Is there a reasonable way to count votes?

I have seen a supposedly learned journal which suggests that there is no ‘fair’ way to select a winner from a set of ballots. Although not emphasised by wikipedia or the UN, this is a commonly held view, often relying on abstruse maths.

A simple example

Suppose that there are two candidates who are the first choice of an equal number of voters: a tie. Then no fair deterministic method can select a winner. But is this a practical problem?

Typical proofs

Condorcet was at least one of the first to note that a set of ballots could imply a 3-way tie, with A being above B on most ballots, B above C on most, and yet C above A on most ballots. Such ties could be relatively robust, in that the addition of another ballot would not necessarily break the tie. For example, if a voters vote ABC, b voters vote BCA and c vote CAB then there is a 3-way tie (A over B over C over A) whenever (a+b)>c etc. If those who voted ABC wish to eliminate C they can do so by voting BAC, in which case B has a clear majority. This is certainly a problem, but it did not stop Condorcet from devising what he regarded as a ‘reasonable’ method. The difficulties lie in breaking n-way ties. The best known impossibility theorems is Arrow’s. It is implicit that the ‘voting system’ is deterministic. The problem is in what it calls “irrelevant alternatives”. But in Condorcet’s 3-way tie, C is ‘relevant’ to the choice between A and B, although Arrow says that it isn’t. So although Arrow’s approach is interesting, the result could be summarised as saying that ties are a problem. The Gibbard-Satterthwaite Theorem extends Arrow by replacing consideration of ‘irrelevant’ alternatives consideration of tactical voting. This is a much more reasonable criterion. As Condorcet and others appreciated, tactical voting can easily arise in any definite method that seeks to break a 3-way tie. But how significant is this?

A simple counter-example

The mathematics above is correct, but we may doubt the interpretation. A simple method is to put the ballots into a hat, pull one out ‘at random’, and to select a winner from it.

Irrelevant alternatives:  the winner depends only on the chosen ballot, not on any alternatives.

Tactical voting: how people vote only matters if their ballot is selected, in which case they would do best to vote honestly.

This method is not very attractive, but it does show that the impossibility theorems need careful interpretation.

A more reasonable counter-example?

I have devised this example to be ‘obviously’ reasonable, without necessarily being the best. First, a candidate is said to have a majority over another if it is ranked higher on more ballots. The method is:

  • If we can divide the candidates into two sets, such that all members of the first set have a majority over all members of the second set, then the second set is eliminated.
  • This is done to yield a minimal candidate set, with ties between its members.
  • A ballot is chosen at random and that member of the candidate set that is ranked highest is selected.

Independence from Irrelevant Alternatives

All members of the candidate set are relevant in the sense that there is no agreed way to choose between them. In this sense the above method meets the requirement for ‘independence from irrelevant alternatives’. The more conventional requirement seems much too strong.

Tactical Voting

Is the method liable to tactical voting? Firstly, as before, if my candidate gets into the candidate set then it pays to rank my candidate first, and the other rankings do not count. Hence the only possibility of tactical voting is in determining the candidate set. As above, I could try to eliminate candidates from the candidate set by exploiting cycles, but I cannot falsely promote candidates into the candidate set, because I would have ranked them highly anyway. In this sense the scope for tactical voting is limited. You vote for the candidate with the best chance of beating your least liked candidate. If this is your priority, then this is arguably an honest vote. It may be that we need to distinguish between types of tactical voting, rather than treat them all as equally disreputable.

Types of tactics

Some voting systems, such as first-past-the-post, waste votes for genuine first-preferences when they are unpopular. Some hold this to be a good thing, but it means that the votes are cast are not a reliable indication of actual preferences. While it would be ‘a good thing’ for a system not to penalize voters who rank all their preference honestly, a compromise would be a system that never penalizes voters for ranking their true first preference first. Not all tactics are equally harmful. In Condorcet’s system problems arise with ties. If, as above, ties are broken by selecting ballots at random, there is no incentive not to rank one’s first preference first. An alternative method would be to eliminate candidates who were most often worst-ranked. This would encourage voters to take account of how often candidates were likely to be given a low ranking, and so might encourage them to rank a more moderate candidate first. While this is tactical voting, it is not obvious that it would be a ‘bad thing’. Finally, some voting systems (such as first-past-the-post and approval voting) call for tactics that depend on subtle assesments of how others were likely to vote. Under alternative vote, for instance, you want to work out in which round candidates wull be eliminated and who will pick up their next preferences. Tactics which depend more on relatively simple factors, such as who is more moderate, seem preferable. The aim here is not to advocate any particular method, but to cast doubt on simplistic interpretations of ASrrow’s theorem.

See also

Condorcet anticipated Arrow, Approval voting as a counter-example to Arrow.

Dave Marsay


About Dave Marsay
Mathematician with an interest in 'good' reasoning.

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