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13
The complexity of computing a Nash equilibrium
, 2006
"... We resolve the question of the complexity of Nash equilibrium by showing that the problem of computing a Nash equilibrium in a game with 4 or more players is complete for the complexity class PPAD. Our proof uses ideas from the recentlyestablished equivalence between polynomialtime solvability of n ..."
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Cited by 227 (14 self)
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We resolve the question of the complexity of Nash equilibrium by showing that the problem of computing a Nash equilibrium in a game with 4 or more players is complete for the complexity class PPAD. Our proof uses ideas from the recentlyestablished equivalence between polynomialtime solvability of normalform games and graphical games, and shows that these kinds of games can implement arbitrary members of a PPADcomplete class of Brouwer functions. 1
Solving Simple Stochastic Games with Few Random Vertices
"... Abstract. We present a new algorithm for solving Simple Stochastic Games (SSGs). This algorithm is based on an exhaustive search of a special kind of positional optimal strategies, the fstrategies. The running time is O ( VR! · (V E  + p)), where V , VR, E  and p  are respectively ..."
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Cited by 18 (6 self)
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Abstract. We present a new algorithm for solving Simple Stochastic Games (SSGs). This algorithm is based on an exhaustive search of a special kind of positional optimal strategies, the fstrategies. The running time is O ( VR! · (V E  + p)), where V , VR, E  and p  are respectively the number of vertices, random vertices and edges, and the maximum bitlength of a transition probability. Our algorithm improves existing algorithms for solving SSGs in three aspects. First, our algorithm performs well on SSGs with few random vertices, second it does not rely on linear or quadratic programming, third it applies to all SSGs, not only stopping SSGs.
Hybrid System Modeling and Event Identification
, 1993
"... Hybrid control systems contain two distinct types of systems, continuous state and discretestate, that interact with each other. Their study is essential in designing sequential supervisory controllers for continuousstate systems, and it is central in designing control systems with high degree of ..."
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Cited by 10 (2 self)
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Hybrid control systems contain two distinct types of systems, continuous state and discretestate, that interact with each other. Their study is essential in designing sequential supervisory controllers for continuousstate systems, and it is central in designing control systems with high degree of autonomy. After an introduction to intelligent autonomous control and its relation to hybrid control, models for the plant, controller, and interface are introduced. The important role of the interface is discussed at length. System theoretic issues are addressed and the concepts of determinism and quasideterminism are introduced and studied. The relation to the theory of logical discrete event systems is shown and discussed. When the system changes, online identification supervisory control is desirable. To meet the demanding computing requirements, event identification is performed using inductive inference algorithms.
Learning to be Autonomous: Intelligent Supervisory Control
 Intelligent Control Systems: Theory and Applications
, 1993
"... . A brief introduction to the main ideas in Autonomous Control Systems is first given and certain important issues in modeling, analysis and design are discussed. Control systems with high degree of autonomy should perform well under significant uncertainties in the system and environment for extend ..."
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Cited by 5 (4 self)
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. A brief introduction to the main ideas in Autonomous Control Systems is first given and certain important issues in modeling, analysis and design are discussed. Control systems with high degree of autonomy should perform well under significant uncertainties in the system and environment for extended periods of time, and they must be able to compensate for certain system failures without external intervention. Highly autonomous control systems evolve from conventional control systems by adding intelligent components, and their development requires interdisciplinary research. A working characterization of intelligent controllers is introduced and it is argued that the supervisory controller discussed here, which can learn events, is indeed intelligent. There are problems in Autonomous Control Hybrid control systems are of great importance in the development of autonomous control and they are discussed extensively. An appropriate hybrid system model is first introduced and it is used to...
A Strongly Polynomial Rounding Procedure Yielding a Maximally Complementary Solution for P*(κ) Linear Complementarity Problems
, 1998
"... We deal with Linear Complementarity Problems (LCPs) with P () matrices. First we establish the convergence rate of the complementary variables along the central path. The central path is parameterized by the barrier parameter , as usual. Our elementary proof reproduces the known result that the var ..."
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Cited by 5 (4 self)
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We deal with Linear Complementarity Problems (LCPs) with P () matrices. First we establish the convergence rate of the complementary variables along the central path. The central path is parameterized by the barrier parameter , as usual. Our elementary proof reproduces the known result that the variables on, or close to the central path fall apart in three classes in which these variables are O(1); O() and O( p ), respectively. The constants hidden in these bounds are expressed in, or bounded by, the input data. All this is preparation for our main result: a strongly polynomial rounding procedure. Given a point with sufficiently small complementarity gap and close enough to the central path, the rounding procedure produces a maximally complementary solution in at most O(n³) arithmetic operations. The result implies that Interior Point Methods (IPMs) not only converge to a complementary solution of P () LCPs but, when furnished with our rounding procedure, they can produce a max...
Nash equilibria: Complexity, symmetries, and approximation
 Computer Science Review
"... Dedicated to Christos Papadimitriou, the eternal adolescent We survey recent joint work with Christos Papadimitriou and Paul Goldberg on the computational complexity of Nash equilibria. We show that finding a Nash equilibrium in normal form games is computationally intractable, but in an unusual way ..."
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Cited by 4 (0 self)
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Dedicated to Christos Papadimitriou, the eternal adolescent We survey recent joint work with Christos Papadimitriou and Paul Goldberg on the computational complexity of Nash equilibria. We show that finding a Nash equilibrium in normal form games is computationally intractable, but in an unusual way. It does belong to the class NP; but Nash’s theorem, showing that a Nash equilibrium always exists, makes the possibility that it is also NPcomplete rather unlikely. We show instead that the problem is as hard computationally as finding Brouwer fixed points, in a precise technical sense, giving rise to a new complexity class called PPAD. The existence of the Nash equilibrium was established via Brouwer’s fixedpoint theorem; hence, we provide a computational converse to Nash’s theorem. To alleviate the negative implications of this result for the predictive power of the Nash equilibrium, it seems natural to study the complexity of approximate equilibria: an efficient approximation scheme would imply that players could in principle come arbitrarily close to a Nash equilibrium given enough time. We review recent work on computing approximate equilibria and conclude by studying how symmetries may affect the structure and approximation of Nash equilibria. Nash showed that every symmetric game has a symmetric equilibrium. We complement this theorem with a rich set of structural results for a broader, and more interesting class of games with symmetries, called anonymous games. 1
InteriorPoint Methodology for Linear Programming: Duality, Sensitivity Analysis and Computational Aspects
 IN OPTIMIZATION IN PLANNING AND OPERATION OF ELECTRIC POWER SYSTEMS
, 1993
"... In this paper we use the interior point methodology to cover the main issues in linear programming: duality theory, parametric and sensitivity analysis, and algorithmic and computational aspects. The aim is to provide a global view on the subject matter. ..."
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Cited by 3 (1 self)
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In this paper we use the interior point methodology to cover the main issues in linear programming: duality theory, parametric and sensitivity analysis, and algorithmic and computational aspects. The aim is to provide a global view on the subject matter.
Measure A
 Power Amplifier’s FifthOrder Interception Point’, RF Design
, 1999
"... Peertopeer streaming has emerged as a killer application in today’s Internet, delivering a large variety of live multimedia content to millions of users at any given time with low server cost. Though successfully deployed, the efficiency and optimality of the current peertopeer streaming protoco ..."
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Cited by 2 (0 self)
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Peertopeer streaming has emerged as a killer application in today’s Internet, delivering a large variety of live multimedia content to millions of users at any given time with low server cost. Though successfully deployed, the efficiency and optimality of the current peertopeer streaming protocols are still less than satisfactory. In this thesis, we investigate optimizing solutions to enhance the performance of the stateoftheart meshbased peertopeer streaming systems, utilizing both theoretical performance modeling and extensive realworld measurements. First, we model peertopeer streaming applications in both the singleoverlay and multioverlay scenarios, based on the solid foundation of optimization and game theories. Using these models, we design efficient and fully decentralized solutions to achieve performance optimization in peertopeer streaming. Then, based on a large volume of live measurements from a commercial largescale peertopeer streaming application, we extensively study the realworld performance of peertopeer streaming over a long period of time. Highlights of our measurement study include ii the topological characterization of largescale streaming meshes, the statistical characterization
Randomized Simplex Algorithms and Random Cubes (Extended Abstract)
, 1999
"... Michael Joswig and Volker Kaibel December 6, 1999 Abstract Despite its eminent practical impact, the general question for the theoretical behavior of the Simplex Algorithm is still open. However, in recent years some progress has been made by investigating randomized pivot rules. A main thre ..."
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Cited by 1 (0 self)
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Michael Joswig and Volker Kaibel December 6, 1999 Abstract Despite its eminent practical impact, the general question for the theoretical behavior of the Simplex Algorithm is still open. However, in recent years some progress has been made by investigating randomized pivot rules. A main thread through the history of the analysis of the Simplex Algorithm has been the study of linear programs on combinatorial cubes.
A Geometric Theory of Outliers and Perturbation
, 2002
"... We develop a new understanding of outliers and the behavior of linear programs under perturbation. Outliers are ubiquitous in scientific theory and practice. We analyze a simple algorithm for removal of outliers from a highdimensional data set and show the algorithm to be asymptotically good. We ex ..."
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Cited by 1 (0 self)
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We develop a new understanding of outliers and the behavior of linear programs under perturbation. Outliers are ubiquitous in scientific theory and practice. We analyze a simple algorithm for removal of outliers from a highdimensional data set and show the algorithm to be asymptotically good. We extend this result to distributions that we can access only by sampling, and also to the optimization version of the problem. Our results cover both the discrete and continuous cases. This is joint work with Santosh Vempala. The complexity