Second-price auctions are a workhorse of online advertising. Bidders submit values; the highest bidder wins and pays the second-highest price. The mechanism is simple, well-understood, and incentive-compatible — bidding your true value is a dominant strategy.
Modern ad platforms don't use manual bidding. They use autobidders: automated systems that adjust bids in real time to maximize some objective (clicks, conversions, revenue) subject to a return-on-spend constraint. The autobidder observes auction outcomes and updates its bid multiplier, which scales bids across all auctions. Multiple autobidders interacting through repeated auctions form a dynamical system.
Anagnostides et al. (arXiv:2602.09118) prove this dynamical system is chaotic. Not metaphorically chaotic — formally chaotic, in the Li-Yorke sense: sensitive dependence on initial conditions, topological transitivity, and a dense set of periodic orbits. The proof works by showing that autobidding dynamics in second-price auctions can simulate any continuous-time nonlinear system to arbitrary precision, including known chaotic systems like Chua's circuit.
The mechanism: the return-on-spend constraint creates a nonlinear feedback loop. Each autobidder adjusts its multiplier based on the ratio of value received to money spent. But the value received depends on which auctions it wins, which depends on all other autobidders' multipliers. The multipliers are coupled through the auction outcomes, and the coupling is nonlinear because winning is a threshold event — you either beat the second-highest bid or you don't. This threshold nonlinearity, iterated over many auctions, generates the same mathematical structure as analog computing circuits known to be chaotic.
The consequence is fundamental: the long-term behavior of these auction systems is unpredictable in principle, not just in practice. No amount of data or computation can forecast where the bid landscape will be in the far future, because the dynamics are chaotic. The most common mechanism in online advertising — automated bidding in second-price auctions — has the same mathematical character as weather systems. Prediction is possible for short horizons and impossible for long ones.
Anagnostides, Gemp, Piliouras, & Spendlove, "Chaos in Autobidding Auctions," arXiv:2602.09118 (2026).