What is the ruling on an oath sworn to do something, when the condition—which is compensation for the one for whom the oath was sworn—is not met? And is it obligatory to fulfill the oath even if compensation is not provided?

The default ruling for an oath is permissibility, provided that what was sworn against is lawful. Not giving compensation does not necessitate doing what you swore to abandon. You can consider the ruling of what you swore to abandon: if doing it is better than abandoning it, then offer expiation and do it. If abandoning it is obligatory or desirable, then avoid it. This is supported by the hadith: "By Allah, if Allah wills, I will not swear an oath and then see something better than it without doing what is better and seeking expiation for it," and the hadith: "If you swear an oath and then see something better than it, then do what is better and offer expiation for your oath."