This is Chance strategy and choice problem 21-7

21.7 (Kemeny Method) John Kemeny described a method for producing a social welfare in the spirit of Condorcet. Given a preference table with m candidates, each possible ordering of the m candidates is given points by each voter according to how many comparisons the given order shares with the voter. For example, suppose that voter p prefers A to B and B to C (and so p prefers A to C). We write this as ABC. Then the ranking ABC receives 3 points from p because it agrees with all three rankings of voter p. The ranking A over C over B receives 2 points from voter p because it agrees with voter p on the ranking A over B and A over C. Here is the preference table for voter p: Points Awarded by Voter p with Ranking ABC Ranking Points ABC N W ACB BAC N BCA CAB CBA The points for each ranking are totaled for all voters, and the ranking receiving the most points is the Kemeny social welfare. Determine the Kemeny social welfare for the following preference tables: Borda versus Condorcet 333 (a) Number 7 5 Ist choice B 2nd choice B B 3rd choice A A (b) Number 3 2 4 1 Ist choice B A 2nd choice A 3rd choice C B (c) Number 7 5 Ist choice A D C 2nd choice B B D 3rd choice A 4th choice D B Note that there are 4! = 24 orders to consider in this case