Inspired by Scarf (J Econ Theory 3:169–181, 1971) and Kajii (J Econ Theory 56:194–205, 1992), we introduce the notion of mixed-strategy α-core for games with infinitely many pure strategies. We first show the nonemptiness of the mixed-strategy α-core for normal-form games with infinitely many pure strategies and obtain the generic continuity property of the mixed-strategy α-core correspondence. Furthermore, we prove the existence of the mixed-strategy α-core for games with nonordered preferences and infinitely many pure strategies and show the generic continuity property of the mixed-strategy α-core correspondence.
Neng-Fa Wang, Zhe Yang
. The mixed-strategy α-core of games with infinitely many pure strategies[J]. Journal of the Operations Research Society of China, 2025
, 13(2)
: 660
-670
.
DOI: 10.1007/s40305-023-00497-x
[1] Bondareva, O.N.: Some applications of linear programming methods to the theory of cooperative games. Probl. Kibem. 10, 119-139 (1963)
[2] Shapley, L.S.: On balanced sets and cores. Naval Res. Log. Q. 14, 453-460 (1967)
[3] Scarf, H.E.: The core of an \begin{document}$ N $\end{document}-person game. Econometrica 35, 50-69 (1967)
[4] Scarf, H.E.: On the existence of a cooperative solution for a general class of \begin{document}$ n $\end{document}-person games. J. Econ. Theory 3, 169-181 (1971)
[5] Kajii, A.: A generalization of Scarf’s theorem without transitivity or completeness. J. Econ. Theory 56, 194-205 (1992)
[6] Askoura, Y.: The weak-core of a game in normal form with a continuum of players. J. Math. Econ. 47, 43-47 (2011)
[7] Askoura, Y., Sbihi, M., Tikobaini, H.: The ex ante \begin{document}$ \alpha $\end{document}-core for normal form games with uncertainty. J. Math. Econ. 49, 157-162 (2013)
[8] Noguchi, M.: Cooperative equilibria of finite games with incomplete information. J. Math. Econ. 55, 4-10 (2014)
[9] Uyanik, M.: On the nonemptiness of the \begin{document}$ \alpha $\end{document}-core of discontinuous games: transferable and nontransferable utilities. J. Econ. Theory 158, 213-231 (2015)
[10] Yang, Z.: Essential stability of \begin{document}$ \alpha $\end{document}-core. Int. J. Game Theory 46(1), 13-28 (2017)
[11] Yang, Z., Song, Q.P.: A weak \begin{document}$ \alpha $\end{document}-core existence theorem of generalized games with infinitely many players and pseudo-utilities. Math. Soc. Sci. 116, 40-46 (2022)
[12] Al-Najjar, N.: Strategically stable equilibria in games with infinitely many pure strategies. Math. Soc. Sci. 29, 151-164 (1995)
[13] Zhou, Y.H., Yu, J., Xiang, S.W.: Essential stability in games with infinitely many pure strategies. Int. J. Game Theory 35, 493-503 (2007)
[14] Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. Springer, Berlin (2006)
[15] Fort, M.K., Jr.: Points of continuity of semicontinuous functions. Publ. Math. Debrecen 2, 100-102 (1951)
[16] Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)
