A Generalization of the AL method for Fair Allocation of Indivisible Objects

We consider the assignment problem in which agents express ordinal preferences over $m$ objects and the objects are allocated to the agents based on the preferences. In a recent paper, Brams, Kilgour, and Klamler (2014) presented the AL method to compute an envy-free assignment for two agents. The AL method crucially depends on the assumption that agents have strict preferences over objects. We generalize the AL method to the case where agents may express indifferences and prove the axiomatic properties satisfied by the algorithm. As a result of the generalization, we also get a $O(m)$ speedup on previous algorithms to check whether a complete envy-free assignment exists or not. Finally, we show that unless P=NP, there can be no polynomial-time extension of GAL to the case of arbitrary number of agents

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.Comment: extended version of a paper presented at AAMAS 201

Asserting Fairness through AI, Mathematics and Experimental Economics. The CREA Project Case Study.

4noopenThis is an account of Analytical-Experimental Workgroup role in a two-year EU funded project. A restricted group of economists and mathematicians has interacted with law researchers and computer scientist (in the proposal’s words) “to introduce new mechanisms of dispute resolution as a helping tool in legal procedures for lawyers, mediators and judges, with the objective to reach an agreement between the parties”. The novelty of the analysis is to allow different skills (by legal, experimental, mathematical and computer scientists) work together in order to find a reliable and quick methodology to solve conflict in bargaining through equitable algorithms. The variety of specializations has been the main challenge and, finally, the project’s strength.openMarco Dall'Aglio, Daniela Di Cagno, Vito Fragnelli, Francesca MarazziDall'Aglio, Marco; Di Cagno, Daniela Teresa; Fragnelli, Vito; Marazzi, Francesc

Weighted Proportional Allocations of Indivisible Goods and Chores: Insights via Matchings

We study the fair allocation of indivisible goods and chores under ordinal valuations for agents with unequal entitlements. We show the existence and polynomial time computation of weighted necessarily proportional up to one item (WSD-PROP1) allocations for both goods and chores, by reducing it to a problem of finding perfect matchings in a bipartite graph. We give a complete characterization of these allocations as corner points of a perfect matching polytope. Using this polytope, we can optimize over all allocations to find a min-cost WSD-PROP1 allocation of goods or most efficient WSD-PROP1 allocation of chores. Additionally, we show the existence and computation of sequencible (SEQ) WSD-PROP1 allocations by using rank-maximal perfect matching algorithms and show incompatibility of Pareto optimality under all valuations and WSD-PROP1. We also consider the Best-of-Both-Worlds (BoBW) fairness notion. By using our characterization, we show the existence and polynomial time computation of Ex-ante envy free (WSD-EF) and Ex-post WSD-PROP1 allocations under ordinal valuations for both chores and goods.Comment: Accepted at AAMAS 202

Divorcing made easy

We discuss the proportionally fair allocation of a set of indivisible items to k agents. We assume that each agent specifies only a ranking of the items from best to worst. Agents do not specify their valuations of the items. An allocation is proportionally fair if all agents believe that they have received their fair share of the value according to how they value the items. We give simple conditions (and a fast algorithm) for determining whether the agents rankings give sufficient information to determine a proportionally fair allocation. An important special case is a divorce situation with two agents. For such a divorce situation, we provide a particularly simple allocation rule that should have applications in the real world

Divorcing made easy

We discuss the proportionally fair allocation of a set of indivisible items to k agents. We assume that each agent specifies only a ranking of the items from best to worst. Agents do not specify their valuations of the items. An allocation is proportionally fair if all agents believe that they have received their fair share of the value according to how they value the items. We give simple conditions (and a fast algorithm) for determining whether the agents rankings give sufficient information to determine a proportionally fair allocation. An important special case is a divorce situation with two agents. For such a divorce situation, we provide a particularly simple allocation rule that should have applications in the real world