Abstract
Using the existence results in Kajii (J Econ Theory 56:194–205, 1992), we identify a class of n-person noncooperative games containing a dense residual subset of games whose cooperative equilibria are all essential. Moreover, we show that every game in this collection possesses an essential component of the \(\alpha \)-core by proving the connectivity of minimal essential subsets of the \(\alpha \)-core.
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Notes
\(u:X\longrightarrow {\mathbb {R}}\) is said to be upper pseudocontinuous at \(z_0\in X\) if for all \(z\in X\) such that \(u(z_0) <u(z)\), we have \(\lim \sup _{y \longrightarrow z_0}u(y) <u(z)\); u is said to be lower pseudocontinuous at \(z_0\in Z\) if \({-}f\) is upper pseudocontinuous at \(z_0\); u is said to be pseudocontinuousif it is both upper and lower pseudocontinuous.
We say that a function \(\phi : X \times X \longrightarrow {\mathbb {R}}\) is generalized t-quasi-transfer continuous if, whenever \(\phi (x, z) > t\) for some \((x, z) \in X \times X\), there exists a neighborhood \(U_z\) of z and a well-behaved correspondence \(\xi : U_z \rightrightarrows X\) such that \(\phi (\xi (z'), z') > t\) for any \(z'\in U_z\) . We say that \(\phi \) is generalized positively quasi-transfer continuous if it is generalized t-quasi-transfer continuous for any \(t > 0\).
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This research was supported by National Natural Science Foundation of China (Nos. 11501349 and 71301094), the Chen Guang Project sponsored by the Shanghai Municipal Education Commission and Shanghai Education Development Foundation (No. 13CG35), and the open project of Key Laboratory of Mathematical Economics (SUFE), Ministry of Education (No. 201309KF02).
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Yang, Z. Essential stability of \(\alpha \)-core. Int J Game Theory 46, 13–28 (2017). https://doi.org/10.1007/s00182-015-0515-5
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DOI: https://doi.org/10.1007/s00182-015-0515-5