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A variational inequality framework for network games: Existence, uniqueness, convergence and sensitivity analysis

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  • Parise, Francesca
  • Ozdaglar, Asuman

Abstract

We provide a unified variational inequality framework for the study of fundamental properties of the Nash equilibrium in network games. We identify several conditions on the underlying network (in terms of spectral norm, infinity norm and minimum eigenvalue of its adjacency matrix) that guarantee existence, uniqueness, convergence and continuity of equilibrium in general network games with multidimensional and possibly constrained strategy sets. We delineate the relations between these conditions and characterize classes of networks that satisfy each of these conditions.

Suggested Citation

  • Parise, Francesca & Ozdaglar, Asuman, 2019. "A variational inequality framework for network games: Existence, uniqueness, convergence and sensitivity analysis," Games and Economic Behavior, Elsevier, vol. 114(C), pages 47-82.
    • Handle: RePEc:eee:gamebe:v:114:y:2019:i:c:p:47-82
    DOI: 10.1016/j.geb.2018.11.012
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    1. Yann Bramoull? & Rachel Kranton & Martin D'Amours, 2014. "Strategic Interaction and Networks," American Economic Review, American Economic Association, vol. 104(3), pages 898-930, March.
    2. Belhaj Mohamed & Deroïan Frédéric, 2014. "Competing Activities in Social Networks," The B.E. Journal of Economic Analysis & Policy, De Gruyter, vol. 14(4), pages 1431-1466, October.
    3. Allouch, Nizar, 2015. "On the private provision of public goods on networks," Journal of Economic Theory, Elsevier, vol. 157(C), pages 527-552.
    4. Zenou, Yves & Jackson, Matthew O. & Rogers, Brian, 2016. "Networks: An economic perspective," CEPR Discussion Papers 11452, C.E.P.R. Discussion Papers.
    5. Yann Bramoullé & Rachel Kranton, 2015. "Games Played on Networks," AMSE Working Papers 1530, Aix-Marseille School of Economics, France.
    6. William Novshek, 1985. "On the Existence of Cournot Equilibrium," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 52(1), pages 85-98.
    7. Ui, Takashi, 2016. "Bayesian Nash equilibrium and variational inequalities," Journal of Mathematical Economics, Elsevier, vol. 63(C), pages 139-146.
    8. Kukushkin, Nikolai S., 2004. "Best response dynamics in finite games with additive aggregation," Games and Economic Behavior, Elsevier, vol. 48(1), pages 94-110, July.
    9. Coralio Ballester & Antoni Calvó-Armengol & Yves Zenou, 2006. "Who's Who in Networks. Wanted: The Key Player," Econometrica, Econometric Society, vol. 74(5), pages 1403-1417, September.
    10. Ying-Ju Chen & Yves Zenou & Junjie Zhou, 2018. "Multiple Activities in Networks," American Economic Journal: Microeconomics, American Economic Association, vol. 10(3), pages 34-85, August.
    11. Melo, Emerson, 2018. "A Variational Approach to Network Games," ETA: Economic Theory and Applications 268732, Fondazione Eni Enrico Mattei (FEEM).
    12. Daron Acemoglu & Asuman Ozdaglar & Alireza Tahbaz-Salehi, 2015. "Networks, Shocks, and Systemic Risk," NBER Working Papers 20931, National Bureau of Economic Research, Inc.
    13. Belhaj, Mohamed & Bramoullé, Yann & Deroïan, Frédéric, 2014. "Network games under strategic complementarities," Games and Economic Behavior, Elsevier, vol. 88(C), pages 310-319.
    14. Vives, Xavier, 1990. "Nash equilibrium with strategic complementarities," Journal of Mathematical Economics, Elsevier, vol. 19(3), pages 305-321.
    15. Dubey, Pradeep & Haimanko, Ori & Zapechelnyuk, Andriy, 2006. "Strategic complements and substitutes, and potential games," Games and Economic Behavior, Elsevier, vol. 54(1), pages 77-94, January.
    16. Cornes, Richard & Hartley, Roger, 2012. "Fully aggregative games," Economics Letters, Elsevier, vol. 116(3), pages 631-633.
    17. Stella Dafermos, 1988. "Sensitivity Analysis in Variational Inequalities," Mathematics of Operations Research, INFORMS, vol. 13(3), pages 421-434, August.
    18. Emerson Melo, 2018. "A Variational Approach to Network Games," Working Papers 2018.05, Fondazione Eni Enrico Mattei.
    19. Bulow, Jeremy I & Geanakoplos, John D & Klemperer, Paul D, 1985. "Multimarket Oligopoly: Strategic Substitutes and Complements," Journal of Political Economy, University of Chicago Press, vol. 93(3), pages 488-511, June.
    20. Milgrom, Paul & Roberts, John, 1990. "Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities," Econometrica, Econometric Society, vol. 58(6), pages 1255-1277, November.
    21. Jackson, Matthew O. & Zenou, Yves, 2015. "Games on Networks," Handbook of Game Theory with Economic Applications,, Elsevier.
    22. Durlauf, Steven N., 2004. "Neighborhood effects," Handbook of Regional and Urban Economics, in: J. V. Henderson & J. F. Thisse (ed.), Handbook of Regional and Urban Economics, edition 1, volume 4, chapter 50, pages 2173-2242, Elsevier.
    23. Martin Jensen, 2010. "Aggregative games and best-reply potentials," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 43(1), pages 45-66, April.
    24. Kukushkin, Nikolai S., 1994. "A fixed-point theorem for decreasing mappings," Economics Letters, Elsevier, vol. 46(1), pages 23-26, September.
    25. Jayash Koshal & Angelia Nedić & Uday V. Shanbhag, 2016. "Distributed Algorithms for Aggregative Games on Graphs," Operations Research, INFORMS, vol. 64(3), pages 680-704, June.
    26. Acemoglu, Daron & Jensen, Martin Kaae, 2013. "Aggregate comparative statics," Games and Economic Behavior, Elsevier, vol. 81(C), pages 27-49.
    27. Bramoulle, Yann & Kranton, Rachel, 2007. "Public goods in networks," Journal of Economic Theory, Elsevier, vol. 135(1), pages 478-494, July.
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    Cited by:

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    3. Bayer, Péter & Herings, P. Jean-Jacques & Peeters, Ronald, 2021. "Farsighted manipulation and exploitation in networks," Journal of Economic Theory, Elsevier, vol. 196(C).
    4. Péter Bayer & György Kozics & Nóra Gabriella Szőke, 2020. "Best-Response Dynamics in Directed Network Games," CEU Working Papers 2020_1, Department of Economics, Central European University.
    5. Xiaowei Hu & Peng Li, 2021. "Relief and Stimulus in A Cross-sector Multi-product Scarce Resource Supply Chain Network," Papers 2101.09373, arXiv.org, revised Nov 2022.
    6. Bayer, Péter & Kozics, György & Szőke, Nóra Gabriella, 2023. "Best-response dynamics in directed network games," Journal of Economic Theory, Elsevier, vol. 213(C).
    7. Péter Bayer & György Kozics & Nóra Szőke, 2019. "Best-Response Dynamics in Directed Network Games," CEU Working Papers 2019_3, Department of Economics, Central European University.
    8. Emerson Melo, 2022. "On the uniqueness of quantal response equilibria and its application to network games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 74(3), pages 681-725, October.
    9. Shi, Xiangyu, 2024. "Helping behavior in networked organizations," European Economic Review, Elsevier, vol. 165(C).
    10. Hu, Xiaowei & Li, Peng, 2022. "Relief and stimulus in a cross-sector multi-product scarce resource supply chain network," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 168(C).
    11. Yang Sun & Wei Zhao & Junjie Zhou, 2023. "Structural Interventions In Networks," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 64(4), pages 1533-1563, November.
    12. Xu, Jin & Zenou, Yves & Zhou, Junjie, 2022. "Equilibrium characterization and shock propagation in conflict networks," Journal of Economic Theory, Elsevier, vol. 206(C).
    13. Kun Jin & Yevgeniy Vorobeychik & Mingyan Liu, 2021. "Multi-Scale Games: Representing and Solving Games on Networks with Group Structure," Papers 2101.08314, arXiv.org.
    14. Luke A. Boosey & Christopher Brown, 2021. "Contests with Network Externalities: Theory & Evidence," Working Papers wp2021_07_02, Department of Economics, Florida State University.
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    16. Zenou, Yves & Bochet, Olivier & Faure, Mathieu & Long, Yan, 2020. "Perceived Competition in Networks," CEPR Discussion Papers 15582, C.E.P.R. Discussion Papers.
    17. P'eter Bayer & Gyorgy Kozics & N'ora Gabriella SzH{o}ke, 2021. "Best-response dynamics in directed network games," Papers 2101.03863, arXiv.org.

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    More about this item

    Keywords

    Network games; Variational inequalities; Strong monotonicity; Uniform P-function; Nash equilibrium; Existence and uniqueness; Best response dynamics; Sensitivity analysis;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D85 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Network Formation

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