Internet Advertising and Optimal Auction Design Michael Schwarz

1. Internet Advertising and Optimal Auction Design Michael Schwarz Yahoo! Research Keynote Address KDD July 2008

2. Humorous History of Market Design Wife auctions, Babylon, 5th century BC Market design, matching theory, second half 20th century, US Moving from a metaphor to reality, Everywhere, now • Note: Vickrey (1961) did not invent Vickrey auction • Gale, Shapley (1962) did not invent deferred acceptance algorithm Over time mechanism design moved from being primarily a metaphor describing markets to a tool that shapes them. Indeed, mechanisms can be viewed as models describing more or less everything in the economy---e.g. a worker negotiating with a few employers can be modeled as a seller conducting an auction; young man and woman complex journey towards finding life partners can be modeled as a deferred acceptance algorithm etc. Literal interpretation of the words mechanism design becomes increasingly appropriate--- FCC conducting a spectrum auction and medical residency match are a few examples where mechanism design is no longer a metaphor of reality but rather it is a force that shapes reality of the market place with clear and rigid rules. This in turn gave rise to a number of interesting algorithmic and data mining problems that are of both theoretical and practical importance.

3. “Designed Mechanisms” v. “Metaphors” in the Internet Age • Until recently there was a sharp distinction between situation were mechanism is a "metaphor (or a model)" vs. "designed mechanisms". In the former case the underlying rules of the game are complex and implicit---the economic reality only roughly resembles the simple rules of mechanism design models. In the later case the rules tend to be fairly simple and explicit. • Recently, a new trend emerged---mechanisms that are designed (in a sense that the rules of the game are explicitly specified in a market run by a computer program), yet the rules of the market place are complex and as long as market participants are concerned the rules are implicit because they are not fully observable by market participants. • The market for sponsored search is perhaps the first example of such marketplace-- the mechanism used for selling sponsored search advertisement is better described by words "pricing mechanism" than an auction. In essence, when machine learning meets mechanism design we end up with a "designed mechanism" that shares some features of unstructured environment of the off line world. As mechanism becomes enriched with tweaks based on complex statistical models the rules become complex enough to be impossible to communicate to market participants.

6. History • Generalized First-Price Auctions 1997 auction revolution by Overture (then GoTo) • Pay per-click for a particular keyword • Links arranged in descending order of bids • Pay your bid Problem. Generalized First-Price Auction is unstable, because it generally does not have a pure strategy equilibrium, and bids can be adjusted dynamically

9. History (continued) • Google’s (2002) generalized second-price auction (GSP) • Pay the bid of the next highest bidder • Later adopted by Yahoo!/Overture and others

10. GSP and the Generalized English Auction • N≥2 slots and K = N +1 advertisers • αi is the expected number of clicks in position i • sk is the value per click to bidder k • A clock shows the current price; continuously increases over time • A bid is the price at the time of dropping out • Payments are computed according to GSP rules • Bidders’ values are private information

11. Strategy can be represented by pi(k,h,si) si is the value per click of bidder i, pi is the price at which he drops out k is the number of bidders remaining (including bidder i), and h=(bk+1,…,bk) is the history of prices at which bidders K, K-1, …, k+1 have dropped out If bidder i drops out next he pays bk+1 (unless the history is empty, then set bk+1≡0).

12. Theorem. In the unique perfect Bayesian equilibrium of the generalized English auction with strategies continuous in si, an advertiser with value si drops out at price pi(k,h, si)= si -(si-bk+1) αk /αk-1 In this equilibrium, each advertiser's resulting position and payoff are the same as in the dominant-strategy equilibrium of the game induced by VCG. This equilibrium is ex post: the strategy of each bidder is a best response to other bidders' strategies regardless of their realized values. The above is from Edelman, Ostrovsky and Schwarz Internet Advertising and the Generalized Second Price Auction: Selling Billions of Dollars Worth of Keywords, AER, March, 2007

13. The Intuition of the Proof First, with k players remaining and the next highest bid equal to bk+1, it is a dominated strategy for a player with value s to drop out before price p reaches the level at which he is indifferent between getting position k and paying bk+1 per click and getting position k-1 and paying p per click. Next, if for some set of types it is not optimal to drop out at this "borderline" price level, we can consider the lowest such type, and then once the clock reaches this price level, a player of this type will know that he has the lowest per-click value of the remaining players. But then he will also know that the other remaining players will only drop out at price levels at which he will find it unprofitable to compete with them for the higher positions.

14. Optimal Mechanism Assume that bidder values are iid draws from a distribution that satisfies the following regularity condition: (1-F(v))/f(v) is a decreasing function of v. Proposition. Generalized English auction with a reserve price v* is an optimal mechanism, where v* denote the solution of (1-F(v))/f(v) =v Note: The optimal mechanism design in multi-unit auctions remains an open problem. Note: Reserve price does not depend on the rate of decline in CTR, on the number of positions and on number of bidders From Edelman and Schwarz Optimal Auction Design in a Multi-unit Environment: The Case of Sponsored Search Auctions

15. Percent increase in search engine revenue when search engines set optimal reserve prices

16. Total increase in each advertiser's payment, when reserve price is set optimally versus at $0.10

17. Theorem. Reserve price causes an equal increase in total payments of all advertiser whose value are above reserve price.