View source for Core in the theory of games

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The set of all non-dominated outcomes, that is, the set  $  C $
of outcomes such that a [[Domination|domination]]  $  s \succ _ {K} c $
cannot hold for any outcomes  $  s \in S $, 
$  c \in C $
and coalition  $  K \in \mathfrak R _ {i} $. 
One defines in this respect:

1) The core. The set  $  c ( v) $
of [[imputation]]s that are not dominated by any other imputation; the core coincides with the set of imputations satisfying  $  \sum _ {i \in S }  x _ {i} \geq  v ( S) $
for any coalition  $  S $. 
If  $  c ( v) \neq \emptyset $
and a von Neumann–Morgenstern solution (see [[Solution in game theory]]) exists, then  $  c ( v) $
is contained in any von Neumann–Morgenstern solution.

2) The kernel. The set  $  k ( v) $
of individually rational configurations  $  ( x, \mathfrak B ) $(
see [[Stability in game theory]]) such that the following inequality holds for any  $  i, j \in B \in \mathfrak B $:

$$ 
\left ( \max _ {S \in \tau _ {ij} }  e ( S, x) -
\max _ {S \in \tau _ {ji} }  e ( S, x) \right ) x _ {j}  \leq   0,
$$

where  $  e ( S, x) = v ( S) - \sum _ {k \in S }  x _ {k} $
and  $  \tau _ {ij} $
is the set of coalitions containing the player  $  i $
and not containing the player  $  j $. 
The kernel  $  k ( v) $
is contained in an  $  M _ {1}  ^ {i} $-
bargaining set.

3) The nucleolus. The minimal imputation  $  n ( v) $
relative to the quasi-order  $  \prec _  \nu  $
defined on the set of imputations by:  $  x \prec _  \nu  y $
if and only if the vector  $  \theta ( x, v) = ( \theta _ {1} ( x, v) \dots \theta _ {n} ( x, v)) $, 
where

$$ 
\theta _ {i} ( x, v)  =  \max _ {\begin{array}{c}
{} \\
 | \mathfrak U | = i 
\end{array}
 } \ 
\min _ {\begin{array}{c}
{} \\
 S \in \mathfrak U 
\end{array}
 }  e
( S, x) ,
$$

lexicographically precedes  $  \theta ( y, v) $. 
The nucleolus  $  n ( v) $
exists and is unique for any game with a non-empty set of imputations. In a cooperative game the nucleolus is contained in the kernel.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Vorob'ev,   "The present state of the theory of games"  ''Russian Math. Surveys'' , '''25''' :  2  (1970)  pp. 77–136  ''Uspekhi Mat. Nauk'' , '''25''' :  2  (1970)  pp. 103–107</TD></TR>
</table>

====Comments====
The Russian word ( "yadro" ) is the same for all three notions defined above, but these notions may be distinguished by prefixing with the corresponding English letter ( "c-yadro"  for core,  "k-yadro"  for kernel and  "n-yadro"  for nucleolus). These three notions do not share many properties.

See [[#References|[a1]]], [[#References|[a7]]] for core, [[#References|[a2]]] for kernel and [[#References|[a3]]] for nucleolus. [[#References|[a4]]], [[#References|[a5]]] are general references. [[#References|[a6]]] deals also with mathematical economics and the role of the concept of the core of a game in that setting.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  O.N. Bondareva,   "Certain applications of the methods of linear programming to the theory of cooperative games"  ''Probl. Kibernet'' , '''10'''  (1963)  pp. 119–139  (In Russian)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Maschler,   M. Davis,   "The kernel of a cooperative game"  ''Naval Res. Logist. Quart.'' , '''12'''  (1965)  pp. 223–259</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Schmeidler,   "The nucleolus of a characteristic function game"  ''SIAM J. Appl. Math.'' , '''17'''  (1969)  pp. 1163–1170</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Owen,   "Game theory" , Acad. Press  (1982)</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Szép,   F. Forgó,   "Introduction to the theory of games" , Reidel  (1985)  pp. 171; 199</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Rosenmüller,   "Cooperative games and markets" , North-Holland  (1981)</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  L.S. Shapley,   "On balanced sets and cores"  ''Naval Res. Logist. Quart.'' , '''14'''  (1967)  pp. 453–460</TD></TR>
</table>