Computing Walrasian Equilibria: Fast Algorithms and Structural Properties

Leme, Renato Paes; Wong, Sam Chiu-wai

doi:10.1137/1.9781611974782.42

Cited by 3 publications

(3 citation statements)

References 20 publications

(40 reference statements)

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“…By the well known First Welfare Theorem (Nisan, Roughgarden, Tardos, & Vazirani, 2007, Theorem 11.13), whenever they exist, Walrasian equilibria maximize social welfare over all possible outcomes. It is also known (Kelso & Crawford, 1982;Leme & Wong, 2017) that a market with gross substitutes valuations always admits a Walrasian Equilibrium that can be computed in polynomial time and with a polynomial number of calls to the value oracle. Moreover, a market with unit-demand buyers (as Γ ) is a special case of a market with gross substitutes valuations in which the value oracle trivially requires polynomial time to be implemented, and therefore always admits a Walrasian Equilibrium that can be computed in polynomial time.…”

Section: Buyermentioning

confidence: 99%

Pricing Problems with Buyer Preselection

Bilò

Flammini

Monaco

et al. 2022

jair

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We investigate the problem of preselecting a subset of buyers (also called agents) participating in a market so as to optimize the performance of stable outcomes. We consider four scenarios arising from the combination of two stability notions, namely market envy-freeness and agent envy-freeness, with the two state-of-the-art objective functions of social welfare and seller’s revenue. When insisting on market envy-freeness, we prove that the problem cannot be approximated within n 1−ε (with n being the number of buyers) for any ε > 0, under both objective functions; we also provide approximation algorithms with an approximation ratio tight up to subpolynomial multiplicative factors for social welfare and the seller’s revenue. The negative result, in particular, holds even for markets with single-minded buyers. We also prove that maximizing the seller’s revenue is NP-hard even for a single buyer, thus closing a previous open question. Under agent envy-freeness and for both objective functions, instead, we design a polynomial time algorithm transforming any stable outcome for a market involving any subset of buyers into a stable outcome for the whole market without worsening its performance. This result creates an interesting middle-ground situation where, if on the one hand buyer preselection cannot improve the performance of agent envy-free outcomes, on the other one it can be used as a tool for simplifying the combinatorial structure of the buyers’ valuation functions in a given market. Finally, we consider the restricted case of multi-unit markets, where all items are of the same type and are assigned the same price. For these markets, we show that preselection may improve the performance of stable outcomes in all of the four considered scenarios, and design corresponding approximation algorithms.

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Section: Buyermentioning

confidence: 99%

Pricing Problems with Buyer Preselection

Bilò

Flammini

Monaco

et al. 2022

jair

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“…The efficiency guarantee for the minimal Walrasian prices was considered in [16] for both matching markets and matroid based valuations. 2 Interestingly, the question of finding a Walrasian equilibrium without ties in the demand for the case of gross substitutes valuations with a unique optimum was also asked by [21] in the quite different context of efficiently computing a Walrasian equilibrium given access to an aggregate demand oracle. For a survey on gross substitutes valuations, see [20].…”

Section: Related Workmentioning

confidence: 99%

“…Independently, Paes Leme and Wong[21] defined robust Walrasian pricing where there is no overlap between the demand sets of different buyers, and showed that for Gross substitute valuations with unique optima, such prices exist. Viewed from our perspective, this gives static prices that achieve optimal welfare for any order of arrival.…”

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confidence: 99%

The Invisible Hand of Dynamic Market Pricing

Cohen-Addad

Eden

Feldman

et al. 2016

Proceedings of the 2016 ACM Conference on Economics and Computation

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Walrasian prices, if they exist, have the property that one can assign every buyer some bundle in her demand set, such that the resulting assignment will maximize social welfare. Unfortunately, this assumes carefully breaking ties amongst different bundles in the buyer demand set. Presumably, the shopkeeper cleverly convinces the buyer to break ties in a manner consistent with maximizing social welfare. Lacking such a shopkeeper, if buyers arrive sequentially and simply choose some arbitrary bundle in their demand set, the social welfare may be arbitrarily bad. In the context of matching markets, we show how to compute dynamic prices, based upon the current inventory, that guarantee that social welfare is maximized. Such prices are set without knowing the identity of the next buyer to arrive. We also show that this is impossible in general (e.g., for coverage valuations), but consider other scenarios where this can be done. We further extend our results to Bayesian and bounded rationality models.

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