Chapter 2 Computation of equilibria in finite games

Handbook of Computational Economics

Volumn 1, Issue , 1996, Pages 87-142

  McKelvey, Richard D ^a,b   McLennan, Andrew ^a,b  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^b UNIVERSITY OF MINNESOTA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 70350102387     PISSN: 15740021     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1574-0021(96)01004-0     Document Type: Review

References (81)

1. mimeo. Azhar S., McLennan A., and Reif J.H. Computation of equilibria in noncooperative games (1992), University of Minnesota
2. Banks J.S., and Sobel J. Equilibrium selection in signalling games. Econometrica 55 (1987) 647-661
3. Ben-Or M., Kozen D., and Reif J. The complexity of elementary algebra and geometry. Journal of Computer and System. Sciences 32 (1986) 251-264
4. Ben-Or M., Feig E., Kozen D., and Tiwari P. A fast parallel algorithm for determining all roots of a polynomial with real roots. SIAM Journal of Computation 17 (1988) 1081-1092
5. Blair J.R.S., and Mutchler D. Pure strategy equilibria in the presence of imperfect information: NP-hard problems. University of Tennessee, Dept. Computer Science. Tech. Report CS-93-220 (1993)
6. Blume L., and Zame W.R. The algebraic geometry of perfect and sequential equilibrium. Econometrica 62 (1994) 783-794
7. Brown G.W., and von Neumann J. Solutions of games by differential equations. In: Kuhn H.W., and Tucker A.W. (Eds). Contributions to the theory of games. Annals of Mathematical Studies Number 24 (1950), Princeton Univ. Press, Princeton, NJ 73-79
8. Canny J. Some algebraic and geometric computations in PSPACE. ACM symposium on theory of computing (1988) 460-467
9. Canny J., and Emeris I. An efficient algorithm for the sparse mixed resultant. Proceedings, AAEEC (1993) 89-104
10. Cho I. A refinement of sequential equilibrium. Econometrica 55 (1987) 1367-1389
11. Cho I., and Kreps D.M. Signalling games and stable equilibrium. Quarterly Journal of Economics 102 (1987) 179-221
12. Collins G.E. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. Second Gl conference on automata theory and formal languages. Lecture notes in computer science Vol. 33 (1975), Springer, Berlin 134-183
13. Dickhaut J., and Kaplan T. A program for finding Nash equilibria. The Mathematica Journal 1 (1991) 87-93
14. Doup T.M. Simplicial algorithms on the simplotope. Lecture notes in economics and mathematical systems, #318 (1988), Springer, Berlin
15. Doup T.M., and Talman A.J.J. A new simplicial variable dimension algorithm to find equilibria on the product space of unit simplices. Mathematical Programming 37 (1987) 319-355
16. Doup T.M., and Talman A.J.J. A continuous deformation algorithm on the product space of unit simplices. Mathematics of Operations Research 12 (1987) 485-521
17. Drexler F.J. A homotopy method for the calculation of all zeros of polynomial ideals. In: Wacker H. (Ed). Continuation methods (1978), Academic Press, New York
18. Eaves B.C. The linear complementarity problem. Management Science 17 (1971) 612-634
19. Eaves B.C. Homotopies for computation of fixed points. Mathematical Programming 3 (1972) 1-22
20. El-Gamal M., McKelvey R.D., and Palfrey T. A Bayesian sequential experimental study of learning in games. Journal of the American Statistical Society 42 (1993) 428-435
21. Garcia C.B., Lemke C.E., and Luethi H. Simplicial approximation of an equilibrium point for non-cooperative n-person games. In: Hu T.C., and Robinson S.M. (Eds). Mathematical programming (1973), Academic Press, New York 227-260
22. Garcia C.B., and Zangwill W.I. Finding all solutions to polynomial systems and other systems of equations. Mathematical Programming 16 (1979) 159-176
23. Garcia C.B., and Zangwill W.I. Global calculation methods for finding all solutions to polynomial systems of equations in n variables. Extremal methods and systems analysis (1980), Springer, Heidelberg/New York
24. Gul F., Pearce D., and Stacchetti E. A bound on the proportion of pure strategy equilibria in generic games. Mathematics of Operations Research 18 (1993) 548-552
25. Harker P.T., and Pang J. Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Mathematical Programming 48 (1990) 161-220
26. Hironaka H. Triangulation of algebraic sets. AMS Symposium in Pure Mathematics 29 (1975) 165-185
27. Hirsch M.D., Papadimitriou C.H., and Vavasis S.A. Exponential lower bounds for finding Brouwer fixed points. Journal of Complexity 5 (1989) 379-416
28. Huber B., and Sturmfels B. A polyhedral method for solving sparse polynomial systems. Mathematics of Computation 64 (1995) 1541
29. Kalai E., and Samet D. Persistent equilibria. International Journal of Game Theory 13 (1984) 129-144
30. Kohlberg E., and Mertens J.F. On the strategic stability of equilibria. Econometrica 54 (1986) 1003-1037
31. mimeo. Kohlberg E., and Reny P.J. An interpretation of consistent assessments (1993)
32. Koller D., and Megiddo N. The complexity of two person zero-sum games in extensive form. Games and Economic Behavior 4 (1992) 528-552
33. forthcoming. Koller D., and Megiddo N. Finding mixed strategies with small supports in extensive games. International Journal of Game Theory (1996)
34. forthcoming. Koller D., Megiddo N., and von Stengel B. Efficient solutions of extensive two-person games. Games and Economic Behavior (1996)
35. Kostreva M.M. Nonconvexity in noncooperative game theory. International Journal of Game Theory 18 (1989) 247-259
36. Kostreva M.M., and Kinard L.A. A differential homotopy approach for solving polynomial optimization problems and noncooperative games. Computers and Mathematics with Applications 21 (1991) 135-143
37. Kreps D.M., and Wilson R. Sequential equilibria. Econometrica 50 (1982) 863-894
38. Kuhn H.W. Extensive games and the problem of information. In: Kuhn H.W., and Tucker A.W. (Eds). Contribution to the theory of games Vol. II (1953) 193-216
39. Kuhn H.W., and MacKinnon J.G. Sandwich method for finding fixed points. Journal of Optimization Theory and Applications 17 (1975) 189-204
40. Lemke C.E. Bimatrix equilibrium points and mathematical programming. Management Science 11 (1965) 681-689
41. Lemke C.E., and Howson Jr. J.T. Equilibrium points in bimatrix games. SIAM Journal of Applied Mathematics 12 (1964) 413-423
42. Mangasarian O.L. Equilibrium points in bimatrix games. Journal of the Society for Industrial and Applied Mathematics 12 (1964) 778-780
43. Mathiesen L. An algorithm based on a sequence of linear complementarity problems applied to a Walrasian equilibrium model: An example. Mathematical Programming 37 (1987) 1-18
44. McKelvey R.D. A Liapunov function for Nash equilibria. California Institute of Technology, Social Science Working Paper #953 (1996)
45. forthcoming. McKelvey R.D., and McLennan A. The maximal generic number of totally mixed Nash equilibria. Journal of Economic Theory (1996), Department of Economics, University of Minnesota
46. McKelvey R.D., and Palfrey T.R. An experimental analysis of the centipede game. Econometrica 60 (1992) 803-836
47. McKelvey R.D., and Palfrey T.R. Quantal response equilibria for normal form games. Games and Economic Behavior 10 (1995) 6-38
48. McKelvey R.D., and Palfrey T.R. Quantal response equilibria for extensive form games. California Institute of Technology, Social Science Working Paper #947 (1995)
49. McLennan A. Justifiable beliefs in sequential equilibrium. Econometrica 53 (1985) 889-904
50. Merrill O.H. Applications and extensions of an algorithm that computes fixed points of certain upper semi-continuous point to set mappings. PhD thesis (1972), University of Michigan
51. Mertens J. Stable equilibria, a reformulation. Mathematics of Operations Research 14 (1989) 575-625
52. Myerson R.B. Refinements of the Nash equilibrium concept. International Journal of Game Theory 7 (1978) 73-80
53. Murty K.G. Computational complexity and linear pivot methods. Mathematical Programming Study 7 (1978) 61-73
54. Nash J.F. Noncooperative games. Annals of Mathematics 54 (1951) 289-295
55. Rosen J.B. Existence and uniqueness of equilibrium points for concave N-person games. Econometrica 33 (1965) 520-534
56. Rosenmüller J. On a generalization of the Lemke-Howson algorithm to noncooperative N-person games. SIAM Journal of Applied Mathematics 1 (1971) 73-79
57. Saigal R. On the convergence rate of algorithms for solving equations that are based on methods of complementary pivoting. Mathematics of Operations Research 2 (1977) 108-124
58. Scarf H. The approximation of fixed points of a continuous mapping. SIAM Journal of Applied Mathematics 15 (1967) 1328-1343
59. Scarf H. The computation of economic equilibria (1973), Yale Univ. Press, New Haven, CT
60. Schanuel S.H., Simon L.K., and Zame W.R. The algebraic geometry of games and the tracing procedure. In: Selten R. (Ed). Game equilibrium models, Vol. II: Methods, morals and markets (1991), Springer, Berlin
61. Selten R. Reexamination of the perfectness concept for equilibrium points in extensive games. International Journal of Game Theory 4 (1975) 25-55
62. Shapley L. A note on the Lemke-Howson algorithm. Mathematical Programming Study 1 (1974) 175-189
63. Shapley L.S. On balanced games without sidepayments. In: Hu T.C., and Robinson S.M. (Eds). Mathematical programming (1973), Academic Press, New York 261-290
64. Shapley L. On the accessibility of fixed points. In: Moeschlin O., and Pallaschke D. (Eds). Game theory and mathematical economics (1981), North-Holland, Amsterdam 367-377
65. Smale S. A convergent process of price adjustment and global Newton methods. Journal of Mathematical Economics 3 (1976) 107-120
66. Talman A.J.J., and Yang Z. A simplicial algorithm for computing proper equilibria of finite games. Center Discussion Paper 9418 (1994), Tilburg University
67. Tarski A. A decision method for elementary algebra and geometry (1951), Univ. of California Press, Berkeley, CA
68. Todd M.J. On the computational complexity of piecewise-linear homotopy algorithms. Mathematical Programming 24 (1982) 216-224
69. Todd M.J. The computation of fixed points and applications (1976), Springer, Berlin
70. van den Elzen A.H., and Talman A.F.F. Finding a Nash equilibrium in noncooperative n-person games by solving a sequence of linear stationary point problems. ZOR-methods and Models of Operations Research 35 (1994) 27-43
71. van der Laan G., and Talman A.J.J. A restart algorithm for computing fixed points without an extra dimension. Mathematical Programming 17 (1979) 74-84
72. van der Laan G., and Talman A.J.J. On the computation of fixed points in the product space of unit simplices and an application to noncooperative n-person games. Mathematics of Operations Research 7 (1982) 1-13
73. van der Laan G., and Talman A.J.J. Simplicial approximation of solutions to the nonlinear complementarity problem with lower and upper bounds. Mathematical Programming 38 (1987) 1-15
74. van der Laan G., Talman A.J.J., and van der Heyden L. Simplicial variable dimension algorithms for solving the nonlinear complementarity problem on a product of unit simplices using a general labelling. Mathematics of Operations Research 12 (1987) 377-397
75. van der Waerden B.L. Modern algebra (1949), Ungar, New York
76. forthcoming. von Stengel B. Efficient computation of behavior strategies. Games and Economic Behavior (1996)
77. Wilson R. Computing equilibria of n-person games. SIAM Journal of Applied Mathematics 21 (1971) 80-87
78. Wilson R. Computing equilibria of two-person games from the extensive form. Management Science 18 (1972) 448-460
79. Wilson R. Computing simply stable equilibria. Econometrica 60 (1992) 1039-1070
80. Yamamoto Y. A path-following procedure to find a proper equilibrium of finite games. International Journal of Game Theory 22 (1993) 49-59
81. Zangwill W.I., and Garcia C.B. Pathways to solutions, fixed points, and equilibria (1981), Prentice-Hall, Englewood Cliffs, NJ