GSP with General Independent Click-through-Rates

The popular generalized second price (GSP) auction for sponsored search is built upon a separable model of click-through-rates that decomposes the likelihood of a click into the product of a “slot effect” and an “advertiser effect”—if the first slot is twice as good as the second for some bidder, then it is twice as good for everyone. Though appealing in its simplicity, this model is quite suspect in practice. A wide variety of factors including externalities and budgets have been studied that can and do cause it to be violated. In this paper we adopt a view of GSP as an iterated second price auction (see, e.g., Milgrom [2010]) and study how the most basic violation of separability—position dependent, arbitrary public click-through-rates that do not decompose—affects results from the foundational analysis of GSP [Varian, 2007; Edelman et al., 2007]. For the two-slot setting we prove that for arbitrary click-through-rates, for arbitrary bidder values, an efficient pure-strategy equilibrium always exists; however, without separability there always exist values such that the VCG outcome and payments cannot be realized by any bids, in equilibrium or otherwise. The separability assumption is therefore necessary in the two-slot case to match the payments of VCG but not for efficiency. We moreover show that without separability, generic existence of efficient equilibria is sensitive to the choice of tie-breaking rule, and when there are more than two slots, no (bid-independent) tie-breaking rule yields the positive result. In light of this we suggest alternative mechanisms that trade the simplicity of GSP for better equilibrium properties when there are three or more slots.