  The nucleolus offers a desirable payoff-sharing solution in cooperative games
thanks to its attractive properties - it always exists and lies in the core (if
the core is non-empty), and is unique. Although computing the nucleolus is very
challenging, the Kohlberg criterion offers a powerful method for verifying
whether a solution is the nucleolus in relatively small games (i.e., the number
of players $n \leq 20$). This, however, becomes more challenging for larger
games because of the need to form and check the balancedness of possibly
exponentially large sets of coalitions, each set could be of an exponentially
large size. We develop a simplifying set of the Kohlberg criteria that involves
checking the balancedness of at most $(n-1)$ sets of coalitions. We also
provide a method for reducing the size of these sets and a fast algorithm for
verifying the balancedness.
