# 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:
 [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

show all references

##### 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
 [1] Yannick Viossat. Game dynamics and Nash equilibria. Journal of Dynamics & Games, 2014, 1 (3) : 537-553. doi: 10.3934/jdg.2014.1.537 [2] Boris Kruglikov, Martin Rypdal. A piece-wise affine contracting map with positive entropy. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 393-394. doi: 10.3934/dcds.2006.16.393 [3] Vladimír Špitalský. Transitive dendrite map with infinite decomposition ideal. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 771-792. doi: 10.3934/dcds.2015.35.771 [4] Stefan Haller, Tomasz Rybicki, Josef Teichmann. Smooth perfectness for the group of diffeomorphisms. Journal of Geometric Mechanics, 2013, 5 (3) : 281-294. doi: 10.3934/jgm.2013.5.281 [5] Sheri M. Markose. Complex type 4 structure changing dynamics of digital agents: Nash equilibria of a game with arms race in innovations. Journal of Dynamics & Games, 2017, 4 (3) : 255-284. doi: 10.3934/jdg.2017015 [6] Ali Naimi-Sadigh, S. Kamal Chaharsooghi, Marzieh Mozafari. Optimal pricing and advertising decisions with suppliers' oligopoly competition: Stakelberg-Nash game structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020028 [7] Moez Kallel, Maher Moakher, Anis Theljani. The Cauchy problem for a nonlinear elliptic equation: Nash-game approach and application to image inpainting. Inverse Problems & Imaging, 2015, 9 (3) : 853-874. doi: 10.3934/ipi.2015.9.853 [8] Xue-Yan Wu, Zhi-Ping Fan, Bing-Bing Cao. Cost-sharing strategy for carbon emission reduction and sales effort: A nash game with government subsidy. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1999-2027. doi: 10.3934/jimo.2019040 [9] Georg Ostrovski, Sebastian van Strien. Payoff performance of fictitious play. Journal of Dynamics & Games, 2014, 1 (4) : 621-638. doi: 10.3934/jdg.2014.1.621 [10] Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295 [11] Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001 [12] J. C. Robinson. A topological time-delay embedding theorem for infinite-dimensional cocycle dynamical systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 731-741. doi: 10.3934/dcdsb.2008.9.731 [13] Lasse Kliemann, Elmira Shirazi Sheykhdarabadi, Anand Srivastav. Price of anarchy for graph coloring games with concave payoff. Journal of Dynamics & Games, 2017, 4 (1) : 41-58. doi: 10.3934/jdg.2017003 [14] Janos Kollar. The Nash conjecture for threefolds. Electronic Research Announcements, 1998, 4: 63-73. [15] William Geller, Bruce Kitchens, Michał Misiurewicz. Microdynamics for Nash maps. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1007-1024. doi: 10.3934/dcds.2010.27.1007 [16] Michael C. Sullivan. Invariants of twist-wise flow equivalence. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 475-484. doi: 10.3934/dcds.1998.4.475 [17] Michael C. Sullivan. Invariants of twist-wise flow equivalence. Electronic Research Announcements, 1997, 3: 126-130. [18] Simon Hoof. Cooperative dynamic advertising via state-dependent payoff weights. Journal of Dynamics & Games, 2019, 6 (3) : 195-209. doi: 10.3934/jdg.2019014 [19] Georgios Konstantinidis. A game theoretic analysis of the cops and robber game. Journal of Dynamics & Games, 2014, 1 (4) : 599-619. doi: 10.3934/jdg.2014.1.599 [20] Jiahua Zhang, Shu-Cherng Fang, Yifan Xu, Ziteng Wang. A cooperative game with envy. Journal of Industrial & Management Optimization, 2017, 13 (4) : 2049-2066. doi: 10.3934/jimo.2017031

2019 Impact Factor: 1.366