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)


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
  • Nash Equilibria
  • Polymatrix Games
  • PPAD
  • Brouwer Fixed Points


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