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The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions [article]

Paul W. Goldberg, Christos H. Papadimitriou, Rahul Savani
2011 arXiv   pre-print
We show that the widely used homotopy method for solving fixpoint problems, as well as the Harsanyi-Selten equilibrium selection process for games, are PSPACE-complete to implement.  ...  A further application of our techniques yields the result that it is PSPACE-complete to compute any of the equilibria that could be found via the classical Lemke-Howson algorithm, a complexity-theoretic  ...  This paper is a complexity-theoretic critique of the tracing procedure and the homotopy method: we show that finding the solutions they prescribe requires the power of PSPACE.  ... 
arXiv:1006.5352v2 fatcat:tnukvlwxgrf5zokw47hdxic6ri

The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions

Paul W. Goldberg, Christos H. Papadimitriou, Rahul Savani
2011 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science  
We show that the widely used homotopy method for solving fixpoint problems, as well as the Harsanyi-Selten equilibrium selection process for games, are PSPACEcomplete to implement.  ...  A further application of our techniques yields the result that it is PSPACE-complete to compute any of the equilibria that could be found via the classical Lemke-Howson algorithm, a complexity-theoretic  ...  Papadimitriou was supported by an NSF grant and by a Google grant.  ... 
doi:10.1109/focs.2011.26 dblp:conf/focs/GoldbergPS11 fatcat:fc7o54gmfrbxlpxz73tqzmumci

The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions

Paul W. Goldberg, Christos H. Papadimitriou, Rahul Savani
2013 ACM Transactions on Economics and Computation  
We show that the widely used homotopy method for solving fixpoint problems, as well as the Harsanyi-Selten equilibrium selection process for games, are PSPACEcomplete to implement.  ...  A further application of our techniques yields the result that it is PSPACE-complete to compute any of the equilibria that could be found via the classical Lemke-Howson algorithm, a complexity-theoretic  ...  Papadimitriou was supported by an NSF grant and by a Google grant.  ... 
doi:10.1145/2465769.2465774 fatcat:uqaj6yrr7vabnj6s2w6ieijtem

How Do You Like Your Equilibrium Selection Problems? Hard, or Very Hard? [chapter]

Paul W. Goldberg
2010 Lecture Notes in Computer Science  
In the talk I will give an overview of this topic, and a summary of recent progress showing that the equilibria that are found by the Lemke-Howson algorithm, as well as related homotopy methods, are PSPACE-complete  ...  Thus we show that there are no short cuts to the Lemke-Howson solutions, subject only to the hardness of PSPACE. I mention some open problems.  ...  Homotopy methods The survey paper [8] discusses homotopy methods in detail, including the Lemke-Howson algorithm, which falls within this framework.  ... 
doi:10.1007/978-3-642-16170-4_2 fatcat:v5lgj7akdfhehgnwf5o6n4ur24

A Survey of PPAD-Completeness for Computing Nash Equilibria [article]

Paul W. Goldberg
2011 arXiv   pre-print
The most well-known such problem is that of computing a Nash equilibrium of a game. Other examples include the search for market equilibria, and envy-free allocations in the context of cake-cutting.  ...  I also give an overview of some recent work that uses these ideas to show PSPACE-completeness for the computation of specific equilibria found by homotopy methods.  ...  Related algorithms include the linear tracing procedure and homotopy methods [39, 41, 42, 43, 44] discussed below: in these papers the motivation is equilibrium selection -in cases where multiple equilibria  ... 
arXiv:1103.2709v2 fatcat:xopz6wzhwnbqln4krtc6v5iwze

Structure Learning for Approximate Solution of Many-Player Games

Zun Li, Michael Wellman
2020 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
We experimentally demonstrate the efficacy of both methods in reaching quality solutions and uncovering hidden structure, on both perfectly and approximately structured game instances.  ...  Our first algorithm, K-Roles, exploits symmetry by learning a role assignment for players of the game through unsupervised learning (clustering) methods.  ...  Acknowledgment This work was supported in part by the US Army Research Office under MURI W911NF-18-1-0208.  ... 
doi:10.1609/aaai.v34i02.5586 fatcat:in26jvhfpjcwpgvrmm4lchjphi

Multiagent systems: algorithmic, game-theoretic, and logical foundations

2009 ChoiceReviews  
a sample Nash equilibrium 91 4.2.2 An LCP formulation and the Lemke-Howson algorithm 93 4.2.3 Searching the space of supports 101 4.2.4 Beyond sample equilibrium computation 104 4.3 Computing Nash equilibria  ...  Solution Concepts of Normal-Form Games 89 4.1 Computing Nash equilibria of two-player, zero-sum games 89 4.2 Computing Nash equilibria of two-player, general-sum games 91 4.2.1 Complexity of computing  ...  We will call the first type of belief "mere belief" (or sometimes "belief" for short) and the second type of belief "certainty."  ... 
doi:10.5860/choice.46-5662 fatcat:pr2pmv7k2bad3pp5bxgogecgnq

Finding All Nash Equilibria of a Finite Game Using Polynomial Algebra [article]

Ruchira S. Datta
2006 arXiv   pre-print
We then explain how to find all the complex roots of the polynomial systems for arbitrary generic games, by polyhedral homotopy continuation starting from the solutions to the specially constructed games  ...  We describe the use of Groebner bases to solve these polynomial systems and to learn geometric information about how the solution set varies with the payoff functions.  ...  We would like to express our gratitude to the following for generously taking the time to personally discuss with us the use of their software packages: Andrew McLennan and Ted Turocy (Gambit [18] ),  ... 
arXiv:math/0612462v1 fatcat:2lng5b4majabfa5jt2qyehczfu

Finding all Nash equilibria of a finite game using polynomial algebra

Ruchira S. Datta
2009 Economic Theory  
We then explain how to find all the complex roots of the polynomial systems for arbitrary generic games, by polyhedral homotopy continuation starting from the solutions to the specially constructed games  ...  We describe the use of Gröbner bases to solve these polynomial systems and to learn geometric information about how the solution set varies with the payoff functions.  ...  Readers may be familiar with the homotopy continuation method under the guise of "tracing procedures", such as the techniques of Lemke-Howson or Govindan and Wilson.  ... 
doi:10.1007/s00199-009-0447-z fatcat:tr6n7jjz5vgwleky56t2mxgeky

Computing Nash Equilibria in Bimatrix Games: GPU-Based Parallel Support Enumeration

Safraz Rampersaud, Lena Mashayekhy, Daniel Grosu
2014 IEEE Transactions on Parallel and Distributed Systems  
The algorithm achieves significant speedups relative to the OpenMP-based parallel implementation of the support enumeration method running on conventional multicore machines.  ...  Computing Nash equilibria is a very important problem in strategic analysis of markets, conflicts and resource allocation.  ...  ACKNOWLEDGMENT This research was supported in part by NSF grants DGE-0654014 and CNS-1116787.  ... 
doi:10.1109/tpds.2014.2307887 fatcat:gdyxywzwenchrok7jfbrtds4xq

Computing Nash equilibria in bimatrix games: GPU-based parallel support enumeration

Safraz Rampersaud, Lena Mashayekhy, Daniel Grosu
2012 2012 IEEE 31st International Performance Computing and Communications Conference (IPCCC)  
The algorithm achieves significant speedups relative to the OpenMP-based parallel implementation of the support enumeration method running on conventional multicore machines.  ...  Computing Nash equilibria is a very important problem in strategic analysis of markets, conflicts and resource allocation.  ...  ACKNOWLEDGMENT This research was supported in part by NSF grants DGE-0654014 and CNS-1116787.  ... 
doi:10.1109/pccc.2012.6407775 dblp:conf/ipccc/RampersaudMG12 fatcat:thxdpchrpffohittfjq7lpscxy

Uniqueness of Stationary Equilibrium Payoffs in Coalitional Bargaining

Hulya Eraslan, Andrew McLennan
2011 Social Science Research Network  
The theory of the fixed point index is used to show that stationary equilibrium expected payoffs of this coalitional bargaining game are unique.  ...  each other agent either approves or rejects the proposal, and the proposal is implemented if the set of approving agents is a winning coalition for the proposer.  ...  based on linear pivoting such as the simplex algorithm for linear programming, the Lemke-Howson algorithm (Lemke and Howson (1964) ) for two player games, or the Lemke (1965) algorithm for linear complementary  ... 
doi:10.2139/ssrn.1860085 fatcat:5kwv35rserd2lnrl4yehrnfbf4

Uniqueness of stationary equilibrium payoffs in coalitional bargaining

Hülya Eraslan, Andrew McLennan
2013 Journal of Economic Theory  
The theory of the fixed point index is used to show that stationary equilibrium expected payoffs of this coalitional bargaining game are unique.  ...  each other agent either approves or rejects the proposal, and the proposal is implemented if the set of approving agents is a winning coalition for the proposer.  ...  based on linear pivoting such as the simplex algorithm for linear programming, the Lemke-Howson algorithm (Lemke and Howson (1964) ) for two player games, or the Lemke (1965) algorithm for linear complementary  ... 
doi:10.1016/j.jet.2013.09.007 fatcat:lunm6owobfdehisavh7geahvoe

Equilibria, fixed points, and complexity classes

Mihalis Yannakakis
2009 Computer Science Review  
A B S T R A C T Many models from a variety of areas involve the computation of an equilibrium or fixed point of some kind.  ...  There are certain common computational principles underlying different types of equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP.  ...  Newton-based, and homotopy methods (some of these assume differentiability and use also the derivatives of the function).  ... 
doi:10.1016/j.cosrev.2009.03.004 fatcat:lhr6iqyodfgond7vjrochf44la

Toward Large-Scale Agent Guidance in an Urban Taxi Service [article]

Lucas Agussurja, Hoong Chuin Lau
2012 arXiv   pre-print
Nash equilibrium policy is proposed as the solution to the game, where no driver has the incentive to singly deviate from it.  ...  Empirical result shows that both formulations improve the efficiency of the service significantly.  ...  The most well-known of such methods is the Lemke-Howson algorithm. See (Herings and Peeters, 2009) for a recent survey, and (Goldberg et al., 2011) for a discussion on the complexity of such methods.  ... 
arXiv:1210.4849v1 fatcat:iu5vs2sdlvhpjklatgt5b6d5d4
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