The Ran on this sugya is pretty confusing, so i thought it would be a good idea to summarize it. In a situation where there is an owner of a bath house who is not allowed to provide benefit to a bather, and the bath house is managed by a third party, it depends on the level of involvement of the owner. The gemara says that if the owner is somewhat removed from any direct involvement with the bath house, the mudar may bathe there. The main issue in the gemara is understanding whether the heter for the mudar to bathe there is based on the assumption that he is not receiving his benefit from the owner, rather from the manager, or is it because the original neder that was imposed by the owner never meant to include this case. The obvious distinction between the 2 approaches, is a bath house that was given over to a manager after the neder was made. Based on the approach that it is not called a benefit from the owner it is mutar since at the time of benefit the owner is removed from any involvement, but based on the approach that it is called a benefit just that it was not included in the neder, in this case since it was in the hands of the owner at the time of the neder it was certainly included in the neder. The Ran in the mishna at first implies like the first approach "דלאו מדידיה מהני אלא מהשוכר" and then quotes the rashba who says like the second approach "כל שאין לו בהן תפיסת יד, אין האיסור חל עליהם". A third possible approach is that even if the madir wants to assur the mudar, and it is considered benefit from the madir, the madir does not have the power to do so.
Ran asks: Since we find that an owner can make a house hekdesh and assur it on the renter, he should certainly be able to assur the house on someone else who is invited by the renter. So, why does our mishna allow the mudar to bathe in the bath house even though it is under control of the manager?
Answers brought in Ran:
1. Tosafos - Although the owner has rights to assur on the bather, he did not intend to include the bathhouse in the neder since he has no involvement in it. 2nd approach above.
2. Rabbeinu Tam - Hekdesh can uproot the lien of the manager/renter but a private konam cannot. In our mishna which speaks about a konam, the owner can't assur the manager/renter. So long as the manager/renter lien remains the owner is powerless to assur the mudar. (Ran disagrees and holds tha konam can uproot the lien). 3rd approach above.
3. Acheirim - When the manager/renter has already paid the entire year of rent, the owner has no power to uproot his lien and therefore is powerless to assur on the mudar. (Ran disagrees and holds that advance payment does not give the renter/manager a stronger lien). 3rd approach above.
4. Only hekdesh of kedushas mizbeiach can uproot the manager/renters lien, but kedushas bedek habayis like being makdish a house cannot. Therefore, if "a house" was rented, the owner can be makdish it since the manager/renter has no lien on "this house", but if "this house" was rented, the owner cannot remove the lien of the manager/renter. (Ran points out that this does not help to explain our case, because a konam is like kedushas mizbeiach since there is no possibility of redemption).
5. The case here is not a standard rental, but rather a sale that is paid by giving x amount of dollars per year to the original owner. The renter/manager is really an "owner" and the original owner has no power to break his lien/ownership. 3rd approach above.
6. Ran - Although the owner can technically break the lien of the manager/renter and assur on him and on everyone else for that matter; so long as he does make a konam against the manager/renter so that the lien stays in tact, the manager/renter is considered the "owner" in the sense that all benefit being provided to the mudar is from the manager/renter, not from the actual owner of the property (This is like the first approach above, that the neder was binding against the mudar benefiting from anything owned by the madir, but the bathhouse is considered benefit to the mudar from the manager/renter, not from the madir). 1st approach above.