Cherry on the Cake: Fairness is NOT an Optimization Problem

Fair cake-cutting is a mathematical subfield that studies the problem of fairly dividing a resource among a number of participants. The so-called ``cake,'' as an object, represents any resource that can be distributed among players. This concept is connected to supervised multi-label classification: any dataset can be thought of as a cake that needs to be distributed, where each label is a player that receives its share of the dataset. In particular, any efficient cake-cutting solution for the dataset is equivalent to an optimal decision function. Although we are not the first to demonstrate this connection, the important ramifications of this parallel seem to have been partially forgotten. We revisit these classical results and demonstrate how this connection can be prolifically used for fairness in machine learning problems. Understanding the set of achievable fair decisions is a fundamental step in finding optimal fair solutions and satisfying fairness requirements. By employing the tools of cake-cutting theory, we have been able to describe the behavior of optimal fair decisions, which, counterintuitively, often exhibit quite unfair properties. Specifically, in order to satisfy fairness constraints, it is sometimes preferable, in the name of optimality, to purposefully make mistakes and deny giving the positive label to deserving individuals in a community in favor of less worthy individuals within the same community. This practice is known in the literature as cherry-picking and has been described as ``blatantly unfair.''