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17
COMPUTATION OF EQUILIBRIA in Finite Games
, 1996
"... We review the current state of the art of methods for numerical computation of Nash equilibria for nitenperson games. Classical path following methods, such as the LemkeHowson algorithm for two person games, and Scarftype fixed point algorithms for nperson games provide globally convergent metho ..."
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Cited by 120 (1 self)
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We review the current state of the art of methods for numerical computation of Nash equilibria for nitenperson games. Classical path following methods, such as the LemkeHowson algorithm for two person games, and Scarftype fixed point algorithms for nperson games provide globally convergent methods for finding a sample equilibrium. For large problems, methods which are not globally convergent, such as sequential linear complementarity methods may be preferred on the grounds of speed. None of these methods are capable of characterizing the entire set of Nash equilibria. More computationally intensive methods, which derive from the theory of semialgebraic sets are required for finding all equilibria. These methods can also be applied to compute various equilibrium refinements.
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 90 (12 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.
Finding Mixed Strategies with Small Supports in Extensive Form Games
 International Journal of Game Theory
, 1995
"... The complexity of algorithms that compute strategies or operate on them typically depends on the representation length of the strategies involved. One measure for the size of a mixed strategy is the number of strategies in its supportthe set of pure strategies to which it gives positive probabili ..."
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Cited by 23 (1 self)
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The complexity of algorithms that compute strategies or operate on them typically depends on the representation length of the strategies involved. One measure for the size of a mixed strategy is the number of strategies in its supportthe set of pure strategies to which it gives positive probability. This paper investigates the existence of "small" mixed strategies in extensive form games, and how such strategies can be used to create more efficient algorithms. The basic idea is that, in an extensive form game, a mixed strategy induces a small set of realization weights that completely describe its observable behavior. This fact can be used to show that for any mixed strategy ¯, there exists a realizationequivalent mixed strategy ¯ 0 whose size is at most the size of the game tree. For a player with imperfect recall, the problem of finding such a strategy ¯ 0 (given the realization weights) is NPhard. On the other hand, if ¯ is a behavior strategy, ¯ 0 can be constructed from...
Lossless abstraction of imperfect information games
 Journal of the ACM
, 2007
"... Abstract. Finding an equilibrium of an extensive form game of imperfect information is a fundamental problem in computational game theory, but current techniques do not scale to large games. To address this, we introduce the ordered game isomorphism and the related ordered game isomorphic abstractio ..."
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Cited by 21 (9 self)
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Abstract. Finding an equilibrium of an extensive form game of imperfect information is a fundamental problem in computational game theory, but current techniques do not scale to large games. To address this, we introduce the ordered game isomorphism and the related ordered game isomorphic abstraction transformation. For a multiplayer sequential game of imperfect information with observable actions and an ordered signal space, we prove that any Nash equilibrium in an abstracted smaller game, obtained by one or more applications of the transformation, can be easily converted into a Nash equilibrium in the original game. We present an algorithm, GameShrink, for abstracting the game using our isomorphism exhaustively. Its complexity is Õ(n2), where n is the number of nodes in a structure we call the signal tree. It is no larger than the game tree, and on nontrivial games it is drastically smaller, so GameShrink has time and space complexity sublinear in the size of the game tree. Using GameShrink, we find an equilibrium to a poker game with 3.1 billion nodes—over four orders of magnitude more than in the largest poker game solved previously. To address even larger games, we introduce approximation methods that do not preserve equilibrium, but nevertheless yield (ex post) provably closetooptimal strategies.
Perfect Recall and Pruning in Games with Imperfect Information
 Computational Intelligence
, 1996
"... Games with imperfect information are an interesting and important class of games. They include most card games (e.g., bridge and poker), as well as many economic and political models. Here, we investigate algorithms for solving imperfect information games expressed in their extensive (gametree) for ..."
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Cited by 12 (0 self)
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Games with imperfect information are an interesting and important class of games. They include most card games (e.g., bridge and poker), as well as many economic and political models. Here, we investigate algorithms for solving imperfect information games expressed in their extensive (gametree) form. In particular, we consider algorithms for the simplest form of solution  a purestrategy equilibrium point. We introduce to the artificial intelligence (AI) community a classic algorithm due to Wilson, that finds a purestrategy equilibrium point in oneplayer games with perfect recall. Wilson's algorithm, which we call IMPminimax, runs in time linear in the size of the gametree searched. In contrast to Wilson's result, Koller & Meggido have shown that finding a purestrategy equilibrium point in oneplayer games without perfect recall is NPhard. Here, we provide another contrast to Wilson's result  we show that in games with perfect recall, finding a purestrategy equilibrium p...
Mixed integer linear programming for exact finitehorizon planning in decentralized POMDPs
 in: Proceedings of the 17th International Conference on Automated Planning and Scheduling, 2007
"... We consider the problem of finding an nagent jointpolicy for the optimal finitehorizon control of a decentralized Pomdp (DecPomdp). This is a problem of very high complexity (NEXPhard in n≥2). In this paper, we propose a new mathematical programming approach for the problem. Our approach is bas ..."
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Cited by 12 (1 self)
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We consider the problem of finding an nagent jointpolicy for the optimal finitehorizon control of a decentralized Pomdp (DecPomdp). This is a problem of very high complexity (NEXPhard in n≥2). In this paper, we propose a new mathematical programming approach for the problem. Our approach is based on two ideas: First, we represent each agent’s policy in the sequenceform and not in the treeform, thereby obtaining a very compact representation of the set of jointpolicies. Second, using this compact representation, we solve this problem as an instance of combinatorial optimization for which we formulate a mixed integer linear program (MILP). The optimal solution of the MILP directly yields an optimal jointpolicy for the DecPomdp. Computational experience shows that formulating and solving the MILP requires significantly less time to solve benchmark DecPomdp problems than existing algorithms. For example, the multiagent tiger problem for horizon 4 is solved in 72 secs with the MILP whereas existing algorithms require several hours to solve it. 1
A Forward Monte Carlo Method for Solving Influence Diagrams Using Local Computation
, 2000
"... The main goal of this paper is to describe a new Monte Carlo method for solving influence diagrams using local computation. We propose a forward Monte Carlo sampling technique that draws independent and identically distributed observations. Methods that have been proposed in this spirit sample from ..."
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Cited by 10 (2 self)
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The main goal of this paper is to describe a new Monte Carlo method for solving influence diagrams using local computation. We propose a forward Monte Carlo sampling technique that draws independent and identically distributed observations. Methods that have been proposed in this spirit sample from the entire distribution. However, when the number of variables is large, the state space of all variables is exponentially large, and the sample size required for good estimates may be too large to be practical. In this paper, we develop a forward Monte Carlo method, which generates observations from only a small set of chance variables for each decision node in the influence diagram. We use methods developed for exact solution of influence diagrams to limit the number of chance variables sampled at any time. Because influence diagrams model each chance variable with a conditional probability distribution, the forward Monte Carlo solution method lends itself very well to influencediagram representations.
Computing Normal Form Perfect Equilibria For Extensive TwoPerson Games
 ECONOMETRICA
, 2001
"... This paper presents an algorithm for computing an equilibrium of an extensive twoperson game with perfect recall. The method is computationally e#cient by virtue of using the sequence form, whose size is proportional to the size of the game tree. The equilibrium is traced on a piecewise linear p ..."
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Cited by 10 (1 self)
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This paper presents an algorithm for computing an equilibrium of an extensive twoperson game with perfect recall. The method is computationally e#cient by virtue of using the sequence form, whose size is proportional to the size of the game tree. The equilibrium is traced on a piecewise linear path in the sequence form strategy space from an arbitrary starting vector. If the starting vector represents a pair of completely mixed strategies, then the equilibrium is normal form perfect. Computational experiments compare the sequence form and the reduced normal form, and show that only the sequence form is tractable for larger games.