On the 1-Nucleolus for Classes of Cooperative Games

This paper analyzes the 1-nucleolus and, in particular, its relation to the nucleolus and compromise value. It is seen that the 1-nucleolus of a cooperative game can be characterized using a combination of standard bankruptcy rules for associated bankruptcy problems. In particular, for any zero-normalized balanced game, the 1-nucleolus coincides with the Aumann-Maschler rule (Aumann and Maschler, 1985) in this sense. From this result, not only necessary conditions on a compromise stable game are derived such that the 1nucleolus and the nucleolus coincide, but also necessary and sufficient conditions such that the 1-nucleolus and the compromise value of exact games coincide.