Republican presidential candidates, from left, John Kasich, Mike Huckabee, Jeb Bush, Marco Rubio, Donald Trump, Ben Carson, Carly Fiorina, and Ted Cruz take the stage during the CNBC Republican presidential debate at the University of Colorado, Wednesday, Oct. 28, 2015, in Boulder, Colo. (AP Photo/Mark J. Terrill)

Republican presidential candidates spent a lot of time this week hammering out a list of demands for television networks, in hopes of improving the quality of upcoming debates. Which is interesting, because there’s no evidence that the debates -- while certainly frustrating for the candidates at times -- are hurting the GOP’s chances of winning back the White House next year.

If anything, in fact, they might be helping -- unless you’re Rand Paul.

Presidential primary debates, believe it or not, weren’t designed to be ratings bonanzas for television networks. They’re supposed to help the political parties who sponsor them. Theoretically, the debates should do this in two ways: By giving primary voters more information to help them select a candidate, and by introducing the candidates to voters outside the party who might be persuaded to vote for them in the general election. Ideally, that introduction would help the party’s candidates win swing voters, not hurt them.

On the first point, the Republican primary debates appear to be succeeding. They’ve produced greater awareness of lesser-known candidates, including the rise of Carly Fiorina, which shows GOP voters are responding to what they see the candidates saying to each other on screen (or at least to news reports and commentary around the debates).

The debates also are not exacting any significant damage to candidates’ reputations with political independents, according to Gallup data provided to the Washington Post. Since July, Gallup has been tracking candidate favorable ratings in daily national surveys, and it kindly broke down how Republicans have fared in three time periods: 1) Before the first debate 2) between the second and third debates, and 3) between the second and third debates. (We’ll look at the impact of the third debate a bit later).

The chart below shows "net favorable" ratings for each candidate at these three time periods - the percent rating each favorable minus the percent unfavorable (the higher the better). The flattish lines indicate candidates’ popularity did not shift much after debates. On average, Republican candidates’ net favorable ratings among independents shifted 0.2 points, basically no change at all. One big bonus of Gallup’s tracking survey is sample size -- results are based on at least 1,800 independents interviewed during each time period for almost all candidates.

Source: Gallup. Daily polls aggregated to across month before first debate and periods between debates 1/2 and 2/3.

Don't squint too hard to spot movement -- the chart below zeroes in on the statistically significant changes. The debates appeared to have helped Ben Carson and Carly Fiorina, whose net favorable ratings among independents rose by +8 and +7 respectively from before the first debate to after the second. The story is worst for Rand Paul, whose popularity has sunk over the same period by negative-11 on net favorability, from +5 to -6.

Source: Gallup. Daily polls aggregated to across month before first debate and periods between debates 1/2 and 2/3.

None of these are gigantic shifts in opinion – for example, all these changes are smaller than if a candidate’s favorable-unfavorable split went from 50-50 to 56-44 (a swing of +12).  Carson also started off as the most popular by this metric and Trump the least popular. That’s still the same.

While debate season hasn’t altered candidates’ positive/negative images very much, they have become better-known to political independents. Before the first debate, 49 percent of political independents had a positive or negative impression the average Republican candidate tested in Gallup’s polling. That was up to 57 percent after two debates. The biggest rises in familiarity were for Fiorina (+22) and Carson (+19), while others were are up by single digits. Despite the higher ratings, there is still a massive range in familiarity with candidates, ranging from only 33 percent who are familiar with Ohio Gov. John Kasich to 87 percent familiar with Trump.

Source: Gallup. Daily polls aggregated to across month before first debate and periods between debates 1/2 and 2/3

And what about that third debate, last week on CNBC, which so enraged Republican candidates and pundits? A different set of polls -- from Quinnipiac University -- suggests it also did not hurt GOP candidates with independents.

Quinnipiac found that the GOP candidates analyzed above gained, on average, two points of net favorability among independent registered voters from the time after the second debate to after the third debate. (It also found a much larger boost from the debates as a whole than Gallup -- an average net favorability increase of 5.9 percent from before Debate No. 1 to after Debate No. 3.) The bigger shifts could be a result of smaller sample size, as Quinnipiac’s surveys included several hundred independents, while Gallup surveyed more than 1,800 at each time period for almost all candidates. In better news for Paul, Quinnipiac showed he recovered some ground after the latest debate.

If there’s any complaint for Republicans in the debates, then, it might be that they’re not helping the field enough. Gallup still shows independents view the GOP candidates, on average, more unfavorably than favorably. Quinnipiac shows the candidates are view just slightly more favorably than not. In both cases, though, most candidates' ratings are at least holding steady – even as they're not nearly as high as the party probably would like them to be.

The Gallup polls were conducted among national samples of U.S. adults interviewed on landline and cellular phones. Sample sizes range somewhat for different candidates as each was not asked of the full sample. For each candidate, favorable ratings were measured among at least 1,800 self-identified independents during the pre-debate period (July 8-Aug. 6); there were at least 2,300 respondents for all candidates between the first and second debates (Aug. 7-Sept. 16), and at least 1,150 interviews for each candidate between the second and third debates (Sept. 17-Oct. 28).