Everything about Cooperative Game totally explained
A
cooperative game is a
game where groups of players ("coalitions") may enforce cooperative behaviour, hence the game is a competition between
coalitions of players, rather than between individual players. It is like a
coordination game, when players choose the strategies by a
consensus decision-making process.
Recreational games are rarely cooperative, because they usually lack mechanisms by which coalitions may enforce coordinated behaviour on the members of the coalition. Such mechanisms, however, are abundant in real life situations, such as contract law.
Mathematical definition
A cooperative game is given by specifying a value for every coalition. Formally, the game is a finite set of players
, called the
grand coalition, and a
characteristic function is a vertex of the
core of
. Any vertex of the
core can be constructed in this way by choosing an appropriate
permutation .
Submodular and
supermodular set functions are also studied in
combinatorial optimization. Many of the results in have analogues in, where
submodular functions were first presented as generalizations of
matroids. In this context, the
core of a convex cost game is called the
base polyhedron, because its elements generalize base properties of
matroids.
However, the optimization community generally considers
submodular functions to be the discrete analogues of convex functions, because the minimization of both types of functions is computationally tractable. Unfortunately, this conflicts directly with
Shapley's original definition of
supermodular functions as "convex".
Further Information
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