Rafael Frongillo

Enrique Nueve, Bao Nguyen, Rafael Frongillo et al.

Prediction markets rely on liquidity to convert trades into informative prices, yet existing mechanisms fix liquidity ex ante. This restriction enforces a static trade-off between price responsiveness and worst-case loss despite inherently nonstationary trading conditions. We propose a fundamentally different approach that treats liquidity selection itself as an online learning problem. Our mechanism mixes a family of cost-function markets via learnable weights, yielding a single adaptive market that preserves no-arbitrage, bounded worst-case loss, expressiveness, and positive upside. We introduce a hybrid structural risk signal, a per-round objective that quantifies the trade-off between price impact and inventory risk, and show that standard online learning algorithms achieve switching-regret guarantees relative to the best sequence of liquidity regimes in hindsight. Simulations demonstrate that the mechanism adaptively shifts liquidity across regimes in response to both order flow and inventory dynamics. Our results establish a principled framework for adaptive liquidity, connecting prediction market design with online learning.

Rafael Frongillo

The Determinant Mutual Information (DMI) mechanism of Kong (2020, 2024) is dominantly truthful within the class of *consistent* reporting strategies, those that apply the same single-task strategy to every task. In settings where agents see multiple tasks before reporting, such as peer grading or peer review, it is natural to consider *joint-task* strategies that may condition reports on the full signal vector. Perhaps surprisingly, we show that the DMI mechanism preserves truthful reporting as a best response among all joint-task strategies when other agents play consistent strategies, so that truthfulness remains a Bayes--Nash equilibrium in the joint-task class. Without the restriction of peers to consistent strategies, however, both dominant truthfulness and informed truthfulness fail against joint-task peer strategies.

Rafael Frongillo, Haipeng Luo, Nishant A. Mehta et al.

U-calibration studies online forecasting algorithms whose predictions can be consumed by any unknown downstream agent, guaranteeing sublinear regret simultaneously for all proper loss functions. Existing U-calibration algorithms achieve worst-case optimal $O(\sqrt{T})$ regret for every bounded proper loss, but they fail to adapt to easier losses: as we show, even for smooth losses such as squared loss, they incur $Ω(\sqrt{T})$ regret instead of the optimal $O(\log T)$ regret. In this work, we show that this limitation is not inherent. Specifically, we design a single forecast algorithm that simultaneously achieves $\tilde O(\sqrt{T})$ regret for every bounded proper loss and $O(\log T)$ regret for every bounded smooth proper loss. More generally, our algorithm also attains logarithmic regret for losses that are smooth relative to the log-barrier, which include several non-Lipschitz examples. Our approach is based on a novel variant of Follow-the-Perturbed-Leader (FTPL) in which perturbations are applied directly in the prediction space using self-concordant noise. The resulting analysis also departs substantially from prior FTPL analyses due to the complex nature of this noise and may be of independent interest.