Arrow’s Impossibility Theorem very specifically applies only to rank-order voting systems. All candidates must be ranked in order, with no two candidates being given equal rankings. Has there been any analysis of how the results might change if ties are allowed, both in the voters’ rankings and in the final societal preference? Does this change allow for a system that meets Arrow’s criteria of unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives?
However, "independence of irrelevant alternatives" is dubious because the range of a voter's preference scale is sensitive to the introduction of an extremely good or bad candidate. For example, a voter who in a two-candidate election, might vote:
- Bush: 10
- Gore: 0
could, in a three-candidate election, equally sincerely vote:
- Bush: 10
- Gore: 8
- Zombie Josef Stalin: 0
All candidates must be ranked in order, with no two candidates being given equal rankings.
This is not correct. Arrow always considers the possibility of voters being indifferent on some alternatives. This may be often omitted or hidden in popular descriptions because it just complicates some descriptions (and in particular on the mathematical level).
From Arrow’s A difficulty in the concept of social welfare (boldface mine):
It is assumed further that the choice is made in this way: […] the chooser considers […] all possible pairs of alternatives, say x and y, and for each pair he makes one and only one of three decisions: x is preferred to y, x is indifferent to y, or y is preferred to x.