FIXP-membership via Convex Optimization: Games, Cakes, and Markets

We introduce a new technique for proving membership of problems in FIXP – the class capturing the complexity of computing a fixed-point of an algebraic circuit. Our technique constructs a “pseudogate” which can be used as a black box when building FIXP circuits. This pseudogate, which we term the “OPT-gate”, can solve most convex optimization problems. Using the OPT-gate, we prove new FIXP-membership results, and we generalize and simplify several known results from the literature on fair division, game theory and competitive markets. In particular, we prove complexity results for two classic problems: computing a market equilibrium in the Arrow-Debreu model with general concave utilities is in FIXP, and computing an envy-free division of a cake with general valuations is FIXP-complete. We further showcase the wide applicability of our technique, by using it to obtain simplified proofs and extensions of known FIXP-membership results for equilibrium computation for various types of strategic games, as well as the pseudomarket mechanism of Hylland and Zeckhauser.

[1]  R. Aharoni,et al.  Fractionally Balanced Hypergraphs and Rainbow KKM Theorems , 2020, Combinatorica.

[2]  Kristoffer Arnsfelt Hansen,et al.  Strong Approximate Consensus Halving and the Borsuk-Ulam Theorem , 2021, ICALP.

[3]  Aris Filos-Ratsikas,et al.  On the Complexity of Equilibrium Computation in First-Price Auctions , 2021, EC.

[4]  Kousha Etessami,et al.  The complexity of computing a (quasi-)perfect equilibrium for an n-player extensive form game , 2021, Games Econ. Behav..

[5]  Miquel Oliu-Barton,et al.  New Algorithms for Solving Zero-Sum Stochastic Games , 2020, Math. Oper. Res..

[6]  Mihalis Yannakakis,et al.  Computational Complexity of the Hylland-Zeckhauser Scheme for One-Sided Matching Markets , 2020, ITCS.

[7]  F. Echenique,et al.  Constrained Pseudo-Market Equilibrium , 2019, American Economic Review.

[8]  R. Selten.  Reexamination of the perfectness concept for equilibrium points in extensive games , 1975, Classics in Game Theory.

[9]  Paul G. Spirakis,et al.  Computing Exact Solutions of Consensus Halving and the Borsuk-Ulam Theorem , 2019, ICALP.

[10]  Paul W. Goldberg,et al.  The Hairy Ball Problem is PPAD-Complete , 2019, ICALP.

[11]  László A. Végh,et al.  A strongly polynomial algorithm for linear exchange markets , 2018, STOC.

[12]  M. Pycia,et al.  A Pseudo-Market Approach to Allocation with Priorities , 2018, American Economic Journal: Microeconomics.

[13]  Kristoffer Arnsfelt Hansen,et al.  Computational Complexity of Proper Equilibrium , 2018, EC.

[14]  Ruta Mehta,et al.  Settling the complexity of Leontief and PLC exchange markets under exact and approximate equilibria , 2017, STOC.

[15]  M. Yannakakis,et al.  The Complexity of Non-Monotone Markets , 2017 .

[16]  John Fearnley,et al.  Inapproximability Results for Approximate Nash Equilibria , 2016, WINE.

[17]  Aviad Rubinstein,et al.  Settling the Complexity of Computing Approximate Two-Player Nash Equilibria , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[18]  Haris Aziz,et al.  A Discrete and Bounded Envy-Free Cake Cutting Protocol for Any Number of Agents , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[19]  Kurt Mehlhorn,et al.  An Improved Combinatorial Polynomial Algorithm for the Linear Arrow-Debreu Market , 2015, SODA.

[20]  Ruta Mehta,et al.  Dichotomies in Equilibrium Computation and Membership of PLC Markets in FIXP , 2016, Theory Comput..

[21]  Ariel D. Procaccia.  Cake Cutting Algorithms , 2016, Handbook of Computational Social Choice.

[22]  V. Vazirani,et al.  A Complementary Pivot Algorithm for Market Equilibrium under Separable, Piecewise-Linear Concave Utilities , 2015, SIAM J. Comput..

[23]  Paul G. Spirakis,et al.  Computing Approximate Nash Equilibria in Polymatrix Games , 2014, Algorithmica.

[24]  Kurt Mehlhorn,et al.  A combinatorial polynomial algorithm for the linear Arrow-Debreu market , 2012, Inf. Comput..

[25]  Kristoffer Arnsfelt Hansen,et al.  The Complexity of Approximating a Trembling Hand Perfect Equilibrium of a Multi-player Game in Strategic Form , 2014, SAGT.

[26]  Vijay V. Vazirani,et al.  On Computability of Equilibria in Markets with Production , 2013, SODA.

[27]  M. Slater.  Lagrange Multipliers Revisited , 2014 .

[28]  F. John.  Extremum Problems with Inequalities as Subsidiary Conditions , 2014 .

[29]  W. Karush.  Minima of Functions of Several Variables with Inequalities as Side Conditions , 2014 .

[30]  Ariel D. Procaccia,et al.  Cake cutting: not just child's play , 2013, CACM.

[31]  Xiaotie Deng,et al.  Algorithmic Solutions for Envy-Free Cake Cutting , 2012, Oper. Res..

[32]  Kristoffer Arnsfelt Hansen,et al.  Exact Algorithms for Solving Stochastic Games , 2012, ArXiv.

[33]  Mihalis Yannakakis,et al.  DATE AND TIME ! ! ! A Complementary Pivot Algorithm for Market Equilibrium under Separable , Piecewise-Linear Concave Utilities , 2012 .

[34]  Paul W. Goldberg,et al.  A Survey of PPAD-Completeness for Computing Nash Equilibria , 2011, ArXiv.

[35]  Kristoffer Arnsfelt Hansen,et al.  The Computational Complexity of Trembling Hand Perfection and Other Equilibrium Refinements , 2010, SAGT.

[36]  Ariel D. Procaccia.  Thou Shalt Covet Thy Neighbor's Cake , 2009, IJCAI.

[37]  Nikhil R. Devanur,et al.  An Improved Approximation Scheme for Computing Arrow-Debreu Prices for the Linear Case , 2003, FSTTCS.

[38]  Xiaotie Deng,et al.  Settling the complexity of computing two-player Nash equilibria , 2007, JACM.

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

[40]  Nikhil R. Devanur,et al.  Market equilibrium via a primal--dual algorithm for a convex program , 2008, JACM.

[41]  Mihalis Yannakakis,et al.  Equilibria, Fixed Points, and Complexity Classes , 2008, STACS.

[42]  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).

[43]  Gerhard J. Woeginger,et al.  On the complexity of cake cutting , 2007, Discret. Optim..

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

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

[46]  Kamal Jain,et al.  A polynomial time algorithm for computing an Arrow-Debreu market equilibrium for linear utilities , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[47]  Vijay V. Vazirani,et al.  An Auction-Based Market Equilibrium Algorithm for the Separable Gross Substitutability Case , 2004, APPROX-RANDOM.

[48]  Amin Saberi,et al.  Approximating Market Equilibria , 2003, RANDOM-APPROX.

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

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

[51]  Hervé Moulin,et al.  A New Solution to the Random Assignment Problem , 2001, J. Econ. Theory.

[52]  E. Damme,et al.  Non-Cooperative Games , 2000 .

[53]  F. Su.  Rental Harmony: Sperner's Lemma in Fair Division , 1999 .

[54]  Jack M. Robertson,et al.  Cake-cutting algorithms - be fair if you can , 1998 .

[55]  Atila Abdulkadiroglu,et al.  RANDOM SERIAL DICTATORSHIP AND THE CORE FROM RANDOM ENDOWMENTS IN HOUSE ALLOCATION PROBLEMS , 1998 .

[56]  Steven J. Brams,et al.  Fair division - from cake-cutting to dispute resolution , 1998 .

[57]  A. Mas-Colell,et al.  Microeconomic Theory , 1995 .

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

[59]  Marie-Françoise Roy,et al.  Real algebraic geometry , 1992 .

[60]  R. B. Bapat,et al.  A constructive proof of a permutation-based generalization of Sperner's lemma , 1989, Math. Program..

[61]  D. Gale.  Equilibrium in a discrete exchange economy with money , 1984 .

[62]  E. Damme.  Refinements of the Nash Equilibrium Concept , 1983 .

[63]  Douglas R. Woodall,et al.  Dividing a cake fairly , 1980 .

[64]  W. Stromquist.  How to Cut a Cake Fairly , 1980 .

[65]  C. Berge.  Topological Spaces: including a treatment of multi-valued functions , 2010 .

[66]  R. Zeckhauser,et al.  The Efficient Allocation of Individuals to Positions , 1979, Journal of Political Economy.

[67]  H. Varian.  Equity, Envy and Efficiency , 1974 .

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

[69]  D. Foley.  Resource allocation and the public sector , 1967 .

[70]  O. Mangasarian,et al.  The Fritz John Necessary Optimality Conditions in the Presence of Equality and Inequality Constraints , 1967 .

[71]  J. Goodman.  Note on Existence and Uniqueness of Equilibrium Points for Concave N-Person Games , 1965 .

[72]  A. M. Fink,et al.  Equilibrium in a stochastic $n$-person game , 1964 .

[73]  Masayuki Takahashi.  Equilibrium points of stochastic non-cooperative $n$-person games , 1964 .

[74]  L. McKenzie,et al.  ON THE EXISTENCE OF GENERAL EQUILIBRIUM FOR A COMPETITIVE MARKET , 1959 .

[75]  D. Gale.  The law of supply and demand , 1955 .

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

[77]  L. McKenzie,et al.  On Equilibrium in Graham's Model of World Trade and Other Competitive Systems , 1954 .

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

[79]  Gerard Debreu,et al.  A Social Equilibrium Existence Theorem* , 1952, Proceedings of the National Academy of Sciences.

[80]  K. Fan.  Fixed-point and Minimax Theorems in Locally Convex Topological Linear Spaces. , 1952, Proceedings of the National Academy of Sciences of the United States of America.

[81]  I. Glicksberg.  A FURTHER GENERALIZATION OF THE KAKUTANI FIXED POINT THEOREM, WITH APPLICATION TO NASH EQUILIBRIUM POINTS , 1952 .

[82]  J. Nash.  Equilibrium Points in N-Person Games. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[83]  H. Steinhaus,et al.  Sur la division pragmatique , 1949 .

[84]  E. Rowland.  Theory of Games and Economic Behavior , 1946, Nature.

[85]  S. Kakutani.  A generalization of Brouwer’s fixed point theorem , 1941 .

[86]  P. Hall.  On Representatives of Subsets , 1935 .

[87]  Karol Borsuk.  Drei Sätze über die n-dimensionale euklidische Sphäre , 1933 .

[88]  Bronisław Knaster,et al.  Ein Beweis des Fixpunktsatzes für n-dimensionale Simplexe , 1929 .

[89]  J. Neumann.  Zur Theorie der Gesellschaftsspiele , 1928 .

[90]  E. Sperner.  Neuer beweis für die invarianz der dimensionszahl und des gebietes , 1928 .

[91]  L. Brouwer.  Über Abbildung von Mannigfaltigkeiten , 1911 .

[92]  H. Poincaré,et al.  Sur les courbes définies par les équations différentielles(III) , 1885 .

[93]  K. Schittkowski,et al.  NONLINEAR PROGRAMMING , 2022 .