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Generalized binary utility functions and fair allocations

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  • Camacho, Franklin
  • Fonseca-Delgado, Rigoberto
  • Pino Pérez, Ramón
  • Tapia, Guido

Abstract

The problem of finding envy-free allocations of indivisible goods cannot always be solved; therefore, it is common to study some relaxations such as envy-free up to one good (EF1) and envy-free up to any positively valued good (EFX). Another property of interest for the efficiency of an allocation is the Pareto Optimality (PO). Under additive utility functions for goods, it is possible to find EF1 and PO allocations using the Nash social welfare. However, finding an allocation that maximizes the Nash social welfare is a computationally costly problem. Maximizing the utilitarian social welfare subject to EF1 constraints is an NP-complete problem for the case where three or more agents participate. In this work, we propose a restricted case of additive utility functions called generalized binary utility functions. The proposed utilities are a generalization of binary and identical utilities simultaneously. In this scenario, we present a polynomial-time algorithm that maximizes the utilitarian social welfare and, at the same time, produces an EF1 and PO allocation for goods as well as for chores. Moreover, a slight modification of our algorithm gives a better allocation: one which is EFX.

Suggested Citation

  • Camacho, Franklin & Fonseca-Delgado, Rigoberto & Pino Pérez, Ramón & Tapia, Guido, 2023. "Generalized binary utility functions and fair allocations," Mathematical Social Sciences, Elsevier, vol. 121(C), pages 50-60.
    • Handle: RePEc:eee:matsoc:v:121:y:2023:i:c:p:50-60
    DOI: 10.1016/j.mathsocsci.2022.10.003
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    References listed on IDEAS

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    2. Soroush Ebadian & Dominik Peters & Nisarg Shah, 2022. "How to Fairly Allocate Easy and Difficult Chores," Post-Print hal-03834514, HAL.
    3. Siddharth Barman & Sanath Kumar Krishnamurthy & Rohit Vaish, 2018. "Greedy Algorithms for Maximizing Nash Social Welfare," Papers 1801.09046, arXiv.org.
    4. Eric Budish, 2011. "The Combinatorial Assignment Problem: Approximate Competitive Equilibrium from Equal Incomes," Journal of Political Economy, University of Chicago Press, vol. 119(6), pages 1061-1103.
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    Cited by:

    1. Suksompong, Warut & Teh, Nicholas, 2023. "Weighted fair division with matroid-rank valuations: Monotonicity and strategyproofness," Mathematical Social Sciences, Elsevier, vol. 126(C), pages 48-59.
    2. Warut Suksompong & Nicholas Teh, 2023. "Weighted Fair Division with Matroid-Rank Valuations: Monotonicity and Strategyproofness," Papers 2303.14454, arXiv.org, revised Sep 2023.
    3. Tzeh Yuan Neoh & Nicholas Teh, 2025. "Understanding EFX Allocations: Counting and Variants," Papers 2504.03951, arXiv.org.
    4. Hao Guo & Weidong Li & Bin Deng, 2023. "A Survey on Fair Allocation of Chores," Mathematics, MDPI, vol. 11(16), pages 1-28, August.

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