Some Recent Algorithms for Finding the Nucleolus of Structured Cooperative Games

Nucleolus is one of the fundamental solution concepts in cooperative game theory. There has been considerable progress in locating the nucleolus in the last three years. The paper motivates through examples how the recent algorithms work efficiently for certain structured class of coperative games. Though the data of a cooperative game grows exponentially in size with the number of players, assignment games, and balanced connected games, grow only polynomially in size, on the number of players. The algorithm for assignment games is based on an efficient graph theoretic algorithm which counts the longest paths to each vertex and trimming of cycles to quickly arrive at the lexicographic geometric centre. Connected games are solved by the technique of feasible direction, initiated in the assignment case. The sellers market corner of the core for assignment games has its counterpart, the lexmin vertex in balanced connected games. Nucleolus has also been characterized via a set of anxioms based on subgame consistency. This is exploited for standard tree games to arrive at an efficient and intuitively explainable algorithm. Improvements on the pivoting manipulations to locate coalitions with constant excess are possible and the paper initially discusses such an algorithm at the beginning.