We consider a market where a set of objects is sold to a set of buyers, each equipped with a valuation function for the objects. The goal of the auctioneer is to determine reasonable prices together with a stable allocation. One definition of "reasonable" and "stable" is a Walrasian equilibrium, which is a tuple consisting of a price vector together with an allocation satisfying the following desirable properties: (i) the allocation is market-clearing in the sense that as much as possible is sold, and (ii) the allocation is stable in the sense that every buyer ends up with an optimal set with respect to the given prices. Moreover, "buyer-optimal" means that the prices are smallest possible among all Walrasian prices. In this paper, we present a combinatorial network flow algorithm to compute buyer-optimal Walrasian prices in a multi-unit matching market with additive valuation functions and buyer demands. The algorithm can be seen as a generalization of the classical housing market auction and mimics the very natural procedure of an ascending auction. We use our structural insights to prove monotonicity of the buyer-optimal Walrasian prices with respect to changes in supply or demand.

中文翻译：

---

基于流量的升序拍卖计算买方最优瓦尔拉斯价格

我们考虑一个市场，其中一组对象被出售给一组买家，每个买家都配备了对象的估值功能。拍卖师的目标是确定合理的价格和稳定的分配。“合理”和“稳定”的一个定义是瓦尔拉斯均衡，它是由价格向量和满足以下理想属性的分配组成的元组：(i) 在某种意义上，分配是市场出清的可能被出售，并且（ii）分配是稳定的，因为每个买家最终都会得到关于给定价格的最优集合。此外，“买方最优”意味着价格在所有瓦尔拉斯价格中可能是最小的。在本文中，我们提出了一种组合网络流算法来计算多单元匹配市场中具有附加估值函数和买方需求的买方最优瓦尔拉斯价格。该算法可以看作是对经典房地产市场拍卖的概括，并模仿了升序拍卖的非常自然的过程。我们使用我们的结构洞察力来证明买方最优瓦尔拉斯价格相对于供需变化的单调性。