## Finding Mixed Strategies with Small Supports in Extensive Form Games (1995)

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Venue: | International Journal of Game Theory |

Citations: | 24 - 1 self |

### BibTeX

@ARTICLE{Koller95findingmixed,

author = {Daphne Koller and Nimrod Megiddo},

title = {Finding Mixed Strategies with Small Supports in Extensive Form Games},

journal = {International Journal of Game Theory},

year = {1995},

volume = {25},

pages = {73--92}

}

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### Abstract

The complexity of algorithms that compute strategies or operate on them typically depends on the representation length of the strategies involved. One measure for the size of a mixed strategy is the number of strategies in its support---the set of pure strategies to which it gives positive probability. This paper investigates the existence of "small" mixed strategies in extensive form games, and how such strategies can be used to create more efficient algorithms. The basic idea is that, in an extensive form game, a mixed strategy induces a small set of realization weights that completely describe its observable behavior. This fact can be used to show that for any mixed strategy ¯, there exists a realization-equivalent mixed strategy ¯ 0 whose size is at most the size of the game tree. For a player with imperfect recall, the problem of finding such a strategy ¯ 0 (given the realization weights) is NP-hard. On the other hand, if ¯ is a behavior strategy, ¯ 0 can be constructed from...

### Citations

533 | The Linear Complementarity Problem
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(Show Context)
Citation Context ...an be used to find equilibria in two-person extensive-form games. In this case, the problem of finding equilibria in a normal-form game can be described as a linear complementarity problem (LCP) (see =-=[2]-=-). There are a number of standard algorithms for finding such equilibria. One possibility is to enumerate all the possible supports for a mixed strategy pair (a support for each of the two players), a... |

197 |
Equilibrium points of bimatrix games
- Lemke, Howson
- 1964
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Citation Context ...ble" size of these strategies may reduce the space and time complexities of algorithms handling them. This idea was first utilized by Wilson [14] in his modified version of the Lemke-Howson algor=-=ithm [9]-=-. This algorithm searches the space of mixed strategies pairs for an equilibrium of a general two-person game. Wilson's variant represents the strategies encountered in the search sparsely; i.e., it m... |

89 | Fast Algorithms for Finding Randomized Strategies in Game Trees
- Koller, Meggido, et al.
- 1994
(Show Context)
Citation Context ...hts were first introduced by Koller and Megiddo [4] in the context of perfect recall games. In their algorithm, and in the more recent ones of von Stengel [12] and of Koller, Megiddo, and von Stengel =-=[6]-=-, the realization weights corresponding to equilibrium mixed strategies are computed directly. This relies on the fact that, for games with perfect recall, realization weights can be described using a... |

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 ...ayer. Thus, it also can be represented in linear space. A mixed strategy assigns a probability to each pure strategy. The number of pure strategies is usually exponential in the size of the game tree =-=[7, 4]-=-. Thus, the size of a mixed strategy may be exponential in the size of the tree. In many useful mixed strategies, however, only a small number of pure strategies receive positive probabilities. For a ... |

31 | AND NIMROD MEGIDDO: Constructing small sample spaces satisfying given constraints - KOLLER - 1994 |

30 |
Computing Equilibria of N-Person Games
- Wilson
- 1971
(Show Context)
Citation Context ... better than h i () thensis not an equilibrium. 6 We could apply this scheme to any of the algorithms for solving N-player normal form games, for example, the algorithms of Rosenmuller [10] or Wilson =-=[13]-=-. This would result in an algorithm for finding equilibria in N-player extensive-form games. We now show how this scheme can be used to find equilibria in two-person extensive-form games. In this case... |

28 |
Extensive Games
- Kuhn
- 1950
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Citation Context ...ayer. Thus, it also can be represented in linear space. A mixed strategy assigns a probability to each pure strategy. The number of pure strategies is usually exponential in the size of the game tree =-=[7, 4]-=-. Thus, the size of a mixed strategy may be exponential in the size of the tree. In many useful mixed strategies, however, only a small number of pure strategies receive positive probabilities. For a ... |

28 |
On a Generalization of the Lemke-Howson Algorithm to Noncooperative N-Person Games
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- 1971
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Citation Context ... that payoff is better than h i () thensis not an equilibrium. 6 We could apply this scheme to any of the algorithms for solving N-player normal form games, for example, the algorithms of Rosenmuller =-=[10]-=- or Wilson [13]. This would result in an algorithm for finding equilibria in N-player extensive-form games. We now show how this scheme can be used to find equilibria in two-person extensive-form game... |

18 |
Computing Equilibria of Two-person Games from the Extensive Form
- WILSON
(Show Context)
Citation Context ...ified with a sparse representation. The relatively "manageable" size of these strategies may reduce the space and time complexities of algorithms handling them. This idea was first utilized by Wilson =-=[14]-=- in his modified version of the Lemke-Howson algorithm [9]. This algorithm searches the space of mixed strategies pairs for an equilibrium of a general two-person game. Wilson's variant represents the... |

6 | Using fast matrix multiplication to find basic solutions. Theoretical Computer Science 205
- Beling, Megiddo
- 1998
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Citation Context ...algorithm requires O(jSupp()j \Delta jT j 2 ) arithmetic operations for this conversion. 4 We can speed up this construction somewhat using a faster basis-crashing algorithm due to Beling and Megiddo =-=[1]-=-, resulting in the following theorem: Theorem 3.1 Given a mixed strategysin sparse representation, it is possible to construct a small equivalent strategys0 using O(jSupp()j \Delta jT j 1:62 ) arithme... |

6 | Fast algorithms for nding randomized strategies in game trees - Koller, Megiddo, et al. - 1994 |

5 |
LP representation and efficient computation of behavior strategies
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Citation Context ...the observable behavior of . Realization weights were first introduced by Koller and Megiddo [4] in the context of perfect recall games. In their algorithm, and in the more recent ones of von Stengel =-=[12]-=- and of Koller, Megiddo, and von Stengel [6], the realization weights corresponding to equilibrium mixed strategies are computed directly. This relies on the fact that, for games with perfect recall, ... |

5 |
von Stengel: Fast algorithms for finding randomized strategies in game trees, 28th ACM Symposium of theory of computing ("STOC
- Koller, Megiddo, et al.
- 1994
(Show Context)
Citation Context ...hts were first introduced by Koller and Megiddo [4] in the context of perfect recall games. In their algorithm, and in the more recent ones of von Stengel [12] and of Koller, Megiddo, and von Stengel =-=[6]-=-, the realization weights corresponding to equilibrium mixed strategies are computed directly. This relies on the fact that, for games with perfect recall, realization weights can be described using a... |

3 |
Equivalence of information patterns and essentially indeterminate games
- Dalkey
- 1953
(Show Context)
Citation Context ...is reduction of the space of pure strategies forms the basis for the reduced normal form of an extensive game. Further reductions for a particular given payoff function have been considered by Dalkey =-=[3]-=- and by Swinkels [11], but these provide no additional savings in the context of a generic game. Unfortunately, the savings provided by the reduced normal form are often limited. Consider a game where... |

2 |
Extensive games and the problem of information. In: Contribution to the Theory of Games
- HW
- 1953
(Show Context)
Citation Context ...so [4]). In many interesting problems, behavior strategies play an important role. A player is said to have perfect recall if she remembers throughout the play everything she has known and done. Kuhn =-=[8]-=- showed that for a player with perfect recall, every mixed strategy has an equivalent behavior strategy. Thus, for such a player, it suffices to investigate only behavior strategies. In [4], we also m... |

2 | Lp representation and e cient computation of behavior strategies - Stengel - 1993 |

1 |
Subgames and the reduced normal form
- Swinkels
- 1989
(Show Context)
Citation Context ...space of pure strategies forms the basis for the reduced normal form of an extensive game. Further reductions for a particular given payoff function have been considered by Dalkey [3] and by Swinkels =-=[11]-=-, but these provide no additional savings in the context of a generic game. Unfortunately, the savings provided by the reduced normal form are often limited. Consider a game where the player has paral... |

1 |
Extensive games
- HW
- 1950
(Show Context)
Citation Context ...ayer. Thus, it also can be represented in linear space. A mixed strategy assigns a probability to each pure strategy. The number of pure strategies is usually exponential in the size of the game tree =-=[7,4]-=-. Thus, the size of a mixed strategy may be exponential in the size of the tree. In many useful mixed strategies, however, only a small number of pure strategies receive positive probabilities. For a ... |

1 |
Howson Jr JT (1964) Equilibrium points in bimatrix games
- CW
(Show Context)
Citation Context ...ble" size of these strategies may reduce the space and time complexities of algorithms handling them. This idea was first utilized by Wilson [14] in his modified version of the Lemke-Howson algorithm =-=[9]-=-. This algorithm searches the space of mixed strategies pairs for an equilibrium of a general two-person game. Wilson's variant represents the strategies encountered in the search sparsely; i.e. it ma... |

1 | On a generalization of the Lemke-Howson algorithm to noncooperative N-person games - Rosenmiiller - 1971 |

1 | Using fast matrix multiplication to nd basic solutions - Beling, Megiddo - 1993 |