The largest study contrasting approval voting with plurality voting in an important actual election was for the French Presidential Election (non-runoff round) of 2002. Here is an abbreviated English-language version of their paper:
Jean-Francois Laslier & Karine Vander Straeten: Approval Voting: An Experiment during the French 2002 Presidential Election. And the full French-language version is #2003-007 here (or see local copy) and here's what seems to be a French-language pdf slideshow on this.
They conducted an approval-voting pseudo-election in parallel with the genuine French (plurality-style) election, using several thousand voters, all of whom were also actual voters in the real election. Here are their results (listed in order of finish in the official plurality election).
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As you can see, with Approval Voting the election results – not only the top two, but in fact all the way down the line – would have been dramatically different. This illustrates the tremendous distortions caused by strategic-voting effects, "vote-splitting" effects, and "cloned candidate" effects with plurality voting; such bad effects tend to be even more shriekingly obvious in elections like this one (we expect similar effects in the Iowa 2008 Presidential Caucuses) which include many viable candidates.
Under the French rules this (plurality) election was close enough that by law, a second "runoff" election needed to be held between the top two plurality finishers Chirac and Le Pen. The fact that Plurality (in the election above) had produced a tremendously distorted result ranking Chirac and Le Pen as "close," was proven by the fact that in the runoff, Chirac beat Le Pen by an enormous margin of 82.2% to 17.8% (far greater than the largest landslide ever in any US presidential election, which was only about 60%). As you can see, even Approval Voting's (less distorted) results still were somewhat distorted as far as what you might have expected about Chirac versus Le Pen is concerned. Probably Range Voting would have behaved better still.
Another interesting point is that, even with approval voting, no candidate got more than 37% Approval. Keep that in mind as a counterexample in case somebody claims that "obviously" some candidate will get more than 50% Approval. Voters are not necessarily happy about their choices – and whenever that is so, it will clearly be seen with approval voting.
| Number of approved candidates (average = 3.15) | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | >10 |
| 36 | 287 | 569 | 783 | 492 | 258 | 94 | 40 | 16 | 6 | 1 | 5 |
If the French 2002 election had been held with Approval Voting, the top two instead would have been Chirac versus Jospin, and quite plausibly the popular former president Jospin (who suffered a lot of vote-splitting in the above election) would have won that runoff, changing history. (Pre-election polls had shown that a hypothetical Jospin-Chirac runoff would be too close to call, but it was clear Le Pen would lose big in a runoff with either.)
Another interesting possibility is that some Chirac voters may have intentionally voted for Le Pen figuring that way Chirac would beat Le Pen in the runoff, whereas Chirac was much less likely to defeat Jospin in the runoff. If so that was a case where voting for the worst candidate in that voter's view, was actually superior strategy to voting for her best choice. That's an example of one common runoff pathology which never arises with approval voting.