This paper addresses significant gaps in the study of fuzzy games by exploring the existence (nonemptiness) and essential stability of the \(\alpha \) -core for fuzzy games without assuming compactness. We demonstrate the nonemptiness of the \(\alpha \) -core for fuzzy games under relaxed conditions, thereby extending the results established by Yang and Wang (2018) and highlighting the limitations of previous frameworks that depend on compactness. Additionally, we construct two classes of fuzzy game spaces to illustrate that most of these games are essential (or weakly essential) through set-valued analysis. Furthermore, we provide an example showing that not all fuzzy games with noncompact conditions guarantee an essential equilibrium point, emphasizing the complexity of essential equilibrium existence. Finally, we investigate the existence of essential components of the \(\alpha \) -core within fuzzy games, affirming their existence even in the absence of compactness assumptions.
