## Pure Nash Equilibria: Hard and Easy Games

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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 ...the class NC2 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 (=-=Nash, 1951-=-; Aumann, 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, & Saf... |

622 | C.: Worst-case equilibria
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Citation Context ...xts. However, 360sPure Nash Equilibria: Hard and Easy Games there are important generalizations that appear to be much more realistic for practical applications. These concepts are bounded treewidth (=-=Robertson & Seymour, 1986-=-) and bounded hypertree width (Gottlob, Leone, & Scarcello, 2002b) (see also Section 5), which are suitable measures of the degree of cyclicity of a graph and of a hypergraph, respectively. In particu... |

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Citation Context ... each player p, the evaluation of her utility function for that particular combination. This representation is said to be in standard normal form (SNF) (see, for instance, Osborne & Rubinstein, 1994; =-=Owen, 1982-=-). Note that, if there are many players, this representation may be very space consuming, particularly if some players are not interested in all other players, but only in some subset of them. Moreove... |

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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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Citation Context ...f games defined Rosenthal (1973), called congestion games, by showing that a pure Nash equilibrium can be computed in polynomial time in the symmetric network case, while the problem is PLS-complete (=-=Johnson, Papadimitriou, & Yannakakis, 1998-=-) in general. Our goal is to study fundamental questions such as the existence of pure Nash, Pareto, and strong Nash equilibria, the computation of such equilibria, and to find arguably realistic rest... |

164 |
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Citation Context ...h H(G) is acyclic. Note that there are several definitions of hypergraph acyclicity (Fagin, 1983). Here we refer to the broadest (i.e., the most general) one, also known as α-acyclicity (Fagin, 1983; =-=Beeri, Fagin, Maier, & Yannakakis, 1983-=-) (see Section 2). Each acyclic-graph game is also an acyclic-hypergraph game, but not vice-versa. As an extreme example, let G be a game with player set P in which the utility of each action for each... |

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... |

144 | A comparison of structural CSP decomposition methods
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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 ... |

133 | Tree-width: Algorithmic Techniques and Results
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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... |

131 | Complexity results about nash equilibria
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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... |

129 |
Nash and Correlated Equilibria: Some Complexity Considerations
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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 ...mes there are important generalizations that appear to be much more realistic for practical applications. These concepts are bounded treewidth (Robertson & Seymour, 1986) and bounded hypertree width (=-=Gottlob, Leone, & Scarcello, 2002-=-b) (see also Section 5), which are suitable measures of the degree of cyclicity of a graph and of a hypergraph, respectively. In particular, each acyclic graph (hypergraph) has treewidth (hypertree wi... |

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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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Alternation
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Citation Context ...n. Specifically, we next show an implementation of this task by an alternating Turing machine M with a logarithmic-space working tape. Therefore, the problem is in ALOGSPACE, which is equal to PTIME (=-=Chandra, Kozen, & Stockmeyer, 1981-=-). 386sPure Nash Equilibria: Hard and Easy Games The machine M for deciding whether x is not a strong Nash equilibrium works as follows: • guess a player q; • guess a strategy stq for q that is differ... |

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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 ... where no change of strategies of whatever coalition (i.e., group of players) can simultaneously increase the utility for all players in the coalition. A pure Nash equilibrium is Pareto optimal (e.g. =-=Maskin, 1985-=-) if the game admits no other pure Nash equilibrium for which each player has a strictly higher utility. A Pareto-optimal Nash equilibrium is also called a Pareto Nash Equilibrium. 358sPure Nash Equil... |

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Citation Context ... Nash equilibria in some particular classes of games, and the various contributions on different kinds of concise game representations (e.g. Koller & Milch, 2001; Vickrey, 2002; Kearns et al., 2001b; =-=Leyton-Brown & Tennenholtz, 2003-=-; Gal & Pfeffer, 2004; Kearns & Mansour, 2002). We recall that a preliminary version of the present work has been presented at the 9th ACM Conference on Theoretical Aspects of Rationality and Knowledg... |

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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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23 | Nash propagation for loopy graphical games - Ortiz, Kearns - 2002 |

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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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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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