Coalitional game

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A game in which the coalitions of actions $\mathcal{K}_A$ and the coalitions of interests $\mathcal{K}_I$ are different (generally, intersecting) families of subsets of the set of players $P$ and in which the preference for each of the coalitions of interests $K \in \mathcal{K}_I$ is described by its pay-off function $H_K$ (see Games, theory of). Only the case $\mathcal{K}_I \subseteq \mathcal{K}_A$ has been investigated.

It is natural to consider $\mathcal{K}_A$ as a simplicial complex with vertex set $P$. Certain topological properties of $\mathcal{K}_A$ have a game-theoretic sense; in particular, if $\mathcal{K}_A$ is zero-dimensional, then the game turns out to be a non-cooperative game.

The play of a coalitional game can be interpreted as a coordinated choice of coalitional strategies (cf. Strategy (in game theory)) by the players (at the "coalition conference" ) for each coalition of action after which, in the situation $s$ thus formed, each coalition of interests $K$ receives the pay-off $H_K(s)$.

Optimality in a coalitional game can, in its own way, be regarded as a "localization of conflicts" , that is, as a stability of the situation $s$ in the sense that conditions of the following form prevail: The coalition of interests $K$ is not interested in the departure from its coalition strategy in $s$, even if some coalition of action $K'$ departs from its strategy. Equilibrium in the sense of Nash is covered by this principle.


[1] N.N. Vorob'ev, "Coalitional games" Teor. Veroyatn. Primenen. , 12 : 2 (1967) pp. 289–306 (In Russian) (English summary)


The notions explained in the article above do not occur in the Western literature and are particular to the author and his school.

How to Cite This Entry:
Coalitional game. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by N.N. Vorob'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article