In an economy with several agents who have different utility functions, it is common to define a welfare function, defined as an aggregate of the utility functions of the different agents. Then we can ask if and how this welfare function can be maximized.
AFAIK, the most common welfare function is the sum of utilities, also called utilitarian welfare. A more general function is a weighted-sum, in which each agent has a different weight. Such functions are well-studied. For example, it can be proved (e.g. Varian, 1976) that an allocation is Pareto-efficient, if and only if it maximizes a weighted-sum welfare function, for some choice of weights.
I am looking for references in which social welfare is measured using the median, or a similar statistic (e.g. a certain percentile), instead of a weighted sum. Particularly, I am looking for references about existence of an allocation which maximizes the medial social welfare in various settings.