# Coalitional game

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.

#### References

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

#### Comments

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: http://encyclopediaofmath.org/index.php?title=Coalitional_game&oldid=38560