I tend to agree with Eliezer that this is not really about fairness, but insofar as we're playing the "what's fair?" game...

Utilities of different players are classically treated as incomparable ... thus we'd like the "fair point" to be invariant under affine recalibrations of utility scales.

Proclaiming incomparability has always struck me as elevating a practical problem (it's difficult to compare utilities) to the level of a conceptual problem (it's impossible to compare utilities). At a practical level, we compare utilities all the time. To take a somewhat extreme example, it seems pretty obvious that a speck of dust in Adam's eye is less bad than Eve being tortured.

The implication of this is that I actively do not want the fair point to be invariant to affine tranformations of the utility scales. If one person is getting much more utility than someone else, that is relevant information to me and I do not want it thrown away.

NB: In the event that I did think that utility was incomparable in the way "classically" assumed, then wouldn't the solution need to be invariant to monotone transformations of the utility function? Why should affine invariance suffice?