Topological essentiality in infinite games

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

Topological essentiality in infinite games

Yonghui Zhou ^1, , Jian Yu ^2, and Long Wang ^3,

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

Abstract
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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.

Keywords: Nash map, topological essentiality, payoff- wise essentiality., Infinite game, perfectness.

Mathematics Subject Classification: Primary: 91A10, 60B10; Secondary: 46G1.

Citation: Yonghui Zhou, Jian Yu, Long Wang. Topological essentiality in infinite games. Journal of Industrial and Management Optimization, 2012, 8 (1) : 179-187. doi: 10.3934/jimo.2012.8.179

References:

[1]	N. Al-Najjar, Strategically stable equilibria in games with infinitely many pure strategies, Math. Soc. Sci., 29 (1995), 151-164. doi: 10.1016/0165-4896(94)00765-Z.
[2]	P. Billingsley, "Convergence of Probability Measures," John Wiley & Sons, Inc., New York-London-Sydney, 1968.
[3]	K. Fan, Fixed-point and minimax theorems in locally convex linear spaces, Proc. Natl. Acad. Sci. USA, 38 (1952), 121-126. doi: 10.1073/pnas.38.2.121.
[4]	D. Fudenberg and D. Levine, Subgame perfect equilibria of finite- and infinite-horizon games, J. Economic Theory, 31 (1983), 251-268.
[5]	D. Fudenberg and D. Levine, Limit games and limit equilibria, J. Economic Theory, 38 (1986), 261-279. doi: 10.1016/0022-0531(86)90118-3.
[6]	I. Glicksberg, A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points, Proc. Amer. Math. Soc., 3 (1952), 170-174.
[7]	S. Govindan and R. Wilson, Essential equilibria, Proc. Natl. Acad. Sci. USA, 102 (2005), 15706-15711. doi: 10.1073/pnas.0506796102.
[8]	J. Hillas, On the definition of the strategic stability of equilibria, Econometrica, 58 (1990), 1365-1390. doi: 10.2307/2938320.
[9]	J. Jiang, Essential equilibrium points of n-person non-cooperative games. II, Sci. Sinica, 12 (1963), 651-671.
[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), 951-964.
[11]	E. Klein and A. Thompson, "Theory of Correspondences. Including Applications to Mathematical Economics," Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984.
[12]	E. Kohlberg and J. Mertens, On the strategic stability of equilibria, Econometrica, 54 (1986), 1003-1037. doi: 10.2307/1912320.
[13]	A. McLennan, Consistent conditional beliefs in noncooperative game theory, Int. J. of Game Theory, 18 (1989), 175-184. doi: 10.1007/BF01268156.
[14]	J. F. Nash, Jr., Equilibrium points in $n$-person games, Proc. Natl. Acad. Sci. USA, 36 (1950) 48-49. doi: 10.1073/pnas.36.1.48.
[15]	J. Nash, Non-cooperative games, Ann. Math. (2), 54 (1951), 286-295. doi: 10.2307/1969529.
[16]	B. O'Neill, Essential sets and fixed points, Am. J. Math., 75 (1953), 497-509.
[17]	R. Selten, Reexamination of the perfectness concept for equilibrium points in extensive games, Int. J. of Game Theory, 4 (1975), 25-55. doi: 10.1007/BF01766400.
[18]	L. Simon, Local perfection, J. Economic Theory, 43 (1987), 134-156. doi: 10.1016/0022-0531(87)90118-9.
[19]	L. Simon and M. Stinchcombe, Equilibrium refinement for infinite normal-form games, Econometrica, 63 (1995), 1421-1443. doi: 10.2307/2171776.
[20]	A. Tychonoff, Ein fixpunktsatz, Math. Ann., 111 (1935), 767-776.
[21]	E. van Damme, "Stability and Perfection of Nash Equilibria," Second edition, Springer-Verlag, New-York, 1991.
[22]	W. Wu and J. Jiang, Essential equilibrium points of n-person non-cooperative games, Sci. Sinica, 11 (1962), 1307-1322.
[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), 253-256. doi: 10.1016/j.aml.2010.09.014.
[24]	Y. Zhou, J. Yu and S. Xiang, Essential stability in games with infinitely many pure strategies, Int. J. of Game Theory, 35 (2007), 493-503. doi: 10.1007/s00182-006-0063-0.
[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), 489-495. doi: 10.1016/j.aml.2007.05.015.

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References:

[1]	N. Al-Najjar, Strategically stable equilibria in games with infinitely many pure strategies, Math. Soc. Sci., 29 (1995), 151-164. doi: 10.1016/0165-4896(94)00765-Z.
[2]	P. Billingsley, "Convergence of Probability Measures," John Wiley & Sons, Inc., New York-London-Sydney, 1968.
[3]	K. Fan, Fixed-point and minimax theorems in locally convex linear spaces, Proc. Natl. Acad. Sci. USA, 38 (1952), 121-126. doi: 10.1073/pnas.38.2.121.
[4]	D. Fudenberg and D. Levine, Subgame perfect equilibria of finite- and infinite-horizon games, J. Economic Theory, 31 (1983), 251-268.
[5]	D. Fudenberg and D. Levine, Limit games and limit equilibria, J. Economic Theory, 38 (1986), 261-279. doi: 10.1016/0022-0531(86)90118-3.
[6]	I. Glicksberg, A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points, Proc. Amer. Math. Soc., 3 (1952), 170-174.
[7]	S. Govindan and R. Wilson, Essential equilibria, Proc. Natl. Acad. Sci. USA, 102 (2005), 15706-15711. doi: 10.1073/pnas.0506796102.
[8]	J. Hillas, On the definition of the strategic stability of equilibria, Econometrica, 58 (1990), 1365-1390. doi: 10.2307/2938320.
[9]	J. Jiang, Essential equilibrium points of n-person non-cooperative games. II, Sci. Sinica, 12 (1963), 651-671.
[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), 951-964.
[11]	E. Klein and A. Thompson, "Theory of Correspondences. Including Applications to Mathematical Economics," Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984.
[12]	E. Kohlberg and J. Mertens, On the strategic stability of equilibria, Econometrica, 54 (1986), 1003-1037. doi: 10.2307/1912320.
[13]	A. McLennan, Consistent conditional beliefs in noncooperative game theory, Int. J. of Game Theory, 18 (1989), 175-184. doi: 10.1007/BF01268156.
[14]	J. F. Nash, Jr., Equilibrium points in $n$-person games, Proc. Natl. Acad. Sci. USA, 36 (1950) 48-49. doi: 10.1073/pnas.36.1.48.
[15]	J. Nash, Non-cooperative games, Ann. Math. (2), 54 (1951), 286-295. doi: 10.2307/1969529.
[16]	B. O'Neill, Essential sets and fixed points, Am. J. Math., 75 (1953), 497-509.
[17]	R. Selten, Reexamination of the perfectness concept for equilibrium points in extensive games, Int. J. of Game Theory, 4 (1975), 25-55. doi: 10.1007/BF01766400.
[18]	L. Simon, Local perfection, J. Economic Theory, 43 (1987), 134-156. doi: 10.1016/0022-0531(87)90118-9.
[19]	L. Simon and M. Stinchcombe, Equilibrium refinement for infinite normal-form games, Econometrica, 63 (1995), 1421-1443. doi: 10.2307/2171776.
[20]	A. Tychonoff, Ein fixpunktsatz, Math. Ann., 111 (1935), 767-776.
[21]	E. van Damme, "Stability and Perfection of Nash Equilibria," Second edition, Springer-Verlag, New-York, 1991.
[22]	W. Wu and J. Jiang, Essential equilibrium points of n-person non-cooperative games, Sci. Sinica, 11 (1962), 1307-1322.
[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), 253-256. doi: 10.1016/j.aml.2010.09.014.
[24]	Y. Zhou, J. Yu and S. Xiang, Essential stability in games with infinitely many pure strategies, Int. J. of Game Theory, 35 (2007), 493-503. doi: 10.1007/s00182-006-0063-0.
[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), 489-495. doi: 10.1016/j.aml.2007.05.015.

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