"I try to unravel the paradox [of blackmail] and provide a coherent basis for distinguishing legitimate from illegitimate threats. In brief, I argue that the key to the wrongfulness of the blackmail transaction is its triangular structure. The transaction implicitly involves not only the blackmailer and his victim but always a third party as well. This third party may be, for example, the victim's spouse or employer, the authorities or even the public at large. When a blackmailer tries to use his right to release damaging information, he is threatening to tell others. If the blackmail victim pays the blackmailer, it is to avoid the harm that those others would inflict. Thus blackmail is a way that one person requests something in return for suppressing the actual or potential interests of others. To get what he wants, the blackmailer uses leverage that is less his than someone else's. Selling the right to go to the police involves suppressing the state's interests. Selling the right to tell a tort victim who committed the tort involves suppressing the tort victim's interests. And selling the right to inform others of embarrassing (but legal) behavior involves suppressing the interests of those other people."
James Lindgren, Unraveling the Paradox of Blackmail, 84 Colum. L. Rev. 670, 672 (1984)
That still doesn't quite answer the question: If the money is paid, there is no crime to begin with. The transfer isn't selling the right to go to the authorities, it's resolving the crime of non-payment in the first place.
James Lindgren, Unraveling the Paradox of Blackmail, 84 Colum. L. Rev. 670, 672 (1984)