Answers to Practice Problems Median voter theorem, supermajority rule, & bicameralism.
Median Voter Theorem Questions: 2.1-2.4, and 2.8. Located at the end of Hinich and Munger, chapter 2, The Spatial Model of Downs and Black in the course packet. 2.1. The median can be found by putting the ideal points in ascending order and canceling the same number to the left as the right: -2, 2, 6, 9, 9. The median is 6. The mean (or average) is (-2+2+6+9+9)/5 = 4.8. 2.2. With sincere and symmetric preferences everyone votes for the alternative closest to their ideal point. So x 1 and x 5 prefer z=3 and x 2, x 3, and x 4 prefer y = 8.
Median Voter Theorem Questions: 2.1-2.4, and 2.8. Located at the end of Hinich and Munger, chapter 2, The Spatial Model of Downs and Black in the course packet. 2.3. With x 6 = 7 added to the committee, the ideal points are -2, 2, 6, 7, 9, 9 and the median is now the range from 6 to 7 inclusive, [6,7]. Since the voters at 6 and 7 both prefer 8, y = 8 is the winner (note: if they split their vote, the status quo would prevail). 2.4. These voters can be arranged -4, -3, 6, 7, 21. The median is 6. The mean is 5.4. If x 3 = 25, then 25 replaces 7, the median is unchanged and the mean is now 9. Hence, the mean changes more.
Median Voter Theorem Questions: 2.1-2.4, and 2.8. Located at the end of Hinich and Munger, chapter 2, The Spatial Model of Downs and Black in the course packet. 2.8. Suppose the number of voters n is odd and the ideal points are sorted in ascending order. Counting from left to right, there must be (n-1)/2 points followed by one point, followed by (n-1)/2 more points, even if some of the values repeat. Because the one point must be the median and can only hold one value, the median of an odd size population cannot be an interval.
Supermajority Rules Suppose N=7 with the following ideal points: A=2, B=4, C=7, D=8, E=10, F=14, G=16 and the status quo q=0. q A B C D E F G 1. What set of alternatives are in equilibrium under pairwise majority rule (i.e., what is the core)? D=8 because it is in the median location. Alternatively, we can use the same method for other k-majority rule. Four is a majority of 7. Counting four ideal points from the left gives us D. Counting four ideal points from the right gives us D. 2. What set of alternatives beat q under pairwise majority rule? W D (q) = (0, 16) as denoted above. These points are closer to the median voter than they are to q. The fact that G is at the right end point of the winset is irrelevant.
Supermajority Rules Suppose N=7 with the following ideal points: A=2, B=4, C=7, D=8, E=10, F=14, G=16 and the status quo q=0. q A B C D E F G 1. What set of alternatives are in equilibrium under pairwise 2/3rds rule (i.e., what is the core)? [C,E] = [7,10] because 5 is a 2/3rds majority of 7. Counting five points left to right gives us E; counting five points right to left gives us C. For all points in this interval, there is not 5 or more points outside the interval. Hence, none of the points can be defeated. 2. What set of alternatives beat q under pairwise 2/3rds rule? W C (q) = (0, 14) as denoted above. These points are closer to both C and E than they are to q. Making both pivots happy, makes it pass. The fact that G is at the right end point of the winset is irrelevant.
Supermajority Rules Suppose N=7 with the following ideal points: A=2, B=4, C=7, D=8, E=10, F=14, G=16 and the status quo q=0. q A B C D E F G 1. What set of alternatives are in equilibrium under pairwise unanimity rule (i.e., what is the core)? [A,G] = [2,16] because 7 is a unanimity of 7. Counting seven points left to right gives us G; counting seven points right to left gives us A. For all points in this interval, there is not 7 or more points outside the interval. Hence, none of the points can be defeated. 2. What set of alternatives beat q under pairwise unanimity rule? W A (q) = (0, 4) as denoted above. These points are closer to both A and G than they are to q. Making both pivots happy, makes it pass. The fact that B is at the right end point of the winset is irrelevant.
Bicameralism Consider the following single-dimensional spatial voting model with single peaked preferences and symmetric utility. All actors vote sincerely, where H=4 (median of the House), P = 8 (ideal point of the President), S = 10 (median of the Senate), and the status quo q=0. W S (q) x W H (q) W P (q) q H P S
Bicameralism, cont. 1. If all actors vote sincerely (i.e. vote for their most preferred alternative, without considering the consequences for future steps in the game), what set of alternatives could pass both Congress and the President? Why? W H (Q). This is because W H (Q) is the set of alternatives that are preferred by a majority of both houses of Congress (W H (Q) = W H (Q) W S (Q)). Furthermore, W H (Q) is within the preferred to set of the President. Therefore, the President will accept anything within W H (Q). 2. Now assume that members of the House propose strategically (i.e. think about the consequences of their proposal for future steps of the game), what alternative(s) would the House propose? Why? H. If the majority of the House proposes strategically, they would want to pick a proposal that gives them the best possible outcome at the end of the game. Since all the alternatives that can pass are in W H (Q), the majority of the House wants to choose the outcome within W H (Q) that is closest to H. In this case, that s H itself, because at least the member at H and all those to the right (a majority prefer H to Q).
Bicameralism, cont. 3. Now assume that all actors vote strategically. If the President has the power to propose and he/she proposes strategically, what alternative(s) would the President propose? Why? How does this answer differ from the previous one and why? x, as depicted in the figure above. If the President proposes strategically, he/she will want to propose the alternative that gives him/her the best outcome at the end of the game. Since the set of alternatives that could pass Congress are W H (Q), the President chooses the alternative within W H (Q) that is closest to P. That s x in the figure. The question is whether members of Congress could do better at the end of the game by voting differently than they voted in the sincere case. If they could, that might suggest a better location for the President s proposal. If they could not do better, then strategic behavior would be the same as sincere behavior and the President would still propose x. To see if the majority of the House would vote any differently, note that they must chose between q and x. If they voted strategically, all this vote would do is increase that chances of attaining q, which they don t want to do because q is further from H than x is from H. The same is true for all members to the right of H. Hence, a majority of the House will want to vote for x in the choice between q and x (i.e. they would vote the same as the sincere case). Similarly, note that members of the Senate are faced with a choice between q and x and a majority prefer x to q. Hence, they would have no incentive to vote for q, which would only increase the chances of them receiving their less preferred alternative. In this case, strategic voting and sincere voting are observationally equivalent (i.e. individuals vote the same under both motivations, though they propose differently).
Bicameralism, cont. 4. Now assume that all actors vote strategically. If the Senate has the power to propose and proposes strategically, what alternative(s) would the Senate propose? Why? How does this answer differ from the previous ones and why? The Senate would propose x as well, because it is the closest alternative to S that will pass. For the reasons stated in the previous problem, none of the actors can do better than this outcome when voting strategically.