[igf]

I don't know if game theory is the branch of mathematics that can help you out here. Game theory usually rests on matters of "If I do X then my opponent(s) will do Y", but in this situation you don't even know how many opponents you might have.

Your best shot would probably to be just to sit there and collect a shitload of statistics about under exactly what circumstances you're most likely to be the final bidder, and then only bid under those circumstances... and then hope nobody else has been collecting the same statistics and coming to exactly the same conclusion.