5.1 Voting Systems: Theory

1.11_08.1 Consider a three person city where person 1 prefers A > B > C, person 2 prefers B > C > A, and person 3 prefers C > A > B. If the population first votes on A against C, and then the winner (of A vs. C) against B, which option will win?

1. A 2. B 3. C Answer: 2 Explanation: In the first round, C will beat A two to one. In the second round, B will beat C, two to one. This is an example that illustrates how the order of voting can affect who wins when there is a Condorcet Cycle.

1.11_03.1 Suppose we have three choices, ordered A, B, and C, on a line. Which of the following are plausible preference orderings that do not violate single-peaked preferences? A__________B__________C

1. A -> B -> C 2. A -> C -> B 3. B -> A -> C 4. C -> A -> B Answer: 1, 3 Explanation: The key difference is that in the correct answers, there is only one local maximum, while there are two local maxima in the incorrect answers. Specifically, the local maximum for (A -> B -> C) is A, and the local maximum for (B -> A ->C) is B. On the other hand, the local maxima for (A -> B -> C) are A and C, and the local maxima (C -> A -> B) for are also A and C.

1.11_02.1 Which of the following is a key assumption for single-peaked preferences?

1. All choices can be naturally ordered and therefore represented on a line. 2. Agents must be indifferent between at least two alternatives. 3. Agents are never indifferent between two choices. Answer: 1 Explanation: There must be a natural ordering of the choices along a single dimension. On the other hand, agents can still be indifferent between choices in order to have single-peaked preferences, but they do not need to be.

1.11_06.2 What is a Condorcet Winner?

1. An alternative that gains a majority of votes when paired against a majority of the other alternatives 2. An alternative that gains a majority of votes when paired against each of the other alternatives Answer: 2 Explanation: This is the definition of the Condorcet Winner. In other words, this is when one alternative beats every alternative.

1.11_02.2 Which of the following are examples of alternatives that can be represented as points on a single line?

1. Income tax rate 2. Level of spending on public goods 3. Where to locate a public good (specifically, distance from home) Answer: 1, 2, 3 Explanation: All of the above are examples of single-peaked preferences because they can be naturally ordered on a one-dimensional line. Income tax rate is simply a number, as is the level of spending on public goods and the distance from home to a public good; social preferences can similarly be ordered along the left-right spectrum.

1.11_12.1 Imagine the electorate grows by 50%, all with preferences to the left of the ones the existing electorate has. How does the policy, as predicted by the Median Voter Theorem, change?

1. It does not move 2. It moves to the right 3. It moves to the left Answer: 3 Explanation: For instance, if the interval of voters is originally [0,1], then the median voter will be at ½. However, when the interval of voters changes to [-1/2, 1], the median voter becomes ¼, thus moving to the left from ½.

1.11_11.1 In the context of the Meltzer-Richards (1981) model, what does the Median Voter Theorem imply that a highly unequal country will vote to do?

1. Keep the tax rate as is 2. Lower the tax rate 3. Increase the tax rate Answer: 3 Explanation: Since the mean income is significantly larger than the median income in a highly unequal country, the median voter will vote to increase the tax rate because it will affect high income earners much more than it will affect him or herself.

1.11_10.2 Which assumption does the Borda Count violate?

1. No dictatorship 2. Pareto optimality 3. Independence of irrelevant alternatives 4. Universal domain Answer: 3 Explanation: As illustrated in the lecture, it is possible that when using the Borda Count, an irrelevant alternative X can shift the winner from some alternative A to another alternative B.

1.11_04.1 Which of the following are assumptions of the Median Voter Theorem?

1. Preferences are single-peaked over a one-dimensional policy. 2. There are two candidates who announce their policies at the same time and commit to them. 3. Voting is done by majority rule. Answer: 1, 2, 3 Explanation: All of the above are key assumptions of the Median Voter Theorem.

1.11_08.2 How do Condorcet Cycles increase the power of the person in charge of scheduling votes?

1. They can affect the outcome through the order of votes. 2. They can affect the rules of the voting. 3. They can affect the overall preferences of voters. Answer: 1 Explanation: This is the idea of agenda-setting, precisely that setting the agenda is a powerful ability because it can affect the final outcome of voting.

1.11_03.2 Suppose voters are considering a tax rate of either 20% or 30%. What can we say about voters who have single-peaked preferences and whose bliss point is 35%?

1. They will vote for the tax rate of 20% 2. They will vote for the tax rate of 30% 3. They are indifferent between the two tax rates 4. We cannot determine which tax rate they will vote for Answer: 2 Explanation: If a voter has single-peaked preferences and a bliss point of 35%, then they must prefer 35% to 30% and prefer 30% to 20%, and would thus choose a tax rate of 30% over a tax rate of 20%.

1.11_06.1 Fill in the blank: A Condorcet Cycle occurs when there is a violation of the __________ property in the social preference ordering.

1. Transitivity 2. Commutativity 3. Additivity Answer: 1 Explanation: This is the definition of the Condorcet Cycle. For example, a Condorcet Cycle would occur when a social preference ordering leads to A beating B and B beating C, but then C beating A, thus violating transitivity. In other words, this is when no alternative beats every other alternative.

1.11_07.1 True or False: If a Condorcet Cycle occurs, then there is no Condorcet Winner.

1. True 2. False Answer: 1 Explanation: This is a main theorem as discussed in the lecture. Since every alternative beats at least one other alternative (Condorcet Cycle), there is no alternative that beats every other alternative (Condorcet Winner).

1.11_07.2 True or False: If there is no Condorcet Winner, then there must be a Condorcet Cycle.

1. True 2. False Answer: 1 Explanation: This is a main theorem as discussed in the lecture. Since there is no alternative that beats every other alternative (Condorcet Winner), every alternative beats at least one other alternative (Condorcet Cycle).

1.11_05.1 True or False: If politicians care about a certain policy in addition to getting elected, then their chosen policy will still always be that which is preferred by the median voter.

1. True 2. False Answer: 2 Explanation: Although the same driving force leading to the Median Voter Theorem will probably hold, the politician will also take into account his or her preferred policy in this case (whereas under the Median Voter Theorem, we assume politicians care only about winning and they don't also care about policy). The exact distribution of the preferences of the population and the policies supported by the politicians will affect how closely the Median Voter Theorem will apply. The intensity of a politician's preference for a certain policy will also matter for how closely the MVT will apply.

1.11_05.2 True or False: Results of simple majority elections, as discussed in the context of the Median Voter Theorem, reflect the intensity of preferences of the voters.

1. True 2. False Answer: 2 Explanation: Intensity of preferences is not taken into account by the Median Voter Theorem. Only the most preferred policy of each person is taken into account. There are ways to incorporate intensities of preferences into the analysis (for example, campaign contributions and making voting costly), but the simple majority election we analyze in the lecture does not incorporate this preference intensity dimension.

1.11_01.1 True or False: Voting is the only way to aggregate preferences in order for a society to make a decision.

1. True 2. False Answer: 2 Explanation: Voting is only one way, albeit the most common way, of aggregating preferences to make a social decision. Having an auction, for instance, would be another example of a way to aggregate preferences.

1.11_04.2 Fill in the blank: The Median Voter Theorem states that if preferences are single-peaked and there are two candidates who commit in advance to policies and only care about winning, then the candidates ________________.

1. Will each choose their personally preferred policy. 2. Will each choose the policy that is preferred by the median voter. 3. Will choose the median policy of their personally preferred policies. 4. Will choose the median policy under consideration. Answer: 2 Explanation: This is the main result of the Median Voter Theorem.

1.11_10.1 Consider a society with three individuals voting over three candidates. Person 1 prefers X to Y to Z. Person 2 prefers Z to Y to X. Person 3 prefers Y to X to Z. Under the Borda Count Rule, who would win?

1. X 2. Y 3. Z Answer: 2 Explanation: The Borda Count works by having each voter rank the candidates and then adding up the ranks. The candidate with the lowest total rank wins. In this example, X's ranks add up to 6, Y's ranks add up to 5, and Z's ranks add up to 7. Thus, Y will win.

1.11_01.2 Fill in the blank: Preferences are said to be single-peaked when the choice alternatives can be represented as points on a line and the individual's utility has ____________________.

1. at least one minimum at some point in the line 2. a single maximum at some point in the line such that preferences slope upward on either side of this point 3. at least one maximum at some point in the line 4. a single maximum at some point in the line such that preferences slope down on either side of this point Answer: 4 Explanation: This is the definition of single-peaked preferences. Preferences are single-peaked if the alternatives can be represented as points on a line, and each utility function has a maximum at some point on the line and slopes away from the maximum on either side. Therefore, there cannot be more than one local maximum.

1.11_01.3 Which of the following utility functions represent single-peaked preferences over the [0,10] interval?

1. https://drive.google.com/file/d/1ZRxueeAwTh1hj7sC3NCfnFIxJd2jwbUk/view?usp=sharing 2. https://drive.google.com/file/d/1nbnZkdSGFLm9xMfpsJKdHwacdHMmN9iC/view?usp=sharing 3. https://drive.google.com/file/d/1MLCo4-oDWvAG1l7nKPMZbVGHXJNvoulL/view?usp=sharing Answer: 1, 2 Explanation: The correct answers have only one maximum point, while the incorrect answer (the sine curve) has more than one local maximum.

1.11_09.1 Fill in the blank: An individual's preferences are considered "rational" if they are __________.

1. transitive, but not necessarily complete 2. complete, but not necessarily transitive 3. complete and transitive Answer: 3 Explanation: This is the definition of "rational" preferences. An individual has complete preferences if the individual can always choose between any two alternatives. An individual has transitive preferences if when the individual prefers A to B and B to C, the individual prefers A to C.

1.11_04.3 Suppose voters are considering a tax rate between 20% and 40%. Voters' bliss points are uniformly distributed across this range, meaning that there is an equal number of voters that prefer any tax rate between 20% and 40% . What tax rate does the Median Voter Theorem predict will be chosen? Enter your answer as an integer (e.g., 73)

Answer: 30 Explanation: The Median Voter Theorem states that the policy that will be chosen is the one preferred by the median voter, which is 30% in this example. If there is a uniform distribution of voters from 20% to 40%, then the median is at (20%+40%)/2 = 30%.

1.11_09.2 Match each of the following four assumptions of Arrow's Impossibility Theorem with their definitions. Use each of the possible responses once. i. Individuals have rational preferences over all alternatives, but beyond that they can have any ordering: ________________ ii. If everyone prefers A to B, then the social decision rule must prefer A to B: ______________________ iii. Society's ordering between two alternatives is completely independent of all other alternatives: ________________ iv. There is no individual whose preferences solely determine the final social ranking: ___________________

Answer: Universal Domain, Pareto Optimality, Independence of Irrelevant Alternatives, Non-dictatorship Explanation: These are the definitions of the four assumptions for Arrow's Impossibility Theorem.