If vJ is the perceived value of getting the job, pJ the perceived probability of getting the job, vI the value of the literal payment for going through the interview process, and vT the perceived value of one's time (and any other cost if it exists, such as travel) required to go through the process, we can represent the expected value of going through the interview process as:
EV = pJ * vJ + vI - vT
I also assume that the candidate would choose where and whether to apply based on EV for their various options.
One one end, if your pJ is close to 1 (you're highly qualified for the job and will likely get it), the result is dominated by pJ * vJ which is ~= vJ; vI and vT matter little. This means that if you will probably get the job, it doesn't really matter much whether the interview is paid (and it also doesn't matter as much how much time it takes). For top candidates, the difference in pJ * vJ for different companies should be the dominant factor, i.e. they will apply for the best jobs.
On the other end, if your pJ is close to 0 (you're applying on a long shot), then vI and vT become much more significant factors. If your chance to get the job is really low, then the interview being paid makes it significantly more attractive, and it also matters more how much time it takes. The companies that pay for interviews, and companies that are easy to apply and interview for are likely the ones with the highest EV for the poorer candidates.
Basically the worse of a candidate you are for the job, the more important it is for the interview to be paid, because with a low enough probability of getting the job, this payment becomes a big factor in the expected value of doing the interview.
Of course it's not as simple as that, because people are not machines chasing pure financial interest and have feelings about how you treat them. Also, a highly qualified candidate is more likely to have a job that is closer to the one they're applying for, while an unqualified candidate might have a much worse job, making vJ higher for the less qualified candidate. But it is likely that the relative difference in pJ is much greater between a qualified and unqualified candidate than the relative difference in vJ. The candidate's own perception of the probability of being hired (pJ) might also be unrealistic in either direction, and I'm assuming it is a good predictor of the true probability of being hired. But I think in general the rule should hold, paid interviews would decrease the quality of the candidate pool.