Some voting paradoxes: Consider for a moment the following scenario: suppose there are four candidates. Suppose further (as an example for us to follow) there are 10 voters. Suppose also that each voter has a personal linear ranking of the four candidates (i.e., each voter actually has an opinion on each of the candidates, and that no voter has a circular ordering in their preferences). Let's denote the four candidates by A, B, C, D, and preferences by ">" so A > B means candidate A is preferred to candidate B. Now let us imagine that the actual preferences among the 10 voters are as follows: # voters | ranking ---------|--------------- 1 | B > C > A > D 2 | A > B > C > D 2 | A > D > B > C 2 | D > C > B > A 3 | C > D > B > A Let's see what the results of various election procedures will be. 1.) Each voter votes for 1 candidate. The results will be A : C : D : B with vote tallies 4 : 3 : 2 : 1 2.) Each voter votes for 2 candidates, with totals according to total number of votes received: D : C : A : B with vote tallies 7 : 6 : 4 : 3 3.) Each voter votes for 3 candidates: B : C : D : A 10 : 8 : 7 : 5 4.) Each voter expresses their ranking, candidates get 4, 3, 2, or 1 points according to their ranking: C : D : B : A 27 : 26 : 24 : 23 Now let us imagine that the actual preferences among the 10 voters are as follows: # voters | ranking ---------|--------------- 2 | A > B > C > D 3 | A > D > C > B 2 | B > C > D > A 3 | D > B > C > A Let's see what the results of various election procedures will be. 1.) Each voter votes for 1 candidate. The results will be A : D : B : C with vote tallies 5 : 3 : 2 : 0 2.) Each voter votes for 2 candidates, with totals according to total number of votes received: B : D : A : C with vote tallies 7 : 6 : 5 : 2 3.) Each voter votes for 3 candidates: C : D : B : A 10 : 8 : 7 : 5 4.) Each voter expresses their ranking, candidates get 4, 3, 2, or 1 points according to their ranking: D : B : A : C 27 : 26 : 25 : 22 All right. Let's try another one. We have three candidates, A, B, and C, for a position. The selection committee votes, and finds the ranking A > B > C. But, before any offer is made, candidate C withdraws. Could that mean we shouldn't offer the position to A, but rather should vote again? Suppose we have 13 members on the committee, and their preferences among the candidates are: # voters | ranking ---------|--------------- 6 | A > C > B 4 | B > C > A 3 | C > B > A In the initial vote, the results will be A > B > C with vote tallies of 6 : 4 : 3. After C withdraws, the preferences will be: # voters | ranking ---------|--------------- 6 | A > B 4 | B > A 3 | B > A Taking a new vote, the results will be B > A with vote tallies of 7 : 6 . . . Perhaps we should have used the "points" method in the first place (3 for first, 2 for second, 1 for third). In that case, the results would have been: C > A > B with points of 29 : 25 : 24