[hgomersall]

He's not willing to do it. He explicitly does not want to do it and is only going to do it to lower overall utility by more than the £10 can provide to either party. That is, it only makes sense if he values his finger by more than £10.

[jjk166]

We're dealing with two different definitions of "willing." You're using willing to mean a desire to do it. I'm using willing to mean he will choose to do it. It's like I desire to quit my job, but I don't actually quit, because I'm not willing. I desire to be fit, but I'm not willing to give up desert.

It doesn't matter how much the mugger claims he doesn't want to lose his fingers, the fact that he will nevertheless choose a path that causes him to lose his fingers of his own volition means that he has set the utility as being lower.

Yes, the premise only makes sense if he values his finger more than £10, but he demonstrably does not, ergo the premise is contradicted.