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Solving Simple Stochastic Games with Few Random Vertices

Hugo Gimbert, Florian Horn, Roberto Amadio
2009 Logical Methods in Computer Science
First they run in polynomial time when the number of random vertices is fixed, so the problem of solving simple stochastic games is fixed-parameter tractable when the parameter is the number of random  ...  Simple stochastic games are two-player zero-sum stochastic games with turn-based moves, perfect information, and reachability winning conditions.  ...  First, they perform much better on SSGs with few random vertices, as they run in polynomial time when the number of random vertices is logarithmic in the size of the game: it follows that the problem of  ...

Simple Stochastic Games with Few Random Vertices Are Easy to Solve [chapter]

Hugo Gimbert, Florian Horn
Foundations of Software Science and Computational Structures
We present a new algorithm for solving Simple Stochastic Games (SSGs). This algorithm is based on an exhaustive search of a special kind of positional optimal strategies, the f-strategies.  ...  First, our algorithm performs well on SSGs with few random vertices, second it does not rely on linear or quadratic programming, third it applies to all SSGs, not only stopping SSGs.  ...  Simple Stochastic Games In this section we give formal definitions of an SSG, values and optimal strategies.  ...

New Results on Simple Stochastic Games [chapter]

Decheng Dai, Rong Ge
2009 Lecture Notes in Computer Science
We study the problem of solving simple stochastic games, and give both an interesting new algorithm and a hardness result.  ...  of simple stochastic games.  ...  Fast Algorithm for SSGs with few random vertices An interesting case for solving simple stochastic games is when there are a few random vertices.  ...

Finding Optimal Strategies of Almost Acyclic Simple Stochatic Games [article]

David Auger
2014 arXiv   pre-print
The optimal value computation for turned-based stochastic games with reachability objectives, also known as simple stochastic games, is one of the few problems in NP ∩ coNP which are not known to be in  ...  However, there are some cases where these games can be easily solved, as for instance when the underlying graph is acyclic.  ...  Definitions and standard results Simple stochastic games are turn-based stochastic games with reachability objectives involving two players named MAX and MIN.  ...

A Simple Algorithm for Solving Qualitative Probabilistic Parity Games [chapter]

Ernst Moritz Hahn, Sven Schewe, Andrea Turrini, Lijun Zhang
2016 Lecture Notes in Computer Science
In this paper, we develop an approach to find strategies that guarantee a property in systems that contain controllable, uncontrollable, and random vertices, resulting in probabilistic games.  ...  We exploit a simple and only mildly adjusted algorithm from the analysis of nonprobabilistic systems, and use it to show that the qualitative analysis of probabilistic games inherits the much celebrated  ...  The algorithm from [9] is based on translating the stochastic parity game into a simple parity game [8] (a parity game without random vertices / a 2-player game) using a so called "gadget" construction  ...

Complexity of Model Checking by Iterative Improvement: The Pseudo-Boolean Framework [chapter]

Henrik Björklund, Sven Sandberg, Sergei Vorobyov
2004 Lecture Notes in Computer Science
no loss of generality, since a game with non-binary choices may be reduced to binary.  ...  Grants "Infinite Games: Algorithms and Complexity" and "Interior-Point Methods for Infinite Games". ⋆⋆ Extended abstract in PSI'03, LNCS 2890, c Springer-Verlag 1 The restriction to two successors is  ...  The fifth algorithm, KLRA, was first proposed in [21] for solving simple stochastic games, with a subexponential upper bound on the expected running time.  ...

The dual problems of coordination and anti-coordination on random bipartite graphs [article]

Matthew I. Jones, Scott D. Pauls, Feng Fu
2021 arXiv   pre-print
We show that with only two strategies, anti-coordination games (2-colorings) and coordination games (uniform colorings) are dual problems that are equally difficult to solve.  ...  To prove this, we construct an isomorphism between the Markov chains arising from the corresponding anti-coordination and coordination games under certain specific individual stochastic decision-making  ...  Introduction An n-coloring of a graph is a labeling of the vertices of the graph with n different colors such that for each pair of vertices connected by an edge, the vertices have different labels.  ...

Solving simple stochastic games with few coin toss positions [article]

Rasmus Ibsen-Jensen, Peter Bro Miltersen
2012 arXiv   pre-print
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.  ...  Our algorithm for solving simple stochastic games with few coin toss positions is the algorithm of Figure 1 . [1] for solving deterministic graphical games.  ...

Comparison of Algorithms for Simple Stochastic Games (Full Version) [article]

Jan Kretinsky and Emanuel Ramneantu and Alexander Slivinskiy and Maximilian Weininger
2020 arXiv   pre-print
Simple stochastic games are turn-based 2.5-player zero-sum graph games with a reachability objective.  ...  We implemented all algorithms in PRISM-games 3.0, thereby providing the first implementation of quadratic programming for solving simple stochastic games.  ...  few random vertices [22, 28] , i.e. games where most actions yield a successor deterministically.  ...

Serious games to assess mild cognitive impairment: 'The game is the assessment'

Kyle Leduc-McNiven, Benjamin White, Huizhe Zheng, Robert D McLeod, Marcia R Friesen
2018 Research and Review Insights
Both games provide players with immediate feedback but engage different algorithms and heuristics to solve the respective problems at hand.  ...  Methods: Two games have been developed with this objective.  ...  It is analogous to a problem solving strategy that a person may employ in solving a combination lock. Figure 15 illustrates 1000 trajectories from simple Tabu search.  ...

Stochastic Mean Payoff Games: Smoothed Analysis and Approximation Schemes [chapter]

Endre Boros, Khaled Elbassioni, Mahmoud Fouz, Vladimir Gurvich, Kazuhisa Makino, Bodo Manthey
2011 Lecture Notes in Computer Science
We consider two-player zero-sum stochastic mean payoff games with perfect information modeled by a digraph with black, white, and random vertices.  ...  These BWR-games games are polynomially equivalent with the classical Gillette games, which include many well-known subclasses, such as cyclic games, simple stochastic games, stochastic parity games, and  ...  Gimbert and Horn [17] have shown that simple stochastic games on k random vertices can be solved in time O(k!(|V ||E| + L)), where L is the maximum bit length of a transition probability.  ...

Algorithms for Parity Games [chapter]

Hartmut Klauck
2002 Lecture Notes in Computer Science
Recall that in a parity game we are given a (finite) graph with vertices labeled by natural numbers.  ...  with finite arenas and other two-player games.  ...  Using an algorithm by Ludwig [117] , which solves simple stochastic games with fan-out 2, and the reductions we get the following corollary.  ...

Approximation Schemes for Stochastic Mean Payoff Games with Perfect Information and Few Random Positions

Endre Boros, Khaled Elbassioni, Mahmoud Fouz, Vladimir Gurvich, Kazuhisa Makino, Bodo Manthey
2017 Algorithmica
We consider two-player zero-sum stochastic mean payoff games with perfect information.  ...  We show that any such game, with a constant number of random positions and polynomially bounded positive transition probabilities, admits a polynomial time approximation scheme, both in the relative and  ...  Since there is a simple path in G from u to C with at most λ stochastic vertices, this probability is at least p λ min . Furthermore, there exists a vertex v ∈ C with π v ≥ π C /|C| ≥ p λ min /n.  ...

Solving Stochastic Games

Liam Mac Dermed, Charles L. Isbell Jr.
2009 Neural Information Processing Systems
We provide the first approximation algorithm which solves stochastic games with cheap-talk to within absolute error of the optimal game-theoretic solution, in time polynomial in 1/ .  ...  Solving multi-agent reinforcement learning problems has proven difficult because of the lack of tractable algorithms.  ...  In this game the only equilibria are stochastic (E.G. the randomized policy of each player passing and exiting with probability 1 2 ).  ...

Optimistic and Topological Value Iteration for Simple Stochastic Games [article]

Muqsit Azeem, Alexandros Evangelidis, Jan Křetínský, Alexander Slivinskiy, Maximilian Weininger
2022 arXiv   pre-print
While value iteration (VI) is a standard solution approach to simple stochastic games (SSGs), it suffered from the lack of a stopping criterion.  ...  In order to compare the new algorithms with the state of the art, we use not only the standard benchmarks, but we also design a random generator of SSGs, which can be biased towards various types of models  ...  Random Generation of Simple Stochastic Games In order to properly evaluate and compare our algorithms, we need a diverse set of benchmarks.  ...
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