The main goal of this work is to settle a conceptual framework for cooperative game theory in which the notion of composition/aggregation of games is the defining structure. This is done via the mathematical theory of algebraic operads: we start by endowing the collection of all cooperative games in any number of players with a operad structure, and we show that it generalizes all the previous notions of sums, products and compositions of games considered by Shapley, Owen and others. Furthermore, we explicitly compute this operad in terms of generators and relations, showing that the Möbius transform induces a canonical isomorphism between the operad of cooperative games and the operad that encodes commutative triassociative algebras. In other words, we prove that any cooperative game is a linear combination of iterated compositions of the 2-player bargaining game and the 2-player dictator games. Once this setup is established, we leverage operadic methods the study of this operad. First, we give some results about the core and the Shapley value of a composite game with respect to the cores and the Shapley values of the component games. Finally, we show that many interesting classes of games (simple, balanced, belief functions, capacities, etc) are stable under composition and thus induce sub-operads.

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