We study the problem of fairly allocating a set of indivisible goods among agents with {\em bivalued submodular valuations} -- each good provides a marginal gain of either $a$ or $b$ ($a < b$) and goods have decreasing marginal gains. This is a natural generalization of two well-studied valuation classes -- bivalued additive valuations and binary submodular valuations. We present a simple sequential algorithmic framework, based on the recently introduced Yankee Swap mechanism, that can be adapted to compute a variety of solution concepts, including max Nash welfare (MNW), leximin and $p$-mean welfare maximizing allocations when $a$ divides $b$. This result is complemented by an existing result on the computational intractability of MNW and leximin allocations when $a$ does not divide $b$. We show that MNW and leximin allocations guarantee each agent at least $\frac25$ and $\frac{a}{b+2a}$ of their maximin share, respectively, when $a$ divides $b$. We also show that neither the leximin nor the MNW allocation is guaranteed to be envy free up to one good (EF1). This is surprising since for the simpler classes of bivalued additive valuations and binary submodular valuations, MNW allocations are known to be envy free up to any good (EFX).

翻译：我们研究了在一组代理间公平分配不可分商品的问题，其中代理具有“双值子模估值”——每件商品提供边际收益为$a$或$b$（$a<b$），且商品呈递减边际收益。这是两类经典估值模型（双值可加估值与二元子模估值）的自然泛化。我们提出基于近期引入的Yankee Swap机制构建的简单顺序算法框架，可适配计算多种解概念，包括当$a$整除$b$时的最大纳什福利（MNW）、字典序最小（leximin）及$p$-平均福利最大化分配。该结果与现有研究中关于当$a$不整除$b$时MNW与leximin分配计算复杂性的结论互为补充。我们证明，当$a$整除$b$时，MNW与leximin分配分别能保证每个代理获得其最大最小份额的$\frac25$与$\frac{a}{b+2a}$。同时，我们表明leximin与MNW分配均无法保证至多一件商品的无嫉妒性（EF1）。这一结论令人意外，因为对于双值可加估值与二元子模估值这类较简化的模型，MNW分配已知可实现至多任意一件商品的无嫉妒性（EFX）。