Shapley Value

Shapley Value is a concept from cooperative game theory that provides a fair distribution of a total "payout" among players based on their individual contributions to the group. It is widely used in economics, decision-making, and machine learning for feature attribution and model interpretability. The Shapley Value ensures that each participant's contribution is valued in a mathematically fair and consistent manner.

Overview

The Shapley Value is calculated by considering all possible coalitions of players and determining the marginal contribution of each player to every coalition. This approach ensures that the value assigned to each player reflects their average contribution across all possible scenarios.

Key features:

• Fairness: Distributes the payout based on each player's marginal contribution.
• Symmetry: Players with equal contributions receive the same value.
• Additivity: Shapley Values for independent games can be summed.

Formula

The Shapley Value for a player is computed by summing their marginal contributions across all possible coalitions, weighted by the likelihood of each coalition. The weights ensure fairness and account for the size of each coalition.

Applications

Shapley Values are used in various fields:

• Economics:
  • Allocating resources or profits among members of a coalition or partnership.
• Machine Learning:
  • Explaining individual predictions by attributing the impact of features in models such as random forests, neural networks, or gradient-boosted trees.
• Decision Analysis:
  • Evaluating the importance of factors in complex systems or decision-making processes.
• Network Analysis:
  • Determining the influence of nodes in a network based on their contributions.

Properties

The Shapley Value satisfies the following properties:

• Efficiency: The total payout is distributed among all players.
• Symmetry: Identical players receive identical values.
• Dummy Player: A player with no marginal contribution receives a value of zero.
• Additivity: The value for combined games is the sum of individual Shapley Values.

Advantages

• Provides a fair and consistent attribution of contributions.
• Widely applicable across domains requiring cooperative decision-making or resource allocation.
• Supports model interpretability in machine learning.

Limitations

• Computationally expensive for large coalitions due to factorial growth in the number of combinations.
• Assumes independence of players, which may not hold in some real-world scenarios.
• Requires a well-defined value function for all subsets of players.

Example

Consider a coalition of three players A, B, and C with the following payouts for coalitions:

Coalition	Payout
{}	0
{A}	10
{B}	20
{C}	30
{A, B}	50
{A, C}	60
{B, C}	70
{A, B, C}	100

The Shapley Values for A, B, and C can be computed by evaluating their contributions to all coalitions. For example:

• Player A contributes 10 to {A}, 30 to {A, B}, and so on.
• Each contribution is averaged across all coalitions to determine their Shapley Value.

See Also

• Game Theory
• Cooperative Game
• SHAP Analysis
• Feature Importance
• Resource Allocation