## Computing Equilibria for Two-Person Games (1999)

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@MISC{Stengel99computingequilibria,

author = {Bernhard von Stengel},

title = {Computing Equilibria for Two-Person Games},

year = {1999}

}

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### Abstract

### Citations

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922 |
Non-Cooperative Games
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- 1951
(Show Context)
Citation Context ... entries of Ay. Any pure strategy i, 1sism, of player 1 is a best response to y iff the ith component of the slack vector 1mu \Gamma Ay is zero. So (2.7) amounts to the following well-known property (=-=Nash, 1951-=-): A strategy x is a best response to y iff it only plays pure strategies that are best responses with positive probability. For player 2, strategy y is a best response to x iff it maximizes (x ? B)y ... |

611 | The Linear Complementarity Problem
- Cottle, Pang
- 1992
(Show Context)
Citation Context ...y simple since it amounts to the combinatorial problem of deciding which pure strategies may have positive probability. There are various solutions methods for LCPs (for a comprehensive treatment see =-=Cottle, Pang, and Stone, 1992-=-). The most important method for finding one solution of the LCP in Theorem 2.4 is the Lemke--Howson algorithm. 2.4. The Lemke--Howson algorithm In their seminal paper, Lemke and Howson (1964) describ... |

479 | A General Theory of Equilibrium Selection in Games - Harsanyi, Selten - 1988 |

465 |
Reexamination of the Perfectness Concept for Equilibrium Points
- Selten
- 1975
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Citation Context ... the mistake probabilities go to zero. At the same time, the equilibrium strategies have to be best responses to these completely mixed strategies. We use this characterization of perfect equilibria (=-=Selten, 1975, p. 50, Theorem 7) as de-=-finition. Definition 3.2. (Selten, 1975.) An equilibrium (x; y) of a bimatrix game is called perfect if there is a continuous function " 7! (x("); y(")) where (x("); y(")) is ... |

287 | Computational geometry: An introduction through randomized algorithms - Mulmuley - 1994 |

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

202 |
Equilibrium points of bimatrix games
- Lemke, Howson
- 1964
(Show Context)
Citation Context ...x of G, and y is a vertex of G 2 . The edges of G are given by fxg \Theta e 2 for vertices x of G 1 and edges e 2 of G 2 , or e 1 \Theta fyg for edges e 1 of G 1 and vertices y of G 2 . Theorem 2.6. (=-=Lemke and Howson, 1964-=-; Shapley, 1974.) Let (A; B) be a nondegenerate bimatrix game and k be a label in I [ J . Then M(k) in (2.13) consists of disjoint paths and loops in the product graph G of G 1 and G 2 . The endpoints... |

189 | A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra, Discrete Comput - Avis, Fukuda - 1992 |

159 | On the complexity of the parity argument and other inefficient proofs of existence - Papadimitriou - 1994 |

154 | A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems - Kojima, Megiddo, et al. - 1991 |

153 | Refinements of the Nash equilibrium concept - Myerson - 1978 |

152 | Stability and Perfection of Nash Equilibria - Damme - 1991 |

151 |
The maximum numbers of faces of a convex polytope
- McMullen
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(Show Context)
Citation Context ..., however. The upper bound is obtained for the so-called dual neighborly polytopes, which have the number of vertices stated in the following result. Theorem 2.12. (Upper bound theorem for polytopes, =-=McMullen, 1970-=-.) The maximum number of vertices of a d-dimensional polytope with k facets is \Phi(d; k) = / k \Gamma b d\Gamma1 2 c \Gamma 1 b d 2 c ! + / k \Gamma b d 2 c \Gamma 1 b d\Gamma1 2 c ! : (2:27) In (2.2... |

129 | Nash and correlated equilibria: Some complexity considerations - Gilboa, Zemel - 1989 |

128 | Representations and solutions for game-theoretic problems - Koller, Pfeffer - 1997 |

126 | Computation of equilibria in finite games
- McKelvey, McLennan
- 1996
(Show Context)
Citation Context ...ce and game theory. The usefulness of algorithms for solving games should be tested further in practice. Many of the described methods are being implemented in the project GAMBIT (see its overview in =-=McKelvey and McLennan, 1996, or-=- its World Wide Web site at http://www.hss.caltech.edu/��gambit/Gambit.html). This is a program for building and solving extensive games interactively or with the help of a command language. For t... |

99 |
Extensive Games and the Problem of Information." in Contributions to the Theory
- Kuhn
- 1953
(Show Context)
Citation Context ...he tree, and is denoted oe i (t). For example, for the leftmost leaf t in Figure 3.4 this sequence is LS for player 1 and l for player 2. The empty sequence is denoted ;. Player i has perfect recall (=-=Kuhn, 1953-=-) iff oe i (s) = oe i (t) for any nodes s; t 2 h and h 2 H i . Then the unique sequence oe i (t) leading to any node t in h will be denoted oe h . Perfect recall means that the player cannot get addit... |

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

93 | Sequential equilibria - Kreps, Wilson - 1982 |

91 |
Bimatrix equilibrium points and mathematical programming
- Lemke
- 1965
(Show Context)
Citation Context ...lgorithm terminates, or another component of x; y; w, or z , whose complement is then chosen as the next entering variable. This process continues until an equilibrium is found. (For more details see =-=Lemke, 1965-=-; Murty, 1988; or Cottle et al., 1992.) Here, Lemke's algorithm can be interpreted as inducing a path in the strategy space X \Theta Y that starts at (p; q) and ends at the computed equilibrium. The c... |

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

75 | The complexity of two-person zero-sum games in extensive form
- Koller, Megiddo
- 1992
(Show Context)
Citation Context ...nces of moves instead of arbitrary combinations of moves as in pure strategies. The realization probabilities of sequences can be characterized by linear equations if the players have perfect recall (=-=Koller and Megiddo, 1992-=-). In turn, the probabilities for sequences define a behavior strategy for each player. This defines the sequence form of the game that is analogous to the normal form but has small size. The solution... |

55 |
Linear programming
- Chvátal
- 1980
(Show Context)
Citation Context ...ay possibly cycle. The solution to this problem is the so-called lexicographic rule for choosing the leaving variable in the case of ties, which is well known in linear programming (see, for example, =-=Chv'atal, 1983-=-, p. 36), and has been suggested by Lemke and Howson (1964) in terms of perturbations. For a general lexicographic treatment of LCPs see Eaves (1971). Consider a system Dr = b of k linearly independen... |

50 |
Exponential lower bounds for finding Brouwer fixed points
- Hirsch, Papadimitriou, et al.
- 1989
(Show Context)
Citation Context ...ntinuous mapping on a simplex. The approximation of such fixed points takes worst-case exponential time in the dimension and the number of digits of accuracy if the mapping is evaluated as an oracle (=-=Hirsch, Papadimitriou, and Vavasis, 1989). All kno-=-wn fixed-point approximations are such oracle algorithms, but a more efficient method might well "look into" the function and use its specific representation, like the function used in the p... |

50 | On total functions, existence theorems, and computational complexity. Theoretical Computer Science 81(2):317–324 - Megiddo, Papadimitriou - 1991 |

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

36 | On the order of eliminating dominated strategies - Gilboa, Kalai, et al. - 1990 |

35 |
A note on the Lemke-Howson algorithm
- Shapley
- 1974
(Show Context)
Citation Context ...x of G 2 . The edges of G are given by fxg \Theta e 2 for vertices x of G 1 and edges e 2 of G 2 , or e 1 \Theta fyg for edges e 1 of G 1 and vertices y of G 2 . Theorem 2.6. (Lemke and Howson, 1964; =-=Shapley, 1974-=-.) Let (A; B) be a nondegenerate bimatrix game and k be a label in I [ J . Then M(k) in (2.13) consists of disjoint paths and loops in the product graph G of G 1 and G 2 . The endpoints of the paths a... |

33 |
Reduction of a game with complete memory to a matrix game. Soviet Mathematics 3:678–681
- Romanovskii
- 1962
(Show Context)
Citation Context ...sing linear programming duality) can analogously be applied to the sequence form. The equilibria of a zero-sum game are the solutions to a linear program that has the same size as the extensive game (=-=Romanovskii, 1962-=-; von Stengel, 1996). The complementary pivoting algorithm by Lemke (1965) is applied by von Stengel, van den Elzen and Talman (1996) to the sequence form of non-zero-sum games, analogous to the algor... |

27 | Equilibrium points of bimatrix games - Mangasarian - 1964 |

27 | Evolutionary stability in extensive two-person games - Selten - 1983 |

24 | Finding mixed strategies with small support in extensive form games - Koller, Megiddo - 1996 |

21 | A note on strategy elimination in bimatrix games
- Knuth, Papadimitriou, et al.
- 1988
(Show Context)
Citation Context ... A = 2 1 1 1 ; B = 1 1 1 2 ; (2:22) where eliminating first the bottom row and then the right column yields a different game than eliminating first the left column and then the top row, for example. (=-=Knuth, Papadimitriou, and Tsitsiklis, 1988-=-, study computational aspects of strategy elimination where they overlook this fact.) It is often assumed that a game is nondegenerate on the grounds that this is true for a generic game. A generic ga... |

18 | Computing Equilibria of Two-person Games from the Extensive Form - WILSON |

18 | Two-person nonzero-sum games and quadratic programming - Mangasarian, Stone - 1964 |

17 | The linear complementarity problem - Eaves - 1971 |

17 | New maximal numbers of equilibria in bimatrix games - Stengel |

17 | A note on the complexity of P-matrix LCP and computing an equilibrium. IBM Research Report 6439. Available at: http://theory.stanford.edu/ megiddo/pdf/plcp.pdf - Megiddo - 1988 |

17 | Game-theoretic aspects of computing - Linial - 1994 |

14 |
On the maximal number of Nash equilibria in an n × n bimatrix game
- Keiding
- 1997
(Show Context)
Citation Context ...is part of at most one equilibrium, so the smaller number of vertices of the polytope P 1 or P 2 is a bound for the number of equilibria. That is, Theorem 2.12 implies the following. Corollary 2.13. (=-=Keiding, 1997-=-.) A nondegenerate bimatrix game has at most minf\Phi(m; n +m); \Phi(n; m + n)g \Gamma 1 equilibria. It is not hard to show that m ! n implies \Phi(m; n + m) ! \Phi(n; m + n). The case m = n = d, that... |

14 | A theorem on the number of Nash equilibria in a bimatrix game
- Quint, Shubik
- 1997
(Show Context)
Citation Context ...ure strategy in C with probability 1=jCj. Then both P 1 and P 2 are equal to the unit cube. Possibly, this is a tighter bound for the number of equilibria than \Phi(d; 2d) \Gamma 1. Conjecture 2.14. (=-=Quint and Shubik, 1994-=-.) A nondegenerate d \Theta d bimatrix game has at most 2 d \Gamma 1 equilibria. This conjecture is a consequence of Corollary 2.13 for ds3 but not for d ? 3. For d = 4, it was shown by Keiding (1997)... |

12 | A Program for Finding Nash Equilibria - Dickhaut, Kaplan - 1993 |

12 | An algorithm for equilibrium points in bimatrix games - Kuhn - 1961 |

11 | Some topics in two-persons games - Parthasarathy, Raghavan - 1971 |

11 | Generic 4 × 4 two person games have at most 15 Nash equilibria - McLennan, Park - 1997 |

10 | Constrained games and linear programming - Charnes - 1953 |

10 | Maximal Nash subsets for bimatrix games - Jansen - 1981 |

10 | An algorithm to determine all equilibrium points of a bimatrix game - Winkels - 1979 |

10 | Enumeration of all extreme equilibria of bimatrix games - Audet, Hansen, et al. |