For a set $M$ of $m$ elements, we define a decreasing chain of classes of normalized monotone-increasing valuation functions from $2^M$ to $\mathbb{R}_{\geq 0}$, parameterized by an integer $q \in [2,m]$. For a given $q$, we refer to the class as $q$-partitioning. A valuation function is subadditive if and only if it is $2$-partitioning, and fractionally subadditive if and only if it is $m$-partitioning. Thus, our chain establishes an interpolation between subadditive and fractionally subadditive valuations. We show that this interpolation is smooth, interpretable , and non-trivial. We interpolate prior results that separate subadditive and fractionally subadditive for all $q \in \{2,\ldots, m\}$. Two highlights are the following:(i) An $\Omega \left(\frac{\log \log q}{\log \log m}\right)$-competitive posted price mechanism for $q$-partitioning valuations. Note that this matches asymptotically the state-of-the-art for both subadditive ($q=2$) [DKL20], and fractionally subadditive ($q=m$) [FGL15]. (ii)Two upper-tail concentration inequalities on $1$-Lipschitz, $q$-partitioning valuations over independent items. One extends the state-of-the-art for $q=m$ to $q<m$, the other improves the state-of-the-art for $q=2$ for $q > 2$. Our concentration inequalities imply several corollaries that interpolate between subadditive and fractionally subadditive, for example: $\mathbb{E}[v(S)]\le (1 + 1/\log q)\text{Median}[v(S)] + O(\log q)$. To prove this, we develop a new isoperimetric inequality using Talagrand's method of control by $q$ points, which may be of independent interest. We also discuss other probabilistic inequalities and game-theoretic applications of $q$-partitioning valuations, and connections to subadditive MPH-$k$ valuations [EFNTW19].

翻译：对于包含 $m$ 个元素的集合 $M$，我们定义了一类从 $2^M$ 到 $\mathbb{R}_{\geq 0}$ 的归一化单调递增估值函数的递减链，该链由整数参数 $q \in [2,m]$ 刻画。对于给定的 $q$，我们将该类别称为 $q$-划分估值。当且仅当估值函数是 $2$-划分时，其为次可加；当且仅当估值函数是 $m$-划分时，其为分数次可加。因此，我们的链建立了次可加与分数次可加估值之间的插值。我们证明该插值是光滑的、可解释的且非平凡的。我们插值了先前将所有 $q \in \{2,\ldots, m\}$ 的次可加与分数次可加估值分离的结果。两个亮点如下：(i) 针对 $q$-划分估值的 $\Omega \left(\frac{\log \log q}{\log \log m}\right)$-竞争比标价机制。注意，这在渐近意义上匹配了次可加 ($q=2$) [DKL20] 和分数次可加 ($q=m$) [FGL15] 的最优结果。(ii) 关于独立物品上 $1$-Lipschitz $q$-划分估值的两个上尾集中不等式。其中一个将 $q=m$ 的最优结果推广至 $q<m$，另一个则将 $q=2$ 的最优结果改进至 $q > 2$。我们的集中不等式衍生出若干推论，在次可加与分数次可加之间建立插值，例如：$\mathbb{E}[v(S)]\le (1 + 1/\log q)\text{Median}[v(S)] + O(\log q)$。为证明此结论，我们利用 Talagrand 的 $q$ 点控制方法发展了一种新的等周不等式，该不等式本身可能具有独立价值。此外，我们还讨论了 $q$-划分估值的其他概率不等式与博弈论应用，及其与次可加 MPH-$k$ 估值 [EFNTW19] 的联系。