论文标题
估计技能分布
Estimation of Skill Distributions
论文作者
论文摘要
在本文中，我们研究了从锦标赛中的成对游戏观察中学习一群代理商的技能分布的问题。这些游戏是在人口随机吸引的代理商中进行的。我们模型中的代理商可以是个人，运动队或华尔街基金经理。正式地，我们假设游戏结果的可能性受Bradley-terry-luce（或多项式Logit）模型的控制，在这种模型中，代理商击败另一个代理的可能性是其技能水平与技能水平的成对总和之间的比率，而技能参数是从不知名的技能密度中得出的。本质上，问题是要从嘈杂的，量化的观察结果中学习分布。我们提出了一种简单且可进行的算法，该算法以$ n^{ - 1+ \ varepsilon} $了解近乎最佳的minimax平均误差缩放，以均为$ n^{ - varepsilon> 0 $，当密度平滑时。我们的方法从非参数统计数据的内核密度估计中进行了学习技能参数的先前工作。此外，我们证明了最小值的下限，这些下限建立了我们算法中使用的技能参数估计技术的最小值。这些界限利用了Fano方法的连续版本以及覆盖参数。我们将算法应用于各种足球联赛和世界杯，板球世界杯和共同资金。我们发现，学习分布的熵提供了对技能的定量度量，这为人们对体育赛事的感知品质（例如足球联盟排名）提供了严格的解释。最后，我们运用我们的方法来评估共同基金的技能分布。我们的结果阐明了2008年大衰退之前的大量低质量基金，以及在金融危机之后由更熟练的资金统治该行业。
In this paper, we study the problem of learning the skill distribution of a population of agents from observations of pairwise games in a tournament. These games are played among randomly drawn agents from the population. The agents in our model can be individuals, sports teams, or Wall Street fund managers. Formally, we postulate that the likelihoods of game outcomes are governed by the Bradley-Terry-Luce (or multinomial logit) model, where the probability of an agent beating another is the ratio between its skill level and the pairwise sum of skill levels, and the skill parameters are drawn from an unknown skill density of interest. The problem is, in essence, to learn a distribution from noisy, quantized observations. We propose a simple and tractable algorithm that learns the skill density with near-optimal minimax mean squared error scaling as $n^{-1+\varepsilon}$, for any $\varepsilon>0$, when the density is smooth. Our approach brings together prior work on learning skill parameters from pairwise comparisons with kernel density estimation from non-parametric statistics. Furthermore, we prove minimax lower bounds which establish minimax optimality of the skill parameter estimation technique used in our algorithm. These bounds utilize a continuum version of Fano's method along with a covering argument. We apply our algorithm to various soccer leagues and world cups, cricket world cups, and mutual funds. We find that the entropy of a learnt distribution provides a quantitative measure of skill, which provides rigorous explanations for popular beliefs about perceived qualities of sporting events, e.g., soccer league rankings. Finally, we apply our method to assess the skill distributions of mutual funds. Our results shed light on the abundance of low quality funds prior to the Great Recession of 2008, and the domination of the industry by more skilled funds after the financial crisis.
