## Computing Equilibria in Multi-Player Games (2004)

Venue: | In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA |

Citations: | 51 - 3 self |

### BibTeX

@INPROCEEDINGS{Papadimitriou04computingequilibria,

author = {Christos H. Papadimitriou and Tim Roughgarden},

title = {Computing Equilibria in Multi-Player Games},

booktitle = {In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA},

year = {2004},

pages = {82--91},

publisher = {SIAM}

}

### Years of Citing Articles

### OpenURL

### Abstract

We initiate the systematic study of algorithmic issues involved in finding equilibria (Nash and correlated) in games with a large number of players; such games, in order to be computationally meaningful, must be presented in some succinct, game-specific way. We develop a general framework for obtaining polynomial-time algorithms for optimizing over correlated equilibria in such settings, and show how it can be applied successfully to symmetric games (for which we actually find an exact polytopal characterization), graphical games, and congestion games, among others. We also present complexity results implying that such algorithms are not possible in certain other such games. Finally, we present a polynomial-time algorithm, based on quantifier elimination, for finding a Nash equilibrium in symmetric games when the number of strategies is relatively small.

### Citations

10921 | Computers and Intractability: A Guide to the Theory of NP-Completeness - Garey, Johnson |

1271 | Combinatorial Optimization: Algorithms and Complexity - Papadimitriou, Steiglitz - 1982 |

1139 | Geometric Algorithms and Combinatorial Optimizations - Grotschel, Lovász, et al. - 1993 |

807 |
Noncooperative games
- Nash
- 1951
(Show Context)
Citation Context ...the player's strategy. Symmetric games have been widely studied since the dawn of game theory; together with zero-sum games, they form one of the most classical subclasses of games. For example, Nash =-=[32]-=- proved that every symmetric game must have a symmetric equilibrium---an equilibrium in which all players play the same strategy. Several of the papers in the first volume of Contributions to the Theo... |

640 | Evolutionary Game Theory
- Weibull
- 1995
(Show Context)
Citation Context ...e famous examples they have provided: the Prisoner's Dilemma, Chicken, coordination games, and so on. More recently, symmetric games have played 1 a central role in evolutionary game theory (see e.g. =-=[45]-=-). Despite this long history, little is known about the complexity of computing equilibria of symmetric games. Multi-player symmetric games admit a compact representation. Specifically, a symmetric ga... |

633 |
Equilibrium points in n-person games
- Nash
- 1950
(Show Context)
Citation Context ... each player i, picking a strategy from S i according to the distribution p i maximizes i's expected payoff, assuming that each player j 6= i picks a strategy according to the distribution p j . Nash =-=[-=-31] showed that every game admits a Nash equilibrium. In a subsequent paper [32], he showed that every symmetric game admits a symmetric Nash equilibrium, meaning a Nash equilibrium in which p 1 = ... |

622 | C.: Worst-case equilibria
- Koutsoupias, Papadimitriou
- 1999
(Show Context)
Citation Context ...ial-time algorithm for computing a Nash equilibrium is arguably the "holy grail" in this research area (see [34]), and much progress has been made on this [5, 10, 20, 27, 28, 41] and related=-= problems [7, 8, 9, 18, 29]-=- by the theoretical computer science community in the last few years. Multi-player Games. In this paper, we will study the complexity of computing equilibria in games with many players. While two-play... |

506 | É.: How bad is selfish routing
- Roughgarden, Tardos
- 2002
(Show Context)
Citation Context ... in the theoretical computer science literature are in essence applying this philosophy to concrete applications, such as load-balancing (see [6, 12] and the references therein), network routing (see =-=[40]-=-), facility location [43], and congestion games [11]. In this paper, we are aiming for a more systematic investigation---what properties of a compact representation permit polynomialtime algorithms fo... |

392 |
A class of games possessing pure-strategy Nash equilibria
- Rosenthal
- 1973
(Show Context)
Citation Context ...g Nash [20, 28] and correlated equilibria [19] for graphical games defined on trees. Congestion Games. Congestion games are an abstraction of network routing games and were first defined by Rosenthal =-=[38, 39]-=-. In a congestion game, there is a ground set of elements, and players choose a strategy from a prescribed collection of subsets of the ground set. The cost of an element is a function of the number p... |

375 | A polynomial algorithm in linear programming - Khachiyan - 1979 |

366 |
Subjectivity and correlation in randomized strategies
- Aumann
- 1974
(Show Context)
Citation Context ...ibria. Correlated Equilibria. While we also give algorithms for computing Nash equilibria, our widest-ranging theory concerns correlated equilibria. Correlated equilibria were first defined by Aumann =-=[1]-=-, and we will describe them in detail in Section 2. For now, suffice it to say that every Nash equilibrium is a correlated equilibrium and that the set of all correlated equilibria of a game can be de... |

293 | Correlated equilibrium as an expression of Bayesian rationality
- Aumann
- 1987
(Show Context)
Citation Context ...A popular concrete example of a correlated equilibrium is a traffic signal that recommends "red" (stop) or "green" (go) to drivers (see e.g. [33]). For more applications of correla=-=ted equilibria, see [2, 13, 14]-=-. 3 Explicit Descriptions of Correlated Equilibria In this section, our ambition will be to explicitly describe the correlated equilibria of a game that is represented compactly. Put differently, we w... |

291 |
the internet
- Papadimitriou, games
- 2001
(Show Context)
Citation Context ...ng n strategic players from which no player has a unilateral incentive to deviate. A polynomial-time algorithm for computing a Nash equilibrium is arguably the "holy grail" in this research =-=area (see [34]-=-), and much progress has been made on this [5, 10, 20, 27, 28, 41] and related problems [7, 8, 9, 18, 29] by the theoretical computer science community in the last few years. Multi-player Games. In th... |

250 |
Game Theory
- Owen
- 1982
(Show Context)
Citation Context ... theorem [31] implies that this system is always feasible. A popular concrete example of a correlated equilibrium is a traffic signal that recommends "red" (stop) or "green" (go) t=-=o drivers (see e.g. [33]-=-). For more applications of correlated equilibria, see [2, 13, 14]. 3 Explicit Descriptions of Correlated Equilibria In this section, our ambition will be to explicitly describe the correlated equilib... |

224 | S.P.: Graphical models for game theory
- Kearns, Littman, et al.
- 2001
(Show Context)
Citation Context ... a unilateral incentive to deviate. A polynomial-time algorithm for computing a Nash equilibrium is arguably the "holy grail" in this research area (see [34]), and much progress has been mad=-=e on this [5, 10, 20, 27, 28, 41]-=- and related problems [7, 8, 9, 18, 29] by the theoretical computer science community in the last few years. Multi-player Games. In this paper, we will study the complexity of computing equilibria in ... |

174 | On the computational complexity and geometry of the first-order theory of the reals - Renegar - 1992 |

156 | Multi-Agent Influence Diagrams for Representing and Solving Games
- Koller, Milch
- 2001
(Show Context)
Citation Context ...k n numbers required for the standard representation, exponentially so if k = O(n). Graphical Games. Graphical Games were first proposed by Kearns, Littman, and Singh [20]. (See also Koller and Milch =-=[23]-=- and the references therein for related concepts.) In a graphical game, the players are the vertices of a graph, and the payoff of each player only depends on its strategy and those of its neighbors. ... |

136 | Congestion games with player specific payoff functions - Milchtaich - 1996 |

131 | Complexity results about nash equilibria
- Conitzer, Sandholm
- 2003
(Show Context)
Citation Context ... a unilateral incentive to deviate. A polynomial-time algorithm for computing a Nash equilibrium is arguably the "holy grail" in this research area (see [34]), and much progress has been mad=-=e on this [5, 10, 20, 27, 28, 41]-=- and related problems [7, 8, 9, 18, 29] by the theoretical computer science community in the last few years. Multi-player Games. In this paper, we will study the complexity of computing equilibria in ... |

112 | Regret in the on-line decision problem
- Foster, Vohra
- 1999
(Show Context)
Citation Context ...A popular concrete example of a correlated equilibrium is a traffic signal that recommends "red" (stop) or "green" (go) to drivers (see e.g. [33]). For more applications of correla=-=ted equilibria, see [2, 13, 14]-=-. 3 Explicit Descriptions of Correlated Equilibria In this section, our ambition will be to explicitly describe the correlated equilibria of a game that is represented compactly. Put differently, we w... |

99 | Nash equilibria in competitive societies, with applications to facility location, traffic routing and auctions
- Vetta
- 2002
(Show Context)
Citation Context ...ter science literature are in essence applying this philosophy to concrete applications, such as load-balancing (see [6, 12] and the references therein), network routing (see [40]), facility location =-=[43]-=-, and congestion games [11]. In this paper, we are aiming for a more systematic investigation---what properties of a compact representation permit polynomialtime algorithms for computing equilibria? T... |

91 | Playing large games using simple strategies
- Lipton, Markakis, et al.
- 2003
(Show Context)
Citation Context ... a unilateral incentive to deviate. A polynomial-time algorithm for computing a Nash equilibrium is arguably the "holy grail" in this research area (see [34]), and much progress has been mad=-=e on this [5, 10, 20, 27, 28, 41]-=- and related problems [7, 8, 9, 18, 29] by the theoretical computer science community in the last few years. Multi-player Games. In this paper, we will study the complexity of computing equilibria in ... |

86 | Calibrated learning and correlated equilibrium
- Foster, Vohra
- 1997
(Show Context)
Citation Context ...A popular concrete example of a correlated equilibrium is a traffic signal that recommends "red" (stop) or "green" (go) to drivers (see e.g. [33]). For more applications of correla=-=ted equilibria, see [2, 13, 14]-=-. 3 Explicit Descriptions of Correlated Equilibria In this section, our ambition will be to explicitly describe the correlated equilibria of a game that is represented compactly. Put differently, we w... |

79 | Convergence time to Nash equilibria
- Even-Dar, Kesselman, et al.
- 2003
(Show Context)
Citation Context |

61 | A polynomial-time nash equilibrium algorithm for repeated games. Decision Support Systems
- Littman, Stone
- 2005
(Show Context)
Citation Context ...ial-time algorithm for computing a Nash equilibrium is arguably the "holy grail" in this research area (see [34]), and much progress has been made on this [5, 10, 20, 27, 28, 41] and related=-= problems [7, 8, 9, 18, 29]-=- by the theoretical computer science community in the last few years. Multi-player Games. In this paper, we will study the complexity of computing equilibria in games with many players. While two-play... |

39 | Correlated equilibria in graphical games
- Kakade, Kearns, et al.
- 2003
(Show Context)
Citation Context ...ts strategy and those of its neighbors. Algorithms that run in time polynomial in the obvious compact representation have recently been developed for computing Nash [20, 28] and correlated equilibria =-=[19]-=- for graphical games defined on trees. Congestion Games. Congestion games are an abstraction of network routing games and were first defined by Rosenthal [38, 39]. In a congestion game, there is a gro... |

37 |
Solutions of games by differential equations
- Brown, Neumann
- 1950
(Show Context)
Citation Context ...ame in this case could be difficult. In particular, it has long been known that for games with a constant number of players, there is a polynomial-time reduction from general games to symmetric games =-=[3, 15]-=-, and hence symmetry affords no computational advantage in this case. Acknowledgements Thanks to Grant Schoenebeck for directing us to [15], Vangelis Markakis for notifying us about [26], and Vincent ... |

26 |
The network equilibrium problem in integers", Networks
- Rosenthal
- 1973
(Show Context)
Citation Context ...g Nash [20, 28] and correlated equilibria [19] for graphical games defined on trees. Congestion Games. Congestion games are an abstraction of network routing games and were first defined by Rosenthal =-=[38, 39]-=-. In a congestion game, there is a ground set of elements, and players choose a strategy from a prescribed collection of subsets of the ground set. The cost of an element is a function of the number p... |

25 |
Selfish routing in non-cooperative networks: A survey
- Feldmann, Gairing, et al.
- 2003
(Show Context)
Citation Context ...es that admit a compact representation. Many recent papers in the theoretical computer science literature are in essence applying this philosophy to concrete applications, such as load-balancing (see =-=[6, 12]-=- and the references therein), network routing (see [40]), facility location [43], and congestion games [11]. In this paper, we are aiming for a more systematic investigation---what properties of a com... |

25 | Approximating market equilibria
- Jain, Mahdian, et al.
(Show Context)
Citation Context ...ial-time algorithm for computing a Nash equilibrium is arguably the "holy grail" in this research area (see [34]), and much progress has been made on this [5, 10, 20, 27, 28, 41] and related=-= problems [7, 8, 9, 18, 29]-=- by the theoretical computer science community in the last few years. Multi-player Games. In this paper, we will study the complexity of computing equilibria in games with many players. While two-play... |

23 |
Nash equilibria via polynomial equations
- Lipton, Markakis
- 2004
(Show Context)
Citation Context ...n from the so-called first-order theory of the reals, the details of which we provide in the Appendix. A different application of this idea to games was developed independently by Lipton and Markakis =-=[26]-=-. Since the compact representation of a symmetric game has size poly(n k )) when k = O(log n= log log n), we have the following corollary of Theorem 5.1. Corollary 5.2 The problem of computing a Nash ... |

21 | An improved approximation scheme for computing Arrow-Debreu prices for the linear case
- Devanur, Vazirani
- 2003
(Show Context)
Citation Context |

20 | An efficient exact algorithm for singly connected graphical games
- Littman, Kearns, et al.
- 2001
(Show Context)
Citation Context |

19 | Equilibria in a model with partial rivalry - Konishi, Breton, et al. - 1997 |

15 |
On Symmetric Games
- Gale, Kuhn, et al.
- 1950
(Show Context)
Citation Context ...ame in this case could be difficult. In particular, it has long been known that for games with a constant number of players, there is a polynomial-time reduction from general games to symmetric games =-=[3, 15]-=-, and hence symmetry affords no computational advantage in this case. Acknowledgements Thanks to Grant Schoenebeck for directing us to [15], Vangelis Markakis for notifying us about [26], and Vincent ... |

12 | Selfish routing on the Internet
- Czumaj
- 2004
(Show Context)
Citation Context ...es that admit a compact representation. Many recent papers in the theoretical computer science literature are in essence applying this philosophy to concrete applications, such as load-balancing (see =-=[6, 12]-=- and the references therein), network routing (see [40]), facility location [43], and congestion games [11]. In this paper, we are aiming for a more systematic investigation---what properties of a com... |

10 |
Multiagent Systems
- Shoham, Leyton-Brown
- 2009
(Show Context)
Citation Context ...mes have been extensively studied in the game theory literature (see [22] and the references therein), and their importance has long been recognized by the artificial intelligence community (see e.g. =-=[42]-=-), even less is known about computing equilibria in multiplayer games than in the (still mysterious) special case of two-player games. There is an immediate obstacle to discussing complexity results f... |

7 | Non-cooperative games with many players
- Khan, Sun
- 2002
(Show Context)
Citation Context ...by large networks, such as the Internet, where games are obviously being played by a large number of players. While multi-player games have been extensively studied in the game theory literature (see =-=[22]-=- and the references therein), and their importance has long been recognized by the artificial intelligence community (see e.g. [42]), even less is known about computing equilibria in multiplayer games... |

6 |
Long Lemke-Howson paths
- Savani, Stengel
- 2003
(Show Context)
Citation Context |

4 |
Market equilibrium via a primaldual -type algorithm
- Devanur, Papadimitriou, et al.
- 2002
(Show Context)
Citation Context |

4 | A model of migration. Working paper - Quint, Shubik - 1994 |

2 |
The complexity of pure-strategy equilibria
- Fabrikant, Papadimitriou, et al.
- 2004
(Show Context)
Citation Context ...in essence applying this philosophy to concrete applications, such as load-balancing (see [6, 12] and the references therein), network routing (see [40]), facility location [43], and congestion games =-=[11]-=-. In this paper, we are aiming for a more systematic investigation---what properties of a compact representation permit polynomialtime algorithms for computing equilibria? To illustrate our results, w... |

2 |
Contributions to the Theory of Games, volume 1
- Kuhn, Tucker
- 1950
(Show Context)
Citation Context ... every symmetric game must have a symmetric equilibrium---an equilibrium in which all players play the same strategy. Several of the papers in the first volume of Contributions to the Theory of Games =-=[25]-=- are devoted to symmetric games. Symmetric games' long tenure in the spotlight is due in large part to the famous examples they have provided: the Prisoner's Dilemma, Chicken, coordination games, and ... |