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Gross substitutability: An algorithmic survey

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  • Paes Leme, Renato

Abstract

The concept of gross substitute valuations was introduced by Kelso and Crawford as a sufficient conditions for the existence of Walrasian equilibria in economies with indivisible goods. The proof is algorithmic in nature: gross substitutes is exactly the condition that enables a natural price adjustment procedure – known as Walrasian tatônnement – to converge to equilibrium. The same concept was also introduced independently in other communities with different names: M♮-concave functions (Murota and Shioura), Matroidal and Well-Layered maps (Dress and Terhalle) and valuated matroids (Dress and Wenzel). Here we survey various definitions of gross substitutability and show their equivalence. We focus on algorithmic aspects of the various definitions. In particular, we highlight that gross substitutes are the exact class of valuations for which demand oracles can be computed via an ascending greedy algorithm. It also corresponds to a natural discrete analogue of concave functions: local maximizers correspond to global maximizers.

Suggested Citation

  • Paes Leme, Renato, 2017. "Gross substitutability: An algorithmic survey," Games and Economic Behavior, Elsevier, vol. 106(C), pages 294-316.
    • Handle: RePEc:eee:gamebe:v:106:y:2017:i:c:p:294-316
    DOI: 10.1016/j.geb.2017.10.016
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    Citations

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    Cited by:

    1. Elizabeth Baldwin & Paul W. Goldberg & Paul Klemperer & Edwin Lock, 2019. "Solving Strong-Substitutes Product-Mix Auctions," Economics Papers 2019-W08, Economics Group, Nuffield College, University of Oxford.
    2. Kumar, Ujjwal & Roy, Souvik, 2024. "Local incentive compatibility on gross substitutes and other non-convex type-spaces," Journal of Mathematical Economics, Elsevier, vol. 112(C).
    3. Alfred Galichon & Yu-Wei Hsieh & Maxime Sylvestre, 2023. "Monotone comparative statics for submodular functions, with an application to aggregated deferred acceptance," Papers 2304.12171, arXiv.org, revised Aug 2024.
    4. Marie-Charlotte Brandenburg & Christian Haase & Ngoc Mai Tran, 2021. "Competitive equilibrium always exists for combinatorial auctions with graphical pricing schemes," Papers 2107.08813, arXiv.org, revised Nov 2021.
    5. Ozan Candogan & Saša Pekeč, 2018. "Efficient Allocation and Pricing of Multifeatured Items," Management Science, INFORMS, vol. 64(12), pages 5521-5543, December.
    6. Saurabh Amin & Patrick Jaillet & Haripriya Pulyassary & Manxi Wu, 2023. "Market Design for Capacity Sharing in Networks," Papers 2307.03994, arXiv.org, revised Nov 2024.
    7. Roy, Souvik & Kumar, Ujjwal, 2021. "Local incentive compatibility in non-convex type-spaces," MPRA Paper 110872, University Library of Munich, Germany.
    8. Martin Bichler & Johannes Knörr & Felipe Maldonado, 2023. "Pricing in Nonconvex Markets: How to Price Electricity in the Presence of Demand Response," Information Systems Research, INFORMS, vol. 34(2), pages 652-675, June.
    9. Hadi Hosseini & Zhiyi Huang & Ayumi Igarashi & Nisarg Shah, 2022. "Class Fairness in Online Matching," Papers 2203.03751, arXiv.org.
    10. Eric Balkanski & Renato Paes Leme, 2020. "On the Construction of Substitutes," Mathematics of Operations Research, INFORMS, vol. 45(1), pages 272-291, February.
    11. Martin Bichler & Hans Ulrich Buhl & Johannes Knörr & Felipe Maldonado & Paul Schott & Stefan Waldherr & Martin Weibelzahl, 2022. "Electricity Markets in a Time of Change: A Call to Arms for Business Research," Schmalenbach Journal of Business Research, Springer, vol. 74(1), pages 77-102, March.
    12. Oren Ben-Zwi & Ilan Newman, 2023. "No Ascending Auction can find Equilibrium for SubModular valuations," Papers 2312.00522, arXiv.org.
    13. Ozan Candogan & Markos Epitropou & Rakesh V. Vohra, 2021. "Competitive Equilibrium and Trading Networks: A Network Flow Approach," Operations Research, INFORMS, vol. 69(1), pages 114-147, January.
    14. Hatfield, John William & Kominers, Scott Duke & Nichifor, Alexandru & Ostrovsky, Michael & Westkamp, Alexander, 2019. "Full substitutability," Theoretical Economics, Econometric Society, vol. 14(4), November.
    15. Huang, Chao, 2018. "Independence systems in gross-substitute valuations," Economics Letters, Elsevier, vol. 173(C), pages 135-137.
    16. Daniel Lehmann, 2019. "Revealed Preferences for Matching with Contracts," Papers 1908.08823, arXiv.org, revised Mar 2020.

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    More about this item

    Keywords

    Gross substitutes; Discrete convexity; Walrasian equilibrium;
    All these keywords.

    JEL classification:

    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory
    • D44 - Microeconomics - - Market Structure, Pricing, and Design - - - Auctions

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