different notions of core for a cooperative game

Question

What is the difference between strong $\epsilon$-core and an ordinary $\epsilon$-core for a cooperative game $(N,v)$ ? I was trying to look up the definitions with Google but I didn't succeed.

EDIT

Answer 1

For the weak and strong $\epsilon$-core see the snippet below.

Answer 2

The strong refers here to a generalization of the core concept. To generalize the core one imposes on the value of any proper coalitions, i.e., all coalitions excluding the grand coalition $N$ and the empty set, a cost of $\epsilon$ if it positive, and a bonus of $\epsilon$ if it is negative, then it can be defined as

$$ C_{\epsilon}(N,v):= \{ \vec{x} \in I(N,v)| x(N) = v(N)\;\land\; x(S) \ge v(S) - \epsilon \;\forall\; \emptyset \neq S \subset N \} . $$

with $C_{0}(N,v) = C(N,v)$. If $\epsilon_{1} < \epsilon_{2}$, then $C_{\epsilon_{1}}(N,v) \subseteq C_{\epsilon_{2}}(N,v)$. Hence, the strong $\epsilon$-core shrinks if $\epsilon$ becomes smaller. The least core is the threshold level $\epsilon_{0}$ where $C_{\epsilon_{0}}(N,v)$ is still non-empty, and the nucleolus resides on an accentuated position.

To put it differently, the strong $\epsilon$-core consists of all pre-imputations that give rise to excesses not greater than $\epsilon$ for all proper coalitions.

For the weak $\epsilon$-core you should consult the mentioned reference.

Sorry, for the delayed answer but I had and still have problems with the latex formatting. The curly brackets are not visible for me.