Cooperative Games in Game Theory
The Shapley value is a solution concept in cooperative game theory that assigns a value to each player in a coalition game based on their marginal contributions to each of the possible coalitions.
Consider a game with three players: A, B, and C. The game has eight possible coalitions: {A}, {B}, {C}, {A, B}, {A, C}, {B, C}, {A, B, C}, and {}. For each coalition, the value of the game (i.e., the total payoff) is known. The Shapley value provides a way to divide the total payoff among the three players in a fair and efficient manner.
The Shapley value is based on the idea that each player should be rewarded for their contribution to the coalition. The contribution of a player to a coalition is measured by the increase in the value of the coalition when that player joins. The Shapley value is the average marginal contribution of a player over all possible orderings of the players.
To compute the Shapley value for a player, we consider all possible orderings of the players and compute the player's marginal contribution to each coalition as the difference in the value of the coalition with and without the player. We then take the average of the marginal contributions across all possible orderings.
The Shapley value has several desirable properties, including efficiency, fairness, and additivity. It is also unique and independent of the particular representation of the game.
One limitation of the Shapley value is that it can be computationally expensive to compute, especially for large games. However, there are efficient algorithms for computing the Shapley value for certain classes of games, such as convex games and supermodular games.
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