View source for Core in the theory of games

The set of all non-dominated outcomes, that is, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c0264901.png" /> of outcomes such that a [[Domination|domination]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c0264902.png" /> cannot hold for any outcomes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c0264903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c0264904.png" /> and coalition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c0264905.png" />. One defines in this respect:

1) The core. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c0264906.png" /> of [[imputation]]s that are not dominated by any other imputation; the core coincides with the set of imputations satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c0264907.png" /> for any coalition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c0264908.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c0264909.png" /> and a von Neumann–Morgenstern solution (see [[Solution in game theory]]) exists, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c02649010.png" /> is contained in any von Neumann–Morgenstern solution.

2) The kernel. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c02649011.png" /> of individually rational configurations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c02649012.png" /> (see [[Stability in game theory]]) such that the following inequality holds for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c02649013.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c02649014.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c02649015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c02649016.png" /> is the set of coalitions containing the player <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c02649017.png" /> and not containing the player <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c02649018.png" />. The kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c02649019.png" /> is contained in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c02649020.png" />-bargaining set.

3) The nucleolus. The minimal imputation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c02649021.png" /> relative to the quasi-order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c02649022.png" /> defined on the set of imputations by: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c02649023.png" /> if and only if the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c02649024.png" />, where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c02649025.png" /></td> </tr></table>

lexicographically precedes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c02649026.png" />. The nucleolus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026490/c02649027.png" /> 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>
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====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>