## New Complexity Results about Nash Equilibria (2008)

Citations: | 25 - 13 self |

### BibTeX

@MISC{Conitzer08newcomplexity,

author = {Vincent Conitzer and Tuomas Sandholm},

title = { New Complexity Results about Nash Equilibria},

year = {2008}

}

### OpenURL

### Abstract

### Citations

2289 |
Game Theory
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Citation Context ...m. 15Markov (or stochastic) games constitute an important type of multi-stage games. In such games, there is an underlying set of states, and the game shifts between these states from stage to stage =-=[15, 47, 48]-=-. At every stage, each player’s payoff depends not only on the players’ actions, but also on the state. Furthermore, the probability of transitioning to a given state is determined by the current stat... |

629 |
Equilibrium points in n-person games
- Nash
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Citation Context ...ed strategy σi for player i is a probability distribution over Σi. A special case of a mixed strategy is a pure strategy, where all of the probability mass is on one element of Σi. Definition 3 (Nash =-=[36]-=-) Given a normal-form game, a Nash equilibrium (NE) is vector of mixed strategies, one for each player i, such that no player has an incentive to deviate from her mixed strategy given that the others ... |

502 |
The complexity of computing the permanent
- Valiant
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Citation Context ... instance has. (It is required that solutions can be verified efficiently.) An example problem in #P is counting how many satisfying assignments a CNF formula has. (This problem is in fact #Pcomplete =-=[50]-=-.) Another class is PSPACE, the class of problems that can be solved using only polynomial space. 4 Technically, for the classes we mention here, all we know is that they are no smaller than N P—they ... |

475 |
The complexity of theorem proving procedures
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Citation Context ...at it makes the formula evaluate to true.) However, if we add a fourth clause (x1 ∨ −x2 ∨ x3), then the formula is no longer satisfiable. Satisfiability was the first problem shown to be N P-complete =-=[10]-=-, but many other problems have been shown N P-complete since then (often by reducing satisfiability to them). There are other classes of problems that are even larger 4 than N P, and for which natural... |

472 | Games With Incomplete Information Played by Bayesian
- Harsanyi
- 1967
(Show Context)
Citation Context ...h player i, a utility function ui : Θi × Σ1 × Σ2 × . . . × Σ |A| → R. We emphasize again that we only consider finite games; in particular, we only consider finite type spaces. Definition 7 (Harsanyi =-=[21]-=-) Given a Bayesian game, a Bayes-Nash equilibrium (BNE) is a vector of probability distributions over actions, one distribution (over Σi) for each pair i, θi ∈ Θi, such that no player has an incentive... |

374 |
Polynomial algorithms in linear programming
- Khachiyan
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Citation Context ...nstitute a polynomial-length certificate. This is because given the supports, the remainder of the problem can be solved using linear programming (and linear programs can be solved in polynomial time =-=[23]-=-). 8Corollary 1 Even in symmetric 2-player games, it is N P-complete to determine whether there exists a Pareto-optimal Nash equilibrium. (A distribution over outcomes is Paretooptimal if there is no... |

306 |
The Complexity Of Markov Decision Processes
- Papadimitriou, Tsitsiklis
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Citation Context ...rvable Markov decision process is a Markov decision process in which the current state is not directly observable, but a player may observe noisy signals about the state. Papadimitriou and Tsitsiklis =-=[41]-=- show that computing the optimal policy (strategy) for a POMDP is PSPACE-hard even with a finite horizon. (In fact, they show this for a special kind of POMDP in which the states are partitioned, and ... |

300 |
Word problems requiring exponential time
- Stockmeyer, Meyer
- 1973
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Citation Context ...iven state is determined by the current state and the players’ current actions. It should be noted that PSPACE-hardness results are known for alternatingmove games such as generalized Go [30] or QSAT =-=[49]-=-; however, if we were to formulate such a game as a Markov game, we would require an exponential number of states, so these results do not imply PSPACE-hardness for (straightforwardly represented) Mar... |

235 |
Stochastic Games
- Shapley
- 1953
(Show Context)
Citation Context ...m. 15Markov (or stochastic) games constitute an important type of multi-stage games. In such games, there is an underlying set of states, and the game shifts between these states from stage to stage =-=[15, 47, 48]-=-. At every stage, each player’s payoff depends not only on the players’ actions, but also on the state. Furthermore, the probability of transitioning to a given state is determined by the current stat... |

222 | The complexity of computing a Nash equilibrium
- Daskalakis, Goldberg, et al.
- 2006
(Show Context)
Citation Context ...ational problem whose complexity is wide open” and “together with factoring, [...] the most important concrete open question on the boundary of P today” [39]. A recent sequence of breakthrough papers =-=[6, 7, 11, 13]-=- shows that the problem is PPAD2 To define P formally (which we will not do here), one must also formally define a model of computation. Fortunately, the class of polynomial-time solvable problems is ... |

221 | Graphical models for game theory
- Kearns, Littman, et al.
- 2001
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Citation Context ...r to store. In this case, the computer needs to operate directly on a more concise representation of the game. Examples of such representations (other than the extensive form) include graphical games =-=[22]-=-, action-graph games [5, 29], and multiagent influence diagrams [27]. While changing the way the game is represented does not change it strategically, 11 it does affect the computational complexity of... |

202 |
Acceptable Points in General Cooperative n-Person Games," in Contributions to the Theory of Games IV
- Aumann
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Citation Context ...re player 1 never plays a given x ∈ Σ1. Proof. For any φ, inGɛ(φ), there is a Nash equilibrium where player 1 never plays f if and only if φ is satisfiable. ✷ Definition 5. A strong Nash equilibrium (=-=Aumann, 1959-=-) is a vector of mixed strategies for the players so that no nonempty subset of the players can change their strategies to make all players in the subset better off. Corollary 5. Even in symmetric 2-p... |

196 |
Equilibrium Points of Bimatrix Games
- Lemke, Jr
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Citation Context ...e evidence that the problem is indeed hard, although not nearly as much is known about the class PPAD as about N P. The best-known algorithm for finding a Nash equilibrium, the Lemke-Howson algorithm =-=[28]-=-, has been shown to indeed have exponential running time on some instances (and is therefore not a polynomial-time algorithm) [45]. More recent algorithms for computing Nash equilibria have focused on... |

155 | Multi-agent influence diagrams for representing and solving games
- Koller, Milch
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Citation Context ...a more concise representation of the game. Examples of such representations (other than the extensive form) include graphical games [22], action-graph games [5, 29], and multiagent influence diagrams =-=[27]-=-. While changing the way the game is represented does not change it strategically, 11 it does affect the computational complexity of solving the game [20, 46]. However, as long as the representation c... |

135 | Algorithms, games and the internet
- Papadimitriou
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Citation Context ...g ago, it was dubbed “a most fundamental computational problem whose complexity is wide open” and “together with factoring, [...] the most important concrete open question on the boundary of P today” =-=[39]-=-. A recent sequence of breakthrough papers [6, 7, 11, 13] shows that the problem is PPAD2 To define P formally (which we will not do here), one must also formally define a model of computation. Fortun... |

129 |
Nash and Correlated Equilibria: Some Complexity Considerations
- Gilboa, Zemel
- 1989
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Citation Context ... to guesses in which a particular pure strategy is included in the support. These are not the first results of this nature; Gilboa and Zemel provide a number of NPhardness results in the same spirit (=-=Gilboa and Zemel, 1989-=-). Our reduction demonstrates (sometimes stronger versions of) most of their hardness results, as well as some new ones. Significantly, for the problems that concern an optimization (e.g., maximizing ... |

104 | Settling the complexity of two-player Nash equilibrium
- Chen, Deng
- 2006
(Show Context)
Citation Context ...ational problem whose complexity is wide open” and “together with factoring, [...] the most important concrete open question on the boundary of P today” [39]. A recent sequence of breakthrough papers =-=[6, 7, 11, 13]-=- shows that the problem is PPAD2 To define P formally (which we will not do here), one must also formally define a model of computation. Fortunately, the class of polynomial-time solvable problems is ... |

89 | Playing large games using simple strategies
- Lipton, Markakis, et al.
- 2003
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Citation Context ...sticated game theorists armed with state-of-the-art computing equipment. 3 It should be noted that this is different from the problem of computing an approximate equilibrium (Daskalakis et al., 2007; =-=Lipton et al., 2003-=-), that is, a strategy profile from which individual players have only a small incentive to deviate. The problems that we consider require an exact equilibrium that approximately optimizes some object... |

86 | Simple search methods for finding a Nash equilibrium
- Porter, Nudelman, et al.
- 2008
(Show Context)
Citation Context ...quilibria have focused on guessing which of the players’ pure strategies receive positive probability in the equilibrium: after this guess, only a simple linear feasibility problem needs to be solved =-=[14, 42, 44]-=-. These algorithms clearly require exponentially many guesses, and hence exponential time, on some instances, although they are often quite fast in practice. The interest in the problem of computing a... |

85 | Efficient computation of equilibria for extensive two-person games
- Koller, Megiddo, et al.
- 1996
(Show Context)
Citation Context ...s include CURB sets [4], (iterated) dominance [8, 16], other elimination criteria [9], and correlated equilibria [17, 40]. There is also a significant body of research on solving extensive-form games =-=[18, 19, 25, 26, 34, 43, 51, 52]-=-. Another topic of interest is how the game is represented, that is, in what form the game is presented to the solver. A polynomial-time algorithm for normal-form games is of little use if the normal ... |

76 | Computing correlated equilibria in multi-player games
- Papadimitriou
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Citation Context ...ns. Solution concepts other than Nash equilibrium are also receiving attention: examples include CURB sets [4], (iterated) dominance [8, 16], other elimination criteria [9], and correlated equilibria =-=[17, 40]-=-. There is also a significant body of research on solving extensive-form games [18, 19, 25, 26, 34, 43, 51, 52]. Another topic of interest is how the game is represented, that is, in what form the gam... |

73 | The complexity of two-person zero-sum games in extensive form. Games and Economic Behavior 4(4):528–552
- Koller, Megiddo
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Citation Context ...utomaton when the automata under consideration are bounded [38]; the problem of whether a given player with imperfect recall can guarantee herself a given payoff using pure strategies is N P-complete =-=[25]-=-; and in general, best-responding to an arbitrary strategy can even be noncomputable [24, 35]. In this section, we present a PSPACE-hardness result on the existence of a pure-strategy equilibrium. 15... |

67 | Fudenberg and Jean Tirole. Game Theory - Drew - 1991 |

61 | Pure Nash equilibria: Hard and easy games
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Citation Context ...es [5, 29], and multiagent influence diagrams [27]. While changing the way the game is represented does not change it strategically, 11 it does affect the computational complexity of solving the game =-=[20, 46]-=-. However, as long as the representation can capture any game, the computational problem cannot become any easier than under the straightforward representation. Therefore, our hardness results apply t... |

53 | Computing Nash equilibria of actiongraph games
- Bhat, Leyton-Brown
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Citation Context ... the computer needs to operate directly on a more concise representation of the game. Examples of such representations (other than the extensive form) include graphical games [22], action-graph games =-=[5, 29]-=-, and multiagent influence diagrams [27]. While changing the way the game is represented does not change it strategically, 11 it does affect the computational complexity of solving the game [20, 46]. ... |

49 | Local-effect games
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Citation Context ... the computer needs to operate directly on a more concise representation of the game. Examples of such representations (other than the extensive form) include graphical games [22], action-graph games =-=[5, 29]-=-, and multiagent influence diagrams [27]. While changing the way the game is represented does not change it strategically, 11 it does affect the computational complexity of solving the game [20, 46]. ... |

49 | Mixed-integer programming methods for finding nash equilibria
- Sandholm, Gilpin, et al.
- 2005
(Show Context)
Citation Context ...quilibria have focused on guessing which of the players’ pure strategies receive positive probability in the equilibrium: after this guess, only a simple linear feasibility problem needs to be solved =-=[14, 42, 44]-=-. These algorithms clearly require exponentially many guesses, and hence exponential time, on some instances, although they are often quite fast in practice. The interest in the problem of computing a... |

46 | Efficient computation of behavior strategies
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Citation Context ...s include CURB sets [4], (iterated) dominance [8, 16], other elimination criteria [9], and correlated equilibria [17, 40]. There is also a significant body of research on solving extensive-form games =-=[18, 19, 25, 26, 34, 43, 51, 52]-=-. Another topic of interest is how the game is represented, that is, in what form the game is presented to the solver. A polynomial-time algorithm for normal-form games is of little use if the normal ... |

36 | Three-player games are hard
- Daskalakis, Papadimitriou
- 2005
(Show Context)
Citation Context ...ational problem whose complexity is wide open” and “together with factoring, [...] the most important concrete open question on the boundary of P today” [39]. A recent sequence of breakthrough papers =-=[6, 7, 11, 13]-=- shows that the problem is PPAD2 To define P formally (which we will not do here), one must also formally define a model of computation. Fortunately, the class of polynomial-time solvable problems is ... |

35 |
On the Order of Eliminating Dominated Strategies
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Citation Context ...theoretic solutions. Solution concepts other than Nash equilibrium are also receiving attention: examples include CURB sets (Benisch et al., 2006), (iterated) dominance (Conitzer and Sandholm, 2005a; =-=Gilboa et al., 1993-=-), other elimination criteria (Conitzer and Sandholm, 2005b), and correlated equilibria (Gilboa and Zemel, 1989; Papadimitriou, 2005). There is also a significant body of research on solving extensive... |

32 | Gradient-based algorithms for finding nash equilibria in extensive form games
- Gilpin, Hoda, et al.
- 2007
(Show Context)
Citation Context ...s include CURB sets [4], (iterated) dominance [8, 16], other elimination criteria [9], and correlated equilibria [17, 40]. There is also a significant body of research on solving extensive-form games =-=[18, 19, 25, 26, 34, 43, 51, 52]-=-. Another topic of interest is how the game is represented, that is, in what form the game is presented to the solver. A polynomial-time algorithm for normal-form games is of little use if the normal ... |

32 |
Reduction of a game with complete memory to a matrix game. Soviet Mathematics
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Citation Context |

31 | On the complexity of two-player winlose games
- Abbott, Kane, et al.
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Citation Context ...hence that they can solve problems outside P efficiently. 2complete, even in the two-player case. (An earlier result shows that the problem is no easier if all utilities are required to be in {0, 1} =-=[1]-=-.) This gives some evidence that the problem is indeed hard, although not nearly as much is known about the class PPAD as about N P. The best-known algorithm for finding a Nash equilibrium, the Lemke-... |

29 |
3-NASH is PPAD-complete
- Chen, Deng
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Citation Context |

29 |
Go is polynomial-space hard
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Citation Context ... be noted that PSPACE-hardness results are known for alternating-move games such asAuthor's personal copy V. Conitzer, T. Sandholm / Games and Economic Behavior 63 (2008) 621–641 635 generalized Go (=-=Lichtenstein and Sipser, 1980-=-) or QSAT (Stockmeyer and Meyer, 1973); however, if we were to formulate such a game as a Markov game, we would require an exponential number of states, so these results do not imply PSPACE-hardness f... |

24 |
Non-computable strategies and discounted repeated games
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Citation Context ... given player with imperfect recall can guarantee herself a given payoff using pure strategies is N P-complete [25]; and in general, best-responding to an arbitrary strategy can even be noncomputable =-=[24, 35]-=-. In this section, we present a PSPACE-hardness result on the existence of a pure-strategy equilibrium. 15Markov (or stochastic) games constitute an important type of multi-stage games. In such games... |

23 | Hard-to-solve bimatrix games - Stengel, Savani |

22 |
The Complexity of Computing Best Response Automata in Repeated Games with Mixed Strategies
- Ben-Porath
- 1990
(Show Context)
Citation Context ...e stages. There has already been some research into the complexity of playing repeated and sequential games. For example, determining whether a particular automaton is a best response is N P-complete =-=[3]-=-; it is N P-complete to compute a best-response automaton when the automata under consideration are bounded [38]; the problem of whether a given player with imperfect recall can guarantee herself a gi... |

21 | Lossless abstraction of imperfect information games
- Gilpin, Sandholm
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Citation Context |

21 |
On players with a bounded number of states
- Papadimitriou
- 1992
(Show Context)
Citation Context ...r example, determining whether a particular automaton is a best response is N P-complete [3]; it is N P-complete to compute a best-response automaton when the automata under consideration are bounded =-=[38]-=-; the problem of whether a given player with imperfect recall can guarantee herself a given payoff using pure strategies is N P-complete [25]; and in general, best-responding to an arbitrary strategy ... |

20 | Complexity of (iterated) dominance
- Conitzer, Sandholm
- 2005
(Show Context)
Citation Context ...s undoubtedly not the last word on computing game-theoretic solutions. Solution concepts other than Nash equilibrium are also receiving attention: examples include CURB sets [4], (iterated) dominance =-=[8, 16]-=-, other elimination criteria [9], and correlated equilibria [17, 40]. There is also a significant body of research on solving extensive-form games [18, 19, 25, 26, 34, 43, 51, 52]. Another topic of in... |

17 |
Noncooperative Stochastic Games
- Sobel
- 1971
(Show Context)
Citation Context ...m. 15Markov (or stochastic) games constitute an important type of multi-stage games. In such games, there is an underlying set of states, and the game shifts between these states from stage to stage =-=[15, 47, 48]-=-. At every stage, each player’s payoff depends not only on the players’ actions, but also on the state. Furthermore, the probability of transitioning to a given state is determined by the current stat... |

17 |
The Computational Complexity of Nash Equilibria in Concisely Represented Games
- Schoenebeck, Vadhan
- 2006
(Show Context)
Citation Context ...ms (Koller and Milch, 2001). While changing the way the game is represented does not change it strategically, 11 it does affect the computational complexity of solving the game (Gottlob et al., 2003; =-=Schoenebeck and Vadhan, 2006-=-). However, as long as the representation can capture 11 This is assuming that no strategic information is added or lost in the conversion. For example, converting an extensiveform game to a normal-fo... |

15 | The Expected Number of Nash Equilibria of a Normal Form Game
- McLennan
- 2005
(Show Context)
Citation Context ...ssignments of φ, plus one. Counting the number of satisfying assignments to a CNF 10 The number of equilibria in normal-form games has been studied both in the worst case [33] and in the average case =-=[32]-=-. 12formula is #P-hard [50]. In a sense, the most interesting #P-hardness results are the ones where the corresponding existence problem (does there exist at least one solution?) and search problem (... |

15 | Computing Sequential Equilibria for Two-Player Games
- Miltersen, Sørensen
(Show Context)
Citation Context ...989; Papadimitriou, 2005). There is also a significant body of research on solving extensive-form games (Gilpin et al., 2007; Gilpin and Sandholm, 2007; Koller and Megiddo, 1992; Koller et al., 1996; =-=Miltersen and Sørensen, 2006-=-; Romanovskii, 1962; von Stengel, 1996; von Stengel et al., 2002). Another topic of interest is how the game is represented, that is, in what form the game is presented to the solver. A polynomial-tim... |

14 | A generalized strategy eliminability criterion and computational methods for applying it
- Conitzer, Sandholm
- 2005
(Show Context)
Citation Context ...omputing game-theoretic solutions. Solution concepts other than Nash equilibrium are also receiving attention: examples include CURB sets [4], (iterated) dominance [8, 16], other elimination criteria =-=[9]-=-, and correlated equilibria [17, 40]. There is also a significant body of research on solving extensive-form games [18, 19, 25, 26, 34, 43, 51, 52]. Another topic of interest is how the game is repres... |

13 | Algorithms for rationalizability and CURB sets
- Benisch, Davis, et al.
- 2006
(Show Context)
Citation Context ...is symmetric. This paper is undoubtedly not the last word on computing game-theoretic solutions. Solution concepts other than Nash equilibrium are also receiving attention: examples include CURB sets =-=[4]-=-, (iterated) dominance [8, 16], other elimination criteria [9], and correlated equilibria [17, 40]. There is also a significant body of research on solving extensive-form games [18, 19, 25, 26, 34, 43... |

13 | Progress in approximate Nash equilibria
- Daskalakis, Mehta, et al.
(Show Context)
Citation Context ...e answer to the latter instance. Moreover, we require that this mapping is itself easy to compute. 3 It should be noted that this is different from the problem of computing an approximate equilibrium =-=[12, 31]-=-, that is, a strategy profile from which individual players have only a small incentive to deviate. The problems that we consider require an exact equilibrium that approximately optimizes some objecti... |

12 | Generic 4 × 4 two person games have at most 15 Nash equilibria
- McLennan, Park
- 1999
(Show Context)
Citation Context ...R. We emphasize again that we only consider finite games; in particular, we only consider finite type spaces. 10 The number of equilibria in normal-form games has been studied both in the worst case (=-=McLennan and Park, 1999-=-) and in the average case (McLennan, 2005).Author's personal copy V. Conitzer, T. Sandholm / Games and Economic Behavior 63 (2008) 621–641 633 Definition 7. (See Harsanyi, 1967–1968.) Given a Bayesia... |

11 |
A program for finding Nash equilibria
- Dickhaut, Kaplan
- 1991
(Show Context)
Citation Context ...quilibria have focused on guessing which of the players’ pure strategies receive positive probability in the equilibrium: after this guess, only a simple linear feasibility problem needs to be solved =-=[14, 42, 44]-=-. These algorithms clearly require exponentially many guesses, and hence exponential time, on some instances, although they are often quite fast in practice. The interest in the problem of computing a... |