今天来学习一下Shapley Values,先上概念，以及研究背景（borrowed from Wikipedia）：
The Shapley value is a solution concept in cooperative game theory.
A coalition of players cooperates, and obtains a certain overall gain from that cooperation. Since some players may contribute more to the coalition than others or may possess different bargaining power (for example threatening to destroy the whole surplus), what final distribution of generated surplus among the players should arise in any particular game? Or phrased differently: how important is each player to the overall cooperation, and what payoff can he or she reasonably expect? The Shapley value provides one possible answer to this question.
如果对博弈论以及合作博弈不清楚的，可以先读一下A Unique Method for Machine Learning Interpretability: Game Theory & Shapley Values!，个人觉得写的很不错。贴一个作者的个人简介：
这里借用他的例子来直观的理解Shapley Values：
Three friends – Ram, Abhiraj, and Pranav – go out for a meal. They order and share fries, wine, and pi. It is hard to figure out who should pay how much since they did not eat an equal share. So, we have the following information:
- If Ram is eating alone, he would pay 800
- If Abhiraj is eating alone, he would pay 560
- If Pranav is eating alone, he would pay 700
- If Ram and Abhiraj both eat together, they would pay 800
- If Ram and Pranav both eat together, they would pay 850
- If Abhiraj and Pranav both eat together, they would pay 720
- If Ram, Abhiraj, and Pranav all eat together, they would pay 900
So, it turns out the actual amount all 3 of them pay when they eat together is 900. Now, the task at hand is to figure out how much each of them should pay individually.
这个问题是一个如何公平的分配费用的问题，而Shapley Values被用来公平地分配信用或贡献给每个玩家(参与者)，首先先穷举所有可能的序列，接着根据每种顺序计算每个人相应的边际贡献，再进行求平均得到每个参与者的Shapley Value。具体的如下图所示：
采样每个序列得到：
所以Ram, Abhiraj, Pranav支付的钱分别为：
R
a
m
=
800
+
240
+
180
+
150
+
180
+
800
6
≈
392
Ram=\frac{800 + 240 + 180 + 150 + 180 + 800}{6} \approx 392
Ram=6800+240+180+150+180+800≈392
A
b
h
i
r
a
j
=
0
+
560
+
560
+
50
+
20
+
50
6
≈
207
Abhiraj=\frac{0 + 560 + 560 + 50 + 20 + 50}{6} \approx 207
Abhiraj=60+560+560+50+20+50≈207
P
r
a
n
a
v
=
100
+
100
+
160
+
700
+
700
+
50
6
≈
302
Pranav=\frac{100 + 100 + 160 + 700 + 700 + 50}{6} \approx 302
Pranav=6100+100+160+700+700+50≈302这加起来刚好为900。
前面介绍了Shapley Values的概念及其计算，在A Unique Method for Machine Learning Interpretability: Game Theory & Shapley Values!中已经举例介绍了Shapley Values在Machine Learning Interpretability上的一个应用，这里我不再赘述，我将简单介绍一下Shapley Values在模型可解释上的一些现有工作。
基于Shapley Values的文章很多，这里不再举例了，毕竟笔者也没好好调研过，有兴趣自行研究。
