Stability for Coalition Structures in Terms of the Proportional Partitional Shapley Value

Many coalitional values have been introduced in the literature on cooperative games over the last decades, and especially since 2000. The multiplicity of options suggests the convenience of testing the existence of stable coalition structures, in the sense of Hart and Kurz (1983, Econometrica), when payments are made using some of these values. We recall their concept of \(\gamma\) $\gamma$ –stability and give results for the proportional partitional Shapley value, introduced by Alonso–Meijide et al (2015, Discrete Appl. Math.), which shares the utility of any coalition proportionally to the Shapley value of the involved players in the original game.