可分物品的哲学家与预言家不等式
Philosopher and Prophet Inequalities for Divisible Items
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中文总结 AI 辅助
研究可分资源在线福利最大化问题，给出满足收益递减的单调凹估值近似算法，对最优在线策略有2/3近似，得到针对离线事后最优的紧预言家不等式，还表明固定价格拍卖对离线/预言家基准有1/2近似。
中文摘要 AI 辅助
我们研究可分资源的在线福利最大化问题。n个玩家依次到达，到达时从已知分布中抽取m个可分物品的估值函数，揭示估值后，在单位供应约束下分配不可撤销的分数包。虽然在线福利最大化在不可分物品和组合估值方面已被广泛研究，但资源可分且玩家有多维凹估值时了解较少。我们给出满足收益递减的单调凹估值的近似算法。主要结果是对最优在线策略有2/3的近似，即哲学家基准。算法由在线基准的低维凹松弛引导，并通过新的单项上限在线竞争解决方案进行舍入。我们还得到了针对离线事后最优的紧预言家不等式。我们表明，对每个原始可分物品采用一个线性单位价格的固定价格拍卖对离线/预言家基准有1/2的近似。价格通过聚合奥曼 - 沙普利支持价格获得，这是次模/XOS集函数支持价格的连续类似物，产生简单的物品价格而非离散化产生的依赖副本的价格。对于预言家基准，1/2的因子即使对于具有线性估值的单个物品在信息理论上也是紧的。
英文摘要
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.
