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Algorithms, Games, and the Internet
 In STOC
, 2001
"... If the Internet is the next great subject for Theoretical Computer Science to model and illuminate mathematically, then Game Theory, and Mathematical Economics more generally, are likely to prove useful tools. In this talk I survey some opportunities and challenges in this important frontier. 1. ..."
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Cited by 152 (0 self)
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If the Internet is the next great subject for Theoretical Computer Science to model and illuminate mathematically, then Game Theory, and Mathematical Economics more generally, are likely to prove useful tools. In this talk I survey some opportunities and challenges in this important frontier. 1.
Nash QLearning for GeneralSum Stochastic Games
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2003
"... We extend Qlearning to a noncooperative multiagent context, using the framework of generalsum stochastic games. A learning agent maintains Qfunctions over joint actions, and performs updates based on assuming Nash equilibrium behavior over the current Qvalues. This learning protocol provably conv ..."
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Cited by 137 (0 self)
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We extend Qlearning to a noncooperative multiagent context, using the framework of generalsum stochastic games. A learning agent maintains Qfunctions over joint actions, and performs updates based on assuming Nash equilibrium behavior over the current Qvalues. This learning protocol provably converges given certain restrictions on the stage games (defined by Qvalues) that arise during learning. Experiments with a pair of twoplayer grid games suggest that such restrictions on the game structure are not necessarily required. Stage games encountered during learning in both grid environments violate the conditions. However, learning consistently converges in the first grid game, which has a unique equilibrium Qfunction, but sometimes fails to converge in the second, which has three different equilibrium Qfunctions. In a comparison of offline learning performance in both games, we find agents are more likely to reach a joint optimal path with Nash Qlearning than with a singleagent Qlearning method. When at least one agent adopts Nash Qlearning, the performance of both agents is better than using singleagent Qlearning. We have also implemented an online version of Nash Qlearning that balances exploration with exploitation, yielding improved performance.
Representations and Solutions for GameTheoretic Problems
 Artificial Intelligence
, 1997
"... A system with multiple interacting agents (whether artificial or human) is often best analyzed using gametheoretic tools. Unfortunately, while the formal foundations are wellestablished, standard computational techniques for gametheoretic reasoning are inadequate for dealing with realistic games. ..."
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Cited by 135 (0 self)
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A system with multiple interacting agents (whether artificial or human) is often best analyzed using gametheoretic tools. Unfortunately, while the formal foundations are wellestablished, standard computational techniques for gametheoretic reasoning are inadequate for dealing with realistic games. This paper describes the Gala system, an implemented system that allows the specification and efficient solution of large imperfect information games. The system contains the first implementation of a recent algorithm, due to Koller, Megiddo, and von Stengel. Experimental results from the system demonstrate that the algorithm is exponentially faster than the standard algorithm in practice, not just in theory. It therefore allows the solution of games that are orders of magnitude larger than were previously possible. The system also provides a new declarative language for compactly and naturally representing games by their rules. As a whole, the Gala system provides the capability for automa...
Formulating Dynamic Multirigidbody Contact Problems with Friction as Solvable Linear Complementarity Problems
 NONLINEAR DYNAMICS
, 1997
"... A linear complementarity formulation for dynamic multirigidbody contact problems with Coulomb friction is presented. The formulation, based on explicit Euler integration and polygonal approximation of the friction cone, is guaranteed to have a solution for any number of contacts and contact config ..."
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Cited by 134 (21 self)
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A linear complementarity formulation for dynamic multirigidbody contact problems with Coulomb friction is presented. The formulation, based on explicit Euler integration and polygonal approximation of the friction cone, is guaranteed to have a solution for any number of contacts and contact configuration. A model with the same property is formulated for impact problems with friction and nonzero elasticity coefficient. An explicit Euler scheme based on these formulations is presented and is proved to have uniformly bounded velocities as the stepsize tends to zero for the NewtonEuler formulation in body coordinates.
Efficient Computation of Equilibria for Extensive Twoperson Games
, 1996
"... The Nash equilibria of a twoperson, nonzerosum game are the solutions of a certain linear complementarity problem (LCP). In order to use this for solving a game in extensive form, the game must first be converted to a strategic description such as the normal form. The classical normal form, howev ..."
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Cited by 120 (8 self)
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The Nash equilibria of a twoperson, nonzerosum game are the solutions of a certain linear complementarity problem (LCP). In order to use this for solving a game in extensive form, the game must first be converted to a strategic description such as the normal form. The classical normal form, however, is often exponentially large in the size of the game tree. If the game has perfect recall, a linearsized strategic description is the sequence form. For the resulting small LCP, we show that an equilibrium is found efficiently by Lemke’s algorithm, a generalization of the Lemke–Howson method.
Fast Algorithms for Finding Randomized Strategies in Game Trees
, 1994
"... Interactions among agents can be conveniently described by game trees. In order to analyze a game, it is important to derive optimal (or equilibrium) strategies for the different players. The standard approach to finding such strategies in games with imperfect information is, in general, computation ..."
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Cited by 118 (11 self)
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Interactions among agents can be conveniently described by game trees. In order to analyze a game, it is important to derive optimal (or equilibrium) strategies for the different players. The standard approach to finding such strategies in games with imperfect information is, in general, computationally intractable. The approach is to generate the normal form of the game (the matrix containing the payoff for each strategy combination), and then solve a linear program (LP) or a linear complementarity problem (LCP). The size of the normal form, however, is typically exponential in the size of the game tree, thus making this method impractical in all but the simplest cases. This paper describes a new representation of strategies which results in a practical linear formulation of the problem of twoplayer games with perfect recall (i.e., games where players never forget anything, which is a standard assumption). Standard LP or LCP solvers can then be applied to find optimal randomized strategies. The resulting algorithms are, in general, exponentially better than the standard ones, both in terms of time and in terms of space.
A Semismooth Equation Approach To The Solution Of Nonlinear Complementarity Problems
, 1995
"... In this paper we present a new algorithm for the solution of nonlinear complementarity problems. The algorithm is based on a semismooth equation reformulation of the complementarity problem. We exploit the recent extension of Newton's method to semismooth systems of equations and the fact that ..."
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Cited by 105 (12 self)
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In this paper we present a new algorithm for the solution of nonlinear complementarity problems. The algorithm is based on a semismooth equation reformulation of the complementarity problem. We exploit the recent extension of Newton's method to semismooth systems of equations and the fact that the natural merit function associated to the equation reformulation is continuously differentiable to develop an algorithm whose global and quadratic convergence properties can be established under very mild assumptions. Other interesting features of the new algorithm are an extreme simplicity along with a low computational burden per iteration. We include numerical tests which show the viability of the approach.
Penalty Methods For American Options With Stochastic Volatility
, 1998
"... The American early exercise constraint can be viewed as transforming the two dimensional stochastic volatility option pricing PDE into a differential algebraic equation (DAE). Several methods are described for forcing the algebraic constraint by using a penalty source term in the discrete equations. ..."
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Cited by 98 (19 self)
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The American early exercise constraint can be viewed as transforming the two dimensional stochastic volatility option pricing PDE into a differential algebraic equation (DAE). Several methods are described for forcing the algebraic constraint by using a penalty source term in the discrete equations. The resulting nonlinear algebraic equations are solved using an approximate Newton iteration. The solution of the Jacobian is obtained using an incomplete LU (ILU) preconditioned PCG method. Some example computations are presented for option pricing problems based on a stochastic volatility model, including an exotic American chooser option written on a put and call with discrete double knockout barriers and discrete dividends.
A New Merit Function For Nonlinear Complementarity Problems And A Related Algorithm
, 1997
"... We investigate the properties of a new merit function which allows us to reduce a nonlinear complementarity problem to an unconstrained global minimization one. Assuming that the complementarity problem is defined by a P 0 function we prove that every stationary point of the unconstrained problem ..."
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Cited by 94 (7 self)
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We investigate the properties of a new merit function which allows us to reduce a nonlinear complementarity problem to an unconstrained global minimization one. Assuming that the complementarity problem is defined by a P 0 function we prove that every stationary point of the unconstrained problem is a global solution; furthermore, if the complementarity problem is defined by a uniform P function, the level sets of the merit function are bounded. The properties of the new merit function are compared with those of the MangasarianSolodov's implicit Lagrangian and Fukushima's regularized gap function. We also introduce a new, simple, activeset local method for the solution of complementarity problems and show how this local algorithm can be made globally convergent by using the new merit function.
On Dynamic MultiRigidBody Contact Problems with Coulomb Friction
"... . This paper is summary of a comprehensive study of the problem of predicting the possible acceleration(s) of a set of rigid, threedimensional bodies in contact in the presence of Coulomb friction. We begin with a brief introduction to this problem and a survey of related work and previous approach ..."
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Cited by 89 (18 self)
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. This paper is summary of a comprehensive study of the problem of predicting the possible acceleration(s) of a set of rigid, threedimensional bodies in contact in the presence of Coulomb friction. We begin with a brief introduction to this problem and a survey of related work and previous approaches. This is followed by the introduction of two novel complementarity formulations for the contact problem under two friction laws: Coulomb's Law and an analogous law in which Coulomb's quadratic friction cone is approximated by a pyramid. Under a full column rank assumption on the system Jacobian matrix, we establish the existence and uniqueness of a solution to our new models in the case where the friction coefficients are nonnegative and sufficiently small. For the model based on the friction pyramid law, we also show that the classical Lemke almostcomplementary pivot algorithm and our new feasible interior point method are guaranteed to compute a solution. Extensive computational result...