Selling a perfectly divisible item to potential buyers is a fundamental task with apparent applications to pricing communication bandwidth and cloud computing services. Surprisingly, despite the rich literature on single-item auctions, revenue maximization when selling a divisible item is a much less understood objective. We introduce a Bayesian setting, in which the potential buyers have concave valuation functions (defined for each possible item fraction) that are randomly chosen according to known probability distributions. Extending the sequential posted pricing paradigm, we focus on mechanisms that use linear pricing, charging a fixed price for the whole item and proportional prices for fractions of it. Our goal is to understand the power of such mechanisms by bounding the gap between the expected revenue that can be achieved by the best among these mechanisms and the maximum expected revenue that can be achieved by any mechanism assuming mild restrictions on the behavior of the buyers. Under regularity assumptions for the probability distributions, we show that this revenue gap depends only logarithmically on a natural parameter characterizing the valuation functions and the number of agents. Our results follow by bounding the objective value of a mathematical program that maximizes the ex-ante relaxation of optimal revenue under linear pricing revenue constraints.

翻译：销售完美可分物品给潜在买家是一项基本任务，在通信带宽和云计算服务定价等领域具有明显应用价值。令人惊讶的是，尽管关于单一物品拍卖的文献十分丰富，销售可分物品时的收益最大化目标却远未得到充分理解。我们引入一个贝叶斯框架，其中潜在买家具有凹估值函数（针对每个可能的物品份额定义），这些函数根据已知概率分布随机生成。扩展顺序定价范式，我们聚焦于采用线性定价的机制：对整个物品收取固定价格，对其份额按比例定价。我们的目标是通过界定以下两者之间的差距来理解此类机制的效能：最佳线性定价机制所能实现的期望收益，与在买家行为温和限制条件下任何机制所能实现的最大期望收益。在概率分布满足正则性假设的前提下，我们证明该收益差距仅与表征估值函数和代理人数量的自然参数呈对数关系。我们的结论通过界定数学规划的目标值获得，该规划在满足线性定价收益约束的条件下最大化最优收益的事前松弛。