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 im…
We give polynomial time algorithms for deciding almost-sure and limit-sure reachability in Branching Concurrent Stochastic Games (BCSGs). These are a class of infinite-state imperfect-information stochastic games that generalize both finite-state concurrent stochastic reachability games, as well as branching simple stochastic reachability games.
We study the complexity of computing stationary Nash equilibrium (NE) in n-player infinite-horizon general-sum stochastic games. We focus on the problem of computing NE in such stochastic games when each player is restricted to choosing a stationary policy and rewards are discounted. First, we prove that computing such NE is in PPAD (in addition to clearly being PPAD-hard). Second, we consider turn-based specializations of such games where at each state there is at most a single player that can take actions and show that these (seemingly-simpler) games remain PPAD-hard. Third, we show that und…
This paper combines ideas from Q-learning and fictitious play to define three reinforcement learning procedures which converge to the set of stationary mixed Nash equilibria in identical interest discounted stochastic games. First, we analyse three continuous-time systems that generalize the best-response dynamics defined by Leslie et al. for zero-sum discounted stochastic games. Under some assumptions depending on the system, the dynamics are shown to converge to the set of stationary equilibria in identical interest discounted stochastic games. Then, we introduce three analog discrete-time p…
Stochastic assignment games for Mobility-as-a-Service markets
University of California, Los Angeles（加州大学洛杉矶分校）；Institute for Transport Studies, University of Leeds（利兹大学交通研究所）；NYU Tandon School of Engineering（纽约大学坦登工程学院）
We consider zero-sum stochastic games with perfect information and finitely many states and actions. The payoff is computed by a function which associates to each infinite sequence of states and actions a real number. We prove that if 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 epsilon-subgame-perfect strategies in shift-invariant games, a second contribution of the paper. The techniques can be used to establish a thir…
Value methods for solving stochastic games with partial observability model the uncertainty about states of the game as a probability distribution over possible states. The dimension of this belief space is the number of states. For many practical problems, for example in security, there are exponentially many possible states which causes an insufficient scalability of algorithms for real-world problems. To this end, we propose an abstraction technique that addresses this issue of the curse of dimensionality by projecting high-dimensional beliefs to characteristic vectors of significantly lowe…
We study a mathematical model for revenue management under competition with multiple sellers. The model combines the stochastic knapsack problem, a classic revenue management model, with a non coorperative game model that characterizes the sellers' rational behavior. We are able to establish a dynamic recursive procedure that incorporate the value function with the utility function of the games. The formalization of the dynamic recursion allows us to establish some fundamental structural properties.
Many important real-world settings contain multiple players interacting over an unknown duration with probabilistic state transitions, and are naturally modeled as stochastic games. Prior research on algorithms for stochastic games has focused on two-player zero-sum games, games with perfect information, and games with imperfect-information that is local and does not extend between game states. We present an algorithm for approximating Nash equilibrium in multiplayer general-sum stochastic games with persistent imperfect information that extends throughout game play. We experiment on a 4-playe…
Zero-sum stochastic games have found important applications in a variety of fields, from machine learning to economics. Work on this model has primarily focused on the computation of Nash equilibrium due to its effectiveness in solving adversarial board and video games. Unfortunately, a Nash equilibrium is not guaranteed to exist in zero-sum stochastic games when the payoffs at each state are not convex-concave in the players' actions. A Stackelberg equilibrium, however, is guaranteed to exist. Consequently, in this paper, we study zero-sum stochastic Stackelberg games. Going beyond known exis…
We prove the existence and computability of optimal strategies in weighted limit games, zero-sum infinite-duration games with a Büchi-style winning condition requiring to produce infinitely many play prefixes that satisfy a given regular specification. Quality of plays is measured in the maximal weight of infixes between successive play prefixes that satisfy the specification.
We consider two-player stochastic games played on a finite graph for infinitely many rounds. Stochastic games generalize both Markov decision processes (MDP) by adding an adversary player, and two-player deterministic games by adding stochasticity. The outcome of the game is a sequence of distributions over the states of the game graph. We consider synchronizing objectives, which require the probability mass to accumulate in a set of target states, either always, once, infinitely often, or always after some point in the outcome sequence; and the winning modes of sure winning (if the accumulate…
Many real-world domains contain multiple agents behaving strategically with probabilistic transitions and uncertain (potentially infinite) duration. Such settings can be modeled as stochastic games. While algorithms have been developed for solving (i.e., computing a game-theoretic solution concept such as Nash equilibrium) two-player zero-sum stochastic games, research on algorithms for non-zero-sum and multiplayer stochastic games is limited. We present a new algorithm for these settings, which constitutes the first parallel algorithm for multiplayer stochastic games. We present experimental …
We study a finite-horizon two-person zero-sum risk-sensitive stochastic game for continuous-time Markov chains and Borel state and action spaces, in which payoff rates, transition rates and terminal reward functions are allowed to be unbounded from below and from above and the policies can be history-dependent. Under suitable conditions, we establish the existence of a solution to the corresponding Shapley equation (SE) by an approximation technique. Then, by the SE and the extension of the Dynkin's formula, we prove the existence of a Nash equilibrium and verify that the value of the stochast…
Game-theoretic techniques and equilibria analysis facilitate the design and verification of competitive systems. While algorithmic complexity of equilibria computation has been extensively studied, practical implementation and application of game-theoretic methods is more recent. Tools such as PRISM-games support automated verification and synthesis of zero-sum and (epsilon-optimal subgame-perfect) social welfare Nash equilibria properties for concurrent stochastic games. However, these methods become inefficient as the number of agents grows and may also generate equilibria that yield signifi…
We show that the problem of deciding whether in a multi-player perfect information recursive game (i.e. a stochastic game with terminal rewards) there exists a stationary Nash equilibrium ensuring each player a certain payoff is Existential Theory of the Reals complete. Our result holds for acyclic games, where a Nash equilibrium may be computed efficiently by backward induction, and even for deterministic acyclic games with non-negative terminal rewards. We further extend our results to the existence of Nash equilibria where a single player is surely winning. Combining our result with known g…
Simple stochastic games are turn-based 2.5-player games with a reachability objective. The basic question asks whether one player can ensure reaching a given target with at least a given probability. A natural extension is games with a conjunction of such conditions as objective. Despite a plethora of recent results on the analysis of systems with multiple objectives, the decidability of this basic problem remains open. In this paper, we present an algorithm approximating the Pareto frontier of the achievable values to a given precision. Moreover, it is an anytime algorithm, meaning it can be …
Mila（米拉）；Université de Montréal（蒙特利尔大学）；Facebook AI Research（Facebook人工智能研究院）
The success of adversarial formulations in machine learning has brought renewed motivation for smooth games. In this work, we focus on the class of stochastic Hamiltonian methods and provide the first convergence guarantees for certain classes of stochastic smooth games. We propose a novel unbiased estimator for the stochastic Hamiltonian gradient descent (SHGD) and highlight its benefits. Using tools from the optimization literature we show that SHGD converges linearly to the neighbourhood of a stationary point. To guarantee convergence to the exact solution, we analyze SHGD with a decreasing…
Monte-Carlo counterfactual regret minimization (MCCFR) is the state-of-the-art algorithm for solving sequential games that are too large for full tree traversals. It works by using gradient estimates that can be computed via sampling. However, stochastic methods for sequential games have not been investigated extensively beyond MCCFR. In this paper we develop a new framework for developing stochastic regret minimization methods. This framework allows us to use any regret-minimization algorithm, coupled with any gradient estimator. The MCCFR algorithm can be analyzed as a special case of our fr…
We analyze the convergence properties of the two-timescale fictitious play combining the classical fictitious play with the Q-learning for two-player zero-sum stochastic games with player-dependent learning rates. We show its almost sure convergence under the standard assumptions in two-timescale stochastic approximation methods when the discount factor is less than the product of the ratios of player-dependent step sizes. To this end, we formulate a novel Lyapunov function formulation and present a one-sided asynchronous convergence result.
We solve the stochastic generalized Nash equilibrium (SGNE) problem in merely monotone games with expected value cost functions. Specifically, we present the first distributed SGNE seeking algorithm for monotone games that requires one proximal computation (e.g., one projection step) and one pseudogradient evaluation per iteration. Our main contribution is to extend the relaxed forward-backward operator splitting by Malitsky (Mathematical Programming, 2019) to the stochastic case and in turn to show almost sure convergence to a SGNE when the expected value of the pseudogradient is approximated…
