Solution
= Solution
At most two of $X_1>X_2$, $X_2>X_3$, and $X_3>X_1$ can hold in any outcome. Taking expectations gives $a_{12}+a_{23}+a_{31}\leq2$, and hence \b[$\min\{a_{12},a_{23},a_{31}\}\leq2/3$].
= Solution
At most two of $X_1>X_2$, $X_2>X_3$, and $X_3>X_1$ can hold in any outcome. Taking expectations gives $a_{12}+a_{23}+a_{31}\leq2$, and hence \b[$\min\{a_{12},a_{23},a_{31}\}\leq2/3$].