I have a one dollar bill. there is a crowd of people around me. I hold it up and say that i will auction the one dollar bill off, and the dollar would go to the highest bidder.
The catch? the first AND second highest bidder both have to pay me whatever they bid. For example, if the bidding stops when someone bids 1.00 and the next person bids .95, then I get 1.95, and the winner gets nothing, the second person loses 95 cents.
What would you do if you were at this auction, and there had to be at least one bid? What is the "winning" strategy, assuming that everyone will want to do what is best for them?
(In reply to
re(4): Solution by TomM)
Since we cannot forcast the outcome of a bidding war or at what level it will stop, why shouldn't a bidder make the second bid? Bidding may stop at 10 cents, it may stop at $8000.
Consider two bidders. If both make bids, the chance of each winning is 50%. Sure, in reality this chance depends on a lot of factors, but here in puzzleland I think we can agree that it's near 50/50. Most important, the bid level at which the auction ends is irrelevant to this 50/50 probability.
If each bidder "wants to do what is best for them", they will be bound to act by this probability as they cannot foresee the outcome of the auction. They will surely risk 2 cents, or even 10 cents, for the dollar. At a 50% chance of winning, a bid of 10 cents risks 5 cents for 45 cents. That is, I have a 50% chance of winning with a bid of 10 cents (0.5*90 cent profit), and I have a 50% chance of losing with the same bid (loss = 0.5*10 cents). So, making a bid of 10 cents, on average, I will earn 40 cents per auction - a very good profit.
The logic for making the first bid is the same as the logic for making a second bid.
However, since neither a bidding war nor a sucessful low bid can be guaranteed, the best first bid is 99 cents. A first bid of 99 cents makes a second bid impossible.
re(5): Solution
