Minimax regret treatment rules with finite samples when a quantile is the object of interest

Consider a setup in which a decision maker is informed about the population by a finite sample and based on that sample has to decide whether or not to apply a certain treatment. We work out finite sample minimax regret treatment rules under various sampling schemes when outcomes are restricted onto the unit interval. In contrast to Stoye (2009) where the focus is on maximization of expected utility the focus here is instead on a particular quantile of the outcome distribution. We find that in the case where the sample consists of a fixed number of untreated and a fixed number of treated units, any treatment rule is minimax regret optimal. The same is true in the case of random treatment assignment in the sample with any assignment probability and in the case of testing an innovation when the known quantile of the untreated population equals 1/2. However if the known quantile exceeds 1/2 then never treating is the unique optimal rule and if it is smaller than 1/2 always treating is optimal. We also consider the case where a covariate is included.