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Stochastic mean payoff games: Smoothed analysis and approximation schemes
 In Proceedings of the 38th International Colloquium on Automata, Languages and Programming (ICALP 2011
"... Abstract. In this paper, we consider twoplayer zerosum stochastic mean payoff games with perfect information modeled by a digraph with black, white, and random vertices. These BWRgames games are polynomially equivalent with the classical Gillette games, which include many wellknown subclasses, ..."
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Abstract. In this paper, we consider twoplayer zerosum stochastic mean payoff games with perfect information modeled by a digraph with black, white, and random vertices. These BWRgames games are polynomially equivalent with the classical Gillette games, which include many wellknown subclasses, such as cyclic games, simple stochastic games, stochastic parity games, and Markov decision processes. They can also be used to model parlor games such as Chess or Backgammon. It is a longstanding open question if a polynomial algorithm exists that solves BWRgames. In fact, a pseudopolynomial algorithm for these games with an arbitrary number of random nodes would already imply their polynomial solvability. Currently, only two classes are known to have such a pseudopolynomial algorithm: BWgames (the case with no random nodes) and ergodic BWRgames (in which the game's value does not depend on the initial position) with constant number of random nodes. In this paper, we show that the existence of a pseudopolynomial algorithm for BWRgames with constant number of random vertices implies smoothed polynomial complexity and the existence of absolute and relative polynomialtime approximation schemes. In particular, we obtain smoothed polynomial complexity and derive absolute and relative approximation schemes for BWgames and ergodic BWRgames (assuming a technical requirement about the probabilities at the random nodes).
Asynchronous OmegaRegular Games with Partial Information
"... We address the strategy problem for ωregular twoplayer games with partial information, played on finite game graphs. We consider two different kinds of observability on a general model, a standard synchronous and an asynchronous one. In the asynchronous setting, moves which have no visible effect ..."
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Cited by 4 (2 self)
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We address the strategy problem for ωregular twoplayer games with partial information, played on finite game graphs. We consider two different kinds of observability on a general model, a standard synchronous and an asynchronous one. In the asynchronous setting, moves which have no visible effect for a player are hidden completely from that player. We generalize the usual powerset construction for eliminating partial information to arbitrary, not necessarily observation based, winning conditions, both in the synchronous and in the asynchronous case, and we show that this generalized construction effectively preserves ωregular winning conditions. From this we infer decidability of the strategy problem for arbitrary ωregular winning conditions, in both cases. We also show that our ωregular framework is sufficient for reasoning about synchronous and asynchronous knowledge by proving that any formula of the epistemic temporal specification formalism ETL can be effectively translated into an S1Sformula defining the same specification.
Parity Games with Partial Information Played on Graphs of Bounded Complexity
"... We address the strategy problem for parity games with partial information and observable colors, played on finite graphs of bounded graph complexity. We consider several measures for the complexity of graphs and analyze in which cases, bounding the measure decreases the complexity of the strategy p ..."
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We address the strategy problem for parity games with partial information and observable colors, played on finite graphs of bounded graph complexity. We consider several measures for the complexity of graphs and analyze in which cases, bounding the measure decreases the complexity of the strategy problem on the corresponding classes of graphs. We prove or disprove that the usual powerset construction for eliminating partial information preserves boundedness of the graph complexity. For the case where the partial information is unbounded we prove that the construction does not preserve boundedness of any measure we consider. We also prove that the strategy problem is Exptimehard on graphs with directed pathwidth at most 2 and Pspacecomplete on acyclic graphs. For games with bounded partial information we obtain that the powerset construction, while neither preserving boundedness of entanglement nor of (undirected) treewidth, does preserve boundedness of directed pathwidth. Furthermore, if treewidth is bounded then DAGwidth of the resulting graph is bounded. Therefore, parity games with bounded partial information, played on graphs with bounded directed pathwidth or treewidth can be solved in polynomial time.
A Pumping Algorithm for Ergodic Mean Payoff Stochastic Games with Perfect Information
"... In this paper, we consider twoperson zerosum stochastic mean payoff games with perfect information, or BWRgames, given by a digraph G = (V = VB ∪VW ∪VR, E), with local rewards r: E → R, and three types of vertices: black VB, white VW, and random VR. The game is played by two players, White and Bl ..."
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In this paper, we consider twoperson zerosum stochastic mean payoff games with perfect information, or BWRgames, given by a digraph G = (V = VB ∪VW ∪VR, E), with local rewards r: E → R, and three types of vertices: black VB, white VW, and random VR. The game is played by two players, White and Black: When the play is at a white (black) vertex v, White (Black) selects an outgoing arc (v, u). When the play is at a random vertex v, a vertex u is picked with the given probability p(v, u). In all cases, Black pays White the value r(v, u). The play continues forever, and White aims to maximize (Black aims to minimize) the limiting mean (that is, average) payoff. It was recently shown in [BEGM09] that BWRgames are polynomially equivalent with the classical Gillette games, which include many wellknown subclasses, such as cyclic games, simple stochastic games (SSGs), stochastic parity games, and Markov decision processes. In this paper, we give a new algorithm for solving BWRgames in the ergodic case, that is when the game’s value does not depend on the initial position. Our algorithm solves a BWRgame by reducing it, using a potential transformation, to a canonical form in which the optimal strategies of both players and the value for every initial position are obvious, since a locally optimal move in it is optimal in the whole game. We show that this algorithm is pseudopolynomial when the number of random nodes is constant. We also provide an almost matching lower bound on its running time and show that this bound holds for a wider class of algorithms. Let us add that the general (nonergodic) case is at least as hard as SSGs, for which no pseudopolynomial algorithm is known.
Approximation Schemes for Stochastic Mean Payoff Games with Perfect Information and a Few Random Positions
, 2014
"... ..."
Asynchronous ωregular games with Partial Information
"... We address the strategy problem for ωregular twoplayer games with partial information, played on finite game graphs. We consider two different kinds of observability on a general model, a standard synchronous and an asynchronous one. In the asynchronous setting, moves which have no visible effec ..."
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We address the strategy problem for ωregular twoplayer games with partial information, played on finite game graphs. We consider two different kinds of observability on a general model, a standard synchronous and an asynchronous one. In the asynchronous setting, moves which have no visible effect for a player are hidden completely from that player. We generalize the usual powerset construction for eliminating partial information to arbitrary, not necessarily observation based, winning conditions, both in the synchronous and in the asynchronous case, and we show that this generalized construction effectively preserves ωregular winning conditions. From this we infer decidability of the strategy problem for arbitrary ωregular winning conditions, in both cases. We also show that our ωregular framework is sufficient for reasoning about synchronous and asynchronous knowledge by proving that any formula of the epistemic temporal specification formalism ETL can be effectively translated into an S1Sformula defining the same specification.
Synthesis of Winning Strategies for Interaction under Partial Information
, 2013
"... Interaction is a fundamental concept in computer science. Besides the interaction between human users and computing systems, many computing systems are inherently interactive themselves. The individual computers in a network, for example, interact with each other via a given communication structure ..."
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Interaction is a fundamental concept in computer science. Besides the interaction between human users and computing systems, many computing systems are inherently interactive themselves. The individual computers in a network, for example, interact with each other via a given communication structure according to certain protocols. In a reactive system, one or more computing devices, called controllers, interact with some kind of environment, trying to guarantee a correct behavior of the system. Logic as one of the foundations of computer science is intimately linked to interaction, demonstrated by various kinds of model checking games. Moreover, semantics of alternating computing devices as well as several graph complexity measures are characterized in terms of games. Many of these interactive scenarios take place under certain forms of uncertainty. An individual computer in a network, for example, does not necessarily know all the parameters of the other members of the network or the past message transmissions in the joint computation. The same holds for the controllers in reactive systems which often do not have full information about all the internal states of the other components or the history of past events in the whole system. Furthermore, model checking games for certain logics as well as several graph searching