Online markets are a part of everyday life, and their rules are governed by algorithms. Assuming participants are inherently self-interested, well designed rules can help to increase social welfare. Many algorithms for online markets are based on prices: the seller is responsible for posting prices while buyers make purchases which are most profitable given the posted prices. To make adjustments to the market the seller is allowed to update prices at certain timepoints. Posted prices are an intuitive way to design a market. Despite the fact that each buyer acts selfishly, the seller's goal is often assumed to be that of social welfare maximization. Berger, Eden and Feldman recently considered the case of a market with only three buyers where each buyer has a fixed number of goods to buy and the profit of a bought bundle of items is the sum of profits of the items in the bundle. For such markets, Berger et. al. showed that the seller can maximize social welfare by dynamically updating posted prices before arrival of each buyer. B\'{e}rczi, B\'{e}rczi-Kov\'{a}cs and Sz\"{o}gi showed that the social welfare can be maximized also when each buyer is ready to buy at most two items. We study the power of posted prices with dynamical updates in more general cases. First, we show that the result of Berger et. al. can be generalized from three to four buyers. Then we show that the result of B\'{e}rczi, B\'{e}rczi-Kov\'{a}cs and Sz\"{o}gi can be generalized to the case when each buyer is ready to buy up to three items. We also show that a dynamic pricing is possible whenever there are at most two allocations maximizing social welfare.
翻译:在线市场是日常生活的一部分 { 在线市场是日常生活的一部分 { 其规则受算法的支配。 假设参与者本质上是自利的, 设计周密的规则可以帮助提高社会福利。 许多在线市场的算法都是基于价格的: 卖方负责上市, 而买方则负责购买最有利可图的价款。 为了调整市场, 允许卖方在某些时间点更新价格。 贴现价格是设计市场的一种直观方式。 尽管每个买方都自私行事, 卖方的目标往往被假定为社会福利最大化。 伯杰、 伊登和费尔德曼最近审议了一个市场的情况, 有三个买家只有固定数量的货物可以购买, 买家的利润是捆绑的利润。 对于这些市场, 伯杰等人等人。 表明卖方可以通过在每个买家抵达之前的动态更新价格, 我们的首级 rc}, Brc} 和 Felc\ { { { { { { {c} 和 Sz\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 最多买家的每两次更新价格, 能够显示两个动态价格。