Before revision:  The main result is that the only tail risk measure that satisfies a set of economic axioms proposed by Schmeidler (1989, Econometrica) and the statistical property of elicitability (i.e. there exists an objective function such that minimizing the expected objective function yields the risk measure ; see Gneiting (2011, J. Amer. Stat. Assoc.) ) is median shortfall, which is the median of tail loss distribution.
After revision:  The main result is that the only tail risk measure that satisfies a set of economic axioms proposed by Schmeidler (1989, Econometrica) and the statistical property of elicitability (i.e. there exists an objective function such that minimizing the expected objective function yields the risk measure ) are the mean functional and the median shortfall, which is the median of tail loss distribution.