New Results on Simple Stochastic Games

Dai, Decheng; Ge, Rong

doi:10.1007/978-3-642-10631-6_102

Cited by 5 publications

(5 citation statements)

References 5 publications

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“…The converse direction is far from obvious, and it was in fact shown by Hansen, Koucký and Miltersen [12] that if standard binary representation of behavior probabilities is used, merely exhibiting an (1/4)-optimal strategy requires worst case exponential space in the size of the game. In contrast, a (1/4)-approximation to the value vector obviously only requires polynomial space to describe and it may be possible to compute it in polynomial time, though it is currently not known how to do so [6].…”

Section: M}mentioning

confidence: 99%

The Complexity of Solving Reachability Games Using Value and Strategy Iteration

Hansen

Ibsen-Jensen

Miltersen

2011

Computer Science – Theory and Applications

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Two standard algorithms for approximately solving two-player zerosum concurrent reachability games are value iteration and strategy iteration. We prove upper and lower bounds of 2 m Θ(N ) on the worst case number of iterations needed by both of these algorithms for providing non-trivial approximations to the value of a game with N non-terminal positions and m actions for each player in each position. In particular, both algorithms have doubly-exponential complexity. Even when the game given as input has only one non-terminal position, we prove an exponential lower bound on the worst case number of iterations needed to provide non-trivial approximations.

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Section: M}mentioning

confidence: 99%

The Complexity of Solving Reachability Games Using Value and Strategy Iteration

Hansen

Ibsen-Jensen

Miltersen

2011

Computer Science – Theory and Applications

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“…3. This algorithm proves simultaneously the correctness of multiple algorithms ( [10,14,13,24,17,4]). In Sec.…”

Section: Contributionsmentioning

confidence: 68%

“…Interpreting Hoffman-Karp algorithms as instances of GSIA proves that they work on non-stopping games, while in most article the stopping condition is required. Moreover, it shows that their number of iterations is O(nq r ) on q-SSGs, a complexity exponential in r only, which was known only for algorithms specially designed for this purpose [15,13,17,4].…”

Section: G[σa]mentioning

confidence: 99%

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A Generic Strategy Improvement Method for Simple Stochastic Games

Auger,

de Montjoye,

Strozecki

2021

Preprint

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We present a generic strategy iteration algorithm (GSIA) to find an optimal strategy of a simple stochastic game (SSG). We prove the correctness of GSIA, and derive a general complexity bound, which implies and improves on the results of several articles. First, we remove the assumption that the SSG is stopping, which is usually obtained by a polynomial blowup of the game. Second, we prove a tight bound on the denominator of the values associated to a strategy, and use it to prove that all strategy iteration algorithms are in fact fixed parameter tractable in the number of random vertices. All known strategy iteration algorithms can be seen as instances of GSIA, which allows to analyze the complexity of converge from below by Condon [10] and to propose a class of algorithms generalising Gimbert and Horn's algorithm [14,15]. These algorithms require less than r! iterations in general and less iterations than the current best deterministic algorithm for binary SSGs given by .

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“…The algorithm of Gimbert and Horn runs in time r!n O (1) . The next natural step in this direction is to try to find an algorithm with a better dependence on the parameter r. Thus, Dai and Ge [8] gave a randomized algorithm with expected running time √ r!n O (1) . Chatterjee et al [4] pointed out that a variant of the standard algorithm of strategy iteration devised earlier by the same authors [5] can be applied to find a solution in time 4 r r O(1) n O(1) (they only state a time bound of 2 O(r) n O (1) , but a slightly more careful analysis yields the stated bound).…”

Section: Introductionmentioning

confidence: 99%

Solving Simple Stochastic Games with Few Coin Toss Positions

Ibsen-Jensen

Miltersen

2012

Algorithms – ESA 2012

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Gimbert and Horn gave an algorithm for solving simple stochastic games with running time O(r!n) where n is the number of positions of the simple stochastic game and r is the number of its coin toss positions. Chatterjee et al. pointed out that a variant of strategy iteration can be implemented to solve this problem in time 4 r r O(1) n O(1) . In this paper, we show that an algorithm combining value iteration with retrograde analysis achieves a time bound of O(r2 r (r log r + n)), thus improving both time bounds. While the algorithm is simple, the analysis leading to this time bound is involved, using techniques of extremal combinatorics to identify worst case instances for the algorithm.

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New Results on Simple Stochastic Games

Cited by 5 publications

References 5 publications

The Complexity of Solving Reachability Games Using Value and Strategy Iteration

The Complexity of Solving Reachability Games Using Value and Strategy Iteration

A Generic Strategy Improvement Method for Simple Stochastic Games

Solving Simple Stochastic Games with Few Coin Toss Positions

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