最优多维定价的极值定理
Extreme-Value Theorems for Optimal Multidimensional Pricing
摘要：
我们提供了一个接近最优的，计算效率高的算法的单位需求定价问题，卖方希望价格n个项目，以优化收入对单位需求的买方的项目的价值是独立地从已知的分布。对于任何选择的精度eps>0和项目值在[0，1]内，我们的算法在多项式时间内实现了最多eps的最佳附加误差的收入。对于从单调风险率（MHR）分布采样的值，我们在多项式时间内实现了（1-eps）-分数的最优收入，而对于从规则分布采样的值，在准多项式时间内实现了相同的收入保证。我们的有界分布算法应用概率技术来了解收入分布的统计特性，通过动态规划减少算法的搜索空间。适应这种方法的MHR和定期分布需要证明新的极值定理，这样的分布。作为一个副产品，我们的技术建立近似最优和接近最优的解决方案的结构特性。我们发现，独立分布的价值，根据MHR分布，定价所有项目在同一价格实现了一个恒定的最优收入的比例。此外，对于所有eps >0，g（1/eps）不同的价格足以获得最优收益的（1-eps）分数，其中g（1/eps）是1/eps的二次函数，并且与n无关。类似地，对于所有eps>0和n>0，如果值根据规则分布独立分布，则最多g（1/（eps log n））个不同的价格就足够了，其中g（）是多项式函数。最后，当值为i.i.d.从一些MHR分布，我们表明，如果n是一个足够大的函数1/eps，一个单一的价格足以实现（1-eps）-分数的最佳收入。
英文摘要：
We provide a near-optimal, computationally efficient algorithm for the unit-demand pricing problem, where a seller wants to price n items to optimize revenue against a unit-demand buyer whose values for the items are independently drawn from known distributions. For any chosen accuracy eps>0 and item values bounded in [0,1], our algorithm achieves revenue that is optimal up to an additive error of at most eps, in polynomial time. For values sampled from Monotone Hazard Rate (MHR) distributions, we achieve a (1-eps)-fraction of the optimal revenue in polynomial time, while for values sampled from regular distributions the same revenue guarantees are achieved in quasi-polynomial time. Our algorithm for bounded distributions applies probabilistic techniques to understand the statistical properties of revenue distributions, obtaining a reduction in the search space of the algorithm via dynamic programming. Adapting this approach to MHR and regular distributions requires the proof of novel extreme value theorems for such distributions. As a byproduct, our techniques establish structural properties of approximately-optimal and near-optimal solutions. We show that, for values independently distributed according to MHR distributions, pricing all items at the same price achieves a constant fraction of the optimal revenue. Moreover, for all eps >0, g(1/eps) distinct prices suffice to obtain a (1-eps)-fraction of the optimal revenue, where g(1/eps) is quadratic in 1/eps and independent of n. Similarly, for all eps>0 and n>0, at most g(1/(eps log n)) distinct prices suffice if the values are independently distributed according to regular distributions, where g() is a polynomial function. Finally, when the values are i.i.d. from some MHR distribution, we show that, if n is a sufficiently large function of 1/eps, a single price suffices to achieve a (1-eps)-fraction of the optimal revenue.
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