## Pure Nash Equilibria: Hard and Easy Games

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Citations: | 66 - 3 self |

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@MISC{Gottlob_purenash,

author = {Georg Gottlob and Gianluigi Greco and Francesco Scarcello},

title = {Pure Nash Equilibria: Hard and Easy Games},

year = {}

}

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### Abstract

In this paper we investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has a strong Nash equilibrium is St-complete. We then study practically relevant restrictions that lower the complexity. In particular, we are interested in quantitative and qualitative restrictions of the way each player's move depends on moves of other players. We say that a game has small neighborhood if the " utility function for each player depends only on (the actions of) a logarithmically small number of other players, The dependency structure of a game G can he expressed by a graph G(G) or by a hypergraph I-I(G). Among other results, we show that if jC has small neighborhood and if I-I(G) has botmdecl hypertree width (or if G(G) has bounded treewidth), then finding pure Nash and Pareto equilibria is feasible in polynomial time. If the game is graphical, then these problems are LOGCFL-complete and thus in the class _NC ~ of highly parallelizable problems. 1 Introduction and Overview of Results The theory of strategic games and Nash equilibria has important applications in economics and decision making [31, 2]. Determining whether Nash equilibria exist, and effectively computing

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Citation Context ...ble function Neigh with the above property. 1 The player neighborhood relationship, typically represented as a graph (or a hypergraph), is the central notion in graphical games (Koller & Milch, 2001; =-=Kearns, Littman, & Singh, 2001-=-b), as we will see in more detail in the next section. A first idea towards the identification of tractable classes of games is to restrict the cardinality of Neigh(p) for all players p. For instance,... |

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156 | Multi-Agent Influence Diagrams for Representing and Solving Games
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Citation Context ...olynomial-time computable function Neigh with the above property. 1 The player neighborhood relationship, typically represented as a graph (or a hypergraph), is the central notion in graphical games (=-=Koller & Milch, 2001-=-; Kearns, Littman, & Singh, 2001b), as we will see in more detail in the next section. A first idea towards the identification of tractable classes of games is to restrict the cardinality of Neigh(p) ... |

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Citation Context ...s one on the given input. Figure 2 summarizes our results on the existence of pure Nash equilibria. While various authors have dealt with the complexity of Nash equilibria (e.g. Gilboa & Zemel, 1989; =-=Papadimitriou, 1994-=-b; Koller & Megiddo, 1992, 1996; Conitzer & Sandholm, 2003b), most investigations were dedicated to mixed equilibria and — to the best of our knowledge — all complexity results in the present paper ar... |

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Citation Context ...phs have a very low treewidth (Downey & Fellows, 1995). Hypertree width in turn was fruitfully applied in the context of database queries (Gottlob et al., 2002b) and constraint satisfaction problems (=-=Gottlob, Leone, & Scarcello, 2000-=-). Formal definitions are given in Section 5. Note that both computing the treewidth of a graph and the hypertree width of a hypergraph are NP-hard problems. However, for each (fixed) constant k, it c... |

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Citation Context ...In that context, the existence of a Nash equilibrium is guaranteed by Nash’s famous theorem (Nash, 1951), but it is currently open whether such an equilibrium can be computed in polynomial time (cf., =-=Papadimitriou, 2001-=-). First results on the computational complexity for a twoperson game have been presented by Gilboa and Zemel (1989), while extensions to more general types of games have been provided by Megiddo and ... |

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Citation Context ...ting the treewidth of a graph and the hypertree width of a hypergraph are NP-hard problems. However, for each (fixed) constant k, it can be checked in polynomial time whether a graph has treewidth k (=-=Bodlaender, 1997-=-) and whether a hypergraph has hypertree width k (Gottlob et al., 2002b). We have, for each constant k, the following restricted classes of games: Games of treewidth bounded by k: The games G such tha... |

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Citation Context ...sults on the existence of pure Nash equilibria. While various authors have dealt with the complexity of Nash equilibria (e.g. Gilboa & Zemel, 1989; Papadimitriou, 1994b; Koller & Megiddo, 1992, 1996; =-=Conitzer & Sandholm, 2003-=-b), most investigations were dedicated to mixed equilibria and — to the best of our knowledge — all complexity results in the present paper are novel. We are not aware of any other work considering th... |

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Citation Context ... if the circuit outputs one on the given input. Figure 2 summarizes our results on the existence of pure Nash equilibria. While various authors have dealt with the complexity of Nash equilibria (e.g. =-=Gilboa & Zemel, 1989-=-; Papadimitriou, 1994b; Koller & Megiddo, 1992, 1996; Conitzer & Sandholm, 2003b), most investigations were dedicated to mixed equilibria and — to the best of our knowledge — all complexity results in... |

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Citation Context ...1985). Determining whether Nash equilibria exist, and effectively computing them, are relevant problems that have attracted much research in computer science (e.g. Deng, Papadimitriou, & Safra, 2002; =-=McKelvey & McLennan, 1996-=-; Koller, Megiddo, & von Stengel, 1996). Most work has been dedicated to complexity issues related to mixed equilibria of games with mixed strategies, where the player’s choices are not deterministic ... |

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Citation Context ... hypergraph, respectively. In particular, each acyclic graph (hypergraph) has treewidth (hypertree width) ≤ 1. It was argued that an impressive number of “real-life” graphs have a very low treewidth (=-=Downey & Fellows, 1995-=-). Hypertree width in turn was fruitfully applied in the context of database queries (Gottlob et al., 2002b) and constraint satisfaction problems (Gottlob, Leone, & Scarcello, 2000). Formal definition... |

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Citation Context ...idth (or bounded treewidth) is LOGCFL-complete and thus in the parallel complexity class NC2 (Theorem 6.1). Membership in LOGCFL follows from the membership of bounded hypertree-width CSPs in LOGCFL (=-=Gottlob, Leone, & Scarcello, 2001-=-). Hardness for LOGCFL is shown by transforming (logspace uniform families of) semi-unbounded circuits of logarithmic depth together with their inputs into strategic games, such that the game admits a... |

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Citation Context ...ut. Figure 2 summarizes our results on the existence of pure Nash equilibria. While various authors have dealt with the complexity of Nash equilibria (e.g. Gilboa & Zemel, 1989; Papadimitriou, 1994b; =-=Koller & Megiddo, 1992-=-, 1996; Conitzer & Sandholm, 2003b), most investigations were dedicated to mixed equilibria and — to the best of our knowledge — all complexity results in the present paper are novel. We are not aware... |

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Citation Context ...sults on the existence of pure Nash equilibria. While various authors have dealt with the complexity of Nash equilibria (e.g. Gilboa & Zemel, 1989; Papadimitriou, 1994b; Koller & Megiddo, 1992, 1996; =-=Conitzer & Sandholm, 2003-=-b), most investigations were dedicated to mixed equilibria and — to the best of our knowledge — all complexity results in the present paper are novel. We are not aware of any other work considering th... |

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Citation Context ...ove the tractability result, we establish a relationship between strategic games and the well-known finite domain constraint satisfaction problem (CSP), much studied in the AI and OR literature (e.g. =-=Vardi, 2000-=-; Gottlob et al., 2000). Let us point out that also Vickrey and Koller (2002) recently exploited a mapping to CSP for the different problem of finding approximate mixed equilibria in graphical games. ... |

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