COU 5: Inferring Ideal Point Order from Vote Matrices

Imagine a legislature consisting of four members, labeled 1, 2, 3, and 4. Suppose this legislature makes its choices through binary agendas and, as in the U.S. House, members express support for one or the other alternative on each vote by casting either “yea” or “nay”. Suppose you have the records of ballots submitted by each legislator on each of three votes, “Vote A”, “Vote B”, and “Vote C” as follows:

Vote A
Legislator	Vote
1	yea
2	nay
3	nay
4	yea

Vote B
Legislator	Vote
1	yea
2	yea
3	nay
4	yea

Vote C
Legislator	Vote
1	yea
2	nay
3	nay
4	nay

Instructions

Answer each of the following prompts.

Prompt 1A

Write down the vote matrix implied by Vote A.

Prompt 1B

Write down the vote matrix implied by Vote B.

Prompt 1C

Write down the vote matrix implied by Vote C.

Prompt 2A

Draw a diagram that shows all the possible orderings of the ideal points of legislators 1 through 4 that are consistent ONLY with Vote A. As in the diagrams in the lesson, your diagram should draw ONE policy space for EACH possibility.

Prompt 2B

Describe the orderings that Vote A alone rules out. (For instance, in the lesson, the first vote matrix we looked at in the example ruled out any ordering in which Legislator 5’s ideal point sat between legislator 4’s and 6’s ideal points.)

Prompt 2C

In up to half a page of double-space text, plus a diagram if you think it’s needed, explain the reasoning that you used to go from the vote matrix you wrote in response to Prompt 1A to your answers to Prompts 2A and 2B.

Prompt 3A

Draw a diagram that shows all the possible orderings of the ideal points of legislators 1 through 4 that are jointly consistent with Vote A AND Vote B. As in the diagrams in the lesson, your diagram should draw ONE policy space for EACH possibility.

Prompt 3B

Describe the orderings that are consistent with Vote A alone, but that are ruled out when taking account of both Vote A and Vote B. (For instance, in the example in the lesson any ordering in which either 4’s or 6’s ideal point sat in the middle of three ideal points was consistent with the first vote matrix. But once we added the second vote matrix any order in which 4’s ideal point sat between 5’s and 6’s was ruled out.)

Prompt 3C

In up to half a page of double-space text, plus a diagram if you think it’s needed, explain the reasoning that you used to go from the vote matrices you wrote in response to Prompts 1A and 1B to your answers to Prompts 3A and 3B.

Prompt 4A

Draw a diagram that shows all the possible orderings of the ideal points of legislators 1 through 4 that are jointly consistent with Vote A AND Vote B AND Vote C. As in the diagrams in the lesson, your diagram should draw ONE policy space for EACH possibility.

Prompt 4B

Describe the orderings that are jointly consistent with Vote A and Vote B together, but that are ruled out when taking account of Vote C in addition to Vote A and Vote B. (For instance, in the example in the lesson any ordering in which either 4’s or 6’s ideal point sat in the middle of three ideal points was consistent with the first vote matrix. But once we added the second vote matrix any order in which 4’s ideal point sat between 5’s and 6’s was ruled out.)

Prompt 4C

In up to half a page of double-space text, plus a diagram if you think it’s needed, explain the reasoning that you used to go from the vote matrices you wrote in response to Prompts 1A, 1B, and 1C to your answers to Prompts 4A and 4B.

Prompt 5A

Write down a vote matrix that is inconsistent with all of the orderings you drew in response to Prompt 4A.

Prompt 5B

In up to one page of double-spaced text plus diagrams if needed, explain why the orderings you drew in response to prompt 4A are inconsistent with the vote matrix you wrote in response to Prompt 5A.

Rubric

You can earn up to 14 points on this COU, one point for each of the 14 prompts.

Note that your scores on prompts 2A through 5B are independent of your scores on Prompts 1A, 1B and 1C.

On Prompts 1A, 1B and 1C: There is one right answer to each of these prompts. You get 1 point on each for the right answer and zero points otherwise, with no opportunity for partial credit.
On Prompt 2A: In the correct answer there are exactly eight possible orderings. If you draw exactly eight orderings and they cover exactly the eight possibilities, you get 1 point. If you draws 5, 6 or 7 orderings that are all distinct from one another and are each an ordering that is consistent with Vote A, one-half a point. If you draw fewer than 5 distinct orderings that are consistent with Vote A OR you draw 1 or more ordering that is inconsistent with Vote A, you get zero points.
On Prompt 2B: Vote A by itself implies that the four ideal points are grouped into two “blocks” consisting of two ideal points each. It rules out any order of the four ideal points in which these two blocks of ideal points are “intermixed”, in the sense that the right-most ideal point from one block is to the left of the left-most ideal point from the other block. To get one point on this question, your answer must describe this and specify exactly which legislators belong to each block. No partial credit available.
On Prompt 2C: To get one point your answer must describe the midpoint rule and show how it can be applied to determine both the orders that are consistent with Vote A and the orders that are ruled out by Vote A. You get half a point if you correctly describe the midpoint rule, but your account of its application to finding the answers to Prompts 2A and 2B is partially but not wholly incomplete. You get zero points if you don’t correctly describe the mid-point rule OR if you do correctly describe it but fail to give an at-least partially complete explanation of how to apply it in order to get the answers to Prompts 2A and 2B.
On Prompt 3A: In the correct answer there are exactly four possible orderings. If you draw exactly four orderings and they cover exactly the four possibilities, you get 1 point. If you draw three orderings that are all distinct from one another and are each an ordering that is jointly consistent with Vote A and Vote B, one-half a point. If you draw fewer than three distinct orderings that are each jointly consistent with Vote A and Vote B OR you draw 1 or more ordering that is inconsistent with either Vote A and Vote B or both, you get zero points.
On Prompt 3B: Vote A alone requires that the four ideal points be grouped into two blocks, and rules out “intermixing” of ideal points from each block. Adding Vote B rules out the ideal point of one particular member of one of these two blocks being on on the “inside edge” of this block, in the sense that it is in between an ideal point from its block and an ideal point from the other block. To get one point on this question, your answer must describe this and specify exactly which legislators belong to each block. Then you must specify which legislator’s ideal point is ruled out as being located on the “inside edge” of its block by the addition of Vote B. No partial credit available.
On Prompt 3C: To get one point your answer must describe the midpoint rule and show how it can be applied to determine both the orders that are jointly consistent with Vote A and Vote B and the orders that are ruled out by the combination of Vote A and Vote B. You get half a point if you correctly describe the midpoint rule, but your account of its application to finding the answers to Prompts 3A and 3B is partially but not wholly incomplete. You get zero points if you don’t correctly describe the mid-point rule OR if you do correctly describe it but fail to give an at-least partially complete explanation of how to apply it in order to get the answers to Prompts 3A and 3B.
On Prompt 4A: In the correct answer there are exactly two possible orderings. If you draw exactly two orderings and they cover exactly the two possibilities, you get 1 point. No partial credit is available on this question.
On Prompt 4B: Vote A alone requires that the four ideal points be grouped into two blocks, and rules out “intermixing” of ideal points from each block. Adding Vote B rules out the ideal point of one particular member of one of these two blocks being on on the “inside edge” of this block, in the sense that it is in between an ideal point from its block and an ideal point from the other block. Adding Vote C does the same thing for the other block – i.e. it rules out one particular ideal point from that block being on the “inside edge” of that block. To get one point on this question, your answer must describe this and specify exactly which legislators belong to each block. Then you must specify which legislators’ ideal points are ruled out as being located on the “inside edge” of each block by the additions of Vote B and Vote C. No partial credit available.
On Prompt 4C: To get one point your answer must describe the midpoint rule and show how it can be applied to determine both the orders that are jointly consistent with Vote A and Vote B and Vote C and the orders that are ruled out by the combination of Vote A and Vote B and Vote C. You get half a point if you correctly describe the midpoint rule, but your account of its application to finding the answers to Prompts 4A and 4B is partially but not wholly incomplete. You get zero points if you don’t correctly describe the mid-point rule OR if you do correctly describe it but fail to give an at-least partially complete explanation of how to apply it in order to get the answers to Prompts 4A and 4B.
On Prompt 5A: You can get 1 point on this question ONLY if you got 1 point on Prompt 4A and you write down a vote matrix that is inconsistent with each of the two orders you wrote down in your (correct) answer to Prompt 4A. No partial credit available.
On Prompt 5B: You can get 1 point on this question ONLY if you got 1 point on Prompt 4A. A correct explanation shows that for each of the two possible orderings written in your answer to 4A, each possible midpoint cannot generate the vote matrix you wrote down in Prompt 5A. Because this has to be shown and explained for multiple possible midpoints, your answer (including diagrams) might take up a full page of text. Full credit if your logic is correct but your answer to Prompt 5A is wrong. No partial credit available.