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

Kousha Etessami, Christos Papadimitriou, Aviad Rubinstein, Mihalis Yannakakis

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract / Description of output

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 logd N, and we show it requires at least log2 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(logd N) lower bound for small fixed dimension d ≥ 3.
Original languageEnglish
Title of host publication11th Innovations in Theoretical Computer Science Conference (ITCS 2020)
EditorsThomas Vidick
PublisherSchloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany
Pages18:1 - 18:19
Number of pages19
ISBN (Print)978-3-95977-134-4
Publication statusPublished - 10 Jan 2020
Event11th Annual Innovations in Theoretical Computer Science - Seattle, United States
Duration: 12 Jan 202014 Jan 2020

Publication series

NameLeibniz International Proceedings in Informatics (LIPIcs)
PublisherSchloss Dagstuhl--Leibniz-Zentrum fuer Informatik
ISSN (Print)1868-8969


Conference11th Annual Innovations in Theoretical Computer Science
Abbreviated titleITCS 2020
Country/TerritoryUnited States
Internet address

Keywords / Materials (for Non-textual outputs)

  • 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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