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Concurrent Reachability Games
, 2008
"... We consider concurrent twoplayer games with reachability objectives. In such games, at each round, player 1 and player 2 independently and simultaneously choose moves, and the two choices determine the next state of the game. The objective of player 1 is to reach a set of target states; the objecti ..."
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Cited by 68 (22 self)
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We consider concurrent twoplayer games with reachability objectives. In such games, at each round, player 1 and player 2 independently and simultaneously choose moves, and the two choices determine the next state of the game. The objective of player 1 is to reach a set of target states; the objective of player 2 is to prevent this. These are zerosum games, and the reachability objective is one of the most basic objectives: determining the set of states from which player 1 can win the game is a fundamental problem in control theory and system verification. There are three types of winning states, according to the degree of certainty with which player 1 can reach the target. From type1 states, player 1 has a deterministic strategy to always reach the target. From type2 states, player 1 has a randomized strategy to reach the target with probability 1. From type3 states, player 1 has for every real ε> 0 a randomized strategy to reach the target with probability greater than 1 − ε. We show that for finite state spaces, all three sets of winning states can be computed in polynomial time: type1 states in linear time, and type2 and type3 states in quadratic time. The algorithms to compute the three sets of winning states also enable the construction of the winning and spoiling strategies.
Winning concurrent reachability games requires doublyexponential patience
"... We exhibit a deterministic concurrent reachability game PURGATORYn with n nonterminal positions and a binary choice for both players in every position so that any positional strategy for Player 1 achieving the value of the game within given ɛ < 1/2 must use nonzero behavior probabilities that a ..."
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Cited by 13 (6 self)
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We exhibit a deterministic concurrent reachability game PURGATORYn with n nonterminal positions and a binary choice for both players in every position so that any positional strategy for Player 1 achieving the value of the game within given ɛ < 1/2 must use nonzero behavior probabilities that are less than (ɛ2/(1 − ɛ)) 2n−2. Also, even to achieve the value within say 1 − 2−n/2, doubly exponentially small behavior probabilities in the number of positions must be used. This behavior is close to worst case: We show that for any such game and 0 < ɛ < 1/2, there is an ɛoptimal strategy with all nonzero behavior probabilities being at least ɛ2O(n). As a corollary to our results, we conclude that any (deterministic or nondeterministic) algorithm that given a concurrent reachability game explicitly manipulates ɛoptimal strategies for Player 1 represented in several standard ways (e.g., with binary representation of probabilities or as the uniform distribution over a multiset) must use at least exponential space in the worst case.
Termination Criteria for Solving Concurrent Safety and Reachability Games
, 2009
"... We consider concurrent games played on graphs. At every round of a game, each player simultaneously and independently selects a move; the moves jointly determine the transition to a successor state. Two basic objectives are the safety objective to stay forever in a given set of states, and its dual, ..."
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Cited by 8 (1 self)
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We consider concurrent games played on graphs. At every round of a game, each player simultaneously and independently selects a move; the moves jointly determine the transition to a successor state. Two basic objectives are the safety objective to stay forever in a given set of states, and its dual, the reachability objective to reach a given set of states. We present in this paper a strategy improvement algorithm for computing the value of a concurrent safety game, that is, the maximal probability with which player 1 can enforce the safety objective. The algorithm yields a sequence of player1 strategies which ensure probabilities of winning that converge monotonically to the value of the safety game. Our result is significant because the strategy improvement algorithm provides, for the first time, a way to approximate the value of a concurrent safety game from below. Since a value iteration algorithm, or a strategy improvement algorithm for reachability games, can be used to approximate the same value from above, the combination of both algorithms yields a method for computing a converging sequence of upper and lower bounds for the values of concurrent reachability and safety games. Previous methods could approximate the values of these games only from one direction, and as no rates of convergence are known, they did not provide a practical way to solve these games.
Exact Algorithms for Solving Stochastic Games
, 2012
"... Shapley’s discounted stochastic games, Everett’s recursive games and Gillette’s undiscounted stochastic games are classical models of game theory describing twoplayer zerosum games of potentially infinite duration. We describe algorithms for exactly solving these games. When the number of position ..."
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Cited by 8 (1 self)
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Shapley’s discounted stochastic games, Everett’s recursive games and Gillette’s undiscounted stochastic games are classical models of game theory describing twoplayer zerosum games of potentially infinite duration. We describe algorithms for exactly solving these games. When the number of positions of the game is constant, our algorithms run in polynomial time.
Value iteration
 25 Years of Model Checking
, 2008
"... Abstract. We survey value iteration algorithms on graphs. Such algorithms can be used for determining the existence of certain paths (model checking), the existence of certain strategies (game solving), and the probabilities of certain events (performance analysis). We classify the algorithms acco ..."
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Cited by 4 (0 self)
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Abstract. We survey value iteration algorithms on graphs. Such algorithms can be used for determining the existence of certain paths (model checking), the existence of certain strategies (game solving), and the probabilities of certain events (performance analysis). We classify the algorithms according to the value domain (boolean, probabilistic, or quantitative); according to the graph structure (nondeterministic, probabilistic, or multiplayer); according to the desired property of paths (Borel level 1, 2, or 3); and according to the alternation depth and convergence rate of fixpoint computations. 1
Solving simple stochastic games with few coin toss positions
 In ESA
, 2012
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Stochastic parity games on lossy channel systems
 In Proc. of QEST’13, LNCS
, 2013
"... Abstract. We give an algorithm for solving stochastic parity games with almostsure winning conditions on lossy channel systems, for the case where the players are restricted to finitememory strategies. First, we describe a general framework, where we consider the class of 2 12player games with al ..."
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Cited by 1 (0 self)
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Abstract. We give an algorithm for solving stochastic parity games with almostsure winning conditions on lossy channel systems, for the case where the players are restricted to finitememory strategies. First, we describe a general framework, where we consider the class of 2 12player games with almostsure parity winning conditions on possibly infinite game graphs, assuming that the game contains a finite attractor. An attractor is a set of states (not necessarily absorbing) that is almost surely revisited regardless of the players ’ decisions. We present a scheme that characterizes the set of winning states for each player. Then, we instantiate this scheme to obtain an algorithm for stochastic game lossy channel systems. 1