I'm not an expert in game theory (I just had a half-completed econ degree once upon a time) but your analysis is spot on as far as I can tell. The pareto efficient outcome is for the first agent to bid $1 and for the second agent to quit without bidding. If there is a counter bid, then the first player will have an incentive to bid higher to avoid a loss. The second player would then have to counter bid in order to avoid a loss as well, and so on. At the $20 mark nothing changes: at that point both players are now trying to minimize loss. In any case, any other alternative from the pareto efficient $1 vs $0 bid will cause at least one agent to have a less preferable outcome, and hence that strategy would be chosen by a "logical" player.

Maybe more interesting would be to have each agent pick a bid in advance and reveal the bids simultaneously. Of course, the optimal strategy there would be to just bid as close to $20 as possible and hope for winnings to accumulate over time.

And in all of this it's not clear how the proprietor of this site will make money out of this deal.

Feel free to correct me if someone thinks this is wrong: perhaps there is a reason I never finished that degree. :)