In budget-feasible mechanism design, a buyer wishes to procure a set of items of maximum value from self-interested players. We have a valuation function $v:2^U \to \mathbb{R}_+$, where $U$ is the set of all items, where $v(S)$ specifies the value obtained from set $S$ of items. The entirety of current work on budget-feasible mechanisms has focused on the single-dimensional setting, wherein each player holds a single item $e$ and incurs a private cost $c_e$ for supplying item $e$. We introduce multidimensional budget feasible mechanism design: the universe $U$ is now partitioned into item-sets $\{G_i\}$ held by the different players, and each player $i$ incurs a private cost $c_i(S_i)$ for supplying the set $S_i\subseteq G_i$ of items. A budget-feasible mechanism is a mechanism that is truthful, and where the total payment made to the players is at most some given budget $B$. The goal is to devise a budget-feasible mechanism that procures a set of items of large value. We obtain the first approximation guarantees for multidimensional budget feasible mechanism design. Our contributions are threefold. First, we prove an impossibility result showing that the standard benchmark used in single-dimensional budget-feasible mechanism design, namely the algorithmic optimum is inadequate in that no budget-feasible mechanism can achieve good approximation relative to this. We identify that the chief underlying issue here is that there could be a monopolist which prevents a budget-feasible mechanism from obtaining good guarantees. Second, we devise an alternate benchmark, $OPT_{Bench}$, that allows for meaningful approximation guarantees, thereby yielding a metric for comparing mechanisms. Third, we devise budget-feasible mechanisms that achieve constant-factor approximation guarantees with respect to this benchmark for XOS valuations.

翻译：在预算可行机制设计中，买方希望从自利玩家处采购一组价值最大的物品。我们有一个估值函数 $v:2^U \to \mathbb{R}_+$，其中 $U$ 是所有物品的集合，$v(S)$ 表示从物品集 $S$ 中获得的估值。目前关于预算可行机制的全部工作都集中在单维设定上，即每个玩家持有一件单一物品 $e$，并为提供物品 $e$ 产生私有成本 $c_e$。我们引入了多维预算可行机制设计：此时全集 $U$ 被划分为由不同玩家持有的物品集 $\{G_i\}$，每个玩家 $i$ 为提供物品子集 $S_i\subseteq G_i$ 产生私有成本 $c_i(S_i)$。预算可行机制是一种诚实机制，且支付给玩家的总金额不超过某个给定预算 $B$。目标是设计一个能采购到高估值物品集的预算可行机制。我们首次获得了多维预算可行机制设计的近似保证。我们的贡献有三重。首先，我们证明了一个不可能性结果，表明单维预算可行机制设计中使用的标准基准（即算法最优值）是不充分的，因为没有任何预算可行机制能相对于该基准实现良好的近似。我们指出，这里的主要根本问题在于可能存在垄断者，这会阻止预算可行机制获得良好的保证。其次，我们设计了一个替代基准 $OPT_{Bench}$，该基准允许有意义的近似保证，从而为比较不同机制提供了度量标准。第三，我们设计了预算可行机制，对于 XOS 估值，这些机制相对于该基准能实现常数因子的近似保证。