Impossibility Results in Fairness (from Adventures in Computation)

Here are excerpts from a post at Adventures in Computation that will be of interest to market designers:

Impossibility Results in Fairness as Bayesian Inference

"One of the most striking results about fairness in machine learning is the impossibility result that Alexandra Chouldechova, and separately Jon Kleinberg, Sendhil Mullainathan, and Manish Raghavan discovered a few years ago. These papers say something very crisp. I'll focus here on the binary classification setting that Alex studies because it is much simpler. There are (at least) three reasonable properties you would want your "fair" classifiers to have. They are:

1. False-Positive Rate Balance: The rate at which your classifier makes errors in the positive direction (i.e. labels negative examples positive) should be the same across groups.
2. False-Negative Rate Balance:  The rate at which your classifier makes errors in the negative direction (i.e. labels positive examples negative) should be the same across groups.
3. Predictive Parity: The statistical "meaning" of a positive classification should be the same across groups (we'll be more specific about what this means in a moment)

What Chouldechova and KMR show is that if you want all three, you are out of luck --- unless you are in one of two very unlikely situations: Either you have a perfect classifier that never errs, or the base rate is exactly the same for both populations --- i.e. both populations have exactly the same frequency of positive examples. If you don't find yourself in one of these two unusual situations, then you have to give up on properties 1, 2, or 3.
...
"So why is this result true? The proof in Alex's paper can't be made simpler --- its already a one liner, following from an algebraic identity. But the first time I read it I didn't have a great intuition for why it held. Viewing the statement through the lens of Bayesian inference made the result very intuitive (at least for me). With this viewpoint, all the impossibility result is saying is: "If you have different priors about some event (say that a released inmate will go on to commit a crime) for two different populations, and you receive evidence of the same strength for both populations, then you will have different posteriors as well". This is now bordering on obvious --- because your posterior belief about an event is a combination of your prior belief and the new evidence you have received, weighted by the strength of that evidence.  "
...
"So the mathematical fact is simple --- but its implications remain deep. It means we have to choose between equalizing a decision maker's posterior about the label of an individual, or providing an equally accurate signal about each individual, and that we cannot have both. Unfortunately, living without either one of these conditions can lead to real harm."