The mechanic, precisely
At midnight UTC the date becomes the seed for a pseudo-random draw. Up to eight sponsored slots are filled first, by a draw weighted by bid — a bid twice as large gets roughly twice the chance of being picked, not a guaranteed higher position. The remaining twelve slots are filled by an unweighted shuffle of everything else that is listed.
Because the seed is just the date, the same slate is served to everybody, all day. That is also what makes the page cacheable, which is what makes it free to serve — the whole thing is one static page until midnight.
Why weighted rather than ranked
A straight bid ranking means the highest bidder owns slot one for a month. Readers learn to skip slot one, and the slot becomes worthless to the person who paid most for it. Weighting the draw keeps the page worth reading, which is the only thing that makes any slot on it worth buying.
Over thirty days a high bid is visibly dominant. On any given Tuesday it might not appear at all. Both of those are the system working.
Verifying your own placement
The draw is deterministic: same date, same pool, same result. So "we showed you fairly" is a checkable claim rather than a reassurance. If you think a placement did not run on a day it was paid for, say so and it can be reconstructed exactly.
What unclaimed listings can and cannot do
Seeded listings — the ones marked Unclaimed, built from public data already published on this site — rotate through the free slots. They are never eligible for a sponsored slot, because selling a paid placement for a site whose owner never opted in would imply an endorsement that does not exist.
If one of them is yours, claiming it is free, makes it eligible for a sponsored slot, and also lets you have it removed outright, no questions asked.