Fair Division with Bounded Sharing: Binary and Non-Degenerate Valuations

TL;DR

This paper explores fair division of indivisible objects among agents by allowing bounded sharing of objects, analyzing the complexity under various valuation restrictions and fairness notions.

Contribution

It introduces a novel relaxation allowing bounded sharing to achieve fairness, and analyzes the computational complexity for different valuation types and fairness criteria.

Findings

Complexity results vary with valuation restrictions.

Allowing bounded sharing increases the number of fair solutions.

Certain fairness notions become computationally feasible under specific conditions.

Abstract

A set of objects is to be divided fairly among agents with different tastes, modeled by additive utility-functions. If we consider the objects as indivisible, many instances of the decision problem: ``Is there a fair division of the objects among the agents'' are negative. In addition, this question is hard to solve even for most of the special cases. The latter reasons give us a good motivation to relax the problem for which the running time complexity is better, and the number of positive instances (admitting a fair division) will significantly grow. Whereas many works relax the fairness criteria, this paper introduces another relaxation: an agent is allowed to share a \emph{bounded} number of objects between two or more agents in order to attain fairness. The paper studies various notions of fairness, such as proportionality, envy-freeness, equitability, and consensus. We analyze the run-time complexity of finding a fair allocation with a given number of sharings under several restrictions on the agents' valuations, such as: binary, generalized-binary, and non-degenerate.