Generalized second-price auction

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the common second price auction (GSP) is an incorrect auction mechanism for many items. Each bidder places a bid. The highest bidder gets the first slot, the second highest slot, the second slot and so on, but the highest bidder pays the bid bid by the second highest bidder, the second highest price pays the highest bid price by the third party, and so on. First conceived as a natural extension of the Vickrey auction, it saves some of the desirable properties of the Vickrey auction. This is used primarily in the context of keyword auctions, where sponsored search slots are sold by auction. The first analysis of GSP is in the economic literature by Edelman, Ostrovsky, and Schwarz and by Variants. This is used by Google AdWords technology.

Video Generalized second-price auction

Model formal

Misalkan ada ${\ displaystyle n}  ÂÂ$ penawar dan ${\ displaystyle k & lt; n}  ÂÂ$ slot. Setiap slot memiliki kemungkinan diklik ${\ displaystyle \ alpha _ {i}}  ÂÂ$ . Kita dapat mengasumsikan bahwa slot teratas memiliki kemungkinan lebih besar untuk diklik, jadi:

${\ displaystyle \ alpha_ {1} \ geq \ alpha_ {2} \ geq \ cdots \ geq \ alpha_ {k}. \,}  ÂÂ$

Kita dapat menganggap ${\ displaystyle n-k}  ÂÂ$ slot virtual tambahan dengan click-through-rate nol, jadi, ${\ displaystyle \ alpha _ {i} = 0}  ÂÂ$ untuk ${\ displaystyle i & gt; k}  ÂÂ$ . Sekarang, setiap penawar memiliki nilai intrinsik untuk satu slot ${\ displaystyle v_ {i}}  ÂÂ$ mengajukan tawaran ${\ displaystyle b_ {i}}  ÂÂ$ menunjukkan maksimum yang dia bayarkan untuk slot (yang merupakan tawaran penilaian yang dilaporkannya - pemberitahuan tidak perlu sama dengan penilaian sebenarnya ${\ displaystyle v_ {i}}  ÂÂ$ ). Kami memesan penawar berdasarkan nilainya, misalkan:

${\ displaystyle v_ {1} \ geq v_ {2} \ geq \ cdots \ geq v_ {n}, \,}  ÂÂ$

To determine the mechanism, we need to determine the allocation rule (who gets the slot) and the price paid by each bidder. In the second common price auction, we order bidders with their bids and give top slots to the highest bidder, second top slot to the second highest bidder and so on. So, bidder ${\ displaystyle i}$ get slot ${\ displaystyle i}$ . Each bidder pays the bids from the next highest bidder, so: $Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{   Â ·                          me                          =                  b                         me     Â    Â 1                                {\ displaystyle p_ {i} = b_ {i 1}}  Â} .

Maps Generalized second-price auction

Not true

Ada beberapa was on hand penawaran valuasi sebenarnya bukanlah ekuilibrium Nash. Misalnya, pertimbangkan second slot dengan $Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â{?}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â1Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â1Â\; Â\; Â\; Â\; Â}Â\; Â\; ÂÂ\; Â\; Â\{\backslash \; displaystyle\; \backslash \; alpha1\; =\; 1\}\; Â$ dan ${\ displaystyle \ alpha2 = 0,4} Â Â$ dan tiga penawar dengan valuasi ${\ displaystyle v_ {1} = 7} Â$ , ${\ displaystyle v_ {2} = 6} Â$ dan ${\ displaystyle v_ {3} = 1} Â Â$ . Penawaran 7, 6 dan 1 masing-masing bukan ekuilibrium Nash, karena penawar pertama dapat menurunkan tawaran mereka that 5 dan mendapatkan slot kedua untuk harga 1, meningkatkan utilitas mereka.

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Kesetimbangan GSP

Edelman, Ostrovsky dan Schwarz, yang bekerja di bawah informasi lengkap, menunjukkan bahwa GSP (dalam model yang disajikan di atas) selalu memiliki keseimbangan bebas yang bebas secara lokal, yaitu keseimbangan yang memaksimalkan kesejahteraan sosial, yang diukur sebagai < math xmlns = "http://www.w3.org/1998/Math/MathML" alttext = "{\ displaystyle SW = \ sum _ {i} \ alpha i v _ {\ pi (i)}} ">                    S        W        =                 ?                      saya                                  ?                      saya                                   v                       ?             (            saya             )                              {\ displaystyle SW = \ sum _ {i} \ alpha i v _ {\ pi (i)}}  di mana ${\ displaystyle \ pi (i)}  Â$ adalah slot di mana pemain ${\ displaystyle i} Â$ dialokasikan sesuai dengan tawarannya (permutasi ${\ displaystyle \ pi}  Â$ ditentukan oleh vektor tawaran ${\ displaystyle (b_ {1}, \ dots, b_ {n})}  Â$ ). Lebih jauh lagi, pendapatan dalam keseimbangan bebas envy lokal setidaknya setinggi dalam hasil VCG (jujur).

The limit on the well-being of the Nash equilibrium is given by Caragiannis et al., Proving the bound anarchy price of ${\ displaystyle 1.282}$ . DÃÆ'¼tting et al. and Lucier at al. proves that every Nash equilibrium extracts at least one-half of the honest VCG revenues of all the slots but the first. The computational analysis of this game has been done by Thompson and Leyton-Brown.

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GSP and uncertainty

Gomes and Sweeney proved that an efficient balance may not exist in partial information settings. Caragiannis et al. consider welfare losses on Bayes-Nash equilibrium and prove the price of anarchy tied to 2,927. The limit on income in Bayes-Nash equilibrium is given by Lucier et al. and Caragiannis et al.

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Budget limitations

The impact of budgetary constraints in sponsored search/position auction models is discussed in Ashlagi et al. and in the issue of a more general assignment by Aggarwal et al. and DÃÆ'¼tting et al.

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See also

• Vickrey-Clarke-Groves Auction
• Announce the first price auction
• AdWords
• The auction theory
• Japanese Auction

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References

• S. Lahaie, D. Pennock, A. Saberi, and R. Vohra. Algorithmic Game Theory , chapter "Sponsored search auction", pages 699-716. Cambridge University Press, 2007
• A lecture notes on Keyword Based Ads

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