Second Price Sealed-Bid Auctions (Slightly Technical)


Second price sealed-bid auctions include the type of auction effectively used on eBay. In these auctions each player secretly makes a bid, the person who bids the most wins. The winner, however, pays what the second highest person bid. For example, assume that the following bids are made:

Table 12

Bidder

Amount Bid

Tom

$100

Jane

$90

Sue

$30

Tom wins the auction, but he pays only $90. Neither Jane nor Sue pays anything. Had Jane not bid, Tom would still have won the auction but would have paid only $30. The surprising strategy in a second price sealed-bid auction is that you should bid exactly what the good is worth to you.

Imagine that the item being bid on is worth $100 to you. Let’s first examine why bidding $100 is always better than bidding some amount less than $100, such as $99. If the highest bid exceeds $100, it doesn’t matter whether you bid $99 or $100 because you lose the auction either way. If the highest bid other than yours is below $99, it again doesn’t matter whether you bid $99 or $100. In both cases you win the auction and pay the amount of the second highest bid. (If the second highest bid is $88, you win and pay $88 regardless of whether you had bid $99 or $100.) The only time there is a substantive difference between bidding $99 or $100 occurs when the highest bid made by someone else is between $99 and $100. If the highest bid made by someone else is $99.50, you lose the auction if you bid $99 and win the auction if you bid $100. If you bid $100, you would get the good for $99.50, which is beneficial since the item is worth $100 to you. Consequently, if the highest bid other than yours falls between $100 and $99, you are always better off bidding $100 if the good is worth $100 to you. Thus, bidding $100 is either the same or better than bidding $99 or any other amount below $100.

Similar logic shows you should always bid $100 rather than some amount more than $100, such as $105. If someone else bids over $105, then bidding $100 or $105 both result in your losing the auction and paying nothing. If the highest bid other than yours is lower than $100, then bidding $100 or $105 is identical; in both cases you win, paying the second highest bid. The difference between bidding $100 and $105 manifests itself if the highest bid someone else makes is between $100 and $105. If the highest bid is $104, you lose the auction if you bid $100 and win it if you bid $105. If you bid $105, however, you have to pay $104 for the item, which is more than its worth to you. Thus, bidding $100 is always better or equivalent to bidding over $100.

In a second price sealed-bid auction, the amount you bid determines whether you win; it does not determine how much you pay if you win. In these auctions you want to bid only when the amount you would have to pay if you won is less than the item’s value. If you bid what the item is worth to you, you will win if and only if the second highest bid (the amount you pay if victorious) is less than what the item is worth.

The auctions on eBay are essentially second price sealed-bid auctions. In eBay the winner is always the person who bids the most, and the winner pays what the second highest person bid.[2]

If you know the value of the good, and believe the auction to be honest, the optimal strategy on eBay is to bid what the good is worth to you and never raise your price. If you bid what the item is worth, and someone bids more, you should never outbid her because then if you win, you will necessarily pay more than the item’s value. Many people on eBay seem to follow the strategy of bidding small amounts and then raising their bid amount if outbid. This strategy wastes time, because your bid determines when you win, not how much you pay. To see this, assume again that the good is worth $100 to you. Imagine that there will be many other bids, and the highest is X. If you just bid $100 from the beginning, you will win the auction when X < $100 and will pay X if you win. If you initially bid $20, keep raising your bid when someone else outbids you, but never bid more than $100, then you get the same result as if you just bid $100 from the beginning. You win whenever X < $100, and when you win, you pay X.

If eBay were a first price auction, where the winner always paid what he bid, then you would always want to just barely outbid the second highest bidder. In a first price eBay auction, you would want to visit the auction just before it ended and bid just a little bit more than the second highest bidder. In fact, if everyone were rational, in a first price eBay auction, everyone would wait until the end of the auction to bid. It would be stupid to bid before the end; doing so would give you no advantage and might even hurt you because you could end up bidding far more than everyone else. Remember, in a second price auction it doesn’t matter by how much you outbid everyone, since if you win, you always pay the second highest bid. In a first price auction, by contrast, the winner always wants to just barely outbid the second highest bidder, because the winner pays what he bids. Thus, in first price auctions, you want to wait until the end, when you have as much information as possible about the other bids.

[2]I’m assuming that at least two bids exceed the seller-specified reserve price.




Game Theory at Work(c) How to Use Game Theory to Outthink and Outmaneuver Your Competition
Game Theory at Work(c) How to Use Game Theory to Outthink and Outmaneuver Your Competition
ISBN: N/A
EAN: N/A
Year: 2005
Pages: 260

flylib.com © 2008-2017.
If you may any questions please contact us: flylib@qtcs.net