American Institute of Mathematical Sciences

January  2012, 8(1): 179-187. doi: 10.3934/jimo.2012.8.179

Topological essentiality in infinite games

 1 School of Mathematics and Computer Science, Guizhou Normal University, Guizhou, Guiyang 550001, China 2 Department of Mathematics, Guizhou Uniersity, Guizhou, Guiyang 550025, China 3 Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China

Received  February 2011 Revised  September 2011 Published  November 2011

By constructing a corresponding Nash map, we prove that every infinite game with compact metrizable sets of strategies and continuous payoffs has such a topological essential component that contains a minimal payoff-wise essential set containing a stable set, and deduce that every topological essential equilibrium is payoff-wise essential and so is perfect.
Citation: Yonghui Zhou, Jian Yu, Long Wang. Topological essentiality in infinite games. Journal of Industrial & Management Optimization, 2012, 8 (1) : 179-187. doi: 10.3934/jimo.2012.8.179
References:
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References:
 [1] N. Al-Najjar, Strategically stable equilibria in games with infinitely many pure strategies,, Math. Soc. Sci., 29 (1995), 151.  doi: 10.1016/0165-4896(94)00765-Z.  Google Scholar [2] P. Billingsley, "Convergence of Probability Measures,", John Wiley & Sons, (1968).   Google Scholar [3] K. Fan, Fixed-point and minimax theorems in locally convex linear spaces,, Proc. Natl. Acad. Sci. USA, 38 (1952), 121.  doi: 10.1073/pnas.38.2.121.  Google Scholar [4] D. Fudenberg and D. Levine, Subgame perfect equilibria of finite- and infinite-horizon games,, J. Economic Theory, 31 (1983), 251.   Google Scholar [5] D. Fudenberg and D. Levine, Limit games and limit equilibria,, J. Economic Theory, 38 (1986), 261.  doi: 10.1016/0022-0531(86)90118-3.  Google Scholar [6] I. Glicksberg, A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points,, Proc. Amer. Math. Soc., 3 (1952), 170.   Google Scholar [7] S. Govindan and R. Wilson, Essential equilibria,, Proc. Natl. Acad. Sci. USA, 102 (2005), 15706.  doi: 10.1073/pnas.0506796102.  Google Scholar [8] J. Hillas, On the definition of the strategic stability of equilibria,, Econometrica, 58 (1990), 1365.  doi: 10.2307/2938320.  Google Scholar [9] J. Jiang, Essential equilibrium points of n-person non-cooperative games. II,, Sci. Sinica, 12 (1963), 651.   Google Scholar [10] J. Jiang, Essential component of the set of fixed points of the multivalued mappings and its application to the theory of games,, Sci. Sinica, 12 (1963), 951.   Google Scholar [11] E. Klein and A. Thompson, "Theory of Correspondences. Including Applications to Mathematical Economics,", Canadian Mathematical Society Series of Monographs and Advanced Texts, (1984).   Google Scholar [12] E. Kohlberg and J. Mertens, On the strategic stability of equilibria,, Econometrica, 54 (1986), 1003.  doi: 10.2307/1912320.  Google Scholar [13] A. McLennan, Consistent conditional beliefs in noncooperative game theory,, Int. J. of Game Theory, 18 (1989), 175.  doi: 10.1007/BF01268156.  Google Scholar [14] J. F. Nash, Jr., Equilibrium points in $n$-person games,, Proc. Natl. Acad. Sci. USA, 36 (1950), 48.  doi: 10.1073/pnas.36.1.48.  Google Scholar [15] J. Nash, Non-cooperative games,, Ann. Math. (2), 54 (1951), 286.  doi: 10.2307/1969529.  Google Scholar [16] B. O'Neill, Essential sets and fixed points,, Am. J. Math., 75 (1953), 497.   Google Scholar [17] R. Selten, Reexamination of the perfectness concept for equilibrium points in extensive games,, Int. J. of Game Theory, 4 (1975), 25.  doi: 10.1007/BF01766400.  Google Scholar [18] L. Simon, Local perfection,, J. Economic Theory, 43 (1987), 134.  doi: 10.1016/0022-0531(87)90118-9.  Google Scholar [19] L. Simon and M. Stinchcombe, Equilibrium refinement for infinite normal-form games,, Econometrica, 63 (1995), 1421.  doi: 10.2307/2171776.  Google Scholar [20] A. Tychonoff, Ein fixpunktsatz,, Math. Ann., 111 (1935), 767.   Google Scholar [21] E. van Damme, "Stability and Perfection of Nash Equilibria,", Second edition, (1991).   Google Scholar [22] W. Wu and J. Jiang, Essential equilibrium points of n-person non-cooperative games,, Sci. Sinica, 11 (1962), 1307.   Google Scholar [23] Y. Zhou, J. Yu and L. Wang, A new proof of existence of equilibria in infinite normal form games,, Appl. Math. Lett., 24 (2011), 253.  doi: 10.1016/j.aml.2010.09.014.  Google Scholar [24] Y. Zhou, J. Yu and S. Xiang, Essential stability in games with infinitely many pure strategies,, Int. J. of Game Theory, 35 (2007), 493.  doi: 10.1007/s00182-006-0063-0.  Google Scholar [25] Y. Zhou, J. Yu and S. Xiang, A metric on the space of finite measures with an application to fixed point theory,, Appl. Math. Lett., 21 (2008), 489.  doi: 10.1016/j.aml.2007.05.015.  Google Scholar
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