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We show that any cooperative game can be represented by an assignment ofcostly facilities to players, in which it is intuitively obvious how toallocate the total cost in an equitable manner. This equitable solution turnsout to be the Shapley value of the game, and thus provides as an alternativejustification of the value. Game representations also open the door forextending the Shapley value to situations where not all coalitions can form,provided those that can constitute a "semi-algebra"; or, more generally, a"hierarchy"; or, still more generally, have "full span".

Game Representations and Extensions of the Shapley Value