Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSolving Simple Stochastic Games with Few Random Vertices
165
0
0.0
(
0
)
Added by
Hugo Gimbert
Publication date
2009
fields
Informatics Engineering
and research's language is
English
Authors
Hugo Gimbert
Ask ChatGPT about the research
No Arabic abstract
Simple stochastic games are two-player zero-sum stochastic games with turn-based moves, perfect information, and reachability winning conditions. We present two new algorithms computing the values of simple stochastic games. Both of them rely on the existence of optimal permutation strategies, a class of positional strategies derived from permutations of the random vertices. The permutation-enumeration algorithm performs an exhaustive search among these strategies, while the permutation-improvement algorithm is based on successive improvements, `a la Hoffman-Karp. Our algorithms improve previously known algorithms in several aspects. 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 vertices. Furthermore, our algorithms do not require the input game to be transformed into a stopping game. Finally, the permutation-enumeration algorithm does not use linear programming, while the permutation-improvement algorithm may run in polynomial time.
rate research
Read More
Simple stochastic games are turn-based 2.5-player zero-sum graph games with a reachability objective. The problem is to compute the winning probability as well as the optimal strategies of both players. In this paper, we compare the three known classes of algorithms -- value iteration, strategy iteration and quadratic programming -- both theoretically and practically. Further, we suggest several improvements for all algorithms, including the first approach based on quadratic programming that avoids transforming the stochastic game to a stopping one. Our extensive experiments show that these improvements can lead to significant speed-ups. We implemented all algorithms in PRISM-games 3.0, thereby providing the first implementation of quadratic programming for solving simple stochastic games.
Simple stochastic games are turn-based 2.5-player zero-sum graph games with a reachability objective. The problem is to compute the winning probability as well as the optimal strategies of both players. In this paper, we compare the three known classes of algorithms -- value iteration, strategy iteration and quadratic programming -- both theoretically and practically. Further, we suggest several improvements for all algorithms, including the first approach based on quadratic programming that avoids transforming the stochastic game to a stopping one. Our extensive experiments show that these improvements can lead to significant speed-ups. We implemented all algorithms in PRISM-games 3.0, thereby providing the first implementation of quadratic programming for solving simple stochastic games.
We study stochastic games with energy-parity objectives, which combine quantitative rewards with a qualitative $omega$-regular condition: The maximizer aims to avoid running out of energy while simultaneously satisfying a parity condition. We show that the corresponding almost-sure problem, i.e., checking whether there exists a maximizer strategy that achieves the energy-parity objective with probability $1$ when starting at a given energy level $k$, is decidable and in $NP cap coNP$. The same holds for checking if such a $k$ exists and if a given $k$ is minimal.
We study turn-based stochastic zero-sum games with lexicographic preferences over reachability and safety objectives. Stochastic games are standard models in control, verification, and synthesis of stochastic reactive systems that exhibit both randomness as well as angelic and demonic non-determinism. Lexicographic order allows to consider multiple objectives with a strict preference order over the satisfaction of the objectives. To the best of our knowledge, stochastic games with lexicographic objectives have not been studied before. We establish determinacy of such games and present strategy and computational complexity results. For strategy complexity, we show that lexicographically optimal strategies exist that are deterministic and memory is only required to remember the already satisfied and violated objectives. For a constant number of objectives, we show that the relevant decision problem is in NP $cap$ coNP, matching the current known bound for single objectives; and in general the decision problem is PSPACE-hard and can be solved in NEXPTIME $cap$ coNEXPTIME. We present an algorithm that computes the lexicographically optimal strategies via a reduction to computation of optimal strategies in a sequence of single-objectives games. We have implemented our algorithm and report experimental results on various case studies.
This study investigates simple games. A fundamental research question in this field is to determine necessary and sufficient conditions for a simple game to be a weighted majority game. Taylor and Zwicker (1992) showed that a simple game is non-weighted if and only if there exists a trading transform of finite size. They also provided an upper bound on the size of such a trading transform, if it exists. Gvozdeva and Slinko (2011) improved that upper bound; their proof employed a property of linear inequalities demonstrated by Muroga (1971).In this study, we provide a new proof of the existence of a trading transform when a given simple game is non-weighted. Our proof employs Farkas lemma (1894), and yields an improved upper bound on the size of a trading transform. We also discuss an integer-weight representation of a weighted simple game, improving the bounds obtained by Muroga (1971). We show that our bound on the quota is tight when the number of players is less than or equal to five, based on the computational results obtained by Kurz (2012). Furthermore, we discuss the problem of finding an integer-weight representation under the assumption that we have minimal winning coalitions and maximal losing coalitions.In particular, we show a performance of a rounding method. Lastly, we address roughly weighted simple games. Gvozdeva and Slinko (2011) showed that a given simple game is not roughly weighted if and only if there exists a potent certificate of non-weightedness. We give an upper bound on the length of a potent certificate of non-weightedness. We also discuss an integer-weight representation of a roughly weighted simple game.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
