The VCG Auction in Theory and Practice By Hal R. Varian and - PDF document

American Economic Review: Papers & Proceedings 2014, 104(5): 442–445 http://dx.doi.org/10.1257/aer.104.5.442 ‡ MARKET DESIGN FOR AUCTION MARKETS The VCG Auction in Theory and Practice † By Hal R. Varian and Christopher Harris* It is now common to sell online ads using the most prominent slot, the advertiser with the an auction. Auctions are used for search ads second highest bid gets the second most promi- by Google and Microsoft, for display ads by nent slot, and so on. ( In the actual auction, the bids DoubleClick and other ad exchanges, and for are adjusted by a “quality score,” but we ignore social network ads by Facebook. However, differ- this additional complexity in this exposition. ) ent auction designs are used in each of these cases. Search ads use a Generalized Second Price ( GSP ) II. How the GSP Auction Works auction, display ad exchanges generally use a Vickrey ( second price ) auction, and Facebook v Let s be the value of a click to an advertiser uses a Vickrey-Clarke-Groves ( VCG ) auction. in slot s = 1, … , S , and let x s be the clicks It turns out that these auctions are all closely ( or clickthrough rate ) associated with that slot. related. The VCG auction encompasses the We assume that the slots have been ordered traditional Vickrey auction as a special case. with the most prominent slots fjrst, so that It has the attractive property that bidding the x > x > ⋯ > x 1 2 S . true value is a dominant strategy for all players The GSP auction produces a price for each and the equilibrium revenue should, in theory, slot. These prices must satisfy the revealed be about the same as the GSP auction. However, preference conditions that an advertiser who it also has some drawbacks; see Ausubel and purchases slot s prefers that slot to other slots it Milgrom ( 2006 ) and Rothkopf, Teisberg, and could have purchased: Kahn ( 1990 ) for a list of potential issues. In this note we describe two simple theoreti- ( 1 ) v x − p x ≥ v x − p x s s s s s t t t . cal properties of the VCG ad auction and some of the practical lessons learned in implementing It turns out that, if these inequalities are satis- a VCG auction for contextual ads. fjed for t = s + 1, they are satisfjed for all slots. After some manipulation we fjnd the following I. Search Ad Auctions system of inequalities that characterizes equilib- rium prices. In a search ad auction advertisers submit key- words and bids. When the advertiser’s keyword ( 2 ) v ( x − x ) + p x ≥ p x s s s + 1 s + 1 s + 1 s s matches a user’s query, the advertiser enters an auction. The advertiser with the highest bid gets ≥ v ( x − x ) + p x s + 1 s s + 1 s + 1 s + 1 . We note that these inequalities imply that ‡ Discussant: Al Roth, Stanford University. ( 3 ) ( v − v )( x − x ) ≥ 0, s t s t * Varian: Google MTV-1055, 1600 Amphitheatre Parkway, Mountain View, CA 94043 ( e-mail: hal@google. so that advertisers with higher values get more com ) ; Harris: Google MTV-900, 1600 Amphitheatre Parkway, Mountain View, CA 94043 ( e-mail: ckharris@ prominent slots, which shows that the GSP equi- google.com ) . We thank Gagan Aggarwal, Josh Dillon, Adam libria are effjcient. Juda, and Tim Lupus for helpful discussions on these topics. The same manipulations work in reverse. † Go to http://dx.doi.org/10.1257/aer.104.5.442 to visit That is, we can start with an effjcient assign- the article page for additional materials and author disclo- ment of advertisers to slots, which must satisfy sure statement ( s ) . 442

VOL. 104 NO. 5 THE VCG AUCTION IN THEORY AND PRACTICE 443 inequality ( 3 ) and show that there must exists per click. Writing out the VCG payments in the prices that satisfy the equilibrium inequali- three-slot case, we have: ties ( 2 ) . Thus, this simple position auction has mini versions of the First and Second Welfare ( 9 ) p 1 = ( 1 − ) + ( 2 − x ) + 1 x v 2 x x 2 v 3 x 3 v 4 x 3 , Theorems. There are many prices that satisfy these ( 10 ) p 2 = + ( 2 − x ) + 2 x v 3 x 3 v 4 x 3 , inequalities, but a particularly interesting equi- librium is the one with minimal revenue, where ( 11 ) p 3 = + 3 x v 4 x 3. the right inequalities hold with equality. Writing these conditions out for the three-slot case gives us this system: It is easy to check that this produces the same outcome as the GSP system ( 4 ) – ( 6 ) . Hence, ( 4 ) x = v ( x − x ) + p x the minimum-revenue GSP equilibrium has the p 1 1 2 1 2 2 2 same revenue as the VCG equilibrium, a result noted by Edelman, Ostrovsky, and Schwarz ( 5 ) x = v ( x − x ) + p x p 2 2 3 2 3 3 3 ( 2007 ) and Varian ( 2007 ) , and is a special case of a result derived by Demange and Gale ( 1985 ) ; ( 6 ) p x = v x 3 3 4 3 . Demange, Gale, and Sotomayor ( 1986 ) in a dif- ferent context. See Roth and Sotomayor ( 1990 ) Adding up the payments gives us a lower bound for a unifjed treatment. on revenue to the seller of IV. Broad Match ( 7 ) R = v ( x − x ) + 2 v ( x − x ) L 2 1 2 3 2 3 We said that the ad is eligible for the auction if the user’s query matches the advertiser’s key- + 3 v x 4 3 . word. But what counts as a match? It turns out that search engines use several types of matches We can perform the same sort of manipulations including “exact match” and “broad match.” A keyword [ dog food ] would be an exact match to get an upper bound on revenue: for the query “dog food” but a broad match for ( 8 ) R = v ( x − x ) + 2 v ( x − x ) the query “pet food.” U 1 1 2 2 2 3 A single broad match keyword will gener- ally have different values in auctions associated + 3 v x 3 3 . q to v s with different queries. Accordingly, we use denote the value of the keyword to the advertiser _ III. How the VCG Auction Works in slot s , in the auction for query q . We use v s to denote the expected value of slot s across all the In the VCG auction, each bidder is required broad-match auctions. to pay the cost their presence imposes on the Advertisers who choose broad match have to other bidders, using their stated bids as the value pick a single bid that applies for a whole range they place on the slots. We denote the bid by the of auctions. In the VCG auction, each advertiser b advertiser who occupies slot s by s . If advertiser can state its average value for a broad-matched 1 participates in the auction, the stated value visitor to its website and everything works out b x + b x received by the other advertisers is 2 2 3 3 . neatly. The GSP auction can, in general, be quite If advertiser 1 does not participate in the auction, messy since advertisers can appear in different the other advertisers all move up one position positions in different auctions. However, if the b x + b 3 x + b x and so receive 2 1 2 4 3 . Thus, the impact of broad match on advertiser values is “harm” that advertiser 1 imposes on the other small enough so that the ordering of advertis- advertisers is the difference between these two ers in all broad-match auctions is the same, then b ( x − x ) + b ( x − x ) + b x expressions, 2 1 2 3 2 3 3 4 , everything works out neatly in the GSP auction so this is the amount advertiser 1 is required to as well. If the same advertiser is in the same slot pay. It turns out that in the VCG auction, it is in each auction, then the equilibrium calculation _ v optimal for each advertiser to bid its true value is the same as before, with v s replacing s .