The mathematics behind Arrow are quite convoluted, but there is a simple example that makes the result almost self-evident. Consider a three party race, with Abel, Baker, and Charles as candidates. Now consider pair-wise choices between Abel and Baker, between Baker and Charles, and Abel and Charles respectively. It is possible that each of the candidates would win one of those pair-wise matchings. I.e., just because Abel would beat Baker, and Baker would beat Charles, it is conceivable that Charles would still beat Abel. How would this happen? If 40% of the voters unconditionally prefer Charles, 31% unconditionally prefer Abel and 29% unconditionally prefer Baker, then this result occurs if the majority of Charles' strong supporters would prefer Abel over Baker in a two-way race. Now consider another three way race between Abel, Baker and Charles. It is possible that Abel might win both of his pair-wise comparisons against Baker and Charles, and yet lose the three way race. This could happen with the same split of unconditional preferences (40%, 31% 29%), but with Charles' supporters split evenly if only given a choice between Abel and Baker. In both cases Charles has more unconditional support, yet in one case he wins and in another he loses.

Why is this bad? Why does it imply 'unfairness'? That, of course, is where the mathematics gets blurred into political philosophy, but under the assumption that candidacy is open to anyone who wishes to declare, it is clear that the ultimate result of an election becomes subject to manipulation simply by increasing the number of candidates. It is manipulable by one candidate appealing to a second candidate's supporters based on their dislike of a third candidate. Simply put, it allows something other than the real preferences of the voters control the decision.

Approval voting is a system that, while not perfectly fair -- that is mathematically impossible, after all, is considered by many analysts to be more fair than any of the voting systems in common use. It is less subject to manipulation via secondary considerations than either a simple plurality vote or a top-two-with-runoff system. Approval voting changes the premise from "one-man one vote" to "one-man, N-candidates, N-1 votes". Each voter casts one or more votes, one for each candidate that he approves of. The only way that approval voting can be distorted by strategic voting is if voters who strongly disapprove of the leading candidate cast their votes for every other candidate except the one they disapprove of; and this turns out to be a fairly minor distortion, unlikely to favor any particular result.

If Approval voting were used in New Hampshire, I would cast two approval votes, for Clark and Kerry. I could eliminate considerations of who is more electable or who needs a boost in New Hampshire the most in order to ultimately contribute to selecting a candidate that I really like. But we don't have Approval voting, so I'm going to have to make up my mind.