 Solution

Let $L$ be the set of left-shoe owners and $R$ the right-shoe owners. This is the glove game with pair price ten, so its characteristic function of a coalitional game isAll singleton values are zero. Write ${\ell }_i\ge 0$ for left-owner payoffs and $r_j\ge 0$ for right-owner payoffs. Efficiency gives total $10\min (a,b)$, and each mixed pair imposes ${\ell }_i+r_j\ge 10$.

Suppose $a<b$. Removing any right owner leaves at least $a$ right shoes, so the complement coalition still has value $10a$. Its core inequality and efficiency imply $r_j\le 0$. Thus every right owner receives zero. The pair constraints then force ${\ell }_i\ge 10$, and their total is $10a$, so every left owner receives ten. Conversely this allocation gives each coalition $10|S\cap L|$, at least its value, so it is in the core. The argument is symmetric when $a>b$.

When $a=b=k$, sum all $k^2$ mixed-pair inequalities. Their total left side is $k$ times the grand-coalition payoff, namely $10k^2$. Hence every inequality is equality: ${\ell }_i+r_j=10$ for every pair. Fixing one right owner shows all left payoffs are the same $\theta$, and fixing one left owner shows all right payoffs are $10-\theta$. Individual rationality gives $0\le \theta \le 10$. Such allocations are sufficient: for a coalition with $u$ left owners and $v$ right owners,as is seen by subtracting the right side and factoring either $\theta (u-v)$ or $(10-\theta )(v-u)$.

The core of a glove game is thereforePayoffs are in pounds. The parameter in the balanced case is common to all owners of a given shoe type.