Coincidence of the Shapley value with other solutions satisfying covariance

We identify the necessary and sufficient condition under which the Shapley value coincides with the prenucleolus for general TU games. For 0-normalized 3-person games, the coincidence holds if and only if the game is symmetric or satisfies the PS property (Kar et al., 2009). We also identify the necessary and sufficient coincidence condition in the following allocation problems: the airport games (Littlechild and Owen, 1973), the bidder collusion games (Graham et al., 1990) and the polluted river games (Ni and Wang, 2007). The coincidence between the Shapley value and the CIS and ENSC values is discussed as well.