  Predictive policing systems are increasingly used to determine how to
allocate police across a city in order to best prevent crime. Discovered crime
data (e.g., arrest counts) are used to help update the model, and the process
is repeated. Such systems have been empirically shown to be susceptible to
runaway feedback loops, where police are repeatedly sent back to the same
neighborhoods regardless of the true crime rate.
  In response, we develop a mathematical model of predictive policing that
proves why this feedback loop occurs, show empirically that this model exhibits
such problems, and demonstrate how to change the inputs to a predictive
policing system (in a black-box manner) so the runaway feedback loop does not
occur, allowing the true crime rate to be learned. Our results are
quantitative: we can establish a link (in our model) between the degree to
which runaway feedback causes problems and the disparity in crime rates between
areas. Moreover, we can also demonstrate the way in which \emph{reported}
incidents of crime (those reported by residents) and \emph{discovered}
incidents of crime (i.e. those directly observed by police officers dispatched
as a result of the predictive policing algorithm) interact: in brief, while
reported incidents can attenuate the degree of runaway feedback, they cannot
entirely remove it without the interventions we suggest.
