The main goal of this paper 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 with any number of players with an operad structure, and we show that it generalises all the previous notions of sums, products and compositions of games considered by Owen, Shapley, von Neumann and Morgenstern, and many others. Furthermore, we explicitly compute this operad in terms of generators and relations, showing that the Möbius transform map 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. We show that many interesting classes of games (simple, balanced, capacities a.k.a fuzzy measures and convex functions, totally monotone, etc) are stable under compositions, and thus form suboperads. In the convex case, this gives by the submodularity theorem a new operad structure on the family of all generalized permutahedra. Finally, we focus on how solution concepts in cooperative game theory behave under composition: we study the core of a composite and describe it in terms of the core of its components, and we give explicit formulas for the Shapley value and the Banzhaf index of a compound game.

翻译：本文的主要目标是建立合作对策论的概念框架，其中对策的复合/聚合结构作为核心定义。我们通过代数Operad的数学理论实现这一目标：首先赋予所有玩家数量的合作对策集合以Operad结构，并证明该结构统一了Owen、Shapley、von Neumann与Morgenstern等人此前提出的对策和、积与复合等概念。进一步，我们通过生成元与关系显式计算该Operad，揭示Möbius变换映射在合作对策的Operad与编码交换三结合代数的Operad之间建立了典范同构。换言之，我们证明任何合作对策均可表示为两人讨价还价对策与两人独裁者对策迭代复合的线性组合。研究表明，简单对策、平衡对策、容量（即模糊测度与凸函数）、全单调对策等众多有趣对策类在复合运算下保持封闭，从而构成子Operad。在凸情形下，由次模定理可得广义置换多面体族上的新Operad结构。最后，我们聚焦于合作对策论中解概念在复合下的行为：研究复合对策的核心并以其组分之核心刻画，同时给出复合对策的Shapley值与Banzhaf指数的显式公式。