The maximum theorem and the existence of Nash equilibrium of (generalized) games without lower semicontinuities. (English) Zbl 0761.90110
Summary: We generalize Berge’s Maximum Theorem to the case where the payoff (utility) functions and the feasible action correspondences are not lower semicontinuous. The condition we introduced is called the Feasible Path Transfer Lower Semicontinuity (in short, FPT l.s.c.). By applying our Maximum Theorem to game theory and economics, we are able to prove the existence of equilibrium for the generalized games (the so-called abstract economics) and Nash quilibrium for games where the payoff functions and the feasible strategy correspondences are not lower semicontinuous. Thus the existence theorems given in this paper generalize many existence theorems on Nash equilibrium and equilibrium for the generalized games in the literature.
Keywords:
feasible path transfer lower semicontinuity; abstract economics; Nash equilibrium; generalized gamesReferences:
| [1] | Aubin, J. P., Mathematical Methods of Games and Economic Theory (1979), North-Holland: North-Holland Amsterdam · Zbl 0452.90093 |
| [2] | Baye, M.; Tian, G.; Zhou, J., The Existence of Pure Nash Equilibrium in Games with Nonquasiconcave Payoffs (1990), Texas A & M University, mimeo |
| [3] | Berge, C., Espaces Topologiques et Fonctions Multivoques (1959), Dunod: Dunod Paris · Zbl 0088.14703 |
| [4] | Berge, C., Topological Spaces (1963), Macmillan Co: Macmillan Co New York, (Translated by E. M. Patterson) · Zbl 0114.38602 |
| [5] | Bertrand, J., Edgeworth-Bertrand duopoly revisited, (Henn, R., Operations Research-Verfahren, III (1965), Sonderdruck, Verlag: Sonderdruck, Verlag Anton Hein, Meisenhein), 55-68 · Zbl 0153.21503 |
| [6] | Borglin, A.; Keiding, H., Existence of equilibrium actions and of equilibrium: A note on the ‘new’ existence theorem, J. Math. Econom., 3, 313-316 (1976) · Zbl 0349.90157 |
| [7] | Dasgupta, P.; Maskin, E., The existence of equilibrium in discontinuous economic games. I. Theory, Rev. Econom. Stud., 53, 1-26 (1986) · Zbl 0578.90098 |
| [8] | Debreu, G., A social equilibrium existence theorem, (Proc. Nat. Acad. Sci. U.S.A., 38 (1952)) · Zbl 0047.38804 |
| [9] | Edgeworth, F. M., (Papers Relating to Political Economy, Vol. I (1953), Macmillan & Co: Macmillan & Co London) |
| [10] | Fan, K., Fixed-point and minimax theorems in locally convex topological linear spaces, (Proc. Nat. Acad. Sci. U.S.A., 38 (1952)), 131-136 |
| [11] | Fan, K., Minimax theorem, (Proc. Nat. Acad. Sci. U.S.A., 39 (1953)), 42-47 · Zbl 0050.06501 |
| [12] | Glicksberg, I. L., A further generalization of the Kakutani fixed theorem with applications to Nash equilibrium points, (Proc. Amer. Math. Soc., 38 (1952)), 170-174 · Zbl 0046.12103 |
| [13] | Khan, M. Ali, Equilibrium points of nonatomic games over a Banach space, Trans. Amer. Math. Soc., 29, 737-749 (1986) · Zbl 0594.90104 |
| [14] | Khan, M. Ali; Vohra, R., Equilibrium in abstract economics without ordered preferences and with a measure of agents, J. Math. Econom., 13, 133-142 (1984) · Zbl 0552.90009 |
| [15] | Khan, M. Ali; Papageorgiou, R., On Cournot Nash Equilibria in Generalized Quantitative Games with a Continuum of Players (1985), University of Illinois: University of Illinois Urbana, mimeo |
| [16] | Kim, T.; Prikry, K.; Yannelis, N. C., Equilibrium in Abstract Economies with a Measure Space and with an Infinite Dimensional Strategy Space (1985), University of Minnesota, mimeo |
| [17] | Nash, J., Equilibrium points in \(N\)-person games, (Proc. Nat. Acad. Sci. U.S.A., 36 (1950)), 48-49 · Zbl 0036.01104 |
| [18] | Nash, J., Non-cooperative games, Ann. of Math., 54, 286-295 (1951) · Zbl 0045.08202 |
| [19] | Nikaido, H.; Isoda, K., Note on noncooperative convex games, Pacific J. Math., 4, 65-72 (1955) · Zbl 0171.40903 |
| [20] | Shafer, W.; Sonnenschein, H., Equilibrium in abstract economies without ordered preferences, J. Math. Econom., 2, 345-348 (1975) · Zbl 0312.90062 |
| [21] | Shih, M. H.; Tan, K. K., Generalized quasi-variational inequalities in locally convex topological vector spaces, J. Math. Anal. Appl., 108, 333-343 (1985) · Zbl 0656.49003 |
| [22] | Tian, G., Equilibrium in abstract economies with a non-compact infinite dimensional strategy space, an infinite number of agents and without ordered preferences, Econom. Lett., 33, 203-206 (1990) · Zbl 1375.91148 |
| [23] | Tian, G., Fixed points theorems for mapping with non-compact and non-convex domains, J. Math. Anal. Appl., 158, 161-167 (1991) · Zbl 0735.47037 |
| [24] | Toussaint, S., On the existence of equilibria in economics with infinitely many commodities, J. Econom. Theory, 33, 98-115 (1984) · Zbl 0543.90016 |
| [25] | Walker, M., A generalization of the maximum theorem, Internat. Econom. Rev., 20, 267-270 (1979) · Zbl 0406.90001 |
| [26] | Yannelis, N. C., Equilibria in noncooperative models of competition, J. Econom. Theory, 41, 96-111 (1987) · Zbl 0619.90011 |
| [27] | Yannelis, N. C.; Prabhakar, N. D., Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econom., 12, 223-245 (1983) · Zbl 0536.90019 |
| [28] | Yen, C. L., A minimax inequality and its applications to variational inequalities, Pacific J. Math., 97, 477-481 (1981) · Zbl 0493.49009 |
| [29] | Zhou, J.; Chen, G., Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities, J. Math. Anal. Appl., 132, 213-225 (1988) · Zbl 0649.49008 |
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