Shapley value
The Shapley value is a solution concept in cooperative game theory.
It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Memorial Prize in Economic Sciences for it in 2012.
[3][4] Formally, a coalitional game is defined as: There is a set N (of n players) and a function
(S), called the worth of coalition S, describes the total expected sum of payoffs the members of
The Shapley value is one way to distribute the total gains to the players, assuming that they all collaborate.
as a fair compensation, and then for each actor take the average of this contribution over the possible different permutations in which the coalition can be formed.
An alternative equivalent formula for the Shapley value is: where the sum ranges over all
The Shapley values are given in terms of the synergy function by[6][7] where the sum is over all subsets
This can be interpreted as In other words, the synergy of each coalition is divided equally between all members.
An owner, o, provides crucial capital in the sense that, without him/her, no gains can be obtained.
is so the only coalitions that generate synergy are one-to-one between the owner and any individual worker.
Notably, it is the only payment rule satisfying the four properties of Efficiency, Symmetry, Linearity and Null player.
This means that the labeling of the agents doesn't play a role in the assignment of their gains.
The Shapley value can be defined as a function which uses only the marginal contributions of player
In their 1974 book, Lloyd Shapley and Robert Aumann extended the concept of the Shapley value to infinite games (defined with respect to a non-atomic measure), creating the diagonal formula.
Two approaches were deployed to extend this diagonal formula when the function f is no longer differentiable.
Mertens goes back to the original formula and takes the derivative after the integral thereby benefiting from the smoothing effect.
It has been generalized[14] to apply to a group of agents C as, In terms of the synergy function
Lipovetsky (2006) discussed the use of Shapley value in regression analysis, providing a comprehensive overview of its theoretical underpinnings and practical applications.
[16] Shapley value contributions are recognized for their balance of stability and discriminating power, which make them suitable for accurately measuring the importance of service attributes in market research.
[17] Several studies have applied Shapley value regression to key drivers analysis in marketing research.
Pokryshevskaya and Antipov (2012) utilized this method to analyze online customers' repeat purchase intentions, demonstrating its effectiveness in understanding consumer behavior.
[18] Similarly, Antipov and Pokryshevskaya (2014) applied Shapley value regression to explain differences in recommendation rates for hotels in South Cyprus, highlighting its utility in the hospitality industry.
[19] Further validation of the benefits of Shapley value in key-driver analysis is provided by Vriens, Vidden, and Bosch (2021), who underscored its advantages in applied marketing analytics.
[20] The Shapley value provides a principled way to explain the predictions of nonlinear models common in the field of machine learning.
By interpreting a model trained on a set of features as a value function on a coalition of players, Shapley values provide a natural way to compute which features contribute to a prediction [21] or contribute to the uncertainty of a prediction.
[22] This unifies several other methods including Locally Interpretable Model-Agnostic Explanations (LIME),[23] DeepLIFT,[24] and Layer-Wise Relevance Propagation.
[27] The statistical understanding of Shapley values remains an ongoing research question.
A smooth version, called Shapley curves[28], achieves the minimax rate and is shown to be asymptotically Gaussian in a nonparametric setting.
Confidence intervals for finite samples can be obtained via the wild bootstrap.
