We generalize the problem of online submodular welfare maximization to incorporate various stochastic elements that have gained significant attention in recent years. We show that a non-adaptive Greedy algorithm, which is oblivious to the realization of these stochastic elements, achieves the best possible competitive ratio among all polynomial-time algorithms, including adaptive ones, unless NP$=$RP. This result holds even when the objective function is not submodular but instead satisfies the weaker submodular order property. Our results unify and strengthen existing competitive ratio bounds across well-studied settings and diverse arrival models, showing that, in general, adaptivity to stochastic elements offers no advantage in terms of competitive ratio. To establish these results, we introduce a technique that lifts known results from the deterministic setting to the generalized stochastic setting. The technique has broad applicability, enabling us to show that, in certain special cases, non-adaptive Greedy-like algorithms outperform the Greedy algorithm and achieve the optimal competitive ratio. We also apply the technique in reverse to derive new upper bounds on the performance of Greedy-like algorithms in deterministic settings by leveraging upper bounds on the performance of non-adaptive algorithms in stochastic settings.

翻译：我们推广了在线子模福利最大化问题，以纳入近年来备受关注的各类随机元素。我们证明：除非NP$=$RP，否则一种忽视这些随机元素实现情况的非自适应贪心算法，在所有多项式时间算法（包括自适应算法）中，均能达到可能的最优竞争比。即使目标函数不满足子模性而仅满足较弱的子模序性质，该结论依然成立。我们的结果统一并强化了不同经典场景及多样到达模型下的竞争比界，表明在一般情况下，针对随机元素的自适应性在竞争比方面并无优势。为建立这些结果，我们引入了一种将确定性场景已知结论推广至广义随机场景的技术。该技术具有广泛适用性，可证明在某些特殊情形下，非自适应类贪心算法的性能优于贪心算法且达到最优竞争比。我们亦反向应用该技术，通过利用随机场景中非自适应算法性能的上界，推导出确定性场景中类贪心算法性能的新上界。