Market design and algorithmic criminal justice--by Jung, Kannan, Lee, Pai, Roth and Vohra

When fairness isn't your only goal, your other goals may help you choose among competing definitions of fairness.

Fair Prediction with Endogenous Behavior
Christopher Jung, Sampath Kannan, Changhwa Lee, Mallesh M. Pai, Aaron Roth,and Rakesh Vohra
February 17, 2020

Abstract: There  is  increasing  regulatory  interest  in  whether  machine  learning  algorithms  deployed  in  consequential domains (e.g.  in criminal justice) treat different demographic groups “fairly.”  However, there are several proposed notions of fairness, typically mutually incompatible.  Using criminal justice as an example,  we study a model in which society chooses an incarceration rule.  Agents of different demographic groups differ in their outside options (e.g.  opportunity for legal employment) and decide whether to commit crimes.  We show that equalizing type I and type II errors across groups is consistent with the goal of minimizing the overall crime rate; other popular notions of fairness are not.
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And here's a blog post about the paper by one of the authors:

Fair Prediction with Endogenous Behavior
Can Game Theory Help Us Choose Among Fairness Constraints?

"...The crime-minimizing solution is the one that sets different thresholds on posterior probabilities (i.e. uniform thresholds on signals) so as to equalize false positive rates and false negative rates. In other words, to minimize crime, society should explicitly commit to not conditioning on group membership, even when group membership is statistically informative for the goal of predicting crime.

"Why? Its because although using demographic information is statistically informative for the goal of predicting crime when base rates differ, it is not something that is under the control of individuals --- they can control their own choices, but not what group they were born into. And making decisions about individuals using information that is not under their control has the effect of distorting their dis-incentive to commit crime --- it ends up providing less of a dis-incentive to individuals from the higher crime group (since they are more likely to be wrongly incarcerated even if they don't commit a crime). And because in our model people are rational actors, minimizing crime is all about managing incentives. "