Generalized second-price auction

The generalized second-price auction (GSP) is a non-truthful auction mechanism for multiple items. Each bidder places a bid. The highest bidder gets the first slot, the second-highest, the second slot and so on, but the highest bidder pays the price bid by the second-highest bidder, the second-highest pays the price bid by the third-highest, and so on. First conceived as a natural extension of the Vickrey auction, it conserves some of the desirable properties of the Vickrey auction. It is used mainly in the context of keyword auctions, where sponsored search slots are sold on an auction basis. The first analyses of GSP are in the economics literature by Edelman, Ostrovsky, and Schwarz and by Varian. It is employed by Google's AdWords technology.

Suppose that there are ${\displaystyle n}$ bidders and ${\displaystyle k<n}$ slots. Each slot has a probability of being clicked of ${\displaystyle \alpha _{i}}$. We can assume that top slots have a larger probability of being clicked, so:

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