Optimal Efficiency-Envy Trade-Off via Optimal Transport

• URL: http://arxiv.org/abs/2209.15416v1
• Date: Sun, 25 Sep 2022 00:39:43 GMT
• Title: Optimal Efficiency-Envy Trade-Off via Optimal Transport
• Authors: Steven Yin, Christian Kroer
• Abstract summary: We consider the problem of allocating a distribution of items to $n$ recipients where each recipient has to be allocated a fixed, prespecified fraction of all items. We show that this problem can be formulated as a variant of the semi-discrete optimal transport (OT) problem, whose solution structure in this case has a concise representation and a simple geometric interpretation.
• Score: 33.85971515753188
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider the problem of allocating a distribution of items to $n$ recipients where each recipient has to be allocated a fixed, prespecified fraction of all items, while ensuring that each recipient does not experience too much envy. We show that this problem can be formulated as a variant of the semi-discrete optimal transport (OT) problem, whose solution structure in this case has a concise representation and a simple geometric interpretation. Unlike existing literature that treats envy-freeness as a hard constraint, our formulation allows us to \emph{optimally} trade off efficiency and envy continuously. Additionally, we study the statistical properties of the space of our OT based allocation policies by showing a polynomial bound on the number of samples needed to approximate the optimal solution from samples. Our approach is suitable for large-scale fair allocation problems such as the blood donation matching problem, and we show numerically that it performs well on a prior realistic data simulator.

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