Gradient-Based Algorithms for Finding Nash Equilibria in Extensive Form Games

Gilpin, Andrew R.; Hoda, Samid; Peña, Javier; Sandholm, Tüomas

doi:10.1007/978-3-540-77105-0_9

Cited by 33 publications

(29 citation statements)

References 19 publications

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“…Examples of such representations (other than the extensive form) include graphical games (Kearns et al, 2001), action-graph games (Bhat and Leyton-Brown, 2004;Leyton-Brown and Tennenholtz, 2003), and multiagent influence diagrams (Koller and Milch, 2001). While changing the way the game is represented does not change it strategically, 11 it does affect the computational complexity of solving the game (Gottlob et al, 2003;Schoenebeck and Vadhan, 2006). However, as long as the representation can capture any game, the computational problem cannot become any easier than under the straightforward representation.…”

Section: Conclusion and Future Researchmentioning

confidence: 99%

New complexity results about Nash equilibria

Conitzer

Sandholm

2008

Games and Economic Behavior

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224

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We provide a single reduction that demonstrates that in normal-form games: (1) it is N P-complete to determine whether Nash equilibria with certain natural properties exist (these results are similar to those obtained by Gilboa and Zemel [Gilboa, I., Zemel, E., 1989. Nash and correlated equilibria: Some complexity considerations. Games Econ. Behav. 1, 80-93]), (2) more significantly, the problems of maximizing certain properties of a Nash equilibrium are inapproximable (unless P = N P), and (3) it is #P-hard to count the Nash equilibria. We also show that determining whether a pure-strategy Bayes-Nash equilibrium exists in a Bayesian game is N P-complete, and that determining whether a pure-strategy Nash equilibrium exists in a Markov (stochastic) game is PSPACE-hard even if the game is unobserved (and that this remains N P-hard if the game has finite length). All of our hardness results hold even if there are only two players and the game is symmetric.

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Section: Conclusion and Future Researchmentioning

confidence: 99%

New complexity results about Nash equilibria

Conitzer

Sandholm

2008

Games and Economic Behavior

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224

270

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“…1 Given the further developments in linear programming in the past century [10,11], we now have efficient algorithms for computing equilibria in zero-sum games, even in very large ones such as poker [6,7].…”

Section: Introductionmentioning

confidence: 99%

“…This assumption is similar to a typical bounded-payoff assumption made in the MWU protocol. 7 We assume without loss of generality that the players know the identity of the "row" player and of the "column" player. We make this assumption to allow for protocols that are asymmetric in the order of moves of the players.…”

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confidence: 99%

“…• In Section 3, we "decouple" the EGT algorithm 7 We suspect that we can modify our protocol to work in the case where no upper bound is known, by repeatedly guessing values for |A|max and thereby slowing the protocol's convergence rate by a factor polynomial in |A|max. 8 We can augment our protocols with initial rounds of interaction where both players select strategies at random, or according to a simple no-regret protocol such as the MWU algorithm.…”

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confidence: 99%

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Near-Optimal No-Regret Algorithms for Zero-Sum Games

Daskalakis¹,

Deckelbaum²,

Kim³

2011

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

131

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We propose a new no-regret learning algorithm. When used against an adversary, our algorithm achieves average regret that scales as O "with the number T of rounds. This regret bound is optimal but not rare, as there are a multitude of learning algorithms with this regret guarantee. However, when our algorithm is used by both players of a zero-sum game, their average regret scales as O`l n T T´, guaranteeing a near-linear rate of convergence to the value of the game. This represents an almost-quadratic improvement on the rate of convergence to the value of a game known to be achieved by any no-regret learning algorithm, and is essentially optimal as we show a lower bound of Ω`1 T´. Moreover, the dynamics produced by our algorithm in the game setting are strongly-uncoupled in that each player is oblivious to the payoff matrix of the game and the number of strategies of the other player, has limited private storage, and is not allowed funny bit arithmetic that can trivialize the problem; instead he only observes the performance of his strategies against the actions of the other player and can use private storage to remember past played strategies and observed payoffs, or cumulative information thereof. Here, too, our rate of convergence is nearly-optimal and represents an almost-quadratic improvement over the best previously known strongly-uncoupled dynamics.

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“…That game tree size has required us to use the approximation version of GameShrink discussed in Section 5 . More recently we have also applied other lossy abstraction techniques [Gilpin and Sandholm 2007;Gilpin et al 2007b] and custom equilibrium-finding algorithms [Gilpin et al 2007a] to that problem. These techniques have yielded highly competitive software programs for that game.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Lossless abstraction of imperfect information games

Gilpin

Sandholm

2007

J. ACM

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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 multi-player 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 isÕ(n 2 ), 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 close-to-optimal strategies.

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Gradient-Based Algorithms for Finding Nash Equilibria in Extensive Form Games

Cited by 33 publications

References 19 publications

New complexity results about Nash equilibria

New complexity results about Nash equilibria

Near-Optimal No-Regret Algorithms for Zero-Sum Games

Lossless abstraction of imperfect information games

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