Robust ordinal regression for subsets comparisons with interactions

This paper is devoted to a robust ordinal method for learning the preferences of a decision maker between subsets. The decision model, derived from Fishburn and LaValle (1996) and whose parameters we learn, is general enough to be compatible with any strict weak order on subsets, thanks to the consideration of possible interactions between elements. Moreover, we accept not to predict some preferences if the available preference data are not compatible with a reliable prediction. A predicted preference is considered reliable if all the simplest models (Occam's razor) explaining the preference data agree on it. Following the robust ordinal regression methodology, our predictions are based on an uncertainty set encompassing the possible values of the model parameters. We define a new ordinal dominance relation between subsets and design a procedure to determine whether this dominance relation holds. Numerical tests are provided on synthetic and real-world data to evaluate the richness and reliability of the preference predictions made.