We study a natural combinatorial pricing problem for sequentially arriving buyers with equal budgets. Each buyer is interested in exactly one pair of items and purchases this pair if and only if, upon arrival, both items are still available and the sum of the item prices does not exceed the budget. The goal of the seller is to set prices to the items such that the number of transactions is maximized when buyers arrive in adversarial order. Formally, we are given an undirected graph where vertices represent items and edges represent buyers. Once prices are set to the vertices, edges with a total price exceeding the buyers' budgets are evicted. Any arrival order of the buyers leads to a set of transactions that forms a maximal matching in this subgraph, and an adversarial arrival order results in a minimum maximal matching. In order to measure the performance of a pricing strategy, we compare the size of such a matching to the size of a maximum matching in the original graph. It was shown by Correa et al. [IPCO 2022] that the best ratio any pricing strategy can guarantee lies within $[1/2, 2/3]$. Our contribution to the problem is two-fold: First, we provide several characterizations of subgraphs that may result from pricing schemes. Second, building upon these, we show an improved upper bound of $3/5$ and a lower bound of $1/2 + 2/n$, where $n$ is the number of items.

翻译：我们研究了一个自然组合定价问题，涉及预算均等的顺序到达买家。每个买家仅对一对物品感兴趣，且仅当到达时两个物品均可用且物品价格之和不超过预算时，才会购买这对物品。卖家的目标是为物品设定价格，使得在买家以对抗顺序到达的情况下交易数量最大化。形式上，给定一个无向图，其中顶点表示物品，边表示买家。一旦为顶点设定价格，总价格超过买家预算的边将被移除。无论买家以何种顺序到达，交易集合均构成该子图中的一个极大匹配，而对抗性到达顺序将导致最小极大匹配。为了衡量定价策略的性能，我们将这种匹配的大小与原始图中最大匹配的大小进行比较。Correa 等人[IPCO 2022]指出，任何定价策略能保证的最佳比率位于$[1/2, 2/3]$之间。我们对这一问题的贡献有两方面：首先，我们给出了定价方案可能产生的子图的多种刻画；其次，基于这些刻画，我们证明了改进的上界$3/5$和下界$1/2 + 2/n$，其中$n$为物品数量。