Convergence Analysis for Weighted Joint Strategy Fictitious Play in Generalized Second Price Auction

Abstract

Generalized Second Price (GSP) auction is one of the most commonly used auction mechanisms in sponsored search. As compared to conventional equilibrium analyses on GSP auctions, the convergence analysis on the dynamic behaviors of the bidders can better describe real-world sponsored search systems, and give them a more useful guideline for making improvement. However, most existing works on convergence analysis assume the bidders to be greedy in taking actions, i.e., they only utilize the bid information in the current round of auction when determining the best strategy for the next round. We argue that real-world professional advertisers are more capable and can utilize the information in a longer history to optimize their strategies. Accordingly, we propose modeling their behaviors by a weighted joint strategy fictitious play (wJSFP). In the proposed model, bidders determine their optimal strategies based on their beliefs on other bidders’ bid prices, and the beliefs are updated by considering all the information they have received so far in an iterative manner. We have obtained the following theoretical results regarding the proposed model: 1) when there are only two ad slots, the bid profile of the bidders will definitely converge; when there are multiple slots, there is a sufficient condition that guarantees the convergence of the bid profile; 2) as long as the bid profile can converge, it converges to a Nash equilibrium of GSP. To the best of our knowledge, this is the first time that the joint strategy fictitious play is adopted in such a complex game as sponsored search auctions.