Shapley Value Calculator

Shapley Value Formula:

\[ \phi_i(v) = \sum_{S \subseteq N \setminus \{i\}} \frac{|S|! \times (|N| - |S| - 1)!}{|N|!} \times [v(S \cup \{i\}) - v(S)] \]

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1. What is the Shapley Value?

The Shapley value is a solution concept in cooperative game theory that fairly distributes the total gains of a game among the players. It was named after Lloyd Shapley, who introduced it in 1953 and received the Nobel Prize in Economics for this work in 2012.

2. How Does the Calculator Work?

The calculator uses the Shapley value formula:

\[ \phi_i(v) = \sum_{S \subseteq N \setminus \{i\}} \frac{|S|! \times (|N| - |S| - 1)!}{|N|!} \times [v(S \cup \{i\}) - v(S)] \]

Where:

\( \phi_i(v) \) — Shapley value for player i
\( N \) — Set of all players
\( S \) — Subset of players excluding i
\( v(S) \) — Value of coalition S
\( |S| \) — Size of coalition S
\( |N| \) — Total number of players

Explanation: The formula averages the marginal contribution of each player across all possible coalitions, weighted by the probability of each coalition forming.

3. Importance of Shapley Value

Details: The Shapley value is widely used in economics, political science, and computer science for fair resource allocation, cost sharing, and feature importance analysis in machine learning.

4. Using the Calculator

Tips: Enter the number of players and the values for all possible coalitions (including empty set). The number of coalition values should be 2^n where n is the number of players.

5. Frequently Asked Questions (FAQ)

Q1: What are the properties of Shapley value?
A: It satisfies efficiency (total distributed equals grand coalition value), symmetry, dummy player, and additivity properties.

Q2: How is the empty coalition value used?
A: The value of the empty coalition (v(∅)) should always be 0, as no players contribute nothing.

Q3: What's the computational complexity?
A: The calculation is exponential in the number of players (O(2^n)), making it computationally intensive for large n.

Q4: Can Shapley value be negative?
A: Yes, if a player's marginal contributions are negative in many coalitions, their Shapley value can be negative.

Q5: What are practical applications?
A: Used in cost allocation problems, voting power analysis, network routing, and explaining machine learning model predictions.