Question 1. Suppose ve rational roommates are deciding on a place to have coee together. There are four alternatives, Peet's Coee, Starbucks, Think Coee, and making coee at home. The roommates' preferences over these alternatives are the following: the rst roommate strictly prefers Think to Peet's to Starbucks to home, the second roommate and third roommate have the same preference, they both strictly prefer Think to Starbucks to Peet's to home, the fourth roommate strictly prefers Starbucks to Peet's to Think to home, and the fth roommate strictly prefers Starbucks to Peet's to home to Think.

(i) Is there a Condorcet winner? Please justify your answers.

(ii) What is the group preference and what is the group choice according to the Borda count rule? Please justify your answers.

(iii) Again consider the Borda count rule. Manipulate ONLY ONE roommate's preference above to illustrate that the Borda count rule violates independence of irrelevant alternatives.

(iv) Now consider the following new group decision rule: rst, identify the alternative(s) that at least one roommate ranks below staying home, place such alternative(s) as the lowest ranked for the group; second, remove such alternative(s) from the individual rankings, and the group ranks the remaining alternatives according to plurality rule. (This is essentially plurality rule with the additional requirement that everyone has to be willing to leave home.) What is the group preference and what is the group choice according to this rule? Please justify your answers.

(v) Under the rule specied in (iv), does any roommate have an incentive to vote tactically? That is, given all other four roommates report their true preference rankings, does any one have an incentive to misrepresent his/her true preference ranking? Please justify your answers.

Question 2. Consider a group with 11 members needs to make a choice from three alternatives: L, M and H. The individual preferences are given by: 2 members strictly prefer L to M to H, 4 members strictly prefer M to L to H, 2 members strictly prefer H toM to L, and 3 members strictly prefer H to L toM.

(i) What is the group preference according to the plurality rule?

(ii) What is the group choice according to the plurality rule with runo?

(iii) Are the members' preferences in this group single-peaked preferences? Justify your answers. (That is, for every possible order of the three alternative, determine whether all individuals' preferences are single-peaked - if all individuals' preferences are single-peaked for some order, then the preferences in question are single-peaked; otherwise the preferences in question are not single-peaked.)

Question 3. Consider a situation of electoral competition in which candidates choose a position on government spending to allocate to national defense. The alternatives are various percentage of the (xed) budget (any point in the interval [0; 100]). Voters' preferences are single-peaked, and their ideal percentage is distributed according to the following: 60% of the voters' ideal point is 10 (that is, 60% voters think it's best to spend 10 percent of government budget to defense), 30% of the voters have ideal point of 25, and the rest 10% voters have ideal point of 50. Suppose that all voters care equally about policy dierences to the left and right of their ideal point (they have symmetric single-peaked preferences). Voters are sincere and will vote for the candidate with position closest to his/her ideal point, and if two or more candidates have the same position, they split the votes equally.

(i) Suppose that there are two candidates competing for oce. What position(s) will the two candidates choose?

(ii) Now suppose that there are three candidates but one of them is non-strategic and takes a xed position located at 50. For the two strategic candidates, do the position choices in (i) still constitute a Nash equilibrium? (That is, given the position choice of one candidate, is the other candidate's position his/her optimal choice?)

Question 4. You would like to buy a used car. Suppose there are three types of quality (a): low (a = 0), medium (a = 1) and high (a = 2), each equally likely (i.e. each with probability 13 ). Only the seller knows the exact quality of his/her car. Also suppose (as in class) that for any quality of the car, your utility is 1.5 times the seller's utility, that is, for a car of quality a, the seller's utility from it is a, but your utility from it is 1:5a.

(i) The market price for a used car is p. For what values of p, all three types of sellers want to sell? For what values of p, exactly two types want to sell? For what values of p, only one type wants to sell?

(ii) Is there any price at which trade is mutually agreeable? Namely, given your answers in (i), is there any price at which you want to buy a used car?

(iii) Suppose you have help from a mechanic, so now you can tell whether a car is low quality, but cannot tell between medium and high quality cars. Is there any price at which trade is mutually agreeable? If yes, give an example of such a price; if no, explain why. Justify your answers.