# Second Price Auction (Generalized Second Price)

I am trying to find out why we pay second price, but can not understand it. All that I found it is an explanation that it is a real market price, but why it is ?

May be some example helps me.

For example, I'm selling something and I get a next bids: \$15 (from Ann), \$12 (from John) and \$10 (from Helen). According to GSP I must sell my item to Ann, but the price will be \$12. Where is the sence? Why it is profitable for me to get \$12, if I can get \$15. Why do I need to choose the second price auction?

As I see GSP is not profitable for me as for the seller, but Google use it for ads. Why?

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You want to know why it is strategy-proof? – Michael Greinecker Sep 27 '12 at 16:26
No, I'm trying to find out example which shows me that it is no good idea to pay a first price. – demas Sep 27 '12 at 16:28
What do you mean by that. It is a dominant strategy in a secon price auction to submit your own reservation price. – Michael Greinecker Sep 27 '12 at 16:29
I have updated my question to give more information about my confusion. – demas Sep 27 '12 at 16:35

There are two issues: How the auctioneer can generate revenue and who will end up with the item. From an economic efficiency point of view, the revenue of the auctioneer is secondary, it is a pure transfer. Also, there is a result in auction theory, the revenue equivalence theorem, that says that the expected revenue for a large class of auctions is the same. In particular the expected revenue for first and second price auctions coincide.

But second price auctions are great in terms of allocative efficiency. Since the person with the highest bid gets the good, the bidder who values the item the highest will get the item if we can be sure that she also makes the highest bid. In the second price auction, it is optimal for a bidder to bid her reservation value, no matter what the other bidders do.

The reason is fairly simple. I will not bid more than my reservation value because it will only help me get the item if the second highest bidder posted at least my reservation value- in which case I gain nothing and might lose some. I will also not post below my reservation value, because my posting doesn't influence how much I pay if I get the item. Posting a lower value might only make me not get the item even when would have preferred to get it because the price is below my valuation.

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Thanks. Your explanation helps me. – demas Sep 27 '12 at 17:06

Maybe you can't get \$15. When you auction something, you have to tell the bidders the rules in advance, and they adjust their bids in accordance with the rules. If you advertised an auction where the winner pays his/her bid, Ann might bid a lower number. There is a reference in Wikipedia that you can read free online saying in a first-price sealed-bid auction, bidders should bid$1-\frac 1n$of their true value, so Ann should bid \$10 and you get less. This Wikipedia article proves that in a second-price auction, bidders should bid their true value.

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The main reason why second price auctions are so 'famous' is its simplicity. By simplicity I mean that to predict behavior (bidding) in this class of auctions is relatively easy. The revenue equivalence theorem is another reason, since under some assumptions (mainly bidders' risk neutrality), the auctioneer expects the same revenue with a 2dn price auction or 1st price one (in fact, any auction whose rules assign the item to the bidder with the highest valuation and leaves a zero payoff to the bidder with the lowest. You should keep in mind that the comparison is made in ex-ante rather than ex-post terms.

Coming to your particular example, you may not be able to get $15. Suppose that instead of running a 2nd price auction, you run a 1st price (i.e., you pay what you bid in case you win). As a bidder, you have to choose your bid without knowing what bids your competitors submit. Suppose you are the bidder with the highest valuation. If you bid 15 and you expect everybody else to bid their valuations, then you expect a payoff of zero since you win but you pay your valuation. Thus, you incentives to bid a little bit lower than 15 (say 14) because you would still win the auction (everybody else is supposed to bid her valuation), but now you get a strictly positive payoff (1=15-14). But this implies that you cannot have an equilibrium in which everybody is truthful (i.e., bids her true valuation), which means that you as an auctioneer should expect a price that is never equal to 15. This is due to the trade off faced by bidders between a higher probability of winning and a lower payoff conditional on winning. Given this, the revenue equivalence theorem ensures that in expected terms, you as an auctioneer can do as well using a 2nd as using a 1st price auction, with the advantage that the 2nd price auction is much more easy to analyze form a strategic point of view. - Great answer! You should consider the Game Theory proposal – Merbs Nov 29 '12 at 0:48 Basically you're suggesting to use 1st price auction instead of 2nd price auction. It may seem more profitable using 1st price auction in a single auction, as the example you provided. However you have to look at it over time. When a bidder is participating in a series of auctions, if using 1st price auction, he has to change his bids very frequently, hoping to find the maximum ROI point. For example the \$15 bidder in your example would gradually lower the bids to reduce the cost (while still winning), while the \\$12 bidder gradually increase the bids trying to win. This is not an equilibrium and you'd observe great fluctuation of your income, the sum of which would be probably lower over time than using 2nd price auction (depending on number of bidders and their bids).

Besides, a relatively steady income stream is also important for real-world business.

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