Tractable Computational Methods for Finding Nash Equilibria of Perfect-Information Position Auctions

Due to the economic importance of the position auctions used by search engines to sell advertising, these auctions have received considerable recent study. However, most of this study has been analytic, and these analyses have relied on strong assumptions about the structure of the setting. In this paper, we show that it is feasible to perform computational equilibrium analyses of complex, realistic auction problems like advertising auctions. In particular, we show for the first time that the Nash equilibria of position auctions can be computed exactly, and we do so without relying on any of the assumptions that are necessary for closed-form analysis. We achieve these results by deriving a polynomialsized action graph game representation of the position auction (discretizing bid amounts) and then finding a Nash equilibrium of that game. Our formulation makes it possible to show how equilibrium behavior, revenue and efficiency vary across auction types (Generalized First Price or Generalized Second Price), payment structures (pay-per-click or pay-perimpression), and click-through bias (position does/does not also depend on advertiser’s click-through rate).