Question 9.63

33. The 10 members of a board vote by approval voting on eight candidates for new positions on their board, as indicated in the following table. An indicates an approval vote. For example, Voter 1, in the first column, approves of Candidates , and , and disapproves of , and .

Candidate Voters
1 2 3 4 5 6 7 8 9 10
X X X X X X X
X X X X X X X X
X X
X X X X X X X X X
X X X X X
X X X X X X X X
X X X X X X
X X X X X
  1. Which candidate is chosen for the board if just one of them is to be elected?
  2. Which candidates are chosen if the top four are selected?
  3. Which candidates are elected if 80% approval is necessary and at most four are elected?
  4. Which candidates are elected if 60% approval is necessary and at most four are elected?

33.

(a) Because has the most votes, is chosen for the board.

(b) The top four are , , , and .

(c) Candidates , , and have at least 80% (8 out of 10) approval.

(d) Candidates , , , , and have at least 60% (6 out of 10) approval. But because at most four candidates can be elected, only , , , and are considered.