# Randomized Response

Imagine we want to estimate how many people in a population use a drug by asking them. People have an incentive to not answer truthfully, they’re concerned they might be implicated if there is a leak of that private information. Randomized response is a way to set up such a survey with sensitive questions to protects the respondents from privacy leaks. Although it was invented by Stanley Warner way back in the 60s, it is a good starting point in understanding “privacy-preserving data analysis”, which is a modern research area.

The idea is to give respondents a plausible, “innocent” reason for answering “yes” to the implicating question, so that if a leak does happen, there is plausible deniability from the perspective of the respondent. A basic protocol for randomized response is to ask respondents to perform the following steps:

- Before answering the question, flip a coin.
- If it comes out heads, answer the question truthfully.
- If it comes out tails, then flip another coin.
- If the second flip comes out heads, respond “yes”.
- If the second flip comes out tails, respond “no”.

On average, only half the respondents will be asked to respond truthfully. The other half will simply write the results of the coin flip. There are two important properties of this protocol.

- Even if the analyst discloses the entire database, each respondent is “safe”, because there’s a 50-50 chance that their first coin flip came out heads, and so they responded at random. In other words, disclosing a “yes” answer is unlikely to incriminate anyone, because everyone answers “yes” with probability at least 1/2.
- We know that, on average, half the respondents will answer with the coin flip, while the other half will answer truthfully. This means we can recover a very good approximation of the results of the “full survey”

## Estimating the true response rate

The result of the randomized response is a typical “biased” response, in the sense that the response we get is skewed in a particular way, and our goal is to “undo” this skew. In this case, there is one very simple analysis. If we have $r$ respondents, we know that the number of truthful answers will not be very far from $r/2$, and so we estimate that $r/2$ of the answers are truthful. We then need to estimate the number of “truthful yes”s. We know that the number of “random yes”s is going to be about $r/4$, from getting heads then heads from the coin flips.

Let’s say that our biased survey has $Y$ yes’s and $N$ no’s. We simply “invent” an “estimated true survey” with a new number of responses $Y’ = Y - r/4$ and $N’ = N - r/4$, with total respondents $r’ = Y’ + N’$. Now we estimate that the fraction of people who truthfully would answer yes is $Y’/r’$.

## Risks

The neat thing about this protocol is that it does not require the respondent to trust the survey creator: the properties hold independent of the behavior of each party.
Although it might intuitively seem that it doesn’t matter *at all* whether or not each individual’s responses are leaked, there is an important catch.
This protection against leakage only works if the respondents don’t answer multiple independent surveys asking the same question.
Imagine an extreme case where each respondent answers 1000 of these surveys and respects the protocol each time.

Clearly, if only one of these datasets leak, then each of the respondents are not at high risk.
But each additional leakage of a new dataset of the same questions *does* erode the privacy of the answer because for every individual respondent, we start to be able to accumulate a large number of independent runs of that protocol.

Crucially, the same technique that we used to estimate the response rate for a *group* also works to estimate the true response of an individual.
With a single run, there’s too much noise to make any real statement.
But with many runs, eventually the true answer becomes clear and the respondent’s privacy is effectively ruined.

# Takeaways

Randomized response shows that it’s possible to create protocols that enable “privacy-preserving data analysis”. They’re not a panacea, though, as the risks show.

The risks also teach us that privacy loss is not always a “discrete” phenomenon: privacy losses can compound gradually. If data leaks happen, repeated randomized responses over related data will necessarily erode the privacy. In addition, there is a tradeoff going on between the accuracy we can obtain for some questions, and the privacy loss incurred.

The best, modern answer we have in data analysis for this problem is the notion of *differential privacy*.
Differential privacy *also* involves adding noise to answers, but in a fancier way.

## References

- Warner, S.L. Randomized Response: A Survey Technique for Eliminating Evasive Answer Bias. Journal of the American Statistical Association, 60, 63–69. Taylor & Francis, 1965.