House allocation with fractional endowments

This paper studies a generalization of the well known house allocation problem in which agents may own fractions of different houses summing to an arbitrary quantity, but have use for only the equivalent of one unit of a house. It departs from the classical model by assuming that arbitrary quantities of each house may be available to the market. Justified envy considerations arise when two agents have the same initial endowment, or when an agent is in some sense disproportionately rewarded in comparison to her peers. For this general model, an algorithm is designed to find a fractional allocation of houses to agents that satisfies ordinal efficiency, individual rationality, and no justified envy. The analysis extend to the full preference domain. Individual rationality, ordinal efficiency, and no justified envy conflict with weak strategyproofness. Moreover, individual rationality, ordinal efficiency and strategyproofness are shown to be incompatible. Finally, two reasonable notions of envy-freeness, no justified envy and equal-endowment no envy, conflict in the presence of ordinal efficiency and individual rationality. All of the impossibility results hold in the strict preference domain.house allocation, fractional endowments, fairness, individual rationality

Cake Cutting Algorithms for Piecewise Constant and Piecewise Uniform Valuations

Cake cutting is one of the most fundamental settings in fair division and mechanism design without money. In this paper, we consider different levels of three fundamental goals in cake cutting: fairness, Pareto optimality, and strategyproofness. In particular, we present robust versions of envy-freeness and proportionality that are not only stronger than their standard counter-parts but also have less information requirements. We then focus on cake cutting with piecewise constant valuations and present three desirable algorithms: CCEA (Controlled Cake Eating Algorithm), MEA (Market Equilibrium Algorithm) and CSD (Constrained Serial Dictatorship). CCEA is polynomial-time, robust envy-free, and non-wasteful. It relies on parametric network flows and recent generalizations of the probabilistic serial algorithm. For the subdomain of piecewise uniform valuations, we show that it is also group-strategyproof. Then, we show that there exists an algorithm (MEA) that is polynomial-time, envy-free, proportional, and Pareto optimal. MEA is based on computing a market-based equilibrium via a convex program and relies on the results of Reijnierse and Potters [24] and Devanur et al. [15]. Moreover, we show that MEA and CCEA are equivalent to mechanism 1 of Chen et. al. [12] for piecewise uniform valuations. We then present an algorithm CSD and a way to implement it via randomization that satisfies strategyproofness in expectation, robust proportionality, and unanimity for piecewise constant valuations. For the case of two agents, it is robust envy-free, robust proportional, strategyproof, and polynomial-time. Many of our results extend to more general settings in cake cutting that allow for variable claims and initial endowments. We also show a few impossibility results to complement our algorithms.Comment: 39 page

Organizing time banks: Lessons from matching markets

A time bank is a group of people that set up a common platform to trade services among themselves. There are several well-known problems associated with this type of time banking, e.g., high overhead costs and difficulties to identify feasible trades. This paper constructs a non-manipulable mechanism that selects an individually rational and time-balanced allocation which maximizes exchanges among the members of the time bank (and those allocations are efficient). The mechanism works on a domain of preferences where agents classify services as unacceptable and acceptable (and for those services agents have specific upper quotas representing their maximum needs)

Organizing Time Exchanges: Lessons from Matching Markets

This paper considers time exchanges via a common platform (e.g., markets for exchanging time units, positions at education institutions, and tuition waivers). There are several problems associated with such markets, e.g., imbalanced outcomes, coordination problems, and inefficiencies. We model time exchanges as matching markets and construct a non-manipulable mechanism that selects an individually rational and balanced allocation which maximizes exchanges among the participating agents (and those allocations are efficient). This mechanism works on a preference domain whereby agents classify the goods provided by other participating agents as either unacceptable or acceptable, and for goods classified as acceptable agents have specific upper quotas representing their maximum needs

Essays on Housing Market Problem

In my dissertation, I focus on resource reallocation problem. Specifically, I consider the housing market problem. In this problem, there is a group of agents and a group of objects. Each agent owns at most one object and each object is owned by at most one agent. Agents have preferences over objects. The goal is to reallocate these objects among agents while satisfying desirable properties; Pareto efficiency (not possible to make someone better-off without making someone worse-off), individual rationality (each agent is assigned an object at least as good as her endowment), strategy proofness (no agent has an incentive to lie) and weak-core selection (no group of agents can trade among themselves such that each of them becomes better-off). In addition, I consider this problem while allowing agents to be indifferent between objects. Recently, favorable results have been established for such problems. It has been proved that Pareto efficient, weak-core selecting (hence, individually rational) and strategy proof rules exist for such problems. I consider additional properties for the housing market problem with indifferences. I show that there are rules which, in addition to the aforementioned properties, satisfy no justified-envy for agents with identical endowments and weak group strategy proofness even though Pareto efficiency and group strategy proofness are incompatible under the assumption of indifferences. I achieve this by providing sufficient conditions for weak group strategy proofness. Then, I propose a procedural enhancement which prioritizes the outcome achieved without violating strategy proofness. I show that some of the existing rules do not satisfy this criterion. So, I propose a new mechanism which satisfies this property in addition to other desirable results. Additionally, I present an amended version of sufficient condition for strategy proofness for housing market problem with weak preferences. I also consider random assignment solutions to housing market problem which is referred to as fractional housing market problem in literature. For general and strict preferences, several impossibility results have been established for such problems. I show that for a restricted class of preferences, trichotomous preferences, these impossibility results do not hold

Efficiency in Multiple-Type Housing Markets

We consider multiple-type housing markets (Moulin, 1995), which extend Shapley-Scarf housing markets (Shapley and Scarf, 1974) from one dimension to higher dimensions. In this model, Pareto efficiency is incompatible with individual rationality and strategy-proofness (Konishi et al., 2001). Therefore, we consider two weaker efficiency properties: coordinatewise efficiency and pairwise efficiency. We show that these two properties both (i) are compatible with individual rationality and strategy-proofness, and (ii) help us to identify two specific mechanisms. To be more precise, on various domains of preference profiles, together with other well-studied properties (individual rationality, strategy-proofness, and non-bossiness), coordinatewise efficiency and pairwise efficiency respectively characterize two extensions of the top-trading-cycles mechanism (TTC): the coordinatewise top-trading-cycles mechanism (cTTC) and the bundle top-trading-cycles mechanism (bTTC). Moreover, we propose several variations of our efficiency properties, and we find that each of them is either satisfied by cTTC or bTTC, or leads to an impossibility result (together with individual rationality and strategy-proofness). Therefore, our characterizations can be primarily interpreted as a compatibility test: any reasonable efficiency property that is not satisfied by cTTC or bTTC could be considered incompatible with individual rationality and strategy-proofness. For multiple-type housing markets with strict preferences, our characterization of bTTC constitutes the first characterization of an extension of the prominent TTC mechanis