Efficiency and the core in NTU games in partition function form

The aim of this paper is to establish a necessary and sufficient condition for the non-emptiness of the core in NTU games in partition function form, given an externality scheme f is an element of Ex(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f \in \textsf{Ex}(N) $$\end{document}. We extend the notion of convexity to incorporate externality effects. By introducing a new concept of rationality, called collective rationality, we demonstrate the efficiency of the grand coalition N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ N $$\end{document}. We also identify a sufficient condition for the efficiency of the grand coalition using the property of individual superadditivity.