The Complexity of Two-Team Polymatrix Games with Independent Adversaries

[2409.07398] The Complexity of Two-Team Polymatrix Games with Independent Adversaries Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Computer Science > Computer Science and Game Theory arXiv:2409.07398 (cs) [Submitted on 11 Sep 2024 ( v1 ), last revised 10 Apr 2026 (this version, v2)] Title: The Complexity of Two-Team Polymatrix Games with Independent Adversaries Authors: Alexandros Hollender , Gilbert Maystre , Sai Ganesh Nagarajan View a PDF of the paper titled The Complexity of Two-Team Polymatrix Games with Independent Adversaries, by Alexandros Hollender and 1 other authors View PDF HTML (experimental) Abstract: Adversarial multiplayer games are an important object of study in multiagent learning. In particular, polymatrix zero-sum games are a multiplayer setting where Nash equilibria are known to be efficiently computable. Towards understanding the limits of tractability in polymatrix games, we study the computation of Nash equilibria in such games where each pair of players plays either a zero-sum or a coordination game. We are particularly interested in the setting where players can be grouped into a small number of teams of identical interest. While the three-team version of the problem is known to be PPAD-complete, the complexity for two teams has remained open. Our main contribution is to prove that the two-team version remains hard, namely it is CLS-hard. Furthermore, we show that this lower bound is tight (i.e., CLS-membership) for the setting where one of the teams consists of multiple independent adversaries. By leveraging this result we also obtain a simple algorithm that finds an $\varepsilon$-Nash equilibrium and only has a $1/\varepsilon^2$ dependence in $\varepsilon$ in its running time. On the way to obtaining our main result, we prove hardness of finding any stationary point in the simplest type of non-convex-concave min-max constrained optimization problem, namely for a class of bilinear polynomial objective functions. Comments: Added a section on an algorithm to compute an approximate NE and updated references Subjects: Computer Science and Game Theory (cs.GT) Cite as: arXiv:2409.07398 [cs.GT] (or arXiv:2409.07398v2 [cs.GT] for this version) https://doi.org/10.48550/arXiv.2409.07398 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Sai Ganesh Nagarajan [ view email ] [v1] Wed, 11 Sep 2024 16:38:32 UTC (25 KB) [v2] Fri, 10 Apr 2026 09:29:59 UTC (27 KB) Full-text links: Access Paper: View a PDF of the paper titled The Complexity of Two-Team Polymatrix Games with Independent Adversaries, by Alexandros Hollender and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: cs.GT < prev | next > new | recent | 2024-09 Change to browse by: cs References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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