Suppose five rational roommates are deciding on a place to have coffee together. There are four alternatives, Peet's Coffee, Starbucks, Think Coffee, and making coffee at home. The roommates' preferences over these alternatives are the following: the first 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 fifth 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: first, 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.

What is the group preference and what is the group choice according to this rule? Please justify your answers.

(v) Under the rule specified 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.