Monotonicity and Weighted Prenucleoli: A Characterization Without Consistency

Abstract

A solution on a set of transferable utility (TU) games satisfies strong aggregate monotonicity (SAM) if every player can improve when the grand coalition becomes richer. It satisfies equal surplus division (ESD) if the solution allows the players to improve equally. We show that the set of weight systems generating weighted prenucleoli that satisfy SAM is open which implies that for weight systems close enough to any regular system the weighted prenucleolus satisfies SAM. We also provide a necessary condition for SAM for symmetrically weighted nucleoli. Moreover, we show that the per capita nucleolus on balanced games is characterized by single-valuedness (SIVA), translation and scale covariance (COV), and equal adjusted surplus division (EASD), a property that is comparable but stronger than ESD. These properties together with ESD characterize the per capita prenucleolus on larger sets of TU games. EASD and ESD can be transformed to independence of (adjusted) proportional shifting and these properties may be generalized for arbitrary weight systems p to I(A)S p . We show that the p-weighted prenucleolus on the set of balanced TU games is characterized by SIVA, COV, and IAS p ; and on larger sets by additionally requiring IS p .

Weighted nucleoli and dually essential coalitions

“…Various weighted nucleoli (based on weighted surplus measures) were considered by several authors, but mostly from an axiomatization point of view, see e.g. (Derks and Peters 1998;Derks and Haller 1999;Kleppe 2010;Kleppe et al 2016;Calleja et al 2018). We address issues in connection with their computation.…”