Large-Market Regularization in Second-Price Pacing Equilibria

Second-price pacing is widely used in autobidding, combining the incentive structure of second-price auctions with bidding strategies robust to evolving market conditions. Yet its equilibrium concept, the second-price pacing equilibrium (SPPE), is theoretically delicate: previous studies show that it may be non-unique and natural best-response dynamics may cycle in deterministic markets. We first demonstrate that these pathologies persist even when bidders' valuations are independently and identically distributed with a smooth, positive density. In contrast, our main result shows that both phenomena disappear in sufficiently large stochastic markets. Under mild regularity and an elasticity condition on valuation distributions, growing competition induces cross-spending monotonicity, which yields uniqueness of the ex-ante SPPE for markets with an equilibrium bounded away from zero. Best-response dynamics converges globally to this equilibrium with explicit rates. Finally, in repeated auctions, when all bidders use adaptive pacing based only on their realized spending, the multipliers converge almost surely to the same equilibrium. Numerical experiments illustrate these theoretical findings.