Fair allocation of indivisible goods under conflict constraints

We consider the fair allocation of indivisible items to several agents and add a graph theoretical perspective to this classical problem. Thereby we introduce an incompatibility relation between pairs of items described in terms of a conflict graph. Every subset of items assigned to one agent has to form an independent set in this graph. Thus, the allocation of items to the agents corresponds to a partial coloring of the conflict graph. Every agent has its own profit valuation for every item. Aiming at a fair allocation, our goal is the maximization of the lowest total profit of items allocated to any one of the agents. The resulting optimization problem contains, as special cases, both {\sc Partition} and {\sc Independent Set}. In our contribution we derive complexity and algorithmic results depending on the properties of the given graph. We can show that the problem is strongly NP-hard for bipartite graphs and their line graphs, and solvable in pseudo-polynomial time for the classes of chordal graphs, cocomparability graphs, biconvex bipartite graphs, and graphs of bounded treewidth. Each of the pseudo-polynomial algorithms can also be turned into a fully polynomial approximation scheme (FPTAS).Comment: A preliminary version containing some of the results presented here appeared in the proceedings of IWOCA 2020. Version 3 contains an appendix with a remark about biconvex bipartite graph

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

Developments in Multi-Agent Fair Allocation

Fairness is becoming an increasingly important concern when designing markets, allocation procedures, and computer systems. I survey some recent developments in the field of multi-agent fair allocation

Efficient Fair Division with Minimal Sharing

A collection of objects, some of which are good and some are bad, is to be divided fairly among agents with different tastes, modeled by additive utility-functions. If the objects cannot be shared, so that each of them must be entirely allocated to a single agent, then a fair division may not exist. What is the smallest number of objects that must be shared between two or more agents in order to attain a fair and efficient division? We focus on Pareto-optimal, envy-free and/or proportional allocations. We show that, for a generic instance of the problem -- all instances except of a zero-measure set of degenerate problems -- a fair Pareto-optimal division with the smallest possible number of shared objects can be found in polynomial time, assuming that the number of agents is fixed. The problem becomes computationally hard for degenerate instances, where agents' valuations are aligned for many objects.Comment: Add experiments with Spliddit.org dat

Fair Allocation of goods and chores -- Tutorial and Survey of Recent Results

Fair resource allocation is an important problem in many real-world scenarios, where resources such as goods and chores must be allocated among agents. In this survey, we delve into the intricacies of fair allocation, focusing specifically on the challenges associated with indivisible resources. We define fairness and efficiency within this context and thoroughly survey existential results, algorithms, and approximations that satisfy various fairness criteria, including envyfreeness, proportionality, MMS, and their relaxations. Additionally, we discuss algorithms that achieve fairness and efficiency, such as Pareto Optimality and Utilitarian Welfare. We also study the computational complexity of these algorithms, the likelihood of finding fair allocations, and the price of fairness for each fairness notion. We also cover mixed instances of indivisible and divisible items and investigate different valuation and allocation settings. By summarizing the state-of-the-art research, this survey provides valuable insights into fair resource allocation of indivisible goods and chores, highlighting computational complexities, fairness guarantees, and trade-offs between fairness and efficiency. It serves as a foundation for future advancements in this vital field

Computing an Approximately Optimal Agreeable Set of Items

We study the problem of finding a small subset of items that is \emph{agreeable} to all agents, meaning that all agents value the subset at least as much as its complement. Previous work has shown worst-case bounds, over all instances with a given number of agents and items, on the number of items that may need to be included in such a subset. Our goal in this paper is to efficiently compute an agreeable subset whose size approximates the size of the smallest agreeable subset for a given instance. We consider three well-known models for representing the preferences of the agents: ordinal preferences on single items, the value oracle model, and additive utilities. In each of these models, we establish virtually tight bounds on the approximation ratio that can be obtained by algorithms running in polynomial time.Comment: A preliminary version appeared in Proceedings of the 26th International Joint Conference on Artificial Intelligence (IJCAI), 201

Practical algorithms and experimentally validated incentives for equilibrium-based fair division (A-CEEI)

Approximate Competitive Equilibrium from Equal Incomes (A-CEEI) is an equilibrium-based solution concept for fair division of discrete items to agents with combinatorial demands. In theory, it is known that in asymptotically large markets: 1. For incentives, the A-CEEI mechanism is Envy-Free-but-for-Tie-Breaking (EF-TB), which implies that it is Strategyproof-in-the-Large (SP-L). 2. From a computational perspective, computing the equilibrium solution is unfortunately a computationally intractable problem (in the worst-case, assuming $\textsf{PPAD}\ne \textsf{FP}$). We develop a new heuristic algorithm that outperforms the previous state-of-the-art by multiple orders of magnitude. This new, faster algorithm lets us perform experiments on real-world inputs for the first time. We discover that with real-world preferences, even in a realistic implementation that satisfies the EF-TB and SP-L properties, agents may have surprisingly simple and plausible deviations from truthful reporting of preferences. To this end, we propose a novel strengthening of EF-TB, which dramatically reduces the potential for strategic deviations from truthful reporting in our experiments. A (variant of) our algorithm is now in production: on real course allocation problems it is much faster, has zero clearing error, and has stronger incentive properties than the prior state-of-the-art implementation.Comment: To appear in EC 202

Advances in negotiation theory : bargaining, coalitions, and fairness

Bargaining is ubiquitous in real life. It is a major dimension of political and business activities. It appears at the international level, when governments negotiate on matters ranging from economic issues (such as the removal of trade barriers), to global security (such as fighting against terrorism) to environmental and related issues (such as climate change control). What factors determinethe outcomes of such negotiations? What strategies can help reach an agreement? How should the parties involved divide the gains from cooperation? With whom will one make alliances? The authors address these questions by focusing on a noncooperative approach to negotiations, which is particularly relevant for the study of international negotiations. By reviewing noncooperative bargaining theory, noncooperative coalition theory, and the theory of fair division, they try to identify the connections among these different facets of the same problem in an attempt to facilitate progress toward a unified framework.Economic Theory&Research,Social Protections&Assistance,Environmental Economics&Policies,Scientific Research&Science Parks,Science Education

Advances in Negotiation Theory: Bargaining, Coalitions and Fairness

Bargaining is ubiquitous in real-life. It is a major dimension of political and business activities. It appears at the international level, when governments negotiate on matters ranging from economic issues (such as the removal of trade barriers), to global security (such as fighting against terrorism) to environmental and related issues (e.g. climate change control). What factors determine the outcome of negotiations such as those mentioned above? What strategies can help reach an agreement? How should the parties involved divide the gains from cooperation? With whom will one make alliances? This paper addresses these questions by focusing on a non-cooperative approach to negotiations, which is particularly relevant for the study of international negotiations. By reviewing noncooperative bargaining theory, non-cooperative coalition theory, and the theory of fair division, this paper will try to identify the connection among these different facets of the same problem in an attempt to facilitate the progress towards a unified framework.Negotiation theory, Bragaining, Coalitions, Fairness, Agreements