Shapley Value Calculator

The Shapley Value is an important concept in cooperative game theory, used to fairly distribute a total gain or cost among participants based on their contributions. This calculator allows users to input player contributions and calculate the Shapley value, which represents each player's fair share.

Historical Background

The Shapley Value was introduced by Lloyd Shapley in 1953. It is a solution concept in cooperative game theory that aims to fairly allocate the total payoff to all participants based on their individual contributions. It has since become a fundamental tool in economics, political science, and various fields where collaborative efforts are quantified.

Calculation Formula

The Shapley Value for player \( i \) in a coalition game with \( N \) players is calculated as:

\[ \phii = \sum{S \subseteq N \setminus {i}} \frac{|S|! \, (|N| - |S| - 1)!}{|N|!} \left( v(S \cup {i}) - v(S) \right) \]

Where:

• \( S \) is any subset of players not including \( i \).
• \( v(S) \) is the value of coalition \( S \).
• \( |S| \) is the number of players in coalition \( S \).

Example Calculation

Consider a simple example with three players. If their contributions are given as \( [10, 20, 30] \):

1. Calculate all possible combinations and their respective contributions.
2. Determine the marginal contribution of each player to each possible coalition.
3. Use the Shapley formula to find each player's fair share of the total value.

Importance and Usage Scenarios

The Shapley Value is widely used in fields such as economics, network theory, and business. It is particularly useful for:

• Dividing profits among partners in a business.
• Allocating costs in joint projects.
• Assessing contributions in collaborative networks, such as supply chains.

Common FAQs

1. What is the purpose of the Shapley Value?

   • The Shapley Value is used to fairly allocate gains or costs among participants based on their individual contributions.

2. How is the Shapley Value different from equal division?

   • Unlike equal division, the Shapley Value takes into account the individual contribution of each player, ensuring that participants are rewarded based on their impact.

3. Where is the Shapley Value used in real life?

   • It is used in profit-sharing, cost allocation, political voting power analysis, and resource distribution in networks.

The Shapley Value Calculator helps users determine how to fairly allocate resources or costs among multiple participants, ensuring each party receives a share commensurate with their contribution to the overall effort.