Prediction markets like Kalshi and Polymarket are booming, and so is a fight among regulators, lawmakers, and advocates over their legality. Former New Jersey governor Chris Christie, who currently serves as an advisor to the American Gaming Association, has criticized prediction markets. The political fight in the US over the future of prediction markets like Polymarket and Kalshi has escalated into a full-blown war, and battle lines aren't being neatly drawn along party lines. Instead, conservative Mormons have aligned themselves with Las Vegas bigwigs and MAGA royalty is siding with liberal Democrat lobbyists. One side argues that the platforms are breaking the law by operating as shadow casinos.

This paper presents a unified approach for maximizing continuous DR-submodular functions that encompasses a range of settings and oracle access types. Our approach includes a Frank-Wolfe type offline algorithm for both monotone and non-monotone functions, with different restrictions on the general convex set. We consider settings where the oracle provides access to either the gradient of the function or only the function value, and where the oracle access is either deterministic or stochastic. We determine the number of required oracle accesses in all cases. Our approach gives new/improved results for nine out of the sixteen considered cases, avoids computationally expensive projections in three cases, with the proposed framework matching performance of state-of-the-art approaches in the remaining four cases. Notably, our approach for the stochastic function value-based oracle enables the first regret bounds with bandit feedback for stochastic DR-submodular functions.

Furthermore, we propose an adapted embedding method to incorporate numerical awareness.

In this paper, we specifically investigate the deep equilibrium model (DEQ), an infinite-depth neural network with shared weight matrices across layers.