COU 2: Practice with Single-Peaked Preferences

Each of the following problems presents a diagram showing the ideal points of one or more persons and the locations of a set of policies in a one-dimensional policy space. For each problem, assume that each person with an ideal point displayed in the diagram has single-peaked preferences with ideal point located as depicted in the diagram, and submit the preference ordering for each person.

Note that it will be difficult to see small differences in distances if you look at this on a small screen. So don’t own-goal yourself by trying to do this on your phone.

Each diagram uses labels above the policy space to indicate the locations of policies and labels below the policy space to indicate the locations of persons’ ideal points. So, for instance, in this diagram…

…the policies are A, B and C, the persons are M and N, and the correct answer is:

As you work through the problems you’ll notice that each problem duplicates the previous problem and adds one additional policy or person. This may seem annoying and redundant, but it’s useful because it isolates the skill of coping with additional complexity from the skill of evaluating any one array of policy and ideal point positions. To get full credit on your answer to any problem, you must write the full preference representation for all the persons displayed in that problem, even though that means repeating parts of your answer to the previous problem!

Most importantly: There is a short-cut one can use to quickly determine which persons in a spatial model prefer one alternative to another. We will make heavy use of this shortcut in what comes next in this and the next lesson, and using it will make your work in this COU much easier. So learn it! The short cut is called the midpoint rule: Given any pair of policies, find the midpoint between those policies – meaning the point that is exactly halfway between them. Every person with an ideal point to the left of that midpoint prefers the left-hand policy to the right-hand policy. Every person with an ideal point to the right of that midpoint prefers the right-hand policy to the left-hand policy. And any person with an ideal point exactly at that midpoint finds the two policies equally good.

For instance, here’s a policy space with three policies and three persons.

Start with the comparison between policy A and policy B. The midpoint between A and B is the point exactly halfway between A and B, like this:

Each person whose ideal point is to the right of that midpoint prefers B to A and each person whose ideal point is to the left of that midpoint prefers A to B. So the preferences over A, B and C are partially characterized as:

Now let’s do the comparison between A and C. The midpoint between A and C is here:

Each person whose ideal point is to the right of that midpoint prefers C to A. Each person whose ideal point is to the left of that midpoint prefers A to C. So, we can add the comparison between A and C to the preferences as follows:

Finally, the comparison between B and C. Here’s the midpoint:

Each person to the right of that midpoint prefers C to B and each person to the left of that midpoint prefers B to C. So the full representation of the three persons’ preferences is:

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Rubric

For each problem, you earn 1 point for each person in the problem for whom you write a completely correct representation. You get 0 points for each person in the problem for whom you write no representation or a representation that is incorrect in any way.