Equilibria, fixed points, and complexity classes

Many models from a variety of areas involve the computation of an equilibrium or fixed point of some kind. Examples include Nash equilibria in games; market equilibria; computing optimal strategies and the values of competitive games (stochastic and other games); stable configurations of neural networks; analysing basic stochastic models for evolution like branching processes and for language like stochastic context-free grammars; and models that incorporate the basic primitives of probability and recursion like recursive Markov chains. It is not known whether these problems can be solved in polynomial time. There are certain common computational principles underlying different types of equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP. Representative complete problems for these classes are, respectively, pure Nash equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria in two-player normal form games, and (mixed) Nash equilibria in normal form games with three (or more) players. This paper reviews the underlying computational principles and the corresponding classes.

[1]  Eitan Zemel,et al.  Nash and correlated equilibria: Some complexity considerations , 1989 .

[2]  Marek Kimmel,et al.  Branching processes in biology , 2002 .

[3]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[4]  G. Debreu,et al.  Excess demand functions , 1974 .

[5]  Vaidyanathan Ramaswami,et al.  Introduction to Matrix Analytic Methods in Stochastic Modeling , 1999, ASA-SIAM Series on Statistics and Applied Mathematics.

[6]  Mihalis Yannakakis,et al.  The Analysis of Local Search Problems and Their Heuristics , 1990, STACS.

[7]  E. Allen Emerson,et al.  Tree automata, mu-calculus and determinacy , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[8]  Kousha Etessami,et al.  Quasi-Birth-Death Processes, Tree-Like QBDs, Probabilistic 1-Counter Automata, and Pushdown Systems , 2008, QEST.

[9]  Xiaotie Deng,et al.  Settling the Complexity of Two-Player Nash Equilibrium , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[10]  Kousha Etessami,et al.  On the Complexity of Nash Equilibria and Other Fixed Points , 2010, SIAM J. Comput..

[11]  B. Stengel,et al.  COMPUTING EQUILIBRIA FOR TWO-PERSON GAMES , 1996 .

[12]  Kousha Etessami,et al.  Recursive Concurrent Stochastic Games , 2008, Log. Methods Comput. Sci..

[13]  David S. Johnson.  The NP-completeness column: Finding needles in haystacks , 2007, TALG.

[14]  H. Uzawa.  WALRAS' EXISTENCE THEOREM AND BROUWER'S FIXED-POINT THEOREM , 1962 .

[15]  C. Papadimitriou.  Algorithmic Game Theory: The Complexity of Finding Nash Equilibria , 2007 .

[16]  C. E. Lemke,et al.  Equilibrium Points of Bimatrix Games , 1964 .

[17]  Ker-I Ko.  Computational Complexity of Fixed Points and Intersection Points , 1995, J. Complex..

[18]  Peter Bro Miltersen,et al.  On the Complexity of Numerical Analysis , 2009, SIAM J. Comput..

[19]  Kousha Etessami,et al.  On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract) , 2010, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[20]  Richard J. Lipton,et al.  Nash Equilibria via Polynomial Equations , 2004, LATIN.

[21]  Xi Chen,et al.  The approximation complexity of win-lose games , 2007, SODA '07.

[22]  Rahul Savani,et al.  Hard‐to‐Solve Bimatrix Games , 2006 .

[23]  Christos H. Papadimitriou,et al.  The Game World Is Flat: The Complexity of Nash Equilibria in Succinct Games , 2006, ICALP.

[24]  Anne Condon,et al.  On Algorithms for Simple Stochastic Games , 1990, Advances In Computational Complexity Theory.

[25]  Kousha Etessami,et al.  Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations , 2005, JACM.

[26]  Kousha Etessami,et al.  Recursive Markov Decision Processes and Recursive Stochastic Games , 2005, ICALP.

[27]  Xi Chen,et al.  On the complexity of 2D discrete fixed point problem , 2006, Theor. Comput. Sci..

[28]  Aranyak Mehta,et al.  Playing large games using simple strategies , 2003, EC '03.

[29]  Christopher A. Sikorski.  Optimal solution of nonlinear equations , 1985, J. Complex..

[30]  Christos H. Papadimitriou,et al.  On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence , 1994, J. Comput. Syst. Sci..

[31]  Thomas A. Henzinger,et al.  Concurrent reachability games , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[32]  Christos H. Papadimitriou,et al.  The complexity of pure Nash equilibria , 2004, STOC '04.

[33]  Christos H. Papadimitriou,et al.  The Euclidean Traveling Salesman Problem is NP-Complete , 1977, Theor. Comput. Sci..

[34]  Ronald L. Graham,et al.  Some NP-complete geometric problems , 1976, STOC '76.

[35]  P. Boas.  Machine models and simulations , 1991 .

[36]  J. Esparza,et al.  Model checking probabilistic pushdown automata , 2004, LICS 2004.

[37]  Amin Saberi,et al.  Leontief economies encode nonzero sum two-player games , 2006, SODA '06.

[38]  Kousha Etessami,et al.  Efficient Qualitative Analysis of Classes of Recursive Markov Decision Processes and Simple Stochastic Games , 2006, STACS.

[39]  Paul G. Spirakis,et al.  Efficient Algorithms for Constant Well Supported Approximate Equilibria in Bimatrix Games , 2007, ICALP.

[40]  M. Richter,et al.  Non-computability of competitive equilibrium , 1999 .

[41]  Amin Saberi,et al.  The complexity of equilibria: Hardness results for economies via a correspondence with games , 2008, Theor. Comput. Sci..

[42]  Paul G. Spirakis,et al.  Approximate Equilibria for Strategic Two Person Games , 2008, SAGT.

[43]  Lenore Blum,et al.  Complexity and Real Computation , 1997, Springer New York.

[44]  J. Geanakoplos.  Nash and Walras equilibrium via Brouwer , 2003 .

[45]  Mihalis Yannakakis,et al.  How easy is local search? , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[46]  A. Ehrenfeucht,et al.  Positional strategies for mean payoff games , 1979 .

[47]  R. Anderson.  “Almost” implies “near” , 1986 .

[48]  J. Nash,et al.  NON-COOPERATIVE GAMES , 1951, Classics in Game Theory.

[49]  Peter van Emde Boas,et al.  Machine Models and Simulation , 1990, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[50]  Mihalis Yannakakis,et al.  Simple Local Search Problems That are Hard to Solve , 1991, SIAM J. Comput..

[51]  Herbert E. Scarf,et al.  The Approximation of Fixed Points of a Continuous Mapping , 1967 .

[52]  Xi Chen,et al.  On algorithms for discrete and approximate brouwer fixed points , 2005, STOC '05.

[53]  A. Puri.  Theory of hybrid systems and discrete event systems , 1996 .

[54]  P. Jagers,et al.  Branching Processes: Variation, Growth, and Extinction of Populations , 2005 .

[55]  Xi Chen,et al.  Computing Nash Equilibria: Approximation and Smoothed Complexity , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[56]  Anne Condon,et al.  The Complexity of Stochastic Games , 1992, Inf. Comput..

[57]  James Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part III: Quantifier Elimination , 1992, J. Symb. Comput..

[58]  J. Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I , 1989 .

[59]  Aranyak Mehta,et al.  Progress in approximate nash equilibria , 2007, EC '07.

[60]  J. K. Lenstra,et al.  Local Search in Combinatorial Optimisation. , 1997 .

[61]  John F. Canny,et al.  Some algebraic and geometric computations in PSPACE , 1988, STOC '88.

[62]  Hinrich Schütze,et al.  Book Reviews: Foundations of Statistical Natural Language Processing , 1999, CL.

[63]  Prasoon Tiwari,et al.  A problem that is easier to solve on the unit-cost algebraic RAM , 1992, J. Complex..

[64]  Anna R. Karlin,et al.  Random walks with `back buttons' , 2001, STOC 2000.

[65]  Sylvain Sorin,et al.  Stochastic Games and Applications , 2003 .

[66]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[67]  K. Arrow,et al.  EXISTENCE OF AN EQUILIBRIUM FOR A COMPETITIVE ECONOMY , 1954 .

[68]  Paul W. Goldberg,et al.  The complexity of computing a Nash equilibrium , 2006, STOC '06.

[69]  L. Shapley,et al.  Stochastic Games* , 1953, Proceedings of the National Academy of Sciences.

[70]  R. Rosenthal.  A class of games possessing pure-strategy Nash equilibria , 1973 .

[71]  Uri Zwick,et al.  The Complexity of Mean Payoff Games on Graphs , 1996, Theor. Comput. Sci..

[72]  Berthold Vöcking.  Congestion Games: Optimization in Competition , 2006, ACiD.

[73]  J. Filar,et al.  Competitive Markov Decision Processes , 1996 .

[74]  Herbert E. Scarf,et al.  The Computation of Economic Equilibria , 1974 .

[75]  Christos H. Papadimitriou,et al.  Exponential lower bounds for finding Brouwer fixed points , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[76]  J J Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[77]  Marcin Jurdziński,et al.  Deciding the Winner in Parity Games is in UP \cap co-Up , 1998, Inf. Process. Lett..

[78]  Peter Bro Miltersen,et al.  2 The Task of a Numerical Analyst , 2022 .

[79]  Berthold Vöcking,et al.  On the Impact of Combinatorial Structure on Congestion Games , 2006, FOCS.

[80]  Berthold Vöcking,et al.  Inapproximability of pure nash equilibria , 2008, STOC.

[81]  James Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I: Introduction. Preliminaries. The Geometry of Semi-Algebraic Sets. The Decision Problem for the Existential Theory of the Reals , 1992, J. Symb. Comput..