"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 make up 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 events looks radically different, depending on people's political affiliations.

But 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 rewards for giving the right answer, the partisan divisions become a lot smaller. 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 they answered correctly, their name would be entered into a drawing for a $200 gift certificate from Amazon.com.

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 without the economic rewards.

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, people 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 to report a judgment that reflects your political loyalties (at least if you aren't sure). If this hypothesis is right, much of our apparent disagreements come from people's tendency to give the benefit of the doubt to their preferred political team. Is there a way to test this?

In their second experiment, the researchers did exactly that. Again, 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, 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 group. The differences became pretty small, and hardly the stuff of political polarization.

What's going on here? The researchers think that when people answer factual questions about politics, they engage in a degree of cheerleading at the expense of the truth. In a survey, there is no cost to that.

With economic incentives, of course, the calculus is altered. If you stand to earn some money with an accurate answer, cheerleading becomes much less attractive. And if you will lose real money with an inaccurate answer, you will put a higher premium on accuracy.

What is striking is that this experimant was able to slash polarization with very modest monetary rewards. If the incentives were greater, we could expect that partisan differences would diminish still more.

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 surveys, they know in their hearts that while they are entitled to their opinions, they are not entitled to their own facts.

Cass R. Sunstein, a professor at Harvard Law School, is a Bloomberg View columnist and former administrator of the White House Office of Information and Regulatory Affairs.