The Nash rationing problem

This paper studies the problem of allocating utility losses among n agents with cardinal non-comparable utility functions. This problem is referred to as the Nash rationing problem, as it can be regarded as the translation of the Nash bargaining problem to a rationing scenario. We show that there is no single-valued solution satisfying the obvious reformulation of Nash's axioms, nor a multivalued solution satisfying a certain extension of these axioms. However, there is a multivalued solution that is characterised by an appropriate extension of the axioms. We thus call this mapping the Nash rationing solution. It associates with each rationing problem the set of points that maximises a weighted sum of utilities, in which weights are chosen so that all agents' weighted losses are equal.