Allocating joint costs by means of the nucleolus (Q1076622)

Given a side-payment game v with a finite set of players N, the SCRB vector is the solution \(\tilde x\) defined by \(\tilde x_ j:=[v(N)- v(N\setminus \{i\})]-c\), where \(c\in {\mathfrak R}\) is chosen so that \(\sum_{j\in N}\tilde x_ j=v(N)\). It was used in the Tennessee Valley Authority, and coincides with S. Tijs' \(\tau\)-value solution concept for a certain class of games. A sufficient condition for \(\tilde x\) to be the nucleolus of v is given. A part of this condition, called the pseudo convexity, is shown to be weaker than Sharkey's subconvexity condition, and the subconvexity condition is shown to be weaker than Shapley's convexity condition.