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Tree Polymatrix Games Are PPAD-Hard

Authors Argyrios Deligkas, John Fearnley, Rahul Savani



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Argyrios Deligkas
  • Royal Holloway University of London, UK
John Fearnley
  • University of Liverpool, UK
Rahul Savani
  • University of Liverpool, UK

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Argyrios Deligkas, John Fearnley, and Rahul Savani. Tree Polymatrix Games Are PPAD-Hard. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 38:1-38:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ICALP.2020.38

Abstract

We prove that it is PPAD-hard to compute a Nash equilibrium in a tree polymatrix game with twenty actions per player. This is the first PPAD hardness result for a game with a constant number of actions per player where the interaction graph is acyclic. Along the way we show PPAD-hardness for finding an ε-fixed point of a 2D-LinearFIXP instance, when ε is any constant less than (√2 - 1)/2 ≈ 0.2071. This lifts the hardness regime from polynomially small approximations in k-dimensions to constant approximations in two-dimensions, and our constant is substantial when compared to the trivial upper bound of 0.5.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Exact and approximate computation of equilibria
Keywords
  • Nash Equilibria
  • Polymatrix Games
  • PPAD
  • Brouwer Fixed Points

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References

  1. Siddharth Barman, Katrina Ligett, and Georgios Piliouras. Approximating Nash equilibria in tree polymatrix games. In Proc. of SAGT, pages 285-296, 2015. Google Scholar
  2. Yang Cai and Constantinos Daskalakis. On minmax theorems for multiplayer games. In Proc. of SODA, pages 217-234, 2011. URL: https://doi.org/10.1137/1.9781611973082.20.
  3. Xi Chen and Xiaotie Deng. On the complexity of 2D discrete fixed point problem. Theoretical Computer Science, 410(44):4448-4456, 2009. Google Scholar
  4. Xi Chen, Xiaotie Deng, and Shang-Hua Teng. Settling the complexity of computing two-player Nash equilibria. Journal of the ACM, 56(3):14:1-14:57, 2009. Google Scholar
  5. Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou. The complexity of computing a Nash equilibrium. SIAM Journal on Computing, 39(1):195-259, 2009. Google Scholar
  6. Argyrios Deligkas, John Fearnley, Tobenna Peter Igwe, and Rahul Savani. An empirical study on computing equilibria in polymatrix games. In Proc. of AAMAS, pages 186-195, 2016. Google Scholar
  7. Argyrios Deligkas, John Fearnley, and Rahul Savani. Computing constrained approximate equilibria in polymatrix games. In Proc. of SAGT, pages 93-105, 2017. Google Scholar
  8. Argyrios Deligkas, John Fearnley, and Rahul Savani. Tree polymatrix games are PPAD-hard, 2020. URL: http://arxiv.org/abs/2002.12119.
  9. Argyrios Deligkas, John Fearnley, Rahul Savani, and Paul G. Spirakis. Computing approximate Nash equilibria in polymatrix games. Algorithmica, 77(2):487-514, 2017. Google Scholar
  10. Edith Elkind, Leslie Ann Goldberg, and Paul W. Goldberg. Nash equilibria in graphical games on trees revisited. In Proc. of EC, pages 100-109, 2006. Google Scholar
  11. Kousha Etessami and Mihalis Yannakakis. On the complexity of Nash equilibria and other fixed points. SIAM Journal on Computing, 39(6):2531-2597, 2010. Google Scholar
  12. Michael L. Littman, Michael J. Kearns, and Satinder P. Singh. An efficient, exact algorithm for solving tree-structured graphical games. In Proc. of NIPS, pages 817-823. MIT Press, 2001. Google Scholar
  13. Ruta Mehta. Constant rank two-player games are PPAD-hard. SIAM J. Comput., 47(5):1858-1887, 2018. Google Scholar
  14. Luis E. Ortiz and Mohammad Tanvir Irfan. Tractable algorithms for approximate Nash equilibria in generalized graphical games with tree structure. In Proc. of AAAI, pages 635-641, 2017. Google Scholar
  15. Christos H. Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci., 48(3):498-532, 1994. Google Scholar
  16. Aviad Rubinstein. Settling the complexity of computing approximate two-player Nash equilibria. In Proc. of FOCS, pages 258-265, 2016. Google Scholar
  17. Aviad Rubinstein. Inapproximability of Nash equilibrium. SIAM J. Comput., 47(3):917-959, 2018. Google Scholar
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