High-Precision First-Price Throttling Equilibria via Positive-Tree Continuation

First-price throttling lets budget-constrained advertisers randomize participation without changing bids, but ill-conditioned payment Jacobians complicate high-precision equilibrium computation. To overcome this obstacle, we represent a normalized continuation direction by positive tree polynomials. This yields a deterministic algorithm that computes a rational $\delta$-approximate equilibrium in $\mathrm{poly}(N,\log (1/\delta ))$ time for every finite rational first-price market of encoding length $N$. We then characterize extensions through payment evaluation and derive computable revenue and coordinate-error certificates. Finally, learning results distinguish residual accuracy from participation accuracy, with matching sample bounds for a cyclic boundary family.