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40
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 114 (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...
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 87 (7 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.
Approximate Solutions for Partially Observable Stochastic Games with Common Payoffs
 In Proc. of Int. Joint Conference on Autonomous Agents and Multi Agent Systems
, 2004
"... Partially observable decentralized decision making in robot teams is fundamentally different from decision making in fully observable problems. Team members cannot simply apply singleagent solution techniques in parallel. Instead, we must turn to game theoretic frameworks to correctly model the pro ..."
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Cited by 73 (1 self)
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Partially observable decentralized decision making in robot teams is fundamentally different from decision making in fully observable problems. Team members cannot simply apply singleagent solution techniques in parallel. Instead, we must turn to game theoretic frameworks to correctly model the problem. While partially observable stochastic games (POSGs) provide a solution model for decentralized robot teams, this model quickly becomes intractable. We propose an algorithm that approximates POSGs as a series of smaller, related Bayesian games, using heuristics such as QMDP to provide the future discounted value of actions. This algorithm trades off limited lookahead in uncertainty for computational feasibility, and results in policies that are locally optimal with respect to the selected heuristic. Empirical results are provided for both a simple problem for which the full POSG can also be constructed, as well as more complex, robotinspired, problems.
A continuation method for Nash equilibria in structured games
 In Proceedings of the 18th International Joint Conference on Artificial Intelligence (IJCAI
, 2003
"... We describe algorithms for computing Nash equilibria in structured game representations, including both graphical games and multiagent influence diagrams (MAIDs). The algorithms are derived from a continuation method for normalform and extensiveform games due to Govindan and Wilson; they follow a ..."
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Cited by 45 (0 self)
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We describe algorithms for computing Nash equilibria in structured game representations, including both graphical games and multiagent influence diagrams (MAIDs). The algorithms are derived from a continuation method for normalform and extensiveform games due to Govindan and Wilson; they follow a trajectory through the space of perturbed games and their equilibria. Our algorithms exploit game structure through fast computation of the Jacobian of the game's payoff function. They are guaranteed to find at least one equilibrium of the game and may find more. Our approach provides the first exact algorithm for computing an exact equilibrium in graphical games with arbitrary topology, and the first algorithm to exploit finegrain structural properties of MAIDs. We present experimental results for our algorithms. The running time for our graphical game algorithm is similar to, and often better than, the running time of previous approximate algorithms. Our algorithm for MAIDs can effectively solve games that arc much larger than those that could be solved using previous methods. 1
Bargaining with Limited Computation: Deliberation Equilibrium
 ARTIFICIAL INTELLIGENCE
, 2001
"... We develop a normative theory of interactionnegotiation in particularamong selfinterested computationally limited agents where computational actions are game theoretically treated as part of an agent's strategy. We focus on a 2agent setting where each agent has an intractable individual prob ..."
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Cited by 45 (19 self)
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We develop a normative theory of interactionnegotiation in particularamong selfinterested computationally limited agents where computational actions are game theoretically treated as part of an agent's strategy. We focus on a 2agent setting where each agent has an intractable individual problem, and there is a potential gain from pooling the problems, giving rise to an intractable joint problem. At any time, an agent can compute to improve its solution to its own problem, its opponent's problem, or the joint problem. At a deadline the agents then decide whether to implement the joint solution, and if so, how to divide its value (or cost). We present a fully normative model for controlling anytime algorithms where each agent has statistical performance profiles which are optimally conditioned on the problem instance as well as on the path of results of the algorithm run so far. Using this model, we introduce a solution concept, which we call deliberation equilibrium. It is the perfect Bayesian equilibrium of the game where deliberation actions are part of each agent's strategy. The equilibria differ based on whether the performance profiles are deterministic or stochastic, whether the deadline is known or not, and whether the proposer is known in advance or not. We present algorithms for finding the equilibria. Finally, we show that there exist instances of the deliberationbargaining problem where no pure strategy equilibria exist and also instances where the unique equilibrium outcome is not Pareto efficient.
Gradientbased algorithms for finding nash equilibria in extensive form games
 In Proceedings of the Eighteenth International Conference on Game Theory
, 2007
"... We present a computational approach to the saddlepoint formulation for the Nash equilibria of twoperson, zerosum sequential games of imperfect information. The algorithm is a firstorder gradient method based on modern smoothing techniques for nonsmooth convex optimization. The algorithm requires ..."
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Cited by 32 (13 self)
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We present a computational approach to the saddlepoint formulation for the Nash equilibria of twoperson, zerosum sequential games of imperfect information. The algorithm is a firstorder gradient method based on modern smoothing techniques for nonsmooth convex optimization. The algorithm requires O(1/ɛ) iterations to compute an ɛequilibrium, and the work per iteration is extremely low. These features enable us to find approximate Nash equilibria for sequential games with a tree representation of about 10 10 nodes. This is three orders of magnitude larger than what previous algorithms can handle. We present two heuristic improvements to the basic algorithm and demonstrate their efficacy on a range of realworld games. Furthermore, we demonstrate how the algorithm can be customized to a specific class of problems with enormous memory savings. 1
Finding equilibria in large sequential games of imperfect information
 In ACM Conference on Electronic Commerce
, 2006
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MultiStep MultiSensor HiderSeeker Games
"... We study a multistep hiderseeker game where the hider is moving on a graph and, in each step, the seeker is able to search c subsets of the graph nodes. We model this game as a zerosum Bayesian game, which can be solved in weakly polynomial time in the players ’ action spaces. The seeker’s action ..."
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Cited by 27 (6 self)
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We study a multistep hiderseeker game where the hider is moving on a graph and, in each step, the seeker is able to search c subsets of the graph nodes. We model this game as a zerosum Bayesian game, which can be solved in weakly polynomial time in the players ’ action spaces. The seeker’s action space is exponential in c, and both players’ action spaces are exponential in the game horizon. To manage this intractability, we use a column/constraint generation approach for both players. This approach requires an oracle to determine best responses for each player. However, we show that computing a best response for the seeker is NPhard, even for a singlestep game when c is part of the input, and that computing a best response is NPhard for both players for the multistep game, even if c =1. An integer programming formulation of the best response for the hider is practical for moderate horizons, but computing an exact seeker best response is impractical due to the exponential dependence on both c and the horizon. We therefore develop an approximate best response oracle with bounded suboptimality for the seeker. We prove performance bounds on the strategy that results when column/constraint generation with approximate best responses converges, and we measure the performance of our algorithm in simulations. In our experimental results, column/constraint generation converges to nearminimax strategies for both players fairly quickly. 1
SMOOTHING TECHNIQUES FOR COMPUTING NASH EQUILIBRIA OF SEQUENTIAL GAMES
"... We develop firstorder smoothing techniques for saddlepoint problems that arise in the Nash equilibria computation of sequential games. The crux of our work is a construction of suitable proxfunctions for a certain class of polytopes that encode the sequential nature of the games. An implementatio ..."
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Cited by 25 (7 self)
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We develop firstorder smoothing techniques for saddlepoint problems that arise in the Nash equilibria computation of sequential games. The crux of our work is a construction of suitable proxfunctions for a certain class of polytopes that encode the sequential nature of the games. An implementation based on our smoothing techniques computes approximate Nash equilibria for games that are four orders of magnitude larger than what conventional computational approaches can handle.