How should agents bid in repeated sequential auctions when they are budget constrained? A motivating example is that of sponsored search auctions, where advertisers bid in a sequence of generalized second price (GSP) auctions. These auctions, specifically in the context of sponsored search, have many idiosyncratic features that distinguish them from other models of sequential auctions: First, each bidder competes in a large number of auctions, where each auction is worth very little. Second, the total bidder population is often large, which means it is unrealistic to assume that the bidders could possibly optimize their strategy by modeling specific opponents. Third, the presence of a virtually unlimited supply of these auctions means bidders are necessarily expense constrained.

Motivated by these three factors, we first frame the generic problem as a discounted Markov Decision Process for which the environment is independent and identically distributed over time. We also allow the agents to receive income to augment their budget at a constant rate. We first provide a structural characterization of the associated value function and the optimal bidding strategy, which specifies the extent to which agents underbid from their true valuation due to long term budget constraints. We then provide an explicit characterization of the optimal bid shading factor in the limiting regime where the discount rate tends to zero, by identifying the limit of the value function in terms of the solution to a differential equation that can be solved efficiently. Finally, we proved the existence of Mean Field Equilibria for both the repeated second price and GSP auctions with a large number of bidders.