To figure out who would take the test for $100 but not for $80, you would subtract the share of people who responded to the $80 reward from those who took it for $100. The infection rates for the group could be computed in a similar fashion, to reveal the pattern about how testing changes and the infection rate changes as rewards increase.

There are likely to be people who decline to take the test even at the highest reward. But the researchers can approximate their infection rate by extrapolating from the information they have. If they know how the participation rate and the odds of testing positive change as the reward rises from $160 to $180 to $200, they can deduce what the results would be at $220 and $300.

The underlying assumption is that people’s behavior changes smoothly as incentives change. If that holds, the researchers could construct a decent estimate of the infection rate in all or most of the population.

As the virus becomes more and more prevalent across countries, such testing will become critical. “What is the probability that if I go to the store I will become infected?” asked Marc Suchard, a professor of biomathematics at U.C.L.A. “You can’t nail that down without knowing the underlying rate of infection.”

Notably, as Mr. Atkeson found out, it’s hard to tell what share of infected people die from the virus if you don’t know how many have had it. You could get the same number of deaths with a high mortality rate and a low infection rate or the other way around, if the virus propagated very fast but killed fewer people.

Different findings would call for different strategies. A slow spread with a high death rate might suggest the containment policies deployed around the world are working, and would support maintaining a strict lockdown. The alternative could suggest that more people have some resistance to the virus and support relaxing quarantines across U.S. cities and states.

“Distinguishing between these two scenarios is key,” Mr. Mogstad said.

And yet nobody has measured this statistic. Countries testing for the virus have mostly focused on testing people suspected of being sick — people with symptoms and those who have come into contact with them.