Pricing Multi-Unit Markets

We study the power and limitations of posted prices in multi-unit markets, where agents arrive sequentially in an arbitrary order. We prove upper and lower bounds on the largest fraction of the optimal social welfare that can be guaranteed with posted prices, under a range of assumptions about the designer's information and agents' valuations. Our results provide insights about the relative power of uniform and non-uniform prices, the relative difficulty of different valuation classes, and the implications of different informational assumptions. Among other results, we prove constant-factor guarantees for agents with (symmetric) subadditive valuations, even in an incomplete-information setting and with uniform prices

Market Pricing for Matroid Rank Valuations

In this paper, we study the problem of maximizing social welfare in combinatorial markets through pricing schemes. We consider the existence of prices that are capable to achieve optimal social welfare without a central tie-breaking coordinator. In the case of two buyers with rank valuations, we give polynomial-time algorithms that always find such prices when one of the matroids is a simple partition matroid or both matroids are strongly base orderable. This result partially answers a question raised by D\"uetting and V\'egh in 2017. We further formalize a weighted variant of the conjecture of D\"uetting and V\'egh, and show that the weighted variant can be reduced to the unweighted one based on the weight-splitting theorem for weighted matroid intersection by Frank. We also show that a similar reduction technique works for M${}^\natural$-concave functions, or equivalently, gross substitutes functions

MARKET PRICING FOR MATROID RANK VALUATIONS\ast

In this paper, we study the problem of maximizing social welfare in combinatorial markets through pricing schemes. We consider the existence of prices that are capable of achieving optimal social welfare without a central tie-breaking coordinator. In the case of two buyers with matroid rank valuations, we give polynomial-time algorithms that always find such prices when one of the matroids is a partition matroid or both matroids are strongly base orderable. This result partially answers a question raised by Du"\tting and Ve'\gh [Private communication, 2017]. We further formalize a weighted variant of the conjecture of Du"\tting and Ve'\gh, and show that the weighted variant can be reduced to the unweighted one based on the weight-splitting theorem for weighted matroid intersection by Frank. We also show that a similar reduction technique works for M\natural -concave functions or, equivalently, for gross substitutes functions

Cost sharing over combinatorial domains

We study the problem of designing cost-sharing mechanisms for combinatorial domains. Suppose that multiple items or services are available to be shared among a set of interested agents. The outcome of a mechanism in this setting consists of an assignment, determining for each item the set of players who are granted service, together with respective payments. Although there are several works studying specialized versions of such problems, there has been almost no progress for general combinatorial cost-sharing domains until recently [9]. Still, many questions about the interplay between strategyproofness, cost recovery, and economic efficiency remain unanswered.The main goal of our work is to further understand this interplay in terms of budget balance and social cost approximation. Towards this, we provide a refinement of cross-monotonicity (which we term trace-monotonicity) that is applicable to iterative mechanisms. The trace here refers to the order in which players become finalized. On top of this, we also provide two parameterizations (complementary to a certain extent) of cost functions, which capture the behavior of their average cost-shares.Based on our trace-monotonicity property, we design an Iterative Ascending Cost-Sharing Mechanism, which is applicable to the combinatorial cost-sharing setting with symmetric submodular valuations. Using our first cost function parameterization, we identify conditions under which our mechanism is weakly group-strategyproof, O(1)-budget-balanced, and O(Hn)-approximate with respect to the social cost. Furthermore, we show that our mechanism is budget-balanced and Hn-approximate if both the valuations and the cost functions are symmetric submodular; given existing impossibility results, this is best possible.Finally, we consider general valuation functions and exploit our second parameterization to derive a more fine-grained analysis of the Sequential Mechanism introduced by Moulin. This mechanism is budget balanced by construction, but in general, only guarantees a poor social cost approximation of n. We identify conditions under which the mechanism achieves improved social cost approximation guarantees. In particular, we derive improved mechanisms for fundamental cost-sharing problems, including Vertex Cover and Set Cover