Voting matters - Issue 5, Paper 2

An example showing that Condorcet infringes a precept of preferential voting systems

C H E Warren

Hugh Warren is a member of the ERS Council.

It is one of the precepts of preferential voting systems that a later preference should neither help nor harm an earlier preference. The purpose of this paper is to show that the Condorcet system of preferential voting infringes this precept.

Consider an election for one seat in which there are 3 candidates:

1. A is a Catholic Conservative White
2. B is a Protestant Labour White
3. C is a Catholic Labour Asian

1. 17 want Labour, they prefer a White to an Asian, and they are indifferent as to sect, so they vote BC.
2. 16 want Labour, they prefer an Asian to a White, and they are indifferent as to sect, so they vote CB.
3. 15 want a Catholic, they prefer Labour to Conservative and they are indifferent as to race, so they vote CA.
4. 17 want a Catholic, they prefer Conservative to Labour, and they are indifferent as to race, so they vote AC.
5. 16 want a White, they prefer Conservative to Labour, and they are indifferent as to sect, so they vote AB.
6. 15 want a White, they prefer Labour to Conservative, and they are indifferent as to sect, so they vote BA.
7. 1, whom we shall call Voter X, wants primarily a Conservative, and wants also an Asian and a Protestant, so is undecided whether to vote AC or AB, but settles for AC.
8. 1, whom we shall call Voter Y, wants primarily a Protestant, and wants also a Conservative and an Asian, so is undecided whether to vote BA or BC, but settles for BA.
9. 1, whom we shall call Voter Z, wants primarily an Asian, and wants also a Protestant and a Conservative, so is undecided whether to vote CB or CA, but settles for CB.

   AB 16
   AC 18
   BA 16
   BC 17
   CA 15
   CB 17

1. C beats B by 50-49
2. A beats C by 50-49
3. B beats A by 50-49

(Incidentally, the Single Transferable Vote, which amounts to the commonly called Alternative Vote in this case, would exclude the lowest, C, and hence would elect B.)

If the paradox is resolved by electing A, then, if instead of voting AC Voter X had voted AB, Candidate B would have beaten Candidate C, and accordingly by the Condorcet method Candidate B would have been elected. Therefore changing the second preference of Voter X from C to B works to the detriment of his first preference A.

If the paradox is resolved by electing B, then, if instead of voting BA Voter Y had voted BC, Candidate C would have beaten Candidate A, and accordingly by the Condorcet method Candidate C would have been elected. Therefore changing the second preference of Voter Y from A to C works to the detriment of his first preference B.

If the paradox is resolved by electing C, then, if instead of voting CB Voter Z had voted CA, Candidate A would have beaten Candidate B, and accordingly by the Condorcet method Candidate A would have been elected. Therefore changing the second preference of Voter Z from B to A works to the detriment of his first preference C.

Therefore, no matter how the paradox is resolved, the precept that a later preference should not harm an earlier preference is infringed.