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The Complexity of Solving Stochastic Games on Graphs
"... We consider some wellknown families of twoplayer zerosum perfectinformation stochastic games played on finite directed graphs. Generalizing and unifying results of Liggett and Lippman, Zwick and Paterson, and Chatterjee and Henzinger, we show that the following tasks are polynomialtime (Turing) ..."
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Cited by 18 (3 self)
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We consider some wellknown families of twoplayer zerosum perfectinformation stochastic games played on finite directed graphs. Generalizing and unifying results of Liggett and Lippman, Zwick and Paterson, and Chatterjee and Henzinger, we show that the following tasks are polynomialtime (Turing) equivalent. – Solving stochastic parity games, – Solving simple stochastic games, – Solving stochastic terminalpayoff games with payoffs and probabilities given in unary, – Solving stochastic terminalpayoff games with payoffs and probabilities given in binary, – Solving stochastic meanpayoff games with rewards and probabilities given in unary, – Solving stochastic meanpayoff games with rewards and probabilities given in binary, – Solving stochastic discountedpayoff games with discount factor, rewards and probabilities given in binary. It is unknown whether these tasks can be performed in polynomial time. In the above list, “solving ” may mean either quantitatively solving a game (computing the values of its positions) or strategically solving a game (computing an optimal strategy for each player). In particular, these two tasks are polynomialtime equivalent for all the games listed above. We also consider a more refined notion of equivalence between quantitatively and strategically solving a game. We exhibit a linear time algorithm that given a simple stochastic game or a terminalpayoff game and the values of all positions of that game, computes a pair of optimal strategies. Consequently, for any restriction one may put on the simple stochastic game model, quantitatively solving is polynomialtime equivalent to strategically solving the resulting class of 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 10 (2 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.
Discounted stochastic games poorly approximate undiscounted ones
, 2011
"... The purpose of this note is to summarize recent results [6, 4, 5] on stochastic games by the author and his collaborators. These results have appeared or will appear in proceedings of computer science conferences. The intended reader of this note has an interest in finite stochastic games, but no pa ..."
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Cited by 2 (0 self)
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The purpose of this note is to summarize recent results [6, 4, 5] on stochastic games by the author and his collaborators. These results have appeared or will appear in proceedings of computer science conferences. The intended reader of this note has an interest in finite stochastic games, but no particular interest in computation, that is, the computational aspects of the results obtained have been weeded out. We consider twoplayer zerosum finite (but infinite duration) stochastic games G with N positions and at most m actions available to each of the two players in each position. The reward to Player I when Player I plays i and Player II plays j in position k is denoted ak ij. Transition probabilites are denoted pkl ij. We assume stopping probabilitites are 0, i.e., for all k, i, j we have ∑ l pkl ij = 1. To be able to state our results as simply as possible, we shall also assume throughout that for all k, l, i, j, we have ak ij, pkl ij ∈ {0, 1}. In particular, we assume deterministic dynamics of nature and nonnegative payoffs. By G we denote the game with limiting average (undiscounted) payoffs, i.e, payoff lim inft→∞ ( ∑ t−1 i=0 ri)/t to Player I, where ri is the reward collected by Player I at stage i. By Gλ we denote the game with payoffs discounted with a discount factor of 1 − λ, i.e., with payoff λ ∑∞
Monomial strategies for concurrent reachability games and other stochastic games?
"... Abstract. We consider twoplayer zerosum finite (but infinitehorizon) stochastic games with limiting average payoffs. We define a family of stationary strategies for Player I parameterized by ε> 0 to be monomial, if for each state k and each action j of Player I in state k except possibly one ..."
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Abstract. We consider twoplayer zerosum finite (but infinitehorizon) stochastic games with limiting average payoffs. We define a family of stationary strategies for Player I parameterized by ε> 0 to be monomial, if for each state k and each action j of Player I in state k except possibly one action, we have that the probability of playing j in k is given by an expression of the form cεd for some nonnegative real number c and some nonnegative integer d. We show that for all games, there is a monomial family of stationary strategies that are εoptimal among stationary strategies. A corollary is that all concurrent reachability games have a monomial family of εoptimal strategies. This generalizes a classical result of de Alfaro, Henzinger and Kupferman who showed that this is the case for concurrent reachability games where all states have value 0 or 1. 1
Author manuscript, published in "Discrete Applied Mathematics (2011)" DOI: 10.1016/j.dam.2011.07.021
, 2011
"... A good edgelabelling of a graph G is a labelling of its edges such that, for any ordered pair of vertices (x,y), there do not exist two paths from x to y with increasing labels. This notion was introduced in [2] to solve wavelength assignment problems for specific categories of graphs. In this pape ..."
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A good edgelabelling of a graph G is a labelling of its edges such that, for any ordered pair of vertices (x,y), there do not exist two paths from x to y with increasing labels. This notion was introduced in [2] to solve wavelength assignment problems for specific categories of graphs. In this paper, we aim at characterizing the class of graphs that admit a good edgelabelling. First, we exhibit infinite families of graphs for which no such edgelabelling can be found. We then show that deciding if a graph G admits a good edgelabelling is NPcomplete, evenifGisbipartite. Finally, wegivelargeclassesofgraphsadmitting agoodedgelabelling: C3freeouterplanargraphs,planargraphsofgirthatleast 6, {C3,K2,3}free subcubic graphs and {C3,K2,3}free ABCgraphs. Keywords: Graph Theory, NPcompleteness, Edgelabelling, Increasing paths. 1.
General Terms
"... There is great interest in understanding media bias and political information seeking preferences. As many media outlets create online personas, we seek to automatically estimate the political preferences of their audience, rather than of the outlet itself. In this paper, we present a novel method f ..."
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There is great interest in understanding media bias and political information seeking preferences. As many media outlets create online personas, we seek to automatically estimate the political preferences of their audience, rather than of the outlet itself. In this paper, we present a novel method for computing preference among an organization’s Twitter followers. We present an application of this technique to estimate political preference of the audiences of U.S. media outlets. We also discuss how these results may be used and extended. Author Keywords Twitter, politics, liberal, conservative, news, journalism
Strategy Improvement for Concurrent Reachability and Safety Games
"... 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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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. First, we present a simple proof of the fact that in concurrent reachability games, for all ε> 0, memoryless εoptimal strategies exist. A memoryless strategy is independent of the history of plays, and an εoptimal strategy achieves the objective with probability within ε of the value of the game. In contrast to previous proofs of this fact, our proof is more elementary and more combinatorial. Second, we present a strategyimprovement (a.k.a. policyiteration) algorithm for concurrent games with reachability objectives. We then present a strategyimprovement algorithm for concurrent games with safety objectives. Our algorithms yield sequences of player1 strategies which ensure probabilities of winning that converge monotonically to the value of the game. Our result is significant because the strategyimprovement algorithm for safety games provides, for the first time, a way to approximate the value of a concurrent safety game from below. 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.