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## Quantitative Measurements

1. Look at the following electorate, which is distributed uniformly (so the probability of having a voter on each position along the axis is the same). A moderate left candidate, X, is located at a point to the left of which 25 percent of voters lie; a near-center candidate, Y, is located at 40 percent point; and an extreme-right candidate, Z, is located at 90 percent (so only 10 percent of voters lie to his right). Each voter chooses the candidate who is closer to his or her preference.

1. Which candidate wins by plurality rule?

2. Any comment?

2. Look at the distribution of voters' preferences in this electorate:
The election is held by absolute majority rule, with a second-round runoff between the two most voted-for alternatives if necessary.

 Number of voters: 8 7 6 ---------------------- First preference X Z Y Y Y X Least preference Z X Z

a. Who wins?

3. Let's now assume that, in the next election, candidate X gains support and candidate Z loses support among voters, so that now the three groups of voters are formed by 10, 5, and 6 voters respectively (from left to right in the figure).

b. With the same electoral rule, who wins?

4. Given the following distribution of voters' preferences, identify the winner by each of the following voting rules:

 Number of voters: 3 2 1 1 First preference X Y Z W Z W W Y W V V Z V Z Y V Last preference Y X X X

1. Plurality rule.

2. Majority runoff.

As explained in Box 13.2:

3. Condorcet pair-wise comparisons.

4. Borda rank-order count.