FIXP

In computer science, FIXP is a complexity class introduced by Kousha Etessami and Mihalis Yannakakis at 2010. ^[1] It represents problems that can be solved by computing a fixed point of a function that satisfies the conditions of Brouwer's fixed point theorem. More formally, FIXP contains search problem s that can be cast as fixed point computation problems for functions represented by algebraic circuits over basis {+,*,-,/,max,min} with rational constants.

They prove that some fundamental problems in economics and game theory are complete for FIXP, in particular:

• Nash equilibrium computation - computing an exact or approximate mixed-strategy Nash equilibrium in a game with three or more players.
• Market equilibrium computation - computing competitive-equilibrium prices for exchange economies with algebraic demand function s.

Proving membership in FIXP

Filos-Ratsikas, Hansen, Høgh and Hollender ^[2] present a general method for proving membership in FIXP. Their method constructs a black-box that they call “OPT-gate”, which can solve most convex optimization problems. Using their technique, they prove FIXP membership of:

• Market equilibrium computation in Arrow-Debreu markets with general concave utilities;
• Computing an envy-free cake cutting with very general valuations is FIXP-complete.

They also prove FIXP membership for Nash equilibrium computation and for the mechanism of Hylland and Zeckhauser ^[3] for fair random assignment.

Relations to other classes

Relation to PPAD

Etessami and Yannakakis describe the relation succinctly by saying that "The piecewise-linear fragment of FIXP equals PPAD". In other words, ^[4] the problems in PPAD are the problems in FIXP in which the input function is piecewise-linear.

Solutions to problems in PPAD are rational numbers, whereas solutions to problems in FIXP are algebraic number s. ^[5]

PPAD is contained in function classes that are in the intersection of NP and co-NP, whereas FIXP is conjectured to be much harder, and lie in the "harder" end of PSPACE. ^[5]

Relation to SRS

Computing an approximate Nash equilibrium to any factor smaller than 1/2 is at least as hard as the square-root sum problem.

References

1. Etessami, Kousha; Yannakakis, Mihalis (January 2010). "On the Complexity of Nash Equilibria and Other Fixed Points". SIAM Journal on Computing. 39 (6): 2531–2597. doi:10.1137/080720826. ISSN   0097-5397.
2. Filos-Ratsikas, Aris; Hansen, Kristoffer A.; Høgh, Kasper; Hollender, Alexandros (2023-04-04). "FIXP-Membership via Convex Optimization: Games, Cakes, and Markets". SIAM Journal on Computing: FOCS21–30. doi:10.1137/22M1472656. ISSN   0097-5397.
3. Hylland, Aanund; Zeckhauser, Richard (1979). "The Efficient Allocation of Individuals to Positions". Journal of Political Economy. 87 (2): 293. doi:10.1086/260757. S2CID   154167284.
4. Fearnley, John; Goldberg, Paul; Hollender, Alexandros; Savani, Rahul (2022-12-19). "The Complexity of Gradient Descent: CLS = PPAD ∩ PLS". Journal of the ACM. 70 (1): 7:1–7:74. arXiv: 2011.01929 . doi:10.1145/3568163. ISSN   0004-5411. S2CID   263706261.
5. Garg, Jugal; Mehta, Ruta; Vazirani, Vijay V.; Yazdanbod, Sadra (2017-06-19). "Settling the complexity of Leontief and PLC exchange markets under exact and approximate equilibria". Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing. STOC 2017. New York, NY, USA: Association for Computing Machinery. pp. 890–901. doi:10.1145/3055399.3055474. ISBN   978-1-4503-4528-6.