Reasoning with incomplete and imprecise preferences

Preferences are present in many real life situations but it is often difficult to quantify them giving a precise value. Sometimes preference values may be missing because of privacy reasons or because they are expensive to obtain or to produce. In some other situations the user of an automated system may have a vague idea of whats he wants. In this thesis we considered the general formalism of soft constraints, where preferences play a crucial role and we extended such a framework to handle both incomplete and imprecise preferences. In particular we provided new theoretical frameworks to handle such kinds of preferences. By admitting missing or imprecise preferences, solving a soft constraint problem becomes a different task. In fact, the new goal is to find solutions which are the best ones independently of the precise value the each preference may have. With this in mind we defined two notions of optimality: the possibly optimal solutions and the necessary optimal solutions, which are optimal no matter we assign a precise value to a missing or imprecise preference. We provided several algorithms, bases on both systematic and local search approaches, to find such kind of solutions. Moreover, we also studied the impact of our techniques also in a specific class of problems (the stable marriage problems) where imprecision and incompleteness have a specific meaning and up to now have been tackled with different techniques. In the context of the classical stable marriage problem we developed a fair method to randomly generate stable marriages of a given problem instance. Furthermore, we adapted our techniques to solve stable marriage problems with ties and incomplete lists, which are known to be NP-hard, obtaining good results both in terms of size of the returned marriage and in terms of steps need to find a solution

Marriage as a Rat Race: Noisy Pre-Marital Investments with Assortative Matching

We study the incentive to invest to improve marriage prospects, in a frictionless marriage market with non-transferable utility. Stochastic returns to investment eliminate the multiplicity of equilibria in models with deterministic returns, and a unique equilibrium exists under reasonable conditions. Equilibrium investment is efficient when the sexes are symmetric. However, when there is any asymmetry, including an unbalanced sex ratio, investments are generically excessive. For example, if there is an excess of boys, then there is parental over-investment in boys and under-investment in girls, and total investment will be excessive.marriage, ex ante investments, gender differences, assortative matching tournament, sex ratio

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

Essays on Microeconomic Theory.

The present work collects three essays on microeconomic theory. In the first essay, I study a model in which a finite number of men and women look for future spouses via random meetings. I ask whether equilibrium marriage outcomes are stable matchings when search frictions are small. The answer is they can but need not be. For any stable matching there is an equilibrium leading to it almost surely. However unstable---even Pareto-dominated---matchings may still arise with positive probability. In addition, inefficiency due to delay may remain significant despite vanishing search frictions. Finally, a condition is identified under which all equilibria are outcome equivalent, stable, and efficient. In the second essay, a joint work Kfir Eliaz, we model a competition between two teams as an all-pay auction with incomplete information. The teams may differ in size and individuals exert effort to increase the performance of one's own team via an additively separable aggregation function. The team with a higher performance wins, and its members enjoy the prize as a public good. The value of the prize is identical to members of the same team but is unknown to the other team. We show that there exists a unique monotone equilibrium in which everyone actively participates, and in this equilibrium a bigger team is more likely to win if the aggregation function is concave, less likely if convex, or equally likely if linear. In the third essay, I study a situation in which a group of people working on a common objective want to share information. Oftentimes information sharing via precise communication is impossible and instead information is aggregated by institutions within which communication is coarse. The paper proposes a unified framework for modeling a general class of such information-aggregating institutions. Within this class, it is shown that institution A outperforms institution B for any common objective if and only if the underlying communication infrastructure of A can be obtained from that of B by a sequence of elementary operations. Each operation either removes redundant communication instruments from B or introduces effective ones to it.PhDEconomicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133250/1/wqg_1.pd

Structure in Stable Matching Problems

In this thesis we provide two contributions to the study of structure in stable matching problems. The first contribution is a short new proof for the integrality of Rothblum’s linear description of the convex hull of incidence vectors of stable matchings in bipartite graphs. The key feature of our proof is to show that extreme points of the formulation must have a 0, 1-component. The second contribution is a computer search procedure for instances of cyclic stable matching problems with three genders as proposed by Knuth. We provide sufficient conditions for the existence of a stable matching in this context. We also investigate bijections of the problem instance vertex set to itself which preserve the set of stable matchings (up to permutation). Such bijections define “symmetric” problem instances. We study this notion of symmetry, and use it to cut down on the number of problem instances in our search. We implemented our proposed computational procedure in Java and end with a discussion of the results running computational experiments using our code on problem instances of size 5

Mathematical Modeling with Differential Equations in Physics, Chemistry, Biology, and Economics

This volume was conceived as a Special Issue of the MDPI journal Mathematics to illustrate and show relevant applications of differential equations in different fields, coherently with the latest trends in applied mathematics research. All the articles that were submitted for publication are valuable, interesting, and original. The readers will certainly appreciate the heterogeneity of the 10 papers included in this book and will discover how helpful all the kinds of differential equations are in a wide range of disciplines. We are confident that this book will be inspirational for young scholars as well

Marriage as a Rat Race: Noisy Pre-Marital Investments with Assortative Matching

We study the incentives of parents to invest in their children when these investments improve their marriage prospects, in a frictionless marriage market with non-transferable utility. Stochastic returns to investment eliminate the multiplicity of equilibria that plagues models with deterministic returns, and ensure that a unique equilibrium often exists. Equilibrium investment is efficient when there is complete symmetry between the sexes. However, when there is any asymmetry between the sexes, investments are generically excessively relative to Pareto-efficiency. Our model can be used for examine several implications of gender differences. For example, if shocks are more variable for boys than for girls, girls will invest more than boys. If there is an excess of boys, then there is parental over-investment in boys and under-investment in girls, and total investment will be excessive