Enter the total number of players and the player’s contribution value into the calculator to determine the Shapley value. This calculator helps to fairly distribute the total gains or costs to players based on their marginal contributions.
Shapley Value Formula
The Shapley value is calculated using the following formula:
φ(i) = Σ (S!(N−S−1)! / N!) * (v(S ∪ {i}) − v(S))Variables:
- φ(i) is the Shapley value for player i
- S is the size of a coalition of players without player i
- N is the total number of players
- v(S) is the value of the coalition S without player i
- v(S ∪ {i}) is the value of the coalition S with player i
To calculate the Shapley value, sum the marginal contributions of player i to all possible coalitions S, weighted by the probability of S occurring.
What is the Shapley Value?
The Shapley value is a solution concept in cooperative game theory. It represents a fair distribution of the total gains (or costs) to the players based on their individual contributions to the total outcome. It is named after Lloyd Shapley, who introduced it in 1953. The Shapley value ensures that each player’s payoff is proportional to their average marginal contribution across all possible coalitions.
How to Calculate the Shapley Value?
The following steps outline how to calculate the Shapley Value:
- First, determine the total number of players (N) in the game.
- Next, determine the player’s contribution value to the coalition.
- Use the formula above to calculate the Shapley value for the player.
- Finally, calculate the Shapley Value (φ(i)) for the player.
- After inserting the variables and calculating the result, check your answer with the calculator above.
Example Problem:
Use the following variables as an example problem to test your knowledge.
Total number of players (N) = 3
Player’s contribution value = 100
