Assume five rational roommates are deciding on the place to have coffee together. There are four alternatives, Peet's Coffee, Starbucks, Think Coffee, and making coffee at home. Roommates' preferences over these alternatives are the following: the first roommate strictly prefers Think to Peet's to Starbucks to home, second roommate and third roommate have 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? Justify your answers.
(ii) What is the group preference and what is the group choice according to Borda count rule? Justify your answers.
(iii) Again consider Borda count rule. Manipulate ONLY ONE roommate's preference above to show that the Borda count rule violates independence of irrelevant alternatives.
(iv) Now consider following new group decision rule: first, identify 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 extra 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? 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 anyone have an incentive to misrepresent his/her true preference ranking? Justify your answers.