The first bid starts at $0.01, and every bid after that is one penny more than the previous bid. You can bid as many times as you like, but there is a twist. Each person is obligated to pay their highest bid to the seller, regardless of whether they win that item. You can see that getting into a bidding war can be very costly! Here is a quick example of how this might work. Say you are selling a textbook you no longer need, and you decide to use a Committed Bids auction. There are three people bidding on it: Hannah, Joe, and Steven. Hannah places the first bid for $0.01, and Joe takes the second bid for $0.02. Hannah bids again for $0.03, then Joe bids again for $0.04. Hannah and Joe continue alternating bids in this way until Hannah places a bid for $24.99. Then Joe places a bid for $25. At this point, Hannah stops bidding. Meanwhile, Steven decides to begin bidding. He submits his first bid for $25.01. Now Joe and Steven alternate bids until Steven places his final bid for $29.99 and then Joe places a bid for $30. At this point Steven stops bidding and there is only one person left, so the auction is finished. Hannah’s highest bid was for $24.99, so this is what she pays you (even though she didn’t get the textbook!) Steven’s highest bid was $29.99, so he pays you this amount. Joe placed the highest bid for $30, so he receives the textbook and pays you $30. Overall, you collect $24.99 + $29.99 + $30 = $84.98. Suppose now that you (the seller) know the number of bidders and the winning bid, but don’t know the maximum each bidder is willing to pay.

1. If there are three people bidding, what is the minimum you will make on the auction? What is the maximum? What is the minimum/maximum if there are 5 people? What happens as the number of bidders increases? (You might consider writing a formula for the minimum and maximum for n bidders. This will require defining some variables!)

2. In your answer above, you probably noticed that the maximum amount the seller earns from the sale increases quite a bit when there are more bidders. However, the minimum amount the seller earns does not increase very much with more bidders. As a seller, under what conditions would you earn very little from the auction? Under what conditions would you make a good profit? Would you choose to sell an item using Committed Bids?