Simplifying the Kohlberg Criterion on the Nucleolus

TL;DR

This paper simplifies the Kohlberg criterion for verifying the nucleolus in cooperative games, reducing computational complexity and enabling faster verification for larger games.

Contribution

It introduces a simplified Kohlberg criterion involving fewer coalition sets, along with a method to reduce set sizes and a fast algorithm for balancedness verification.

Findings

Reduced the number of coalition sets to check from exponential to at most (n-1).

Provided a method to decrease the size of coalition sets for easier verification.

Developed a fast algorithm for checking the balancedness of coalition sets.

Abstract

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.