The "voting paradox" is based on the observation that it's irrational to vote in a large-scale election, such as those in the U.S. for President or the members in either house of Congress, if you place any value on your time and define the benefit from voting as the gain to you from increasing the likelihood that the election result will have a favorable effect on policy outcomes, from your perspective. After all, if you value your time at as little as $10 an hour and there are, say, 100,000 voters in the relevant constituency, the amount of time it's likely to be worth spending (even if Siddhartha Gautama is running against Adolf Hitler) is bound to be well under a second.
What makes it an ostensible paradox, of course, is the fact that lots of people actually do vote. And what makes it not really a paradox is the fact that people don't actually vote for the stipulated reason. Rather, voting is a consumer act, generally of an expressive character, hence no more irrational than waiting on line to see a good movie (though, to be sure, the nature of the consumer good is different - pulling the lever isn't in itself that much fun, other than perhaps in November 2008).
I've thought for many years, however, that voting's being an expressive consumer act, not a calculating one given the collective action problem (a prisoner's dilemma) that makes the supposedly "rational" approach so irrational, is vital to many of the defects in our political process. In its own way, declining to be individually "rational" about voting when everyone else out there is going to determine who wins leads to outcomes potentially as destructive socially as the fact that each of us may benefit from driving long distances and running the car's AC without regard to the local or global environmental impact.
If you're buying a car, you both get to decide what car you end up with and bear most of the consequences of whether it's a good or bad car. (Leaving aside problems such as SUV externalities on other drivers.) Hence, there's some reason to try to make a good choice in terms of the effect it will have on outcomes, which is not to say everyone always does so. But when your individual vote has effectively zero effect on your wellbeing and indeed that of the entire world (even if Gautama is running against Hitler), you may have no reason not to approach the decision in a fundamentally unserious way, so far as true effects on outcomes are concerned.
Otherwise, would it have been a pertinent factor in the 2000 election that Bush seemed like a better drinking buddy type than Gore? Would Palin backers be so blithely indifferent to evidence bearing on her true character if, say, their lives individually depended on it, conditioned on their backing her? For that matter, though I was glad about the 2008 presidential election outcome, would all those first-time Obama voters have materialized if it hadn't been a bit like showing you're into the new soft drink or Lady Gaga video?
In 2010, the voting non-paradox helps explain the possibly upcoming Republican landslide. Voting is fun if you're a Republican and eager for that solidaristic expression of anger that many Dems had fun with in 2008. Not so much fun to vote expressively on the other side these days. Plus voters in the middle, insofar as there are any, don't ask themselves questions such as which party's or candidate's views are closer to mine, or what will happen on the ground if Congress changes hands. They just want to vent.
But real outcomes are affected by the sum of votes, no one of which individually matters more than infinitesimally. And electoral outcomes often are taken as if they reflect the next two years' outcome preferences rather than the moment's expressive preferences.