"Minus One" is a short story by British author J. G. Ballard; it was first published in the June 1963 edition of Science Fantasy (Volume 20, Number 59).
[1] It was later reprinted in the 1967 collection The Disaster Area, and then later in the larger The Complete Short Stories of J. G. Ballard: Volume 1 anthology (2006).
It provides a private prison where the rich can incarcerate "miscreant or unfortunate relatives whose presence would otherwise be a burden or embarrassment".
Security rather than treatment comes first and the asylum boasted that no-one had ever escaped, that is until the disappearance of a patient called Hinton.
Fortuitously the file is then "lost" and the director announces that the disappearance was an administrative error and that Hinton had never really existed.