I think you're missing the point. This is a value-optimisation problem, and 1.) not all value is monetary, and 2.) the perception of value is inherently subjective. That's why optimisations cannot be created via static externally objective criteria, and must be the process of a negotiation, as the NY Times calculator proposes.

If the millionaire readily agreed to pay 100% of the rent, then that would be "fair" because clearly in order to make that agreement, they'd have to be receiving some kind of non-monetary benefit from the arrangement. On the other hand, if they walked away from that proposition, then obviously no optimisation has been achieved, which is why a static "the millionaire pays for everything" policy is not a good idea. Instead, the millionaire should return with a counter-offer, and this should continue until parity is achieved. When all the subjective qualitative factors are included, there's no guarantee that it will be at 50/50 (which may not even be a possible solution, as in my example with my flatmate). That's the beauty of the Times' calculator: it allows people to apply whatever personal subjective criteria they like when making their own valuations, and then finds the dynamic equilibrium between these.