  It has become the default in markets such as ad auctions for participants to
bid in an auction through automated bidding agents (autobidders) which adjust
bids over time to satisfy return-over-spend constraints. Despite the prominence
of such systems for the internet economy, their resulting dynamical behavior is
still not well understood. Although one might hope that such relatively simple
systems would typically converge to the equilibria of their underlying
auctions, we provide a plethora of results that show the emergence of complex
behavior, such as bi-stability, periodic orbits and quasi periodicity. We
empirically observe how the market structure (expressed as motifs)
qualitatively affects the behavior of the dynamics. We complement it with
theoretical results showing that autobidding systems can simulate both linear
dynamical systems as well logical boolean gates.
