## Efficient Computation of Behavior Strategies (1996)

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Venue: | GAMES AND ECONOMIC BEHAVIOR |

Citations: | 47 - 8 self |

### BibTeX

@ARTICLE{Stengel96efficientcomputation,

author = {Bernhard von Stengel},

title = {Efficient Computation of Behavior Strategies},

journal = {GAMES AND ECONOMIC BEHAVIOR},

year = {1996},

volume = {14},

pages = {220--246}

}

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### Abstract

We propose the sequence form as a new strategic description for an extensive game with perfect recall. It is similar to the normal form but has linear instead of exponential complexity, and allows a direct representation and efficient computation of behavior strategies. Pure strategies and their mixed strategy probabilities are replaced by sequences of consecutive choices and their realization probabilities. A zero-sum game is solved by a corresponding linear program that has linear size in the size of the game tree. General two-person games are studied in the paper by Koller, Megiddo, and von Stengel in this journal issue.

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Citation Context ...tegies are mutual best responses. A mixed strategy is a best response to a mixed strategy of the other player if and only if every pure strategy selected with positive probability is a best response (=-=Nash 1951-=-, p. 287). Player 2 can therefore assign positive probabilities to c and d since both give him the same maximum expected payoff \Gamma1. Conversely, against the mixed strategy ( 1 = 2 ; 1 = 2 ) of pla... |

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Citation Context ...l choices c 2 C u produce the equally `optimal' payoff contribution G 1 (oe u c; y) = 0 in (6.5). These arbitrary moves need not be optimal in a subgame, as it is required by subgame perfectness (see =-=Selten 1975-=-, p. 33). To conclude this section, we describe a nonlinear optimization problem whose solutions are the equilibria of an N -person game. In this problem, the components of all realization plans r 1 ;... |

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Citation Context ...lated to backward induction. That notion is used to distinguish certain equilibria as perfect or subgame perfect (see Selten 1975), and in many other definitions of `stability' of an equilibrium (see =-=Kohlberg and Mertens 1986-=-, van Damme 1987). So far, the equilibria that can be computed with the sequence form are arbitrary and need not induce an equilibrium in every subgame, in which case they are not subgame perfect. On ... |

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Citation Context ...rop. 3.4, this is the case if the information set u of player i cannot be reached when fi i is played, that is, if r i (oe u ) = 0. Then u is called irrelevant when playing fi i , otherwise relevant (=-=Kuhn 1953-=-, p. 201). In this case, the behavior at u under fi i is arbitrary and therefore not unique (if u has at least two choices). In particular, more than one pure strategy may define the same realization ... |

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Citation Context ...subject to linear equations analogous to (3.4). Instead of sequences of choices, they considered nodes of the game tree for defining realization plans, but this is not essentially different (see also =-=Koller and Megiddo 1992-=-, Remark 3.8, p. 545). A minor point is that they overlooked the possibility r i (oe u ) = 0 in (3.6), so the behavior strategy fi i cannot always be uniquely reconstructed from the realization plan r... |

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Citation Context ...ctice, the LP (5.6) is very suitable for the simplex algorithm, which computes an optimal pair of primal and dual solutions. Efficient implementations exploit the sparsity of the matrices A, E and F (=-=Chv'atal 1983-=-, p. 112). The running time of the simplex algorithm can be exponential but is usually quite short. It is mostly determined by the number of constraints and very little by the number of variables (Chv... |

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Citation Context ...d strategies of the other players, both strategies define the same probabilities for reaching the nodes of the game tree. Looking at these probabilities, it is easy to see the following (compare also =-=Koller and Megiddo 1995-=-, Lemma 2.5): Proposition 4.1. Mixed strategies are realization equivalent if and only if they have the same realization plan. (For the `only if' part, any node of the game tree must be reachable for ... |

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Citation Context ... her opponent has only perfect memory. They considered the complete inflation of the game obtained by partitioning the information sets of player 2 such that he can distinguish his earlier moves (see =-=Dalkey 1953, p.-=- 226, Okada 1985, p. 90). In the modified game, player 2 also has perfect recall, and his best response �� 2 to a realization plan x can be computed as before. Furthermore, his best response can b... |

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Citation Context ...ound by linear programming, but the vast increase in the description can make the problem computationally intractable and may even force the analyst to abandon the game theoretic approach altogether (=-=Lucas 1972-=-, p. P-9). In this paper, we present a computational method without these disadvantages. It is based on a new strategic description of an extensive game, called the sequence form. Based on the sequenc... |

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