## The Complexity of Stochastic Games (1992)

Venue: | Information and Computation |

Citations: | 151 - 2 self |

### BibTeX

@ARTICLE{Condon92thecomplexity,

author = {Anne Condon},

title = {The Complexity of Stochastic Games},

journal = {Information and Computation},

year = {1992},

volume = {96},

pages = {203--224}

}

### Years of Citing Articles

### OpenURL

### Abstract

We consider the complexity of stochastic games -- simple games of chance played by two players. We show that the problem of deciding which player has the greatest chance of winning the game is in the class NP " co-NP. 1 Introduction We consider the complexity of a natural combinatorial problem, that of deciding the outcome of a special kind of stochastic game. A simple stochastic game (SSG) is a directed graph with three types of vertices, called max, min and average vertices. There is a special start vertex and two special sink vertices, called the 0-sink and the 1-sink. For simplicity, we assume that all vertices have exactly two (not necessarily distinct) neighbors, except for the sink vertices, which have no neighbors. The graph models a game between two players, 0 and 1. In the game, a token is initially placed on the start vertex, and at each step of the game the token is moved from a vertex to one of its neighbors, according to the following rules: At a min vertex, player 0 cho...

### Citations

516 |
Dynamic Programming and and Markov Processes
- Howard
- 1960
(Show Context)
Citation Context ...G, the value of i with respect to strategies oe; �� (oe) equals the minimum of the values of its children. We call �� (oe) an optimal strategy of player 0 with respect to strategy oe. Lemma 4 =-=(Howard [6]) Le-=-t G be a simple stochastic game with n vertices that halts with probability 1 and let oe be any strategy of player 1. Then there is some strategy �� (oe) such that for each vertex i 2 Vmin with ne... |

375 |
A polynomial algorithm in linear programming
- Khachiyan
- 1979
(Show Context)
Citation Context ...+ v(k)). A similar argument to the first case shows that this leads to a contradiction. Hence the optimal solution to the linear programming problem must be the unique solution of the SSG G. Khachian =-=[7]-=- has shown that the linear programming problem is computable in time polynomial in the length of the input, which is polynomial in n in this case, completing the proof for the max and average case. Th... |

237 | Stochastic Games - Shapley - 1953 |

222 |
Computational complexity of probabilistic Turing machines
- Gill
- 1977
(Show Context)
Citation Context .... Lemma 2 The value of a simple stochastic game G with n vertices is of the form p=q where p and q are integers, 0sp; qs4 n\Gamma1 . Proof: The proof of this lemma is very similar to a proof by Gill (=-=[5], Lemma -=-6.6). Let oe and �� be arbitrary strategies of players 0 and 1 respectively. Then the value of 0-valued vertices and the 1-sink vertex with respect to strategies oe and �� can be written as 0/... |

100 |
Games against nature
- Papadimitriou
- 1985
(Show Context)
Citation Context ..., randomness has been added to nondeterministic and alternating machines in novel ways, giving rise to various "game-like" complexity classes. One example is the games against nature of Papa=-=dimitriou [8]-=-, a model that combines the features of nondeterministic and probabilistic Turing machines (Gill [5]). We briefly describe games against nature and some problems that they naturally model, and then sh... |

35 |
Finite State Markov Decision Processes
- Derman
- 1970
(Show Context)
Citation Context ...stricted to SSG's with just (1) average and max vertices, or (2) average and min vertices, or (3) max and min vertices can be solved in polynomial time. Proof: The proof for case (1) is due to Derman =-=[3]-=-. Let G = (V; E) be an SSG with n vertices. From Lemma 8 we assume that G is a stopping game, so that G has a unique solution. We claim that the solution of G is the optimal solution to the following ... |

28 |
Computational models of games
- Condon
- 1989
(Show Context)
Citation Context ...M if there is some 9-subtree T 0 such that for all 8-subtrees T 00 of T 0 , the probability of reaching an accepting leaf of T 00 is ? 1=2, when a path is followed randomly from the root of T 00 . In =-=[2]-=-, we denote the class of languages accepted by log space bounded randomized alternating Turing machines by AUC-SPACE(log(n)). The SSG value problem is log space complete for the class AUC-SPACE(log(n)... |

10 |
Ordered Field Property for Stochastic Games When the Player Who Controls Transitions Changes from State to State
- Filar
- 1981
(Show Context)
Citation Context ...gated since then (see Peters and Vrieze [9] for a survey). The class of stochastic games studied in this paper is somewhat similar to the stochastic games with switching transitions, studied by Filar =-=[4]-=-. We were motivated to study these simple stochastic games while considering the power of the following complexity model: space bounded alternating Turing machines, generalized to allow random as well... |

5 |
Surveys in game theory and related topics, CWI Tract 39. Centrum voor Wiskunde en Informatica
- Peters, Vrieze
- 1987
(Show Context)
Citation Context ...combinatorial problem with this property. The study of stochastic games was initiated by Shapley [10] in 1953 and many variations of the model have been investigated since then (see Peters and Vrieze =-=[9]-=- for a survey). The class of stochastic games studied in this paper is somewhat similar to the stochastic games with switching transitions, studied by Filar [4]. We were motivated to study these simpl... |