We study the core of $2/3$-convex games, a class of transferable-utility (TU) games that generalizes convex games, which requires at least two of any three distinct coalitions satisfy the supermodular inequality. We show that every $2/3$-convex game admits a marginal worth vector in the core, and hence has a nonempty core. Our proof is constructive and yields a linear-time algorithm for finding such a core allocation. We also show that $2/3$-convexity does not imply permutational convexity in general, while the implication holds under an additional superadditivity condition on pairs of singletons. As an application, we introduce a TU model motivated by the OSCE's consensus-minus-two procedure and show that the resulting game is $2/3$-convex but need not be convex.
Convex cooperative games have a nonempty core and admit an efficient construction of core allocations. However, economically natural local frictions may violate convexity. We study the core of $3$-distant convex games, which permit violations of convexity only between closely related coalitions. Although the core of such a game may be empty, we establish a simple necessary and sufficient condition for its nonemptiness. We also develop a strongly polynomial-time combinatorial algorithm for finding a core allocation whenever one exists. Finally, we demonstrate the applicability of the framework to oligopolistic competition with merger synergies and profit allocation of joint production with complementary inputs.