1. 2008 External Research Supported by Computational Analysis of Sponsored-Search Auctions External Research Initiative University of British Columbia David R.M. Thompson, Kevin Leyton-Brown Sponsored search has become a major part of electronic commerce, generating billions of dollars in revenue for the leading search engines. Whenever a user enters a search query, relevant advertisements are shown alongside the search results. These ads are selected through "position auctions," in which the highest bidder gets the highest position and so on. Many variations on position auctions have been tried: auctions where advertisers only pay when a user clicks on their ad (instead of every time the ad is shown), auctions where advertisers pay the minimum amount needed to maintain their position (instead of paying the amount they bid) and auctions where some bids are given more weight than others (to promote greater relevance or higher quality). Economists predict the outcomes of auctions using a game- theoretic model of rational bidding. The most common such prediction is that bids will constitute a "Nash equilibrium," meaning that each bidder will act to maximize his own payoffs given the actions of others. John Nash (subject of the film A Beautiful Mind) won a Nobel Prize for showing that such equilibria always exist. However, finding Nash equilibria is very computationally expensive. The standard ("normal-form") representation of a game grows exponentially in the number of players, and runtimes of the best known algorithms for finding Nash equilibria grow exponentially in the size of that representation. Partly because of these computational obstacles, economists typically make simplifying assumptions and identify equilibria analytically. Position auctions have evolved so quickly and are so complex that economists have not yet been able to analyze them thoroughly. Our contribution is to show how cutting-edge AI techniques for efficiently computing equilibria can be used to analyze realistic position auctions for the first time. To accomplish this, we represent the auctions using the recent "action-graph game" (AGG) representation. This exploits structure in the payoffs of each player, such as the fact that advertisers are only affected by the placement and price of their own ads. AGGs are far more compact than the normal form, allowing us to represent games in RAM that would otherwise take many terabytes to store. These games also don't rely on many of the standard assumptions required for algebraic analysis. In computational experiments, finding Nash equilibria of AGGs has been shown to be much faster than finding equilibria of equivalent normal form games. Using off-the-shelf algorithms for AGGs, we have been able to analyze many position auctions—including variations that had never been analyzed before—and make direct comparisons between them. Our ongoing work in this area aims to build better bidder models by mining real-world bidding data, and to develop better Nash equilibrium-finding algorithms, optimized for solving auction-based games and adapted to provide economically-relevant results.