Fair assignment of indivisible objects under ordinal preferences

We consider the discrete assignment problem in which agents express ordinal preferences over objects and these objects are allocated to the agents in a fair manner. We use the stochastic dominance relation between fractional or randomized allocations to systematically define varying notions of proportionality and envy-freeness for discrete assignments. The computational complexity of checking whether a fair assignment exists is studied for these fairness notions. We also characterize the conditions under which a fair assignment is guaranteed to exist. For a number of fairness concepts, polynomial-time algorithms are presented to check whether a fair assignment exists. Our algorithmic results also extend to the case of unequal entitlements of agents. Our NP-hardness result, which holds for several variants of envy-freeness, answers an open question posed by Bouveret, Endriss, and Lang (ECAI 2010). We also propose fairness concepts that always suggest a non-empty set of assignments with meaningful fairness properties. Among these concepts, optimal proportionality and optimal weak proportionality appear to be desirable fairness concepts.

[1]  Yingqian Zhang,et al.  On the Complexity of Efficiency and Envy-Freeness in Fair Division of Indivisible Goods with Additive Preferences , 2009, ADT.

[2]  William J. Cook,et al.  Combinatorial optimization , 1997 .

[3]  David Manlove,et al.  Algorithmics of Matching Under Preferences , 2013, Bull. EATCS.

[4]  Amin Saberi,et al.  An approximation algorithm for max-min fair allocation of indivisible goods , 2007, STOC '07.

[5]  Steven J. Brams,et al.  The undercut procedure: an algorithm for the envy-free division of indivisible items , 2009, Soc. Choice Welf..

[6]  Kurt Mehlhorn,et al.  Assigning Papers to Referees , 2009, Algorithmica.

[7]  Elisha A. Pazner,et al.  Egalitarian Equivalent Allocations: A New Concept of Economic Equity , 1978 .

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

[9]  M. R. Rao,et al.  Combinatorial Optimization , 1992, NATO ASI Series.

[10]  Raymond E. Miller,et al.  Complexity of Computer Computations , 1972 .

[11]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[12]  Amin Saberi,et al.  An Approximation Algorithm for Max-Min Fair Allocation of Indivisible Goods , 2010, SIAM J. Comput..

[13]  Jérôme Monnot,et al.  On regular and approximately fair allocations of indivisible goods , 2014, AAMAS.

[14]  Peter C. Fishburn,et al.  Fair division of indivisible items between two people with identical preferences: Envy-freeness, Pareto-optimality, and equity , 2000, Soc. Choice Welf..

[15]  Eyke Hüllermeier,et al.  Preferences in AI: An overview , 2011, Artif. Intell..

[16]  Nils J. Nilsson,et al.  Artificial Intelligence , 1974, IFIP Congress.

[17]  Jay Sethuraman,et al.  A solution to the random assignment problem on the full preference domain , 2006, J. Econ. Theory.

[18]  Jörg Rothe,et al.  A survey of approximability and inapproximability results for social welfare optimization in multiagent resource allocation , 2013, Annals of Mathematics and Artificial Intelligence.

[19]  Ariel D. Procaccia,et al.  No agent left behind: dynamic fair division of multiple resources , 2013, AAMAS.

[20]  Hervé Moulin,et al.  A New Solution to the Random Assignment Problem , 2001, J. Econ. Theory.

[21]  Ivona Bezáková,et al.  Allocating indivisible goods , 2005, SECO.

[22]  Theodore P. Hill,et al.  Equitable distribution of indivisible objects , 1988 .

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

[24]  S. Brams,et al.  Efficient Fair Division , 2005 .

[25]  W. Cho.  Probabilistic Assignment : A Two-fold Axiomatic Approach , 2012 .

[26]  H. Varian.  Equity, Envy and Efficiency , 1974 .

[27]  Peter C. Fishburn,et al.  Paradoxes of Fair Division , 2001 .

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

[29]  D. Golovin.  Max-min fair allocation of indivisible goods , 2005 .

[30]  Jérôme Lang,et al.  A General Elicitation-Free Protocol for Allocating Indivisible Goods , 2011, IJCAI.

[31]  W. Bossert,et al.  Ranking Sets of Objects , 2001 .

[32]  P. Gärdenfors.  Assignment Problem Based on Ordinal Preferences , 1973 .

[33]  Ariel D. Procaccia.  Thou Shalt Covet Thy Neighbor's Cake , 2009, IJCAI.

[34]  Abdallah Saffidine,et al.  Axiomatic and Computational Aspects of Scoring Allocation Rules for Indivisible Goods , 2014 .

[35]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[36]  Ariel D. Procaccia,et al.  On Maxsum Fair Cake Divisions , 2012, AAAI.

[37]  Yann Chevaleyre,et al.  Issues in Multiagent Resource Allocation , 2006, Informatica.

[38]  Patrice Perny,et al.  Infinite order Lorenz dominance for fair multiagent optimization , 2010, AAMAS.

[39]  Hervé Moulin,et al.  Fair division and collective welfare , 2003 .

[40]  Haris Aziz,et al.  A note on the undercut procedure , 2013, AAMAS.

[41]  Ulrich Endriss,et al.  Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods , 2010, ECAI.

[42]  Kirk Pruhs,et al.  Divorcing Made Easy , 2012, FUN.

[43]  Sylvain Bouveret,et al.  Characterizing conflicts in fair division of indivisible goods using a scale of criteria , 2016, Autonomous Agents and Multi-Agent Systems.

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

[45]  Jérôme Lang,et al.  Efficiency and envy-freeness in fair division of indivisible goods: logical representation and complexity , 2005, IJCAI.

[46]  Andreas Darmann,et al.  Maximizing Nash Product Social Welfare in Allocating Indivisible Goods , 2014 .

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

[48]  K. Mehlhorn,et al.  Pareto Optimality in House Allocation Problems , 2005, ISAAC.

[49]  Felix Brandt,et al.  On Popular Random Assignments , 2013, SAGT.

[50]  L. B. Wilson,et al.  Assignment Using Choice Lists , 1977 .

[51]  Eric McDermid,et al.  "Almost stable" matchings in the Roommates problem with bounded preference lists , 2012, Theor. Comput. Sci..

[52]  Haris Aziz,et al.  Housing Markets with Indifferences: A Tale of Two Mechanisms , 2012, AAAI.