## DIMACS TR: 96-33

## Stable Families of Coalitions and Normal Hypergraphs

### Authors: E. Boros, V. Gurvich, A. Vasin

**
ABSTRACT
**

The core of a game is defined as the set of outcomes acceptable
for {\em all} coalitions. This is probably the simplest
and most natural concept of cooperative game theory.
However, the core can be empty because there are too many
coalitions. Yet, some players may not like or know each
other, so they cannot form a coalition.
Let $\cK$ be a fixed family of coalitions.
The $\cK$-core is defined as the set of outcomes acceptable for
all the coalitions from $\cK$. The family $\cK$ is called
{\em stable} if the $\cK$-core is not empty for any normal form game
(or equivalently, for any game in generalized characteristic
function form). \\*[\parskip]
\hspace*{1.5em}{\em Normal hypergraphs} can be characterized by several
equivalent properties, e.g. they are
{\em dual} to the {\em clique hypergraphs} of {\em perfect graphs}.
We prove that a family $\cK$ of coalitions is stable
iff $\cK$ as a hypergraph is normal.

Paper Available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1996/96-33.ps.gz

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