“You are entitled to your own opinion, but not to your own facts.” This quotation, often attributed to Daniel Patrick Moynihan, rightly suggests that while it is perfectly legitimate for people to have different political views, they should not be making up the facts.

Yet a lot of survey evidence seems to suggest that Democrats and Republicans sharply disagree about facts, not just opinions.

Among other things, they appear to think that especially bad things — bigger budget deficits, greater unemployment — happened under presidents of the political party they dislike. Sometimes Democrats and Republicans seem to live in parallel historical universes, in which the course of human events looks radically different, depending on people’s political affiliations.

While a lot of research seems to support this conclusion, we now have good reason to believe that it is fundamentally wrong. Recent studies by Yale University’s John Bullock and his co-authors suggest that with respect to facts, Democrats and Republicans disagree a lot less than we might think.

True, surveys reveal big differences. But if people are given economic rewards for giving the right answer, the partisan divisions start to become a lot smaller. Here’s the kicker: With respect to facts, there is a real difference between what people say they believe and what they actually believe.

In their first experiment, Bullock and his colleagues asked Democrats and Republicans a series of questions and told them that for each question that they answered correctly, their name would be entered into a drawing for a $200 gift certificate from Amazon.com. They were also told that the average chance of winning was 100-1, but that if they answered many questions correctly, their chances would be significantly higher.

The factual questions included the change in the unemployment rate under President George W. Bush, the number of U.S. soldiers killed in Iraq from 2003 to 2011, and the percentage of the federal budget that went to the Medicaid program. A control group was asked the same questions, but without the potential economic reward.

In the control group, the difference between Democrats and Republicans was quite large (as expected). But with the small economic incentive, the difference was cut significantly — by 55 percent. Democrats and Republicans didn’t exactly come into accord, but they got a lot closer. When real money is on the line, Democrats and Republicans are far less likely to answer in a partisan fashion, and far more likely to agree with each other.

This experiment didn’t allow people to answer, “I don’t know.” We might hypothesize that the remaining partisan division reflects a natural human reaction, which is to report a judgment that reflects your political loyalties (at least if you aren’t sure). If this hypothesis is right, much of the apparent disagreement between Republicans and Democrats attests to people’s tendency to give the benefit of the doubt to their preferred political team. Is there a way to test this hypothesis?

In their second experiment, Bullock and his colleagues did exactly that. As in the first experiment, they gave people an economic reward for a correct answer, but they also gave people a reward for a “don’t know” answer. (The reward was smaller, amounting to about 25 percent of the reward for a correct answer.) Stunningly, the result was to cut partisan differences even further — to merely 20 percent of what they were in the control condition. The differences between Democrats and Republicans weren’t exactly wiped out, but they became pretty small, and hardly the stuff of real polarization across political divides.

What’s going on here? Bullock and his colleagues think that when people answer factual questions about politics, they engage in a degree of cheerleading, even at the expense of the truth. In a survey setting, there is no cost to doing that.

It might seem disturbing to find such a divergence between what people say and what they actually believe, but in a way, these findings are immensely encouraging. They suggest that with respect to facts, partisan differences are much less sharp than they seem — and that political polarization is often an artifact of the survey setting.

When Democrats and Republicans claim to disagree, they might be reporting which side they are on, not what they really think. Whatever they say in response to survey questions, they know, in their heart of hearts, that while they are entitled to their own opinions, they are not entitled to their own facts.

Cass R. Sunstein, the Robert Walmsley University professor at Harvard Law School, is a Bloomberg View columnist. He is the former administrator of the White House Office of Information and Regulatory Affairs, the co-author of “Nudge” and author of “Simpler: The Future of Government.”