1) Posner claims that because of the winner-take-all distribution of electoral college votes in nearly all of the states, we have mathematically clearer counts of the winner in an election than would be the case in a national popular vote. I would respond: true, but irrelevant. Any accepted count of votes, electoral or popular, is “clear” exactly to the extent to which people accept the math supporting said count. Would a national popular vote result in less acceptance, more demands for recounts and litigation? Probably–for the first election cycle, at least. But would it continue? Or would judges be forced to establish precedents for counting votes, states be forced to upgrade their voting technology and training, parties be forced to adjust their campaign strategies to minimize such close and legally costly outcomes? I think the latter is far more likely, and thus a new understanding of what makes for “certainty of outcome” would emerge relatively quickly.
2) Posner says the Electoral College forces successful presidential candidates to have transregional appeal. The two-fold flaw with this claim is a) it depends upon a rather limited and historically exclusive definition of what consists of “regional” (does the fact that Obama won the large urban areas on both the East and West coasts make him “transregional”? does the fact that Romney won the South but lost Florida, or won the Intermountain West but lost Colorado, mean that he didn’t actually have “regional” appeal?), and b) it runs against the bedrock (and Supreme Court articulated) standard for a representative democracy that what needs to be counted are the votes of citizens (“one person, one vote”), not where those citizens come from.
3) Posner argues that because the math of the Electoral College forces candidates to spend a lot of time in certain swing states to try to win their votes, the result is that citizens in those states which are likely to decide the election receive enough attention and information from the candidates that they become highly informed voters, and we want the decision for the presidency to rest in the hands of highly informed people. But this tautological. One could just as easily say that, with a national popular vote, the candidates would spend a lot of time and money trying to communicate with people in major media markets, with the result that the people in those media markets would be highly informed, and that’s a good thing, because major media markets serve large population centers, and of course we want the election to be in the hands of those population centers where there are lots of highly informed voters. His claim proves nothing.
4) Posner's weakest claim is that the Electoral College fixes some of the undemocratic consequences of the Senate by forcing presidential candidates to often spend lots of time in big states, giving them a level of electoral consequence which better fits the number of citizens who live within them. Well, yes, all that is true…but it would be even more true if you simply had a national popular vote, and allowed the millions of voters in those large states to make their votes matter directly (as is presently not the case with the millions of Republican voters in California or New York, or the millions of Democratic voters in Texas).
5) Posner's final argument is that, since the Electoral College makes clear majorities very likely, it eliminates the need for run-off elections. But the bug in this claim of his is actually a feature: why not have run-off elections for our chief executive? (They have them in France, after all.) He needs to make an argument for his position besides asserting how it supposedly would make our system irredeemably more complicated.
As a final note, Posner adds that the electoral college doesn’t discourage voters from “express[ing their] political preference.” I’m not sure what study he makes use of to support this claim; the simple fact that many people turn out to vote for losing candidates in safe states doesn’t mean that there aren’t any voters who would like to believe that their “single vote may decide an election.” Ultimately, people vote (as I well know!) for all sorts of different reasons, strategic and expressive alike. A national popular vote would allow all of those motivations to have their place, rather than being marginalized or magnified simply depending on where one lives.
One last thing: obviously, my disagreement with Posner here really begins with the simple fact that we very likely hold to fundamentally different theories of government, and presumably our different justifications for those theories differ greatly as well. Basically, I think it’s best to live in societies that are democratic in whatever areas of governance which can plausibly be conducted democratically, whereas he obviously doesn’t see democracy as nearly as normative as I do. So I don't expect to convince anyone who is by nature suspicious of democracy. But at the very least, I'd like people to come up with better arguments in defense of the Electoral College than these superficially smart but actually quite weak ones here.