The article explores stable matchings in the college admissions problem. It shows that in some cases, certain types of outcomes are stable, while others are not. However, when outcomes include match point truncations, they all become stable. It also demonstrates that all true stable matchings can be reached with truncations. Furthermore, in truncations, each Nash equilibrium leads to only one matching, making the core small. While there can be multiple stable matchings at a Nash equilibrium, if a true stable matching exists, it will be the only equilibrium outcome regardless of strategy manipulation. The study establishes that for a stable matching rule to work in a subset of Nash equilibria, every profile in that subset must have just one true stable matching.