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no. 4
The algebraic structures of social organizations: the operad of cooperative games
Laplace Mermoud, Dylan1; Roca i Lucio, Victor2
1Unité de Mathématiques Appliquées, ENSTA, Institut Polytechnique de Paris, Palaiseau, France, CEDRIC, Conservatoire National des Arts et Métiers, 75003 Paris, France
2Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, F-75013 Paris, France
Algebraic Combinatorics, Volume 9 (2026) no. 4, pp. 1067-1117

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.504
Classification: 18M60, 18M70, 18M80, 91A08, 52B05, 28E10, 05E99
Keywords: Cooperative game theory, algebraic operads, fuzzy measures, capacities, evidence theory, polymatroids, generalized permutahedra

Laplace Mermoud, Dylan  1 ; Roca i Lucio, Victor  2

1 Unité de Mathématiques Appliquées, ENSTA, Institut Polytechnique de Paris, Palaiseau, France, CEDRIC, Conservatoire National des Arts et Métiers, 75003 Paris, France
2 Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, F-75013 Paris, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Laplace Mermoud, Dylan; Roca i Lucio, Victor. The algebraic structures of social organizations: the operad of cooperative games. Algebraic Combinatorics, Volume 9 (2026) no. 4, pp. 1067-1117. doi: 10.5802/alco.504
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