An AI developer often has several candidate versions of a system and can deploy a mixture of them, sending each request to a randomly chosen version, as in A/B testing. Each stakeholder (e.g., a safety team, a regulator, or a customer with private test data) accepts a mixture exactly when its expected performance (e.g., its rate of harmful replies) meets the stakeholder's own standard, which the developer can learn only by asking whether a proposed mixture is acceptable. We model this as choosing a lottery over $m$ alternatives for $n$ agents, each of whom accepts exactly the lotteries in an unknown halfspace and answers such binary membership queries. While Choo et al. (ICML 2026) sought a lottery that every agent accepts, none may exist when, say, safety and helpfulness requirements conflict, so we minimize the number of rejecting agents. We show that this problem is APX-hard even when all halfspaces are known and all utilities are $0$ or $1$. We then give exact algorithms that run in polynomial time when $m$ is constant, and in quasi-polynomial time when, for a constant $η$, some optimal lottery has no accepting agent within $η$ of its threshold. When only some agents can be queried, optimizing over a uniform sample of agents gives additive and relative guarantees for the whole population, with a sample size that does not depend on $n$, and a linear program gives a polynomial-time method for any $m$. On the other hand, for $m\ge 3$, a deterministic algorithm that queries only $K$ distinct agents cannot guarantee fewer than $n-K$ extra rejections, and this is achievable; in other words, it can guarantee nothing for the agents it does not ask. Small synthetic experiments indicate that our worst-case bounds are conservative on random instances.
Karen Frilya Celine, Warut Suksompong, Nicholas Teh
When allocating indivisible goods, the maximum Nash welfare rule is known to be the unique welfarist rule satisfying envy-freeness up to one good. We strengthen this characterization by showing that envy-freeness up to one good can be replaced by substantially weaker fairness requirements: envy-freeness up to $k$ goods or proportionality up to $k$ goods, for any fixed $k$. Our characterization holds even for profiles with identical goods.
We study online fair division of indivisible goods among $n$ agents with frequency predictions: before any good arrives, the algorithm learns the multiset of each agent's values for the goods, but neither the arrival order nor which values of different agents belong to the same good. Neoh et al. (ICML 2026) showed that such predictions allow every share-based guarantee that is achievable offline, and left open whether a positive approximation of envy-freeness up to one good (EF1) is possible. We show that deterministic online algorithms can guarantee $1/2$-EF1 for three agents and $(3-\sqrt5)/2$-EF1 for four agents, whereas for five or more agents, no online algorithm can guarantee any positive approximation of EF1, even with $n+1$ goods and identical predictions. Together with the known result for two agents, a positive approximation of EF1 is therefore possible if and only if $n\leq4$. We also rule out any positive approximation of EFX for three or more agents, even when the valuations take only two distinct positive values. Next, we use the meta-algorithm of Neoh et al. (ICML 2026) to compute an epistemic EFX (EEFX) allocation in polynomial time for any $n$, and hence a $1/(n-1)$-EF$(n-2)$ allocation for $n\geq3$, whereas algorithms that fix a picking sequence in advance can guarantee a positive approximation of EF$k$ if and only if $n\leq k+2$. Finally, for noisy predictions, we give algorithms that return EEFX allocations when the predictions are exact, and guarantee $(1-2\varepsilon+O(\varepsilon^2))$-EEFX under underestimates and, for $n\geq3$, $(1-2(n-1)\varepsilon+O(\varepsilon^2))$-EEFX under overestimates; the linear terms are the best possible, and larger overestimates rule out any positive approximation for such algorithms.