Nash equilibria for strategic games were characterized by Peleg and Tijs (1996) as those solutions satisfying the properties of consistency, converse consistency and one-person rationality.
There are other solutions, like the -Nash equilibria, which enjoy nice properties and appear to be interesting substitutes for Nash equilibria when their existence cannot be guaranteed. They can be characterized using an appropriate substitute of one-person rationality. More generally, we introduce the class of "personalized" Nash equilibria and we prove that it contains all of the solutions characterized by consistency and converse consistency.
Patrone, Fioravante and Pieri, Graziano and Tijs, Stef H. and Torre, Anna, On Consistent Solutions for Strategic Games. International Journal of Game Theory, Vol. 27, Iss. 2, August 31, 1998. Available at SSRN: https://ssrn.com/abstract=121328