1,808 research outputs found

    Two-Player Perfect-Information Shift-Invariant Submixing Stochastic Games Are Half-Positional

    Full text link
    We consider zero-sum stochastic games with perfect information and finitely many states and actions. The payoff is computed by a payoff function which associates to each infinite sequence of states and actions a real number. We prove that if the the payoff function is both shift-invariant and submixing, then the game is half-positional, i.e. the first player has an optimal strategy which is both deterministic and stationary. This result relies on the existence of ϵ\epsilon-subgame-perfect equilibria in shift-invariant games, a second contribution of the paper

    Decision Problems for Nash Equilibria in Stochastic Games

    Get PDF
    We analyse the computational complexity of finding Nash equilibria in stochastic multiplayer games with ω\omega-regular objectives. While the existence of an equilibrium whose payoff falls into a certain interval may be undecidable, we single out several decidable restrictions of the problem. First, restricting the search space to stationary, or pure stationary, equilibria results in problems that are typically contained in PSPACE and NP, respectively. Second, we show that the existence of an equilibrium with a binary payoff (i.e. an equilibrium where each player either wins or loses with probability 1) is decidable. We also establish that the existence of a Nash equilibrium with a certain binary payoff entails the existence of an equilibrium with the same payoff in pure, finite-state strategies.Comment: 22 pages, revised versio

    IST Austria Technical Report

    Get PDF
    The theory of graph games is the foundation for modeling and synthesizing reactive processes. In the synthesis of stochastic processes, we use 2-1/2-player games where some transitions of the game graph are controlled by two adversarial players, the System and the Environment, and the other transitions are determined probabilistically. We consider 2-1/2-player games where the objective of the System is the conjunction of a qualitative objective (specified as a parity condition) and a quantitative objective (specified as a mean-payoff condition). We establish that the problem of deciding whether the System can ensure that the probability to satisfy the mean-payoff parity objective is at least a given threshold is in NP ∩ coNP, matching the best known bound in the special case of 2-player games (where all transitions are deterministic) with only parity objectives, or with only mean-payoff objectives. We present an algorithm running in time O(d · n^{2d}·MeanGame) to compute the set of almost-sure winning states from which the objective can be ensured with probability 1, where n is the number of states of the game, d the number of priorities of the parity objective, and MeanGame is the complexity to compute the set of almost-sure winning states in 2-1/2-player mean-payoff games. Our results are useful in the synthesis of stochastic reactive systems with both functional requirement (given as a qualitative objective) and performance requirement (given as a quantitative objective)

    Optimal Strategies in Infinite-state Stochastic Reachability Games

    Full text link
    We consider perfect-information reachability stochastic games for 2 players on infinite graphs. We identify a subclass of such games, and prove two interesting properties of it: first, Player Max always has optimal strategies in games from this subclass, and second, these games are strongly determined. The subclass is defined by the property that the set of all values can only have one accumulation point -- 0. Our results nicely mirror recent results for finitely-branching games, where, on the contrary, Player Min always has optimal strategies. However, our proof methods are substantially different, because the roles of the players are not symmetric. We also do not restrict the branching of the games. Finally, we apply our results in the context of recently studied One-Counter stochastic games

    Game theory

    Get PDF
    game theory

    Three Puzzles on Mathematics, Computation, and Games

    Full text link
    In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
    corecore