## A continuation method for Nash equilibria in structured games (2003)

### Cached

### Download Links

- [ijcai.org]
- [dli.iiit.ac.in]
- [ijcai.org]
- [www.aaai.org]
- [www.aaai.org]
- [www.jair.org]
- [www.cs.ucr.edu]
- [www.cs.ucr.edu]
- [ai.stanford.edu]
- [robotics.stanford.edu]
- [robotics.stanford.edu]
- [jair.org]
- [www.cs.ucr.edu]
- [dags.stanford.edu]
- [www.cs.ucr.edu]
- [www.cs.ucr.edu]
- [ai.stanford.edu]
- [robotics.stanford.edu]
- [robotics.stanford.edu]
- [www.cs.ucr.edu]
- DBLP

### Other Repositories/Bibliography

Venue: | In Proceedings of the 18th International Joint Conference on Artificial Intelligence (IJCAI |

Citations: | 45 - 0 self |

### BibTeX

@INPROCEEDINGS{Blum03acontinuation,

author = {Ben Blum and Christian R. Shelton and Daphne Koller},

title = {A continuation method for Nash equilibria in structured games},

booktitle = {In Proceedings of the 18th International Joint Conference on Artificial Intelligence (IJCAI},

year = {2003},

publisher = {Morgan Kaufmann}

}

### Years of Citing Articles

### OpenURL

### Abstract

We describe algorithms for computing Nash equilibria in structured game representations, including both graphical games and multi-agent influence diagrams (MAIDs). The algorithms are derived from a continuation method for normal-form and extensive-form games due to Govindan and Wilson; they follow a trajectory through the space of perturbed games and their equilibria. Our algorithms exploit game structure through fast computation of the Jacobian of the game's payoff function. They are guaranteed to find at least one equilibrium of the game and may find more. Our approach provides the first exact algorithm for computing an exact equilibrium in graphical games with arbitrary topology, and the first algorithm to exploit fine-grain structural properties of MAIDs. We present experimental results for our algorithms. The running time for our graphical game algorithm is similar to, and often better than, the running time of previous approximate algorithms. Our algorithm for MAIDs can effectively solve games that arc much larger than those that could be solved using previous methods. 1

### Citations

2326 |
A Theory of
- Laffont, Tirole
- 1993
(Show Context)
Citation Context ...babilities to be greater than or equal to ɛ for some small ɛ > 0. This is, in fact, a requirement for GW’s equilibrium characterization to hold. The algorithm thus looks for an ɛ-perfect equilibrium (=-=Fudenberg & Tirole, 1991-=-): a strategy profile σ in which each component is constrained by σs ≥ ɛ, and each agent’s strategy is a best response among those satisfying the constraint. Note that this is entirely different from ... |

1294 | Local computations with probabilities on graphical structures and their application to expert systems - Lauritzen, Spiegelhalter - 1988 |

810 |
Non-Cooperative Games
- NASH
- 1951
(Show Context)
Citation Context ...profile is that it be optimal for each agent, taken individually: no agent should be able to improve its utility by changing its strategy. The fundamental game theoretic notion of a Nash equilibrium (=-=Nash, 1951-=-) satisfies this criterion precisely. A Nash equilibrium is a strategy profile in which no agent can improve its payoff by deviating unilaterally — changing its strategy while all other agents hold th... |

634 |
Probabilistic Networks and Expert Systems
- Cowell, Dawid, et al.
- 1999
(Show Context)
Citation Context ... is said to be calibrated; at this point, the potential of every clique Ci contains precisely the joint distribution over the variables in Ci according to B (for details, we refer to the reference by =-=Cowell et al., 1999-=-). We can use the clique tree algorithm to perform inference over Bσ. Consider the final decision node for agent n. Due to the perfect recall assumption, all of n’s previous decisions and all of their... |

354 |
Influence diagrams
- Howard, Matheson
- 1984
(Show Context)
Citation Context ...be represented more compactly. Multi-agent influence diagrams (MAIDs) (Koller & Milch, 2001) allow a structured representation of games involving time and information by extending influence diagrams (=-=Howard & Matheson, 1984-=-) to the multi-agent case. MAIDs and influence diagrams derive much of their syntax and semantics from the Bayesian network framework. A MAID compactly represents a certain type of extensiveform game ... |

254 | Game Theory - Owen - 1995 |

224 | Graphical models for game theory
- Kearns, Littman, et al.
- 2001
(Show Context)
Citation Context ...resentation of their work. Our methods address both graphical games and MAIDs. Several recent papers have presented methods for finding equilibria in graphical games. Many of the proposed algorithms (=-=Kearns et al., 2001-=-; Littman, Kearns, & Singh, 2002; Vickrey & Koller, 2002; Ortiz & Kearns, 2003) have focused on finding approximate equilibria, in which each agent may in fact have a small incentive to deviate. These... |

198 | On the Strategic Stability of Equilibria - Kohlberg, Mertens - 1986 |

197 |
Equilibrium points of bimatrix games
- Lemke, Howson
- 1964
(Show Context)
Citation Context ...ticular bonus vector is used (in which only a single entry is nonzero), the steps from support cell to support cell that the algorithm takes are identical to the pivots of the Lemke-Howson algorithm (=-=Lemke & Howson, 1964-=-) for two-agent generalsum games, and the two algorithms find precisely the same set of solutions (Govindan & Wilson, 2002). Thus, the continuation method is a strict generalization of the LemkeHowson... |

165 |
Extensive games and the problem of information
- Kuhn
- 1953
(Show Context)
Citation Context ...brium in mixed strategies. In an extensive-form game satisfying perfect recall, any mixed strategy profile can be represented by a payoff-equivalent behavior profile, and hence by a realization plan (=-=Kuhn, 1953-=-). 3. Structured Game Representations The artificial intelligence community has recently introduced structured representations that exploit independence relations in games in order to represent them c... |

155 | Multi-agent influence diagrams for representing and solving games - Koller, Milch - 2003 |

131 | Complexity results about nash equilibria
- Conitzer, Sandholm
- 2003
(Show Context)
Citation Context ...along this more interesting (in this setting) axis of representation size, we require a different reduction. Our proof, like a number of previous hardness proofs for games (e.g., Chu & Halpern, 2001; =-=Conitzer & Sandholm, 2003-=-; Codenotti & Stefankovic, 2005), reduces 3SAT to equilibrium computation. However, in these previous proofs, variables in 3SAT instances are mapped to actions (or sets of actions) 469sBlum, Shelton, ... |

130 | Nash and Correlated Equilibria: Some Complexity Considerations - Gilboa, Zemel - 1989 |

118 | Computation of Equilibria in Finite Games
- McKelvey, McLennan
- 1996
(Show Context)
Citation Context ...would hope that more compact representations might lead to more efficient computation of equilibria than would be possible with standard game-theoretic solution algorithms (such as those described by =-=McKelvey & McLennan, 1996-=-). Unfortunately, even with compact representations, games are quite hard to solve; we present a result showing that finding Nash equilibria beyond a single trivial one is NP-hard in the types of stru... |

85 | Efficient computation of equilibria for extensive two-person games - Koller, Megiddo, et al. - 1996 |

73 | The complexity of two-person zero-sum games in extensive form. Games and Economic Behavior 4(4):528–552
- Koller, Megiddo
- 1992
(Show Context)
Citation Context ...mation set i ∈ In is then written b(a|i). If y is a node in i, then we can also write b(a|y) as an abbreviation for b(a|i). Our methods primarily employ a variant of the sequence form representation (=-=Koller & Megiddo, 1992-=-; von Stengel, 1996; Romanovskii, 1962), which is built upon the behavior strategy representation. In sequence form, a strategy σn for an agent n is represented as a realization plan, a vector of real... |

64 | Run the gamut: A comprehensive approach to evaluating game-theoretic algorithms - Nudelman, Wortman, et al. - 2004 |

57 |
2013) Gambit: Software Tools for Game Theory, Version 13.1.0, http://www.gambit-project.org
- McKelvey, McLennan, et al.
(Show Context)
Citation Context ... the results for our MAID algorithm, MAID cont, to those achieved by converting the game to extensive-form and running both EF cont, the extensive-form version of cont as specified by GW, and Gambit (=-=McKelvey, McLennan, & Turocy, 2004-=-), a standard game theory software package. The time required for conversion to extensive form is not included in our results. We ran our algorithms on two classes of games, with varying sizes. The fi... |

54 | A global newton method to compute Nash equilibria - Govindan, Wilson - 2001 |

54 | Computing Nash equilibria of ActionGraph Games
- Bhat, Leyton-Brown
- 2004
(Show Context)
Citation Context ... Singh, c○2006 AI Access Foundation. All rights reserved.Blum, Shelton, & Koller 2001), multi-agent influence diagrams (MAIDs) (Koller & Milch, 2001), G nets (La Mura, 2000), and action-graph games (=-=Bhat & Leyton-Brown, 2004-=-). Our goal is to describe rational behavior in a game. In game theory, a description of the behavior of all agents in the game is referred to as a strategy profile: a joint assignment of strategies t... |

49 | Local-effect games
- Leyton-Brown, Tennenholtz
- 2003
(Show Context)
Citation Context ...at an adaptation of cont can be used to efficiently solve a new class of structured games called action-graph games (a generalization of local effect games as presented 495sBlum, Shelton, & Koller in =-=Leyton-Brown & Tennenholtz, 2003-=-). We believe that these games, and other structured representations, show great promise as enablers of new applications for game theory. They have several advantages over their unstructured counterpa... |

46 | Efficient computation of behavior strategies - Stengel - 1996 |

41 | Multi-agent algorithms for solving graphical games
- Vickrey, Koller
- 2002
(Show Context)
Citation Context ...raphical games and MAIDs. Several recent papers have presented methods for finding equilibria in graphical games. Many of the proposed algorithms (Kearns et al., 2001; Littman, Kearns, & Singh, 2002; =-=Vickrey & Koller, 2002-=-; Ortiz & Kearns, 2003) have focused on finding approximate equilibria, in which each agent may in fact have a small incentive to deviate. These sorts of algorithms can be problematic: approximations ... |

32 | Game networks - Mura, P - 2000 |

32 |
Reduction of a game with complete memory to a matrix game. Soviet Mathematics
- Romanovskii
- 1962
(Show Context)
Citation Context ...f y is a node in i, then we can also write b(a|y) as an abbreviation for b(a|i). Our methods primarily employ a variant of the sequence form representation (Koller & Megiddo, 1992; von Stengel, 1996; =-=Romanovskii, 1962-=-), which is built upon the behavior strategy representation. In sequence form, a strategy σn for an agent n is represented as a realization plan, a vector of real values. Each value, or realization pr... |

23 | Nash propagation for loopy graphical games
- Ortiz, Kearns
- 2002
(Show Context)
Citation Context .... Several recent papers have presented methods for finding equilibria in graphical games. Many of the proposed algorithms (Kearns et al., 2001; Littman, Kearns, & Singh, 2002; Vickrey & Koller, 2002; =-=Ortiz & Kearns, 2003-=-) have focused on finding approximate equilibria, in which each agent may in fact have a small incentive to deviate. These sorts of algorithms can be problematic: approximations must be crude for reas... |

20 | An efficient exact algorithm for singly connected graphical games
- Littman, Kearns, et al.
- 2001
(Show Context)
Citation Context ...work. Our methods address both graphical games and MAIDs. Several recent papers have presented methods for finding equilibria in graphical games. Many of the proposed algorithms (Kearns et al., 2001; =-=Littman, Kearns, & Singh, 2002-=-; Vickrey & Koller, 2002; Ortiz & Kearns, 2003) have focused on finding approximate equilibria, in which each agent may in fact have a small incentive to deviate. These sorts of algorithms can be prob... |

16 |
2002. “Structure theorems for game trees
- Govindan, Wilson
(Show Context)
Citation Context ...t the algorithm takes are identical to the pivots of the Lemke-Howson algorithm (Lemke & Howson, 1964) for two-agent generalsum games, and the two algorithms find precisely the same set of solutions (=-=Govindan & Wilson, 2002-=-). Thus, the continuation method is a strict generalization of the LemkeHowson algorithm that allows different perturbation rays and games of more than two agents. This process is described in more de... |

16 | Theory of globally convergent probability-one homotopies for non-linear programming - Watson - 2000 |

15 | Non-Cooperative Games,” Annals of Mathematics - Nash - 1951 |

11 | 2004), “Computing Nash Equilibria by Iterated Polymatrix Approximation - Govindan, Wilson |

10 | On the computational complexity of Nash equilibria for (0, 1) bimatrix games
- Codenotti, Stefankovic
- 2005
(Show Context)
Citation Context ...lized a simpler reduction to arrive at this result and several others in the same vein. Other recent hardness results pertain to restricted subclasses of normal-form games (e.g., Chu & Halpern, 2001; =-=Codenotti & Stefankovic, 2005-=-). However, these results apply only to 2-agent normal-form games. While it is true that proving a certain subclass of a class of problems to be NP-hard also proves the entire class to be NP-hard (bec... |

10 | A bound on the proportion of pure strategy equilibria in generic games
- Gül, Pearce, et al.
- 1993
(Show Context)
Citation Context ...aps an arbitrary m-vector w to the point in the space Σ of mixed strategies which is nearest to w in Euclidean distance. Given this operator, the equilibrium characterization is as follows. Lemma 9. (=-=Gül et al., 1993-=-) If σ is a strategy profile of G, then σ = R(V G (σ) + σ) iff σ is an equilibrium. Although we omit the proof, we will give some intuition for why this result is true. Suppose σ is a fully-mixed equi... |

8 | 2001), On the NP-completeness of finding an optimal strategy in games with common payoffs
- Chu, Halpern
(Show Context)
Citation Context ...m (2003) recently utilized a simpler reduction to arrive at this result and several others in the same vein. Other recent hardness results pertain to restricted subclasses of normal-form games (e.g., =-=Chu & Halpern, 2001-=-; Codenotti & Stefankovic, 2005). However, these results apply only to 2-agent normal-form games. While it is true that proving a certain subclass of a class of problems to be NP-hard also proves the ... |

1 |
A Continuation Method for Nash Equilibria in Structured Games
- Bhat, Leyton-Brown
- 2004
(Show Context)
Citation Context ... Singh, c○2006 AI Access Foundation. All rights reserved.sBlum, Shelton, & Koller 2001), multi-agent influence diagrams (MAIDs) (Koller & Milch, 2001), G nets (La Mura, 2000), and action-graph games (=-=Bhat & Leyton-Brown, 2004-=-). Our goal is to describe rational behavior in a game. In game theory, a description of the behavior of all agents in the game is referred to as a strategy profile: a joint assignment of strategies t... |

1 |
Multiagent algorithms for solving structured games. Undergraduate honors thesis
- Vickrey
- 2002
(Show Context)
Citation Context ...rd extensive-form games. Methods for related types of structured games (La Mura, 2000) are also limited to coarse-grained structure, and are currently unimplemented. Approximate approaches for MAIDs (=-=Vickrey, 2002-=-) come without implementation details or timing results. We provide the first exact algorithm that can take advantage of the fine-grained structure of MAIDs. We present experimental results demonstrat... |