Simple posted pricing mechanisms for selling a divisible item
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We study the problem of selling a divisible item to agents who have concave valuation functions for fractions of the item. This is a fundamental problem with apparent applications to pricing communication bandwidth or cloud computing services. We focus on simple sequential posted pricing mechanisms that use linear pricing, i.e., a fixed price for the whole item and proportional prices for fractions of it. We present results of the following form that can be thought of as analogs of the well-known prophet inequality of Samuel-Cahn (1984). For ρ≈ 32%, if there is a linear pricing so that sequential posted pricing sells a ρ-fraction of the item, this results in a ρ-approximation of the optimal social welfare. The value of ρ can be improved to approximately 42% if sequential posted pricing considers the agents in random order. We also show that the best linear pricing yields an expected revenue that is at most O(κ^2) times smaller than the optimal one, where κ is a bound on the curvature of the valuation functions. The proof extends and exploits the approach of Alaei et al. (2019) and bounds the revenue gap by the objective value of a mathematical program. The dependence of the revenue gap on κ is unavoidable as a lower bound of Ω(lnκ) indicates.
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