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.

翻译：我们考虑一个市场，其中一组物品被出售给一组买方，每位买方对物品持有估值函数。拍卖师的目标是确定合理的价格以及稳定的分配。“合理”与“稳定”的一种定义是瓦尔拉斯均衡，它由一个价格向量与一个分配组成，满足以下理想性质：(i) 该分配是市场出清的，即尽可能多地出售物品；(ii) 该分配是稳定的，即每位买方最终获得相对于给定价格最优的一组物品。此外，“买方最优”意味着在所有瓦尔拉斯价格中，该价格尽可能低。本文提出了一种组合网络流算法，用于计算具有可加估值函数和买方需求的多单位匹配市场中的买方最优瓦尔拉斯价格。该算法可视为经典住房市场拍卖的推广，并模拟了升价拍卖的自然过程。我们利用结构洞见证明了买方最优瓦尔拉斯价格关于供给或需求变化的单调性。