Forward looking Nash equilibrium in sponsored search auction

Abstract

Sponsored search auction has created a market model that has turned the horsepower of search engines into a lucrative business. The currently actively studied protocol for sponsored search auctions is the Generalized Second Price (GSP) auction, instead of the sacred writ of the VCG protocol (at least to theoreticians). Understanding the GSP has therefore become the most favorite current topic in the studies of Internet Economics. One justification to such a protocol is the proof of Varian proof that any symmetric Nash equilibrium would generate at least as much payoff to the auctioneer as in the standard VCG protocol. The same result is derived by Edelman, Ostrovsky and Schwarz, for an equivalent concept of Locally-Envy Free Equilibrium. In some sense, the arguments of Varian, and Edelman, et al, justify to the auctioneer the use of the GSP instead of the VCG protocol as the former will potentially bring in higher revenue than the latter. Indeed, despite of a generalized English auction implementation by Edelman, et al., as well as a truthful protocol (laddered auction) designed by Aggarwal, Goel, Motwani, the non-truthful GSP is adopted in reality. Theoretically, the work of Edelman, et al., and Aggarwal, et al., derive revenue equivalence theorems (RET) with respect to the VCG protocol, but would depend on the auctioneer’s goodwill to implement them. Only this time, the auctioneer’s decision did not fall in their favor. In this thesis, I prove an alternative RET that is based on bidding agents’ rational behavior, under the GSP protocol. First, the forward looking attribute is established. Accordingly, an agents would rationally choose a strategy called the forward looking response, which leads to a unique fixed point of this response function, the forward looking equilibrium. The outcome is equivalent in payment to the VCG protocol for every participants, therefore, the same as that of Edelman et al, and Aggarwal et al. The main difference is that the result is fully based on bidding agents’ rational forward looking strategies under the reigning GSP protocol. Furthermore, the forward looking Nash equilibrium as the fixed point of the forward looking responses of bidding agents is proven to be reached by a randomized schedule policy in finite steps with probability one. Simulation results also show that irrational deviations, by vindicative bidders, from the forward looking responses will not destabilize the system. With the forward looking Nash equilibrium as the analytic tool, I further explore cross-market arbitrage behavior of the auctioneers in presence of multiple markets. It is proven that the cross-market equilibrium exists under mild conditions and auctioneers benefit from the cross-market arbitrage behavior, so does the social efficiency.