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Axiomatic Attribution for Multilinear Functions

机译：多线性函数的公理归因

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We study the attribution problem. That is, given a real-valued characteristic function f of n variables and initial and final values r arid s for its independent variables, our objective is to divide the responsibility for the change f(s)-f(r) in the characteristic function among each of its independent variables. We call these assigned responsibilities attributions, and we would like the attributions to form a complete partition of the total change. When r = 0, the attribution problem coincides with a standard cost sharing model from the social choice literature (cf. Moulin [2]), where the characteristic function is the cost function, the independent variables are the demands of the agents, and the attributions are cost shares for the agents.We follow the cost sharing literature in identifying good attribution methods axiomatically (for a classical example, see Friedman and Moulin ([1]). We consider: 1. Additivity - attributions are additive in the characteristic function, 2. Dummy if the characteristic function does not depend on a variable, then its attribution is zero, and 3. Af f ine Scale Invariance - attributions are invariant under simultaneous afline transformation of the characteristic function and the variables. First, we show that when the characteristic function is the sum of a multilinear function and an additively separable one, every attribution method satisfying these axioms is a random order method. Intuitively, a multilinear function is determined by its values on the vertices of a hypercube, so its attributions should depend on these values alone, leading to the space of random order methods. The proof proceeds by using this idea to count dimensions. Second, in our main result, we show that, there is a unique attribution method satisfying these axioms and Anonymity (which requires attributions to be invariant under relabeling of the variables) if and only if the characteristic function is the sum of a multilinear function and an additively separable one. The main technical tool is the use of Stokes' Theorem to compare attribution methods. The resulting method coincides with the classical Aumann-Shapley and Shaplev-Shubik methods, and thus we term it the Airmann-Shnpley-Shubik method. When the characteristic function is multilinear, our result prescribes this method for use; to this end. we provide a computationally efficient implementation. Together, our results single out the class of multilinear characteristic functions as a particularly nice one for attribution problems. We give several examples of natural attribution problems where such functions arise, including pay-per-click advertising, website traffic analysis, portfolio analysis, and performance analysis of sports teams.
机译：我们研究归因问题。也就是说，给定n个变量的实值特征函数f以及其自变量的初始值和最终值r arid s，我们的目标是在特征函数中划分变化f（s）-f（r）的责任各个独立变量之间我们将这些分配的职责归因称为“归因”，我们希望归因能够构成总变更的完整分区。当r = 0时，归因问题与社会选择文献中的标准成本分摊模型一致（参见Moulin [2]），其中特征函数是成本函数，自变量是代理商的需求，归因是代理商的成本分摊。我们遵循成本分摊文献，公理地确定良好的归因方法（有关经典示例，请参见Friedman和Moulin（[1]）。我们考虑：1.可加性-归因是特征函数的累加，2.虚拟，如果特征函数不依赖变量，则其归因为零； 3.仿射尺度不变性-在特征函数和变量同时进行直线变换后，归因不变。当特征函数是一个多线性函数与一个可加可分函数的和时，满足这些公理的每种归因方法都是一种随机顺序方法。多线性函数是由其在超立方体的顶点上的值确定的，因此它的属性应仅取决于这些值，从而导致随机顺序方法的空间。通过使用此想法对尺寸进行计数来进行证明。其次，在我们的主要结果中，我们表明，当且仅当特征函数是多线性函数的和，并且只有当特征函数为可加分的主要技术工具是使用斯托克斯定理比较归因方法。所得方法与经典的Aumann-Shapley方法和Shaplev-Shubik方法相吻合，因此我们将其称为Airmann-Shnpley-Shubik方法。当特征函数为多线性时，我们的结果规定了该方法的使用；为此。我们提供了一种计算有效的实现。在一起，我们的结果将多线性特征函数的类别选为特别适合归因问题的一类。我们列举了出现这种功能的自然归因问题的几个示例，包括按点击付费广告，网站流量分析，投资组合分析和运动队的绩效分析。

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