Republicans have completed their three-day infomercial with a solid performance by Mitt Romney on Thursday, which followed even-stronger performances by Paul Ryan on Wednesday and Ann Romney on Tuesday. Next week, it will be the Democrats’ turn to present their three-day infomercial from Charlotte.

Will the 2012 political conventions make a difference? Recent political history answers this question with a resounding “maybe.”

New York Times blogger Nate Silver, whose politics I don’t share but whose analytical work I respect, has laid out this recent political history with several informative posts. The latest one came out on Thursday. Silver notes that while some political conventions have generated huge bounces for their presidential candidates, such as both Ronald Reagan’s and Jimmy Carter’s conventions in 1980 and Bill Clinton’s in 1992, other conventions have generated only modest improvements for their candidates that mostly or completely faded away within a couple of weeks.

Interestingly, the 2008 Democratic convention in Denver that nominated Barack Obama should be classified as one of the least influential in moving the numbers. Obama gained only a couple of percentage points after his convention. Then after the GOP convention, the high point for the McCain-Palin ticket, Obama was down a point, on average. On Election Day, however, he won by seven points – in part because of post-convention developments on Wall Street.

Silver’s entire discussion is well worth a read. But if want a few takeaways, here they are:

• In the past, presidential challengers usually got larger bounces from their conventions than presidential incumbents did, presumably because the challengers were less familiar to voters. This effect may be diminishing, however.

• The polls right after the challenger’s convention, but before the incumbent’s convention, will probably exaggerate the challenger’s eventual support by a large amount. If Romney-Ryan gets a large bounce from this week’s convention, history suggests we should expect the Democratic convention to erase a good bit of it – but not necessarily all of it.

• Convention bounces are getting smaller because the number of truly undecided voters at this stage in the campaign is smaller than it used to be.

• Because the Obama-Romney polling has been fairly stable for many weeks – the president up a point or two, on average – that suggests that the bounces from the Tampa and Charlotte conventions are unlikely to match the double-digit swings of years past.

Put it all together, and the math works out to be something like this. In today’s environment, presidential candidates ought to be getting about four points or so of bounce from their conventions. If by early next week, Romney-Ryan leads Obama-Biden by a couple of points, the Tampa convention will have essentially done its job marketing a relatively unfamiliar ticket to swing voters. If the Republican ticket has opened up a larger lead than that, the Democrats should be worried. If Romney still trails Obama by early next week, the Republicans should be worried.

Sounds about right to me.

Hood is president of the John Locke Foundation and author of Our Best Foot Forward: An Investment Plan for North Carolina’s Economic Recovery.