Assignment:
Approval voting In this question, we consider the case in which the individuals are called upon to choose candidates for a task, by specifying which candidates they most approve for the task. Let A be a nonempty set of alternatives. A binary relation Pi over A is called (at most) two-leveled if there exists a set B(Pi) ⊆ A satisfying: (i) b ≈Pi c for every b, c ∈ B(Pi), (ii) b ≈Pi c for every b, c ∈ A \ B(Pi), (iii) b Pi c for every b ∈ B(Pi), and c ∈ A \ B(Pi). In words, the individual is indifferent between all the alternatives in B(Pi), he is indifferent between all the alternatives that are not in B(Pi), and he prefers all the elements in B(Pi) to all the elements that are not in B(Pi). The interpretation that we give to such a preference relation is that the individual approves of all the alternatives in B(Pi), and disapproves of all the alternatives not in B(Pi). A twoleveled preference profile P N is a profile of preference relations all of which are two-leveled. Consider a choice function H associating every two-leveled profile with a single alternative, which is declared to be society’s most-preferred alternative. Such a choice function H is called monotonic if for every pair of twoleveled preference profiles P N and QN , if H(P N ) = a, and if every individual i satisfies B(Qi) = B(Pi) or B(Qi) = B(Pi) ∪ {a}, then H(QN ) = a. In other words, if alternative a is chosen under preference profile P N , and if QN is a preference profile that is identical to P N except that some individuals have added a to the set of their approved alternatives, then a is also chosen under QN. A choice function H is called nonmanipulable if for every two-leveled preference profile P N , for every individual i, and for every two-leveled preference relation Qi, it is the case that H(P N )
Pi H(Qi, P−i). Define a choice function H* as follows: a winning alternative is one that is approved by the greatest number of individuals; in other words, a is a winning alternative if it maximizes the value of |{i ∈ N : a ∈ B(Pi)}|. If there are two or more alternatives receiving the greatest number of approval votes, the alternative whose serial number is lowest is chosen. Is this choice function monotonic? Is it manipulable? Justify your answer.
Provide complete and step by step solution for the question and show calculations and use formulas.