79 research outputs found

    Do Prices Coordinate Markets?

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    Walrasian equilibrium prices can be said to coordinate markets: They support a welfare optimal allocation in which each buyer is buying bundle of goods that is individually most preferred. However, this clean story has two caveats. First, the prices alone are not sufficient to coordinate the market, and buyers may need to select among their most preferred bundles in a coordinated way to find a feasible allocation. Second, we don't in practice expect to encounter exact equilibrium prices tailored to the market, but instead only approximate prices, somehow encoding "distributional" information about the market. How well do prices work to coordinate markets when tie-breaking is not coordinated, and they encode only distributional information? We answer this question. First, we provide a genericity condition such that for buyers with Matroid Based Valuations, overdemand with respect to equilibrium prices is at most 1, independent of the supply of goods, even when tie-breaking is done in an uncoordinated fashion. Second, we provide learning-theoretic results that show that such prices are robust to changing the buyers in the market, so long as all buyers are sampled from the same (unknown) distribution

    Welfare and Revenue Guarantees for Competitive Bundling Equilibrium

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    We study equilibria of markets with mm heterogeneous indivisible goods and nn consumers with combinatorial preferences. It is well known that a competitive equilibrium is not guaranteed to exist when valuations are not gross substitutes. Given the widespread use of bundling in real-life markets, we study its role as a stabilizing and coordinating device by considering the notion of \emph{competitive bundling equilibrium}: a competitive equilibrium over the market induced by partitioning the goods for sale into fixed bundles. Compared to other equilibrium concepts involving bundles, this notion has the advantage of simulatneous succinctness (O(m)O(m) prices) and market clearance. Our first set of results concern welfare guarantees. We show that in markets where consumers care only about the number of goods they receive (known as multi-unit or homogeneous markets), even in the presence of complementarities, there always exists a competitive bundling equilibrium that guarantees a logarithmic fraction of the optimal welfare, and this guarantee is tight. We also establish non-trivial welfare guarantees for general markets, two-consumer markets, and markets where the consumer valuations are additive up to a fixed budget (budget-additive). Our second set of results concern revenue guarantees. Motivated by the fact that the revenue extracted in a standard competitive equilibrium may be zero (even with simple unit-demand consumers), we show that for natural subclasses of gross substitutes valuations, there always exists a competitive bundling equilibrium that extracts a logarithmic fraction of the optimal welfare, and this guarantee is tight. The notion of competitive bundling equilibrium can thus be useful even in markets which possess a standard competitive equilibrium

    The Fair Division of Hereditary Set Systems

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    We consider the fair division of indivisible items using the maximin shares measure. Recent work on the topic has focused on extending results beyond the class of additive valuation functions. In this spirit, we study the case where the items form an hereditary set system. We present a simple algorithm that allocates each agent a bundle of items whose value is at least 0.36670.3667 times the maximin share of the agent. This improves upon the current best known guarantee of 0.20.2 due to Ghodsi et al. The analysis of the algorithm is almost tight; we present an instance where the algorithm provides a guarantee of at most 0.37380.3738. We also show that the algorithm can be implemented in polynomial time given a valuation oracle for each agent.Comment: 22 pages, 1 figure, full version of WINE 2018 submissio

    Approximating Nash social welfare under rado valuations

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    We consider the problem of approximating maximum Nash social welfare (NSW) while allocating a set of indivisible items to n agents. The NSW is a popular objective that provides a balanced tradeoff between the often conflicting requirements of fairness and efficiency, defined as the weighted geometric mean of the agents' valuations. For the symmetric additive case of the problem, where agents have the same weight with additive valuations, the first constant-factor approximation algorithm was obtained in 2015. Subsequent work has obtained constant-factor approximation algorithms for the symmetric case under mild generalizations of additive, and O(n)-approximation algorithms for subadditive valuations and for the asymmetric case. In this paper, we make significant progress towards both symmetric and asymmetric NSW problems. We present the first constant-factor approximation algorithm for the symmetric case under Rado valuations. Rado valuations form a general class of valuation functions that arise from maximum cost independent matching problems, including as special cases assignment (OXS) valuations and weighted matroid rank functions. Furthermore, our approach also gives the first constant-factor approximation algorithm for the asymmetric case under Rado valuations, provided that the maximum ratio between the weights is bounded by a constant

    Minimal Envy and Popular Matchings

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    We study ex-post fairness in the object allocation problem where objects are valuable and commonly owned. A matching is fair from individual perspective if it has only inevitable envy towards agents who received most preferred objects -- minimal envy matching. A matching is fair from social perspective if it is supported by majority against any other matching -- popular matching. Surprisingly, the two perspectives give the same outcome: when a popular matching exists it is equivalent to a minimal envy matching. We show the equivalence between global and local popularity: a matching is popular if and only if there does not exist a group of size up to 3 agents that decides to exchange their objects by majority, keeping the remaining matching fixed. We algorithmically show that an arbitrary matching is path-connected to a popular matching where along the path groups of up to 3 agents exchange their objects by majority. A market where random groups exchange objects by majority converges to a popular matching given such matching exists. When popular matching might not exist we define most popular matching as a matching that is popular among the largest subset of agents. We show that each minimal envy matching is a most popular matching and propose a polynomial-time algorithm to find them

    Designated school choice

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    Turkish government changed the high-school placement system for several concerns in 2018. The government as a designer designates and orders schools to each student in terms of location. Then students reveal their preference list over these designated schools. The government desires students to be assigned to as possible as the closest schools. However, students’ preference list is independent from the designation order. In this context, there is an incompatibility between the students’ preferences and the concern of the designer. The thesis will solve this sort of incompatibility. Two-Stage-Generalized-Priority-Mechanism proposed in the thesis finds the set of all possible designer-optimal matchings. At the second-stage, TSGPM yields the best designer-optimal matching in terms of the students’ preference list. At the last part of the thesis, strategic properties of the mechanism will be discusse

    Toward Fair Recommendation in Two-sided Platforms

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