Degrade GSP ’ s Efficiency ?

We consider variants of the cascade model of externalities in sponsored search auctions introduced independently by Aggrawal et al. and Kempe and Mahdian in 2008, where the click-through rate of a slot depends also on the ads assigned to earlier slots. Aggrawal et al. and Kempe and Mahdian give a dynamic programming algorithm for finding the efficient allocation in this model. We give worst-case efficiency bounds for a variant of the classical Generalized Second Price (GSP) auction in this model. Our technical approach is to first consider an idealized version of the model where an unlimited number of ads can be displayed on the same page; here, Aggrawal et al. and Kempe and Mahdian show that a greedy algorithm finds the optimal allocation. The game theoretic analog of this greedy algorithm can be thought of as a variant of the classical GSP auction. We give the first non-trivial worst-case efficiency bounds for GSP in this model. In the more general model with limited slots, greedy algorithms like GSP can compute extremely bad allocations. Nonetheless, we show that an appropriate extension of the greedy algorithm is approximately optimal, and that the worst-case equilibrium inefficiency in the corresponding analog of GSP also remains bounded. In the context of these models, the GSP mechanisms suffer from two forms of suboptimality: that from using a simple allocation rule (the greedy algorithm) rather than an optimal one (based on dynamic programming), and that from the strategic behavior of the bidders (caused by using the GSP’s critical bid pricing rule rather than one leading to a dominant-strategy implementation). Our results show that for this class of problems, the two causes of efficiency loss can be analyzed separately.