Tarski’s Theorem, Supermodular Games, and the Complexity of Equilibria

Authors Kousha Etessami, Christos Papadimitriou, Aviad Rubinstein, Mihalis Yannakakis

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Kousha Etessami
  • School of Informatics, University of Edinburgh, UK
Christos Papadimitriou
  • Dept. of Computer Science, Columbia University, NY, USA
Aviad Rubinstein
  • Dept. of Computer Science, Stanford University, CA, USA
Mihalis Yannakakis
  • Dept of Computer Science, Columbia University, NY, USA


Thanks to Alexandros Hollender for pointing us to [S. R. Buss and A. S. Johnson, 2012].

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Kousha Etessami, Christos Papadimitriou, Aviad Rubinstein, and Mihalis Yannakakis. Tarski’s Theorem, Supermodular Games, and the Complexity of Equilibria. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 18:1-18:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


The use of monotonicity and Tarski’s theorem in existence proofs of equilibria is very widespread in economics, while Tarski’s theorem is also often used for similar purposes in the context of verification. However, there has been relatively little in the way of analysis of the complexity of finding the fixed points and equilibria guaranteed by this result. We study a computational formalism based on monotone functions on the d-dimensional grid with sides of length N, and their fixed points, as well as the closely connected subject of supermodular games and their equilibria. It is known that finding some (any) fixed point of a monotone function can be done in time log^d N, and we show it requires at least log^2 N function evaluations already on the 2-dimensional grid, even for randomized algorithms. We show that the general Tarski problem of finding some fixed point, when the monotone function is given succinctly (by a boolean circuit), is in the class PLS of problems solvable by local search and, rather surprisingly, also in the class PPAD. Finding the greatest or least fixed point guaranteed by Tarski’s theorem, however, requires d ⋅ N steps, and is NP-hard in the white box model. For supermodular games, we show that finding an equilibrium in such games is essentially computationally equivalent to the Tarski problem, and finding the maximum or minimum equilibrium is similarly harder. Interestingly, two-player supermodular games where the strategy space of one player is one-dimensional can be solved in O(log N) steps. We also show that computing (approximating) the value of Condon’s (Shapley’s) stochastic games reduces to the Tarski problem. An important open problem highlighted by this work is proving a Ω(log^d N) lower bound for small fixed dimension d ≥ 3.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Tarski’s theorem
  • supermodular games
  • monotone functions
  • lattices
  • fixed points
  • Nash equilibria
  • computational complexity
  • PLS
  • PPAD
  • stochastic games
  • oracle model
  • lower bounds


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