Assignment:
The following electoral method is used to choose the mayor of Hobbiton: Every resident ranks the candidates from most preferred to least preferred, and places this ranked list in a ballot box. Each candidate receives a number of points equal to the number of residents who rank him or her least preferred. The candidate who thus amasses the greatest number of points is then removed from the list of candidates. If two or more candidates are tied for first place in the number of points, the candidate among them whose social security number is greatest is removed from the list of candidates. This candidate is then ignored in the strict preference relations submitted by the residents, and the process is repeated as often as is necessary, until only one candidate remains, who is declared the new mayor. Assume there are at least three candidates.
(a) Is it possible for the winner of the election to not be the most preferred candidate of any resident? Justify your answer.
(b) Is it possible for the winner of the election to be ranked least preferred by at least half of the residents? Justify your answer.
(c) Is the social choice function described here dictatorial? Justify your answer.
(d) Is it monotonic? Prove why yes, or show by example that it is not monotonic.
(e) Is it manipulable? If yes, provide an example or otherwise prove that it is not.
(f) After the election of the mayor, using the above method, is completed, the local Elections Board checks which candidate would have won had they implemented instead the electoral method used in Whoville. Will the same candidate necessarily be chosen under both electoral methods? Justify your answer.
Provide complete and step by step solution for the question and show calculations and use formulas.