We study the fair allocation of $m$ indivisible goods among $n$ heterogeneous agents arriving online, under maximin share (MMS) fairness. The key challenge lies in making allocation decisions in an adversarial setting without knowing future agents' valuations. No guarantees were known prior to our work in adversarial settings. Our main positive result is an online allocation algorithm for submodular agents that with probability at least $0.4$ guarantees a $1/24$-approximation to MMS to all agents simultaneously. Our algorithm intersperses a greedy-style adaptive sampling-without-replacement protocol with single-good reduction rounds. For XOS valuations, we adapt this single-or-sample approach to guarantee a $c$-MMS for all agents simultaneously with probability $1-O(c)$ for any parameter $c\in O(1/\log n)$. A key technical contribution of our work is in establishing Chernoff-style tail bounds on item allocation events for this adaptive sampling process despite strong positive correlations among the items, a result that may be of independent interest. The approximation versus success probability tradeoffs achieved by our algorithms are asymptotically tight. We show that even for binary additive valuations, no randomized algorithm can ensure $c$-MMS to all agents with probability exceeding $(1-c^2/5)$, for any $c\in (0,1]$. For XOS valuations, stronger lower bounds hold: no randomized algorithm can ensure $c$-MMS to all agents with probability exceeding $(1-c/3)$, for any $c\in (0,1]$. Furthermore, no algorithm can achieve $ω(1/\log \log n)$-MMS with any constant probability. All our lower bounds hold even for simple binary instances whose MMS values and MMS partitions can be computed in linear time.
Yuanchen Brian Tang, Jiaxin Song, Bhaskar Ray Chaudhury, Ruta Mehta
We study the stability of data exchange among competing firms. Unlike tangible goods, data is non-rival: sharing it does not deprive the giver of its use. On one hand, competing firms-- often operating as machine-learning-driven predictors that collect observations from a shared environment to solve similar tasks-- can improve their predictive accuracies from pooling data. On the other hand, they compete for market share, which may depend on their relative predictive performance. This raises the central question of this paper: can there be a stable data exchange in such competitive environments? We formalize this via a model of $n$ firms exchanging fractions of their datasets, with predictive performance feeding into market competition. We adopt core stability as our solution concept, accounting for externalities via worst-case responses by non-participating firms. Our main results reveal a sharp contrast driven by demand structure: (i) in fully rational markets, where customers flock entirely to the top-performing firm, a core-stable exchange always exists for any monotone continuous prediction functions; (ii) in partially rational markets with logit-style demand, core-stable exchanges may fail to exist-and even approximate cores can be empty. Our impossibility results suggest that internal profit-sharing within coalitions can restore stability in partially rational markets. We formalize this intuition by showing that when co-operating coalitions equally redistribute market-share revenues among members, a core-stable data exchange exists. Finally, we propose a natural exchange dynamic and demonstrate empirically that it converges to a core-stable data exchange in fully rational markets on simulated instances derived from real datasets.