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In game theory, a convex game is one in which the incentives for joining a coalition increase as the coalition grows. This paper shows that the core of such a game -- the set of outcomes that cannot be improved on by any coalition of players -- is quite large and has an especially regular structure. Certain other cooperative solution concepts are also shown to be related to the core in simple ways: (1) The value of a convex game is the center of gravity of the extreme points of the core, and (2) the von Neumann-Morgenstern stable set solution of a convex game is unique and coincides with the core. Similar results for the kernel and the bargaining set will be presented in a later paper. Here, it is also shown that convex games are not necessarily the sum of any number of convex measure games.

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