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?
If there is a best bidding strategy it should be the same for each bidder. Also, we don't really know the rules of the auction, but they shouldn't matter as long as they fairly applied to all bidders.
B = bid
W = winnings
L = loss
P = chance of winning the auction
W = 1 dollar - B
L = B
So for any given instance of the auction,
Net Profit, $ = P*W - (1-P)*L,
= P*(1-B) - (1-P)*B
This applies to each bidder.
Total profit = Sum of $ for all auctions. Assuming a constant bidding stragegy, if we maximize $, we maximize Total profit.
Now, let's assume that there is an optimum bid. When the auction opens there is no reason for the first bidder to bid anything but the optimum bid, since any bidder that follows with a higher bid is not playing by the best strategy (as I've defined it above). Also, if the first bidder bids less than the optimim bid, they give that chance to the next bidder.
Now look at the case with two bidders using the same bid, the optimum bid:
P=0.5 (the bidder who goes first gets it)
So, $ = 0.5*(1-B) - (1-0.5)*B
= 0.5 - 0.5*B - 0.5*B
= 0.5 - B
$ is a maximum when B is as small as possible. If the minimum bid is say 1 cent and the two bidders take turns bidding first, then each bidder wins an average of 48 cents per auction. If the optimum bid is 5 cents, each bidder earns an average of 45 cents per auction.
Note, if the players have chosen two different optimum bids before the auction, the player with the higher optimum bid will always win the auction and the other player will always lose. The losing player must then increase his optimum bid until his profits are maximized (this should be at the other players bid, I think). If a bidding war gets started, both players should conclude that at a bids past 50 cents, they are losing in the long run (at 65, one round I win 35 cents, the next I lose 65 cents). If this happens, lower bids will seem more appealing.
In short, the bidders take turns getting a dollar for a penny.
Here's an idea
