Core of a glove game (${\ell }_i=\theta ,r_j=\omega -\theta$)

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.