  Shapley value is a concept in cooperative game theory for measuring the
contribution of each participant, which was named in honor of Lloyd Shapley.
Shapley value has been recently applied in data marketplaces for compensation
allocation based on their contribution to the models. Shapley value is the only
value division scheme used for compensation allocation that meets three
desirable criteria: group rationality, fairness, and additivity. In cooperative
game theory, the marginal contribution of each contributor to each coalition is
a nonnegative value. However, in machine learning model training, the marginal
contribution of each contributor (data tuple) to each coalition (a set of data
tuples) can be a negative value, i.e., the accuracy of the model trained by a
dataset with an additional data tuple can be lower than the accuracy of the
model trained by the dataset only.
  In this paper, we investigate the problem of how to handle the negative
marginal contribution when computing Shapley value. We explore three
philosophies: 1) taking the original value (Original Shapley Value); 2) taking
the larger of the original value and zero (Zero Shapley Value); and 3) taking
the absolute value of the original value (Absolute Shapley Value). Experiments
on Iris dataset demonstrate that the definition of Absolute Shapley Value
significantly outperforms the other two definitions in terms of evaluating data
importance (the contribution of each data tuple to the trained model).
