Optimal Reserve Price Estimation in the Generalized First and Second Price Auctions with Best Response Dynamics

The generalized first and second price auctions (GFP and GSP) are the primary means through which ad auctions are conducted. Several theoretical results characterizing bidding behavior, expected revenue, and optimal reserve price under both mechanisms have been shown to hold under assumptions of valuation distribution symmetry and monotone hazard rate (MHR). From the standpoint of evolutionary game theory, the stability (or uniqueness) of these equilibria are not well understood. That is, assuming that agents are expected utility maximizing, these equilibria may be unattainable using best response dynamics. In this paper, we seek to computationally analyze the properties of these equilibria. Furthermore, we attempt to recover similar characterizations of bidding behavior, expected revenue, and optimal reserve price without the additional assumptions of symmetry and MHR. To that end, we introduce fast algorithms to compute conditional and joint distributions of order statistics which inevitably play a role in rank-based mechanisms such as the GSP and GFP.