The Shapley value averages each player's marginal contribution over uniformly random player orderings. Exactly orderings have immediately before player , giving the formula. The values sum to by telescoping each ordering. In a simple cooperative game this is the probability of being pivotal. For a convex cooperative game, Shapley value belongs to the core of a convex game guarantees a stable allocation as well.
For a player ordering , is the set before player , and the displayed entries form its marginal contribution vector. Their sum is . The Shapley value is the average of these vectors. In a convex cooperative game, increasing marginal contributions make each such vector satisfy every coalition constraint of the core of a cooperative game.
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The Shapley value is a solution concept from cooperative game theory that provides a way to fairly distribute the total gains or payouts of a cooperative game among its players based on their individual contributions. Named after mathematician Lloyd Shapley, it takes into account the contribution of each player to the overall outcome of the coalition they form with other players.