Complexity of Autobidding Equilibria in GSP Sponsored Shopping

Abstract

We revisit autobidding equilibria in GSP sponsored shopping auctions, recently proposed and studied in Dütting et al. On the existence side, we show that autobidding equilibria under both budget constraints (ABE-B) and return-on-investment constraints (ABE-R) always exist and, moreover, always admit equilibria with polynomial-size support over the allocation space. On the computational side, we show that computing an exact equilibrium is in FIXP, and that computing an approximate equilibrium is in PPAD. We also prove that computing an approximate equilibrium is PPAD-hard for both ABE-B and ABE-R. Together, these results provide a fairly comprehensive picture of the computational complexity of autobidding equilibria in GSP sponsored shopping auctions.