Summary: In a strategic coalitional game, we consider the relations among four cores \(\alpha , \beta , \gamma \), and the one we call \(\delta\) which is obtained by slightly weakening the conjectural cooperative equilibria due to Currarini and Marini. We show that the \(\alpha\)-core and the \(\gamma\)-core are refined by the \(\delta\)-core; and, moreover that if every player has a dominant strategy, the \(\beta\)-core is refined by the \(\gamma\)-core, so that the four cores refine themselves in the greek alphabetical order. Two economic games will be considered to show that the refinement of the \(\alpha\)-core can vary from the weakest to the strongest. While the four cores are equal in the pure exchange game, a radical reduction of the \(\alpha\)-core is obtained in the commons game, a simple version of the Cournot game, bringing about a single strategy profile as the \(\delta\)-core.