WASHINGTON — Republicans learned the lesson the hard way: In the new arithmetic of presidential politics, counting on Election Day votes doesn’t always add up to victory.

In 2008, John McCain won the majority of votes cast at the polls that day in the crucial states of North Carolina, Florida, Iowa, and Colorado, but his victory margin was wiped out by support for Barack Obama among the millions of voters who cast ballots early. The result: McCain netted not a single electoral vote in those four key states.