Local Envy-Freeness in House Allocation Problems

International audienceWe study the fair division problem consisting in allocating one item per agent so as to avoid (or minimize) envy, in a setting where only agents connected in a given social network may experience envy. In a variant of the problem, agents themselves can be located on the network by the central authority. These problems turn out to be difficult even on very simple graph structures, but we identify several tractable cases. We further provide practical algorithms and experimental insights

Gerechte Zuordnungen: Kollektive Entscheidungsprobleme aus der Perspektive von Mathematik und theoretischer Informatik

Wir untersuchen verschiedene Fragestellungen der Sozialwahltheorie aus Sicht der Computational Social Choice. Für ein Problem, das in Bezug zu einem Kollektiv von Agenten steht (z.B. Aufteilungen von Ressourcen oder Repräsentantenwahlen), stehen verschiedene Alternativen als Lösung zur Verfügung; ein wesentlicher Aspekt sind dabei die diversen Pr\"aferenzen der Agenten gegenüber den Alternativen. Die Qualität der Lösungen wird anhand von Kriterien aus den Sozialwissenschaften (Fairness), der Spieltheorie (Stabilität) und den Wirtschaftswissenschaften (Effizienz) charakterisiert. In Computational Social Choice werden solche Fragestellungen mit Werkzeugen der Mathematik (z.B. Logik und Kombinatorik) und Informatik (z.B. Komplexitätstheorie und Algorithmik) behandelt. Als roter Faden zieht sich die Frage nach sogenannten "`gerechten Zuordnungen"' durch die Dissertation. Für die Zuordnung von Gütern zu Agenten zeigen wir, wie mithilfe eines dezentralisierten Ansatzes Zuordnungen gefunden werden können, die Ungleichheit minimieren. Wir analysieren das Verhalten dieses Ansatzes für Worst-Case-Instanzen und benutzen dabei eine innovative Beweismethode, die auf impliziten rekursiven Konstruktionen unter Verwendung von Argumenten der Infinitesimalrechnung beruht. Bei der Zuordnung von Agenten zu Aktivitäten betrachten wir das vereinfachte Szenario, in dem die Agenten Präferenzen bezüglich der Aktivitäten haben und die Menge der zulässigen Zuordnungen Beschränkungen bezüglich der Teilnehmerzahlen pro Aktivität unterliegt. Wir führen verschiedene Lösungskonzepte ein und erläutern die Zusammenhänge und Unterschiede dieser Konzepte. Die zugehörigen Entscheidungsprobleme zur Existenz und Maximalität entsprechender Zuordnungen unterziehen wir einer ausführlichen Komplexitätsanalyse. Zuordnungsprobleme können auch als Auktionen aufgefasst werden. Wir betrachten ein Szenario, in dem die Agenten Gebote auf Transformationen von Gütermengen abgeben. In unserem Modell sind diese durch die Existenz von Gütern charakterisiert, die durch die Transformationen nicht verbraucht werden. Von Interesse sind die Kombinationen von Transformationen, die den Gesamtnutzen maximieren. Wir legen eine (parametrisierte) Komplexitätsanalyse dieses Modells vor. Etwas abseits der Grundfragestellung liegen unsere Untersuchungen zu kombinierten Wettkämpfen. Diese interpretieren wir als Wahlproblem, d.h. als Aggregation von Ordnungen. Wir untersuchen die Anfälligkeit für Manipulationen durch die Athleten.We investigate questions from social choice theory from the viewpoint of computational social choice. We consider the setting that a group of agents faces a collective decision problem (e.g., resource allocation or the choice of a representative): they have to choose among various alternatives. A crucial aspect are the agents' individual preferences over these alternatives. The quality of the solutions is measured by various criteria from the fields of social sciences (fairness), game theory (stability) and economics (efficiency). In computational social choice, such problems are analyzed and accessed via methods of mathematics (e.g., logic and combinatoric) and theoretical computer science (e.g. complexity theory and algorithms). The question of so called `fair assignments' runs like a common thread through most parts of this dissertation. Regarding allocations of goods to agents, we show how to achieve allocations with minimal inequality by means of a distributed approach. We analyze the behavior of this approach for worst case instances; therefor we use an innovative proof technique which relies on implicit recursive constructions and insights from basic calculus. For assignments of agents to activities, we consider a simplified scenario where the agents express preferences over activities and the set of feasible assignments is restricted by the number of agents which can participate in a (specific) activity. We introduce several solution concepts and elucidate the connections and differences between these concepts. Furthermore, we provide an elaborated complexity analysis of the associated decision problems addressing existence and maximality of the corresponding solution concepts. Assignment problems can also be seen as auctions. We consider a scenario where the agents bid on transformations of goods. In our model, each transformation requires the existence of a `tool good' which is not consumed by the transformation. We are interested in combinations of transformations which maximize the total utility. We study the computational complexity of this model in great detail, using methods from both classical and parameterized complexity theory. Slightly off topic are our investigations on combined competitions. We interpret these as a voting problem, i.e., as the aggregation of orders. We investigate the susceptibility of these competitions to manipulation by the athletes

Computing Stable Coalitions: Approximation Algorithms for Reward Sharing

Consider a setting where selfish agents are to be assigned to coalitions or projects from a fixed set P. Each project k is characterized by a valuation function; v_k(S) is the value generated by a set S of agents working on project k. We study the following classic problem in this setting: "how should the agents divide the value that they collectively create?". One traditional approach in cooperative game theory is to study core stability with the implicit assumption that there are infinite copies of one project, and agents can partition themselves into any number of coalitions. In contrast, we consider a model with a finite number of non-identical projects; this makes computing both high-welfare solutions and core payments highly non-trivial. The main contribution of this paper is a black-box mechanism that reduces the problem of computing a near-optimal core stable solution to the purely algorithmic problem of welfare maximization; we apply this to compute an approximately core stable solution that extracts one-fourth of the optimal social welfare for the class of subadditive valuations. We also show much stronger results for several popular sub-classes: anonymous, fractionally subadditive, and submodular valuations, as well as provide new approximation algorithms for welfare maximization with anonymous functions. Finally, we establish a connection between our setting and the well-studied simultaneous auctions with item bidding; we adapt our results to compute approximate pure Nash equilibria for these auctions.Comment: Under Revie

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

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