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 used by Google's AdWords technology and Facebook. Suppose that there are bidders and slots. Each slot has a probability of being clicked of . We can assume that top slots have a larger probability of being clicked, so: We can think of additional virtual slots with click-through-rate zero, so, for . Now, each bidder submits a bid indicating the maximum they are willing to pay for a slot. Each bidder also has an intrinsic value for buying a slot . Notice that a player's bid doesn't need to be the same as their true valuation . We order the bidders by their bid, let's say: and charge each bidder a price . The price will be 0 if they didn't win a slot. Slots are sold in a pay-per-click model, so a bidder just pays for a slot if the user actually clicks in that slot. We say the utility of bidder who is allocated to slot is . The total social welfare from owning or selling all slots is given by: The expected total revenue is given by: To specify a mechanism we need to define the allocation rule (who gets which slot) and the prices paid by each bidder. In a generalized second-price auction we order the bidders by their bid and give the top slot to the highest bidder, the second top slot to the second highest bidder and so on. Then, assuming the bids are listed in decreasing order the bidder bidding gets slot for .