Glove game ($v(S)=\omega \min (|S\cap L|,|S\cap R|)$)

Players each own one object of one of two complementary types. A matched pair sells for $\omega >0$, and objects of one type alone have no value. The characteristic function of a coalitional game is therefore the displayed matching count times pair price. The core of a glove game pays the scarce type the entire pair value and the abundant type zero; if type counts are equal, a common split between the two types can vary continuously.

If there are fewer left owners than right owners, deleting any right owner leaves the grand-coalition value unchanged. The complement's core constraint and efficiency force that owner's payoff to be zero. Mixed-pair inequalities then require every left owner to receive at least $\omega$, and efficiency makes all these payoffs exactly $\omega$. Reverse the types when right owners are scarce. If both counts equal $k$, sum all $k^2$ mixed-pair inequalities. Their total is exactly $k$ times the grand-coalition payoff, so each pair sum is $\omega$. Hence all left owners receive a common $\theta \in [0,\omega ]$ and all right owners $\omega -\theta$. Such allocations meet every coalition constraint because $\theta u+(\omega -\theta )v\ge \omega \min (u,v)$. This proves the complete core description.