SECTION A:

Choose Three out of the four short questions below. All questions are marked equally.

1. Prove that when voters are strategic in their voting behavior (no ab-stention allowed and two or more candidates are competing) then for any voter, voting for his least best alternative is a weakly dominated action.

2. Assume that a group of four individuals, N = {1, 2, 3, 4}, is trying to choose between four alternatives, A = {x, y, z, w} Assume that they use Borda rule where they give weights of 4,3,2 and 1 to their 1st, 2nd, 3rd and 4th ranked alternatives respectively. Find an example of profiles of preferences that would violate the Independence of Irrelevant Alternatives axiom.

3. Discuss, in light of the papers by Myerson, Lizzeri and Lizzeri and Per-sico, what are the effects of increased political competition on political outcomes.

4. Suppose you write down a model of the political process and your analysis yields predictions about how political outcomes are affected by the distribution of preferences in society. Discus whether and how Arrow's impossibility Theorem is related to the predictions you have found.

SECTION B:

Choose TWO questions from the three below. All questions are marked equally. All sub questions are equally marked.

5. Assume a citizen candidate model with a runoff system. If a candidate receives more than 50% of the vote in the first round, he wins the election. If no candidate receives more than 50% of the votes then there is a runoff election between the two candidates with the highest vote shares (there is a lottery with equal probability if several candidates have the same vote share). We are checking whether the following configuration of strategies is equilibrium. Three citizens run, one at the median and two symmetrically on both sides of him at (2/3) and -(2/3).

(a) Show that all three candidates receive the same vote share in the first round.

(b) What is the probability that the candidate on the left (with ideal at -(2/3)) wins the election (both rounds). What is the probability that the central candidate wins the election?

(c) Under what conditions on b and c do the three candidates indeed want to run?

(d) Under what conditions, if at all, does this configuration of strate-gies form equilibrium?

(e) Assume that 5c > b > 4c. Find an equilibrium with four candi-dates running.

6. Consider a polity of n citizens who are choosing among the social alter-natives in the set A = {w, x, y, z}. Consider the following aggregation rules:

(i) The constant rule- for any profile of preferences the social pref-erence is always w > x > y > z.

(ii) The minority rule- Alternative v is ranked above alternative u if and only if there is at most half of the citizens that prefer v to u.

(iii) The sequential rule- Look first at only the top ranked alterna¬tives of the citizens. Choose the alternative that most people put on top. If there are several alternatives like that, choose the one which is lowest in alphabetical order. This alternative is ranked first in the social preference. Now take this alternative out of the citizens' rankings and repeat the procedure on the rest of the alternatives to determine the second ranked alternative. Repeat recursively to determine the whole social ranking.

(a) Which of Arrow's criteria is violated by the Constant rule?

(b) Show by example that the Minority rule might lead to an intran¬sitive social preference?

(c) Does the Minority rule violate any other of Arrow's criteria? Prove your answer.
                                                                                    1  2  3  4
                                                                                    w  x  y  x
(d) Suppose that the preferences of the citizens are: x  y  z  y What would be the social ranking under the Sequential rule?
                                                                                    y  z  w  z
                                                                                    z  w  x  w

(e) Which of Arrow's criteria is violated by the Sequential rule? Prove your answer.

7. Consider the political economy of the country Flatland responding to a changing world. Flatland is a country of one dimensional politics. Traditionally, they only care about economics and the policy space has traditionally been the [0,1] interval, where the smaller is the policy the more it favours farmers and the higher is the policy the more it favours the industrialists. Assume that there are three groups in Flatland, Group F are the farmers, Group I are the industrialists and group M make up the middle class of this society. Currently groups F and I each make up 31% of the total voting population and group M making up 38%. Voters have single peaked preferences and share the same policy preferences within each group. The ideal points of the groups are: 1 for group I, 0 for group F and 0.5 for group M. Assume that each group has a representative who can choose whether to run or not and whether to join a party or not. Throughout assume that running is costly but that these costs are very low with respect to the benefits of winning office. Finally, assume that candidates can implement only their ideal point if they win, when they run alone, and only the party platform if they are part of a party. As in lectures, party platforms must be credible and stable.

(a) What is the equilibrium when the three representatives play the citizen candidate model? Explain your answer.

Lately, the citizens of Flatland have been following the dramatic new changes happening in the neighboring country, Country Green. In par-ticular, a newly elected government in country Green has been imple-menting some dramatic new environmental measures. The citizens of country Flatland are now reformulating their preferences in a new and multidimensional world; they have to think both on where they stand on the traditional economics dimension but now also where they stand about the environmental dimension, also a [0,1] interval where higher policies are more favorable to the environment. Groups I and F fear environmental measures that would restrict the way industrialists man¬ufacture and the way farmers grow agricultural products. Therefore, they both prefer the policy 0 on the environmental dimension. Group M actually prefers some environmental measures and its ideal point on this dimension is at 0.5.

(b) Specify the utility functions we have used in lectures (with quadratic loss) to model how these different groups feel about this two-dimensional political space, specifying the ideal points and the weights on the different dimensions that would be consistent with the above story.

(c) Assume that no parties could form. Under what conditions on the weights that the different groups put on the different dimensions of policy will the equilibrium of the citizen candidate model be that the representative of the middle class runs unopposed? Explain your answer.

(d) Assume that the conditions you have found in (c) hold, under what conditions will a party of Industrialists and Farmers arise in equilibrium? Explain your answer (using graphs if necessary).