On Bicycles, and.... what else is there?

After my last post, I commented that my analysis was obviously flawed, because it would lead to the conclusion the winning bidder had a 100% chance of winning. There were two obvious points of weakness in that analysis. The first is I assumed the expectation value of the number of bidders was the number of bidders, not counting my friend. I don't count my friend because his presence wasn't random (I specifically picked this auction because he was there; I didn't pick it at random). However, that assumption is obviously just a guess. But lacking other information, it is the best I can do. The next approximation I made was that the chance a random bidder was better than my friend's bid was the number of bidders who ranked better than my friend. This was 25%. This seemed a reasonable assumption, until I apply it to the case my friend wins the auction. In that case the result would be 0%. That's obviously wrong. So what I assume is all bids can be mapped onto...

A friend of mine big on a condo and lost, ending up 3rd/9. The friend was convinced the bid had been too low. I wasn't so convinced: after all, what if the two people who'd bid higher hadn't bid, and no other higher bidders had shown up? It's an inherently probabilistic process. If the goal was to win at all cost, you'd just bid the most you could possibly spend every single time. So to estimate the chances that bid would have won the bid requires estimating two probability distributions: the probability distribution of the number of bids and the probability distribution of the values of those bids. Without any information available on these, it's necessary to guess. The first approximation is that bids are uncorrelated: whether someone bids, and how much, is independent of how many others do so. The next approximation is that the bids are statistically representative of the probability distributions. So not counting the friend, there were 8 other bids...