# Tree Polymatrix Games are PPAD-hard

Deligkas, Argyrios, Fearnley, John and Savani, Rahul ORCID: 0000-0003-1262-7831
(2020) Tree Polymatrix Games are PPAD-hard. In: 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), 2020-07-08 - 2020-07-12, Beijing.

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 $\epsilon$-fixed point of a 2D LinearFIXP instance, when $\epsilon$ is any constant less than $(\sqrt{2} - 1)/2 \approx 0.2071$. 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 $0.5$.