Sellers in online markets face the challenge of determining the right time to sell in view of uncertain future offers. Classical stopping theory assumes that sellers have full knowledge of the value distributions, and leverage this knowledge to determine stopping rules that maximize expected welfare. In practice, however, stopping rules must often be determined under partial information, based on scarce data or expert predictions. Consider a seller that has one item for sale and receives successive offers drawn from some value distributions. The decision on whether or not to accept an offer is irrevocable, and the value distributions are only partially known. We therefore let the seller adopt a robust maximin strategy, assuming that value distributions are chosen adversarially by nature to minimize the value of the accepted offer. We provide a general maximin solution to this stopping problem that identifies the optimal (threshold-based) stopping rule for the seller for all possible statistical information structures. We then perform a detailed analysis for various ambiguity sets relying on knowledge about the common mean, dispersion (variance or mean absolute deviation) and support of the distributions. We show for these information structures that the seller's stopping rule consists of decreasing thresholds converging to the common mean, and that nature's adversarial response, in the long run, is to always create an all-or-nothing scenario. The maximin solutions also reveal what happens as dispersion or the number of offers grows large.
翻译:在线市场上的卖主面临挑战,要根据不确定的未来报价确定销售的适当时间。 经典停止理论假定卖主对价值分配有充分的了解,并利用这一知识确定停止提供预期福利的规则。 然而,在实践中,停止规则往往必须根据稀缺的数据或专家预测,根据部分信息确定。 认为有一个销售项目并接受从某些价值分配中得出的连续报价的卖主。 决定是否接受报价是不可撤销的,价值分配只是部分已知的。 因此,我们让卖主采取一项强有力的最大额战略,假设价值分配是按性质进行对抗性选择的,以尽量减少所接受的报价的价值。 我们为制止这一问题提供了一个总体的最好解决办法,确定最佳(基于门槛的)规则,停止卖方所有可能的统计信息结构的规则。 然后我们对各种模糊之处进行详细分析,这取决于对共同平均值、分散性(差异或表示绝对偏差)和分配的支持。 我们为这些信息结构显示,卖方停止分配规则的规则是按敌对性质选择的,以尽量减少所接受报价的价值分配的价值价值的价值。 我们为制止规则提供了一种普遍的最强烈的解决办法, 也就是不断使价格走向最接近于共同的极限。