On Approximate Envy-Freeness for Indivisible Chores and Mixed Resources

We study fair allocation of undesirable indivisible items (or chores) and make three contributions: First, we show that determining the existence of an envy-free allocation is NP-complete even when agents have binary additive valuations. Second, we provide a polynomial-time algorithm for computing an allocation that satisfies envy-freeness up to one chore (EF1) under monotone valuations, correcting a existing proof of the same claim in the literature. A straightforward modification of our algorithm can be used to compute an EF1 allocation for doubly monotone instances (wherein each agent can partition the set of items into objective goods and objective chores). Our third and most important result applies to a mixed resources model consisting of indivisible chores and a divisible, desirable heterogenous resource (metaphorically, a cake). We show that there always exists an allocation that satisfies envy-freeness for mixed resources (EFM) in this setting, complementing an analogous recent result of Bei et al. (AAAI 2020). We also show a similar result in the flipped setting consisting of indivisible goods and a divisible "bad" cake.

[1]  Bo Li,et al.  Strategyproof and Approximately Maxmin Fair Share Allocation of Chores , 2019, IJCAI.

[2]  Ioannis Caragiannis,et al.  Fair allocation of combinations of indivisible goods and chores , 2018, ArXiv.

[3]  D. Gale,et al.  Fair Allocation of Indivisible Goods and Criteria of Justice , 1991 .

[4]  Ariel D. Procaccia,et al.  How to Cut a Cake Before the Party Ends , 2013, AAAI.

[5]  Roman Karasev,et al.  ENVY‐FREE DIVISION USING MAPPING DEGREE , 2019, Mathematika.

[6]  Yann Chevaleyre,et al.  Fair Allocation of Indivisible Goods , 2016, Handbook of Computational Social Choice.

[7]  Rohit Vaish,et al.  Best of Both Worlds: Ex-Ante and Ex-Post Fairness in Resource Allocation , 2020, EC.

[8]  D. Foley.  Resource allocation and the public sector , 1967 .

[9]  Ronald L. Rivest,et al.  Introduction to Algorithms, third edition , 2009 .

[10]  Rohit Vaish,et al.  Equitable Allocations of Indivisible Chores , 2019, AAMAS.

[11]  Ruta Mehta,et al.  Competitive Allocation of a Mixed Manna , 2020, SODA.

[12]  Xingyu Chen,et al.  The Fairness of Leximin in Allocation of Indivisible Chores , 2020, ArXiv.

[13]  Yair Zick,et al.  Finding Fair and Efficient Allocations for Matroid Rank Valuations , 2020, SAGT.

[14]  Yair Zick,et al.  The Price of Quota-based Diversity in Assignment Problems , 2020, ACM Trans. Economics and Comput..

[15]  Lirong Xia,et al.  Fair Division through Information Withholding , 2019, AAAI.

[16]  Ariel D. Procaccia,et al.  Spliddit: unleashing fair division algorithms , 2015, SECO.

[17]  Haris Aziz,et al.  Cake Cutting Algorithms for Piecewise Constant and Piecewise Uniform Valuations , 2013, WINE.

[18]  Martin Aleksandrov,et al.  Jealousy-freeness and other common properties in Fair Division of Mixed Manna , 2020, ArXiv.

[19]  Tayfun Sönmez,et al.  Fair Allocation of Vaccines, Ventilators and Antiviral Treatments: Leaving No Ethical Value Behind in Health Care Rationing , 2020, EC.

[20]  Enriqueta Aragones,et al.  A derivation of the money Rawlsian solution , 1995 .

[21]  Elisha Peterson,et al.  N-person envy-free chore division , 2009, 0909.0303.

[22]  Steven J. Brams,et al.  Fair division - from cake-cutting to dispute resolution , 1998 .

[23]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[24]  Xiaohui Bei,et al.  Maximin Fairness with Mixed Divisible and Indivisible Goods , 2020, ArXiv.

[25]  F. Su.  Rental Harmony: Sperner's Lemma in Fair Division , 1999 .

[26]  E. Eisenberg,et al.  CONSENSUS OF SUBJECTIVE PROBABILITIES: THE PARI-MUTUEL METHOD, , 1959 .

[27]  Jack M. Robertson,et al.  Cake-cutting algorithms - be fair if you can , 1998 .

[28]  Eric Budish,et al.  The Combinatorial Assignment Problem: Approximate Competitive Equilibrium from Equal Incomes , 2010, Journal of Political Economy.

[29]  Judd B. Kessler,et al.  Course Match: A Large-Scale Implementation of Approximate Competitive Equilibrium from Equal Incomes for Combinatorial Allocation , 2015, Oper. Res..

[30]  Toby Walsh,et al.  Fair allocation of indivisible goods and chores , 2019, Autonomous Agents and Multi-Agent Systems.

[31]  Ariel D. Procaccia.  Cake Cutting Algorithms , 2016, Handbook of Computational Social Choice.

[32]  Erel Segal-Halevi,et al.  Fairly Dividing a Cake after Some Parts Were Burnt in the Oven , 2017, AAMAS.

[33]  Xiaohui Bei,et al.  Fair Division of Mixed Divisible and Indivisible Goods , 2019, AAAI.

[34]  Hervé Moulin,et al.  Competitive Division of a Mixed Manna , 2017, ArXiv.

[35]  Ruta Mehta,et al.  Fair and Efficient Allocations under Subadditive Valuations , 2021, AAAI.

[36]  Haris Aziz,et al.  Almost Group Envy-free Allocation of Indivisible Goods and Chores , 2019, IJCAI.

[37]  Xin Huang,et al.  An Algorithmic Framework for Approximating Maximin Share Allocation of Chores , 2019, EC.

[38]  Haris Aziz,et al.  A Discrete and Bounded Envy-Free Cake Cutting Protocol for Any Number of Agents , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[39]  Martin Aleksandrov,et al.  Envy-freeness up to one item: Shall we add or remove resources? , 2020, ArXiv.

[40]  Toby Walsh,et al.  Fair assignment of indivisible objects under ordinal preferences , 2013, AAMAS.

[41]  Lirong Xia,et al.  Equitable Allocations of Indivisible Goods , 2019, IJCAI.

[42]  Rohit Vaish,et al.  Greedy Algorithms for Maximizing Nash Social Welfare , 2018, AAMAS.

[43]  Haris Aziz,et al.  Achieving Envy-freeness and Equitability with Monetary Transfers , 2020, AAAI.

[44]  Elchanan Mossel,et al.  On approximately fair allocations of indivisible goods , 2004, EC '04.

[45]  Toby Walsh,et al.  Two Algorithms for Additive and Fair Division of Mixed Manna , 2020, KI.

[46]  Ioannis Caragiannis,et al.  Computing envy-freeable allocations with limited subsidies , 2020, ArXiv.

[47]  Bo Li,et al.  Weighted Maxmin Fair Share Allocation of Indivisible Chores , 2019, IJCAI.

[48]  Erel Segal-Halevi.  Competitive equilibrium for almost all incomes: existence and fairness , 2020, Autonomous Agents and Multi-Agent Systems.

[49]  Rohit Vaish,et al.  Finding Fair and Efficient Allocations , 2017, EC.

[50]  Flip Klijn,et al.  An algorithm for envy-free allocations in an economy with indivisible objects and money , 2000, Soc. Choice Welf..

[51]  W. Stromquist.  How to Cut a Cake Fairly , 1980 .

[52]  Hans Reijnierse,et al.  Envy-free and Pareto efficient allocations in economies with indivisible goods and money , 2002, Math. Soc. Sci..

[53]  Haris Aziz,et al.  A polynomial-time algorithm for computing a Pareto optimal and almost proportional allocation , 2020, Oper. Res. Lett..

[54]  Ariel D. Procaccia,et al.  Optimal Envy-Free Cake Cutting , 2011, AAAI.

[55]  Shira Zerbib,et al.  Envy-free cake division without assuming the players prefer nonempty pieces , 2018, Israel Journal of Mathematics.

[56]  Ruta Mehta,et al.  Dividing Bads is Harder than Dividing Goods: On the Complexity of Fair and Efficient Division of Chores , 2020, ArXiv.

[57]  Hervé Moulin,et al.  Dividing bads under additive utilities , 2018, Soc. Choice Welf..

[58]  Martin Aleksandrov,et al.  Almost Envy Freeness and Welfare Efficiency in Fair Division with Goods or Bads , 2018, ArXiv.

[59]  Vincent Conitzer,et al.  Group Fairness for the Allocation of Indivisible Goods , 2019, AAAI.

[60]  Equipartition of a Segment , 2020, Mathematics of Operations Research.

[61]  Adrian Vetta,et al.  One Dollar Each Eliminates Envy , 2019, EC.

[62]  Evangelos Markakis,et al.  Multiple Birds with One Stone: Beating 1/2 for EFX and GMMS via Envy Cycle Elimination , 2019, AAAI.

[63]  Simina Brânzei,et al.  Algorithms for Competitive Division of Chores , 2019, Mathematics of Operations Research.

[64]  Matthias G. Raith,et al.  Bidding for envy-freeness: A procedural approach to n-player fair-division problems , 2002, Soc. Choice Welf..

[65]  Tuomas Sandholm,et al.  Finding approximate competitive equilibria: efficient and fair course allocation , 2010, AAMAS.

[66]  Endre Boros,et al.  Envy-free Relaxations for Goods, Chores, and Mixed Items , 2020, ArXiv.

[67]  L. Shapley,et al.  On cores and indivisibility , 1974 .

[68]  Toby Walsh,et al.  Algorithms for Max-Min Share Fair Allocation of Indivisible Chores , 2017, AAAI.

[69]  Nisarg Shah,et al.  Fair Division with Subsidy , 2019, SAGT.

[70]  Ruta Mehta,et al.  Approximating Maximin Shares with Mixed Manna , 2020, ArXiv.

[71]  Martino Traxler,et al.  Fair Chore Division for Climate Change , 2002 .