Philosopher and Prophet Inequalities for Divisible Items

We study online welfare maximization with divisible resources. A sequence of $n$ players arrive one by one; upon arrival, each player draws a valuation function over $m$ divisible items from a known distribution, reveals this valuation, and must be allocated an irrevocable fractional bundle subject to unit supply constraints. While online welfare maximization has been extensively studied for indivisible items and combinatorial valuations, much less is known when the resources are divisible and players have multi-dimensional concave valuations. We give approximation algorithms for monotone concave valuations satisfying diminishing returns. Our main result is a $2/3$-approximation to the optimal online policy, also known as the philosopher benchmark. The algorithm is guided by a low-dimensional concave relaxation of the online benchmark and rounds it via a new single-item capped online contention resolution scheme. This Capped-OCRS problem allocates to each realized type no more than its prescribed fractional bundle while preserving a $2/3$-fraction of that bundle in expectation. Its analysis uses a submartingale potential for the remaining side, we show that computing the optimal online policy is #P-hard even for a single divisible item. We also obtain a tight prophet inequality against the offline hindsight optimum. We show that a fixed-price auction with one linear per-unit price for each original divisible item achieves a $1/2$-approximation to the offline/prophet benchmark. The prices are obtained by aggregating Aumann--Shapley supporting prices, a continuous analogue of supporting prices for submodular/XOS set functions, and yield simple item prices rather than copy-dependent prices arising from discretization. The factor $1/2$ for the prophet benchmark is information-theoretically tight even for one item with linear valuations.