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

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LIPIcs.ITCS.2020.18.pdf
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## Acknowledgements

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

## Cite As

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)
https://doi.org/10.4230/LIPIcs.ITCS.2020.18

## Abstract

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
##### Keywords
• Tarski’s theorem
• supermodular games
• monotone functions
• lattices
• fixed points
• Nash equilibria
• computational complexity
• PLS
• stochastic games
• oracle model
• lower bounds

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## References

1. K. J. Arrow and G. Debreu. Existence of an equilibrium for a competitive economy. Econometrica, pages 265-290, 1954.
2. S. R. Buss and A. S. Johnson. Propositional proofs and reductions between NP search problems. Annals of Pure and Applied Logic, 163:1163-1182, 2012.
3. C.-L. Chang, Y.-D. Lyuu, and Y.-W. Ti. The complexity of Tarski’s fixed point theorem. Theoretical Computer Science, 401:228-235, 2008.
4. A. Condon. The complexity of stochastic games. Information and Computation, 96:203-224, 1992.
5. C. Dang, Q. Qi, and Y. Ye. Computational models and complexities of Tarski’s fixed points. Technical report, Stanford University, 2012.
6. C. Dang and Y. Ye. On the complexity of a class of discrete fixed point problems under the lexicographic ordering. Technical Report CY2018-3, City University of Hong Kong, 2018.
7. C. Dang and Y. Ye. On the complexity of an expanded Tarski’s fixed point problem under the componentwise ordering. Theoretical Computer Science, 732:26-45, 2018.
8. C. Dang and Y. Ye. Personal commuication with the authors, March 2019.
9. C. Daskalakis and C. H. Papadimitriou. Continuous Local Search. In Proceedings of 22nd ACM-SIAM Symp. on Discrete Algorithms (SODA'11), pages 790-804, 2011.
10. F. Echenique. Finding all equilibria in games with strategic complements. Journal of Economic Theory, 135(1):514-532, 2007.
11. K. Etessami, C. Papadimitriou, A. Rubinstein, and M. Yannakakis. Tarski’s theorem, supermodular games, and the complexity of equilibria. arXiv preprint, 2019. URL: http://arxiv.org/abs/1909.03210.
12. K. Etessami and M. Yannakakis. On the complexity of Nash equilibria and other fixed points. SIAM Journal on Computing, 39(6):2531-2597, 2010.
13. J. Fearnley, S. Gordon, R. Mehta, and R. Savani. End of Potential Line. arXiv preprint, 2018. URL: http://arxiv.org/abs/1804.03450.
14. J. Fearnley, S. Gordon, R. Mehta, and R. Savani. Unique End of Potential Line. In Proc. of 46th Int. Coll. on Automata, Languages, and Programming (ICALP'19), page 15 pages, 2019. (Full preprint: https://arxiv.org/abs/1811.03841, 90 pages).
15. H. Freudenthal. Simplizialzerlegungen von beschrankter flachheit. Annals of Mathematics, 43:580-582, 1942.
16. D. S. Johnson, C. H. Papadimitriou, and M. Yannakakis. How easy is local search? Journal of Computer and System Sciences, 37:79-100, 1988.
17. P. Milgrom and J. Roberts. Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica, pages 1255-1277, 1990.
18. J. Nash. Non-cooperative Games. Annals of Mathematics, pages 286-295, 1951.
19. N. Nisan, T. Roughgarden, E. Tardos, and V. V. Vazirani, editors. Algorithmic Game Theory. Cambridge University Press, 2007.
20. C. H. Papadimitriou. On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence. Journal of Computer and System Sciences, 48(3):498-532, 1994.
21. L. S. Shapley. Stochastic Games. Proceeding of the National Academy of Sciences, 39(10):1095-1100, 1953.
22. A. Tarski. A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics, 5(2):285-309, 1955.
23. D. M. Topkis. Equilibrium Points in Nonzero-Sum n-Person Submodular Games. SIAM Journal on control and optimization, 17(6):773-787, 1979.
24. D. M. Topkis. Supermodularity and Complementarity. Princeton University Press, 2011.
25. M. Yannakakis. The analysis of local search problems and their heuristics. In Proc. of Symp. on Theoretical Aspects of Computer Science (STACS'90), Springer LNCS 415, pages 298-311, 1990.
26. M. Yannakakis. Chapter 2: Computational Complexity. In E. Aarts and J. K. Lenstra, editors, Local Search in Combinatorial Optimization. John Wiley & Sons, 1997.
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