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 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 has an intrinsic value for one slot submits a bid indicating the maximum he is willing to pay for a slot (which is his bid reported valuation – notice it doesn't need to be the same as his true valuation ). We order the bidders by their value, let's say:
and charge each bidder a price (this 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 when allocated to slot is . The total social welfare from owning or selling slots is given by: where is the bidder allocated to slot . The 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. So, bidder gets slot . Each bidder pays the bid of the next highest bidder, so: .
There are cases where bidding the true valuation is not a Nash equilibrium. For example, consider two slots with and and three bidders with valuations , and . Bidding 7, 6 and 1 respectively is not a Nash equilibrium, since the first bidder could lower his bid to 5 and get the second slot for the price of 1 and increase his utility therefore.
Equilibria of GSP
Edelman, Ostrovsky and Schwarz, working under complete information, show that GSP (in the model presented above) always has an efficient locally-envy free equilibrium, i.e., an equilibrium maximizing social welfare, which is measured as where is the slot in which player is allocated according to his bid (the permutation is defined by the bid vector ). Further, the revenue in any locally-envy free equilibrium is at least as high as in the (truthful) VCG outcome.
Bounds on the welfare at Nash equilibrium are given by Caragiannis et al., proving a Price of Anarchy bound of (golden ratio). Dütting et al. and Lucier at al. prove  that any Nash equilibrium extracts at least one half of the truthful VCG revenue from all slots but the first. Computational analysis of this game have been performed by Thompson and Leyton-Brown.
GSP and uncertainty
The classical results due to Edelman, Ostrovsky and Schwarz  and Varian  hold in the full information setting – when there is no uncertainty involved. Recent results as Gomes and Sweeney  and Caragiannis et al. and also empirically by Athey and Nekipelov  discuss the Bayesian version of the game - where players have beliefs about the other players, but not necessarily know the other players valuations.
Gomes and Sweeney  prove that an efficient equilibrium might not exist in the partial information setting. Caragiannis et al. consider the welfare loss at Bayes-Nash equilibrium and prove a Price of Anarchy bound of 2.927. Bounds on the revenue in Bayes-Nash equilibrium are given by Lucier et al. and Caragiannis et al.
The impact of budget constraints in the sponsored search/position auction model is discussed in Ashlagi et al. and in the more general assignment problem by Aggarwal et al. and Dütting et al.
- Benjamin Edelman, Michael Ostrovsky, and Michael Schwarz: "Internet Advertising and the Generalized Second-Price Auction: Selling Billions of Dollars Worth of Keywords". American Economic Review 97(1), 2007 pp 242-259
- H. R. Varian: "Position auctions. International Journal of Industrial Organization, 2006".
- Caragiannis, Ioannis; Kaklamanis, Christos; Kannelopolous, Panagiotis; Kyropolou, Maria; Lucier, Brendan; Paes Leme, Renato; Tardos, Eva. "Bounding the inefficiency of outcomes in generalized second price auctions". Journal of Economic Theory 156: 343–388.
- Dütting, Paul; Fischer, Felix; Parkes, David C. (2011). "Simplicity-Expressiveness Tradeoffs in Mechanism Design". Proceedings of the 12th ACM Conference on Electronic Commerce (EC'11): 341–350.
- Lucier, Brendan; Renato, Paes Leme; Eva, Tardos (2012). "On Revenue in the Generalized Second Price Auction". Proceedings of the 21st International World Wide Web Conference (WWW'12): 361–370.
- D. R. M. Thompson and K. Leyton-Brown. Computational analysis of perfect-information position auctions. In EC ’09: Proceedings of the tenth ACM conference on Electronic commerce, pages 51–60, New York, NY, USA, 2009. ACM.
- R. D. Gomes and K. S. Sweeney. Bayes-nash equilibria of the generalized second price auction. In EC ’09: Proceedings of the tenth ACM conference on Electronic commerce, pages 107–108, New York, NY, USA, 2009. ACM
- Susan Athey and Denis Nekipelov. A Structural Model of Sponsored Search Advertising Auctions, Ad Auctions Workshop, 2010
- Caragiannis, Ioannis; Kaklamanis, Christos; Kannelopolous, Panagiotis; Kyropolou, Maria. "Revenue guarantees in the generalized second price auction". ACM Transactions on Internet Technology: forthcoming.
- Ashlagi, Itai; Braverman, Mark; Hassidim, Avinatam; Lavi, Ron; Tennenholtz, Moshe. "Position Auctions with Budgets: Existence and Uniqueness". B.E. journal of Theoretical Economics: Forthcoming.
- Aggarwal, Gagan; Muthukrishnan, Muthu; Pal, David; Pal, Martin (2009). "General Auction Mechanism for Search Advertising". Proceedings of the 18th International World Wide Web Conference (WWW'09): 241–250.
- Dütting, Paul; Henzinger, Monika; Weber, Ingmar (2011). "An Expressive Mechanism for Auctions on the Web". Proceedings of the 20th World Wide Web Conference (WWW'11): 127–136.
- S. Lahaie, D. Pennock, A. Saberi, and R. Vohra. Algorithmic Game Theory, chapter "Sponsored search auctions:, pages 699–716. Cambridge University Press, 2007
- Lecture notes on Keyword-Based Advertisement