  A traditional stochastic program under a finite population typically seeks to
optimize efficiency by maximizing the expected profits or minimizing the
expected costs, subject to a set of constraints. However, implementing such
optimization-based decisions can have varying impacts on individuals, and when
assessed using the individuals' utility functions, these impacts may differ
substantially across demographic groups delineated by sensitive attributes,
such as gender, race, age, and socioeconomic status. As each group comprises
multiple individuals, a common remedy is to enforce group fairness, which
necessitates the measurement of disparities in the distributions of utilities
across different groups. This paper introduces the concept of Distributionally
Fair Stochastic Optimization (DFSO) based on the Wasserstein fairness measure.
The DFSO aims to minimize distributional disparities among groups, quantified
by the Wasserstein distance, while adhering to an acceptable level of
inefficiency. Our analysis reveals that: (i) the Wasserstein fairness measure
recovers the demographic parity fairness prevalent in binary classification
literature; (ii) this measure can approximate the well-known Kolmogorov-Smirnov
fairness measure with considerable accuracy; and (iii) despite DFSO's biconvex
nature, the epigraph of the Wasserstein fairness measure is generally
Mixed-Integer Convex Programming Representable (MICP-R). Additionally, we
introduce two distinct lower bounds for the Wasserstein fairness measure: the
Jensen bound, applicable to the general Wasserstein fairness measure, and the
Gelbrich bound, specific to the type-2 Wasserstein fairness measure. We
establish the exactness of the Gelbrich bound and quantify the theoretical
difference between the Wasserstein fairness measure and the Gelbrich bound.
