Rethinking Resource Allocation: Fairness and Computability

In the field of multiagent systems, one important problem is fairly allocating items among a set of agents. The gold standard fairness property is envy-freeness whereby each agent prefers the bundle allocated to them over any other bundle. For indivisible goods, envy-freeness cannot be guaranteed: consider two agents and one item. In the context of indivisible goods, one key fairness notion which has gained significant attention in recent years is the maximin share guarantee (MMS). MMS extends the cut-and-choose protocol to multiple agents by guaranteeing each agent as much value as if they divided items into bundles but chose last. However, MMS is not guaranteed to exist, and even it exists, computing an MMS allocation is computationally hard. Consequently, several approximation techniques were proposed to ensure all agents receive a fraction of their MMS. We propose an orthogonal approximation which aims to guarantee MMS for a fraction of the agents. We construct instances where any optimal approximation algorithm fails to guarantee MMS for most agents. We show how to interpolate between the fraction of agents and the fraction of MMS guaranteed to agents. We prove the existence of allocations which satisfy 2/3 of the agents. Our algorithmic technique immediately implies a polynomial-time algorithm for any number of items when there are less than nine agents. Our results significantly reduce the existence gap of MMS when agents divide the items into 3n/2 bundles. We empirically demonstrate that our algorithm outperforms its worst-case bounds in practice on both synthetic and real-world data. We extend our discussion of maximin share approximations to chores. We extend many of the techniques used for MMS approximations to the chores setting. Using these new techniques, we prove the existence of allocations which satisfy all agents when they expect to divide the workload among 2n/3 agents

Fairness Guarantees in Allocation Problems

Fair division problems have been vastly studied in the past 60 years. This line of research was initiated by the work of Steinhaus in 1948 in which the author introduced the cake cutting problem as follows: given a heterogeneous cake and a set of agents with different valuation functions, the goal is to find a fair allocation of the cake to the agents. In order to study this problem, several notions of fairness are proposed, the most famous of which are proportionality and envy-freeness, introduced by Steinhaus in 1948 and Foley in 1967. The fair allocation problems have been studied in both divisible and indivisible settings. For the divisible setting, we explore the "Chore Division Problem". The chore division problem is the problem of fairly dividing an object deemed undesirable among a number of agents. The object is possibly heterogeneous, and hence agents may have different valuations for different parts of the object. Chore division is the dual problem of the celebrated cake cutting problem. We give the first discrete and bounded envy-free chore division protocol for any number of agents. For the indivisible setting, we use the maximin share paradigm introduced by Budish as a measure of fairness. We improve previous results on this measure of fairness in the additive setting and generalize our results for submodular, fractionally subadditive, as well as subadditive settings. We also model the maxmin share fairness paradigm for indivisible goods with different entitlements. For the indivisible setting, we also consider the most studied notion of fairness, envy-freeness. It is known that envy-freeness cannot be always guaranteed in the allocation of indivisible items. We suggest envy-freeness up to a random item (EFR) property which is a relaxation of envy-freeness up to any item (EFX) and give an approximation guarantee. For this notion, we provide a polynomial-time 0.72-approximation allocation algorithm

Computational Aspects of Game Theory and Microeconomics

The purpose of this thesis is to study algorithmic questions that arise in the context of game theory and microeconomics. In particular, we investigate the computational complexity of various economic solution concepts by using and advancing methodologies from the fields of combinatorial optimization and approximation algorithms. We first study the problem of allocating a set of indivisible goods to a set of agents, who express preferences over combinations of items through their utility functions. Several objectives have been considered in the economic literature in different contexts. In fair division theory, a desirable outcome is to minimize the envy or the envy-ratio between any pair of players. We use tools from the theory of linear and integer programming as well as combinatorics to derive new approximation algorithms and hardness results for various types of utility functions. A different objective that has been considered in the context of auctions, is to find an allocation that maximizes the social welfare, i.e., the total utility derived by the agents. We construct reductions from multi-prover proof systems to obtain inapproximability results, given standard assumptions for the utility functions of the agents. We then consider equilibrium concepts in games. We derive the first subexponential algorithm for computing approximate Nash equilibria in $2$-player noncooperative games and extend our result to multi-player games. We further propose a second algorithm based on solving polynomial equations over the reals. Both algorithms improve the previously known upper bounds on the complexity of the problem. Finally, we study game theoretic models that have been introduced recently to address incentive issues in Internet routing. A polynomial time algorithm is obtained for computing equilibria in such games, i.e., routing schemes and payoff allocations from which no subset of agents has an incentive to deviate. Our algorithm is based on linear programming duality theory. We also obtain generalizations when the agents have nonlinear utility functions.Ph.D.Committee Chair: Lipton, Richard; Committee Member: Ding, Yan; Committee Member: Duke, Richard; Committee Member: Randall, Dana; Committee Member: Vazirani, Vija

Multi-Winner Voting with Approval Preferences

Approval-based committee (ABC) rules are voting rules that output a fixed-size subset of candidates, a so-called committee. ABC rules select committees based on dichotomous preferences, i.e., a voter either approves or disapproves a candidate. This simple type of preferences makes ABC rules widely suitable for practical use. In this book, we summarize the current understanding of ABC rules from the viewpoint of computational social choice. The main focus is on axiomatic analysis, algorithmic results, and relevant applications.Comment: This is a draft of the upcoming book "Multi-Winner Voting with Approval Preferences

Fair Resource Allocation in Macroscopic Evacuation Planning Using Mathematical Programming: Modeling and Optimization

Evacuation is essential in the case of natural and manmade disasters such as hurricanes, nuclear disasters, fire accidents, and terrorism epidemics. Random evacuation plans can increase risks and incur more losses. Hence, numerous simulation and mathematical programming models have been developed over the past few decades to help transportation planners make decisions to reduce costs and protect lives. However, the dynamic transportation process is inherently complex. Thus, modeling this process can be challenging and computationally demanding. The objective of this dissertation is to build a balanced model that reflects the realism of the dynamic transportation process and still be computationally tractable to be implemented in reality by the decision-makers. On the other hand, the users of the transportation network require reasonable travel time within the network to reach their destinations. This dissertation introduces a novel framework in the fields of fairness in network optimization and evacuation to provide better insight into the evacuation process and assist with decision making. The user of the transportation network is a critical element in this research. Thus, fairness and efficiency are the two primary objectives addressed in the work by considering the limited capacity of roads of the transportation network. Specifically, an approximation approach to the max-min fairness (MMF) problem is presented that provides lower computational time and high-quality output compared to the original algorithm. In addition, a new algorithm is developed to find the MMF resource allocation output in nonconvex structure problems. MMF is the fairness policy used in this research since it considers fairness and efficiency and gives priority to fairness. In addition, a new dynamic evacuation modeling approach is introduced that is capable of reporting more information about the evacuees compared to the conventional evacuation models such as their travel time, evacuation time, and departure time. Thus, the contribution of this dissertation is in the two areas of fairness and evacuation. The first part of the contribution of this dissertation is in the field of fairness. The objective in MMF is to allocate resources fairly among multiple demands given limited resources while utilizing the resources for higher efficiency. Fairness and efficiency are contradicting objectives, so they are translated into a bi-objective mathematical programming model and solved using the ϵ-constraint method, introduced by Vira and Haimes (1983). Although the solution is an approximation to the MMF, the model produces quality solutions, when ϵ is properly selected, in less computational time compared to the progressive-filling algorithm (PFA). In addition, a new algorithm is developed in this research called the θ progressive-filling algorithm that finds the MMF in resource allocation for general problems and works on problems with the nonconvex structure problems. The second part of the contribution is in evacuation modeling. The common dynamic evacuation models lack a piece of essential information for achieving fairness, which is the time each evacuee or group of evacuees spend in the network. Most evacuation models compute the total time for all evacuees to move from the endangered zone to the safe destination. Lack of information about the users of the transportation network is the motivation to develop a new optimization model that reports more information about the users of the network. The model finds the travel time, evacuation time, departure time, and the route selected for each group of evacuees. Given that the travel time function is a non-linear convex function of the traffic volume, the function is linearized through a piecewise linear approximation. The developed model is a mixed-integer linear programming (MILP) model with high complexity. Hence, the model is not capable of solving large scale problems. The complexity of the model was reduced by introducing a linear programming (LP) version of the full model. The complexity is significantly reduced while maintaining the exact output. In addition, the new θ-progressive-filling algorithm was implemented on the evacuation model to find a fair and efficient evacuation plan. The algorithm is also used to identify the optimal routes in the transportation network. Moreover, the robustness of the evacuation model was tested against demand uncertainty to observe the model behavior when the demand is uncertain. Finally, the robustness of the model is tested when the traffic flow is uncontrolled. In this case, the model's only decision is to distribute the evacuees on routes and has no control over the departure time

Approval-based apportionment

In the apportionment problem, a fixed number of seats must be distributed among parties in proportion to the number of voters supporting each party. We study a generalization of this setting, in which voters can support multiple parties by casting approval ballots. This approval-based apportionment setting generalizes traditional apportionment and is a natural restriction of approval-based multiwinner elections, where approval ballots range over individual candidates instead of parties. Using techniques from both apportionment and multiwinner elections, we identify rules that generalize the D’Hondt apportionment method and that satisfy strong axioms which are generalizations of properties commonly studied in the apportionment literature. In fact, the rules we discuss provide representation guarantees that are currently out of reach in the general setting of multiwinner elections: First, we show that core-stable committees are guaranteed to exist and can be found in polynomial time. Second, we demonstrate that extended justified representation is compatible with committee monotonicity (also known as house monotonicity)

Multi-Winner Voting with Approval Preferences

From fundamental concepts and results to recent advances in computational social choice, this open access book provides a thorough and in-depth look at multi-winner voting based on approval preferences. The main focus is on axiomatic analysis, algorithmic results and several applications that are relevant in artificial intelligence, computer science and elections of any kind. What is the best way to select a set of candidates for a shortlist, for an executive committee, or for product recommendations? Multi-winner voting is the process of selecting a fixed-size set of candidates based on the preferences expressed by the voters. A wide variety of decision processes in settings ranging from politics (parliamentary elections) to the design of modern computer applications (collaborative filtering, dynamic Q&A platforms, diversity in search results, etc.) share the problem of identifying a representative subset of alternatives. The study of multi-winner voting provides the principled analysis of this task. Approval-based committee voting rules (in short: ABC rules) are multi-winner voting rules particularly suitable for practical use. Their usability is founded on the straightforward form in which the voters can express preferences: voters simply have to differentiate between approved and disapproved candidates. Proposals for ABC rules are numerous, some dating back to the late 19th century while others have been introduced only very recently. This book explains and discusses these rules, highlighting their individual strengths and weaknesses. With the help of this book, the reader will be able to choose a suitable ABC voting rule in a principled fashion, participate in, and be up to date with the ongoing research on this topic

On the Trade-offs between Modeling Power and Algorithmic Complexity

Mathematical modeling is a central component of operations research. Most of the academic research in our field focuses on developing algorithmic tools for solving various mathematical problems arising from our models. However, our procedure for selecting the best model to use in any particular application is ad hoc. This dissertation seeks to rigorously quantify the trade-offs between various design criteria in model construction through a series of case studies. The hope is that a better understanding of the pros and cons of different models (for the same application) can guide and improve the model selection process. In this dissertation, we focus on two broad types of trade-offs. The first type arises naturally in mechanism or market design, a discipline that focuses on developing optimization models for complex multi-agent systems. Such systems may require satisfying multiple objectives that are potentially in conflict with one another. Hence, finding a solution that simultaneously satisfies several design requirements is challenging. The second type addresses the dynamics between model complexity and computational tractability in the context of approximation algorithms for some discrete optimization problems. The need to study this type of trade-offs is motivated by certain industry problems where the goal is to obtain the best solution within a reasonable time frame. Hence, being able to quantify and compare the degree of sub-optimality of the solution obtained under different models is helpful. Chapters 2-5 of the dissertation focus on trade-offs of the first type and Chapters 6-7 the second type