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Pure-Strategy Nash Equilibria of GSP Keyword Auction

Linjing Li, Daniel Zeng, Huimin Zhao

2012
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Journal of the AIS
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Research Article Linjing Li Despite the tremendous commercial success of generalized second-price (GSP) keyword auctions, it still remains a big challenge for an advertiser to formulate an effective bidding strategy. In this paper, we strive to bridge this gap by proposing a framework for studying pure-strategy Nash equilibria in GSP auctions. We first analyze the equilibrium bidding behaviors by investigating the properties and distribution of all pure-strategy Nash equilibria. Our analysis
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... ws that the set of all pure-strategy Nash equilibria of a GSP auction can be partitioned into separate convex polyhedra based on the order of bids if the valuations of all advertisers are distinct. We further show that only the polyhedron that allocates slots efficiently is weakly stable, thus allowing all inefficient equilibria to be weeded out. We then propose a novel refinement method for identifying a set of equilibria named the stable Nash equilibrium set (STNE) and prove that STNE is either the same as or a proper subset of the set of the well-known symmetrical Nash equilibria. These findings free both auctioneers and advertisers from complicated strategic thinking. The revenue of a GSP auction on STNE is at least the same as that of the classical Vickrey-Clarke-Groves mechanism and can be used as a benchmark for evaluating other mechanisms. At the same time, STNE provides advertisers a simple yet effective and stable bidding strategy. Li et al. / PSNE of GSP Auction As discussed above, rational bid revisions must follow the directions indicated by the arrows in various regions. For an arbitrary Nash equilibrium in ℰ 0 ⋆ , if a perturbation forces it out of this region, following the directions indicated by the arrows, the bidding vector will eventually converge to ℰ 0 ⋆ after several rational bid revisions. According to the definition of SNE/LEF, it is easy to see that SNE/LEF is ℰ 0 ⋆ ∪ ℰ 0 3 . The stable set ℰ 0 ⋆ (discussed in the next section) is only a subset of SNE/LEF. Thus, not all equilibria in SNE/LEF are stable in a dynamic environment.

doi:10.17705/1jais.00286
fatcat:ibwlhotoh5ftzgzkbafq7cqufe