Manipulation and stability in the college admissions problem

Roth and Vande Vate (1991) studied the marriage problem and introduced the notion of truncation strategies and showed in an example that the unstable matchings can arise at Nash equilibria in truncations. This paper studies the college admissions problem and shows that all rematching proof or strong equilibria in truncations produce stable matchings, even though the equilibrium profiles are manipulated, and all stable matchings can be achieved in rematching proof or strong equilibria in truncations. It is shown that a preference profile that is rematchinng proof or strong equilibrium for one stable matching mechanism is also rematching proof or strong equilibrium for all stable matching mechanisms. This result shows that there is no difference among all stable matching mechanisms in rematching proof or strong equilibria in truncations, which is in the contrast to the situation in which agents report their preferences in a straightforward manner.