position - Position Auctions Hal R. Varian * . December...

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Unformatted text preview: Position Auctions Hal R. Varian * . December 2005 Revised: March 29, 2006 Abstract I analyze the equilibria of a game based on the ad auction used by Google and Yahoo. This auction is closely related to the assign- ment game studied by Shapley-Shubik, Demange-Gale-Sotomayer and Roth-Sotomayer. However, due to the special structure of preferences, the equilibria of the ad auction can be calculated explicitly and some known results can be sharpened. I provide some empirical evidence that the Nash equilibria of the position auction describe the basic properties of the prices observed in Googles ad auction reasonably accurately. * I received many helpful comments from Marc Berndl, John Lamping, Amit Patel, Rob Shillingsburg, Diane Tang, and Eric Veach. I am particularly grateful to Meredith Goldsmith for her close reading of the paper, which improved the exposition significantly. I also thank Jonathan Rosenberg for allowing me to publish these results. Email contact: hal@sims.berkeley.edu 1 1 NASH EQUILIBRIUM OF POSITION AUCTION 2 I consider the problem of assigning agents a = 1 , . . . , A to slots s = 1 , . . . , S where agent a s valuation for slot s is given by u as = v a x s . We number the slots so that x 1 > x 2 > > x S so that all agents agree on their ordering of the slots, though each agent may value them differently. We also set x s = 0 for all s > S and assume that the number of agents is at least equal to the number of slots plus 1. This problem is motivated by the ad auctions used by Google and Yahoo. In these auctions the agents are advertisers and the slots are positions on a web page. Higher positions receive more clicks, so x s can be interpreted as the clickthrough rate for slot s . The value v a > 0 can be interpreted as the expected profit per click so u as = v a x s indicates the expected profit to advertiser a from appearing in slot s . The slots are sold via an auction. Each agent bids an amount b a , with the best clickthrough rate being assigned to the agent with the highest bid, the second-best slot to the agent with the second highest bid, and so on. Renumbering the agents if necessary, let v s be the value per click of the agent assigned to slot s . The price agent s faces is the bid of the agent immediately below him, so p s = b s +1 . Hence the net profit that agent a can expect to make if he acquires slot s is ( v a- p s ) x s = ( v a- b s +1 ) x s . Googles ad auction generated about $1.5 billion in Q3 2005 so its financial success alone makes it worthy of study. We will also see that position auctions have a nice mathematical structure and a strong relationship to existing literature on two-sided matching models. Edelman et al. [2005] independently examine these auctions and develop related results. However, our treatments are somewhat different and I also add some empirical analysis....
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This note was uploaded on 12/08/2011 for the course CIS 625 taught by Professor Michaelkearns during the Spring '12 term at Pennsylvania State University, University Park.

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position - Position Auctions Hal R. Varian * . December...

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