, Czechanski has 207 and Davis has 129; Alvarez is eliminated. The updated preference table is shown below: # of votes 129 90 87 78 42 1
st
D C C C C 2
ND
C D D D D
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vaccine side
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lives saved by
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55
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- Nonearrow_forwardSuppose that some members of a population are infected with a sexually transmitted virus that causes disease X, and some are infected with another sexually transmitted virus that causes disease Y. Some members of the population may be infected with both viruses. While diseases X and Y are similar in that they are both sexually transmitted diseases, the disease X is much more harmful to its carriers than the disease Y is to its carriers. Condom use is highly effective in preventing the transmission of either disease. Suppose scientists discover a cure for the sexually transmitted disease X. This cure (a vaccination) will eliminate the virus causing the disease X from its carrier before its carrier can suffer any harm from the virus. Everybody is immediately given this vaccination against the disease X. a. What do you expect to happen to the prevalence of the virus causing the disease X in the population? Draw a graph with time on the horizontal axis and prevalence of the virus causing…arrow_forwardSuppose that some members of a population are infected with a sexually transmitted virus that causes disease X, and some are infected with another sexually transmitted virus that causes disease Y. Some members of the population may be infected with both viruses. While diseases X and Y are similar in that they are both sexually transmitted diseases, the disease X is much more harmful to its carriers than the disease Y is to its carriers. Condom use is highly effective in preventing the transmission of either disease. Suppose scientists discover a cure for the sexually transmitted disease X. This cure (a vaccination) will eliminate the virus causing the disease X from its carrier before its carrier can suffer any harm from the virus. Everybody is immediately given this vaccination against the disease X. What do you expect to happen to the rate of condom use in the population over time? Add a curve for the rate of condom use to your graph.arrow_forward
- New answer neededarrow_forwardLactose intolerance causes difficulty digesting dairy products that contain lactose (milk sugar). It is particularly common among people of African and Asian ancestry. In the United States, 82 % of the population is white, 14 % is black, and 4 % is Asian. Moreover, 15 % of whites, 70 % of blacks, and 90 % of Asians are lactose intolerant.. What percent of the entire population is lactose intolerant?arrow_forwardThe climate policy toolkit has, to-date, mostly focused on demand side policies (i.e. policies that affect the pollution demand curve), some of which are ‘command-and control’ and others are market based (e.g. cap-and-trade schemes, carbon taxes). Suppose the government decided to issue tradeable permits for a certain form of pollution, does it matter for economic efficiency whether the government allocates the permits to firms or whether it auctions them off? Why or why not?arrow_forward
- Question 4: Suppose that scientists develop a way to cross tomatoes and potatoes into a new super veggie (after much debate over whether it would be called a totato or a pomato, the latter name won). There are only 6 consumers in the world that would even consider buying a pomato, each of whom wants just one, and so the scientists ask these 6 consumers how much they will pay (WTP) for one pomato. The consumers respond (truthfully) Consumer VWTP Alex 7 Ben 16 Carrie 14 Dan 24 Ellie 8 Frances 10 (1) Draw the market demand curve for pomatoes (2) Suppose the price of pomatoes is p= 10. What is the consumers' surplus? (3) If the price increases to p 16, what is the loss in consumer surplus? How much of this is due to fewer pomatoes being sold and how much is due to the higher price of pomatoes on the units that are sold?arrow_forwardYou are the manager of an organization in America that distributes blood to hospitals in all 50states and the District of Columbia. A recent report indicates that nearly 50 americans contractHIV each year through blood transfusions. Although every pint of blood donated in the unitedstates undergoes a battery of nine different test, existing screening methods can detect only theantibodies produced by the bodyâs immune system-not foreign agents in the blood, newlyinfected HIV donors can pass along the virus through blood that has passed existing screeningtests. Happily, researchers have developed a series of new tests aimed at detecting and removinginfections from donated blood before it is used in transfusions. The obvious benefit of these testsis reduced incidence of infection through blood transfusions. The report indicates that the currentprice of decontaminated blood is $60 per pint. However, if the new screening methods areadopted, the demand and supply for…arrow_forwardPeoples' behavior in behavioral economic games like the Dictator Game generally show that Many people seem to prefer outcomes that benefit others Under conditions of anonymity, most people are completely selfish. People are not generous unless they are in situations in which their partners will be able to reciprocate. There is very little variation in peoples' responses across cultures.arrow_forward
- The town of Fenerbahce has hired world‐class econometricians to investigate their criminal justice system. The researchers report an interesting finding. People who were previously detected and punished by law enforcers gain a better understanding of the functioning of the law enforcement system, and are better able to exploit laws designed to protect defendants against the state. Thus, people who have been detected and punished before are better able to avoid detection and punishment. In particular, theyreport the following facts: ‐ The offense of jay‐walking causes social harm of $100.‐ A person who has never been punished for jaywalking before faces a 0.5probability of being detected while jaywalking.‐ A person who has been punished before for jaywalking faces a probability of0.25 of being detected while jaywalking. The town wants to implement optimal punishment schemes. It is well known thateveryone in Fenerbahce is risk‐neutral and rich (i.e. not judgment proof). Moreover, the…arrow_forwardTen years ago, the town of Easton decided to increase its annual spending on education so that its high school graduates would be able to earn higher wages. Now Easton has asked you to evaluate the effectiveness of the spending increase. Their data show that before the spending increase, the average annual salary of recent high school graduates was $25,000 and that now that average salary has risen to $28,500. Fortunately for your analysis, a neighboring community (Allentown) did not change its annual spending on education. Ten years ago, recent Allentown high school graduates earned an average of $22,500, and now that average is $23,750. (a) Use a difference-in-differences estimator to determine whether Easton’s spending increase caused the wages of their high school graduates to increase. (b) What underlying assumption do you have to make in order for your estimate to be valid? What might cause your underlying assumption to be invalid? (c) This data set contains only two…arrow_forwardFor questions 44 and 45 consider the following: The town of Fenerbahce has hired world-class econometricians to investigate their criminal justice system. The researchers report an interesting finding. People who were previously detected and punished by law enforcers gain a better understanding of the functioning of the law enforcement system, and are better able to exploit laws designed to protect defendants against the state. Thus, people who have been detected and punished before are better able to avoid detection and punishment. In particular, they report the following facts: The offense of jay-walking causes social harm of $100. A person who has never been punished for jaywalking before faces a 0.5 probability of being detected while jaywalking. A person who has been punished before for jaywalking faces a probability of 0.25 of being detected while jaywalking. The town wants to implement optimal punishment schemes. It is well known that everyone in Fenerbahce is risk-neutral and…arrow_forward
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