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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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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.
Games through Nested Fixpoints
, 2009
"... In this paper we consider twoplayer zerosum payoff games on finite graphs, both in the deterministic as well as in the stochastic setting. In the deterministic setting, we consider totalpayoff games which have been introduced as a refinement of meanpayoff games [18, 10]. In the stochastic setti ..."
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In this paper we consider twoplayer zerosum payoff games on finite graphs, both in the deterministic as well as in the stochastic setting. In the deterministic setting, we consider totalpayoff games which have been introduced as a refinement of meanpayoff games [18, 10]. In the stochastic setting, our class is a turnbased variant of liminfpayoff games [15, 16, 4]. In both settings, we provide a nontrivial characterization of the values through nested fixpoint equations. The characterization of the values of liminfpayoff games moreover shows that solving liminfpayoff games is polynomialtime reducible to solving stochastic parity games. We construct practical algorithms for solving the occurring nested fixpoint equations based on strategy iteration. As a corollary we obtain that solving deterministic totalpayoff games and solving stochastic liminfpayoff games is in UP ∩ co−UP.