Ordinal axiomatization of proportional division
The answer hinges on the level of 'atomicity' dictated by the problem. If the decision making unit is non atomic- i.e. can make infinitesmal decisions- then there is always an algorithm corresponding to the competitive allocation such that it is known also to be a 'cardinal' value allocation under certain conditions- vide Aumann &amp; Shapley. However this would not be robust to even a small perturbation towards atomicity. My understanding is that this field is interesting because of how small changes in the lower limit to what is within the agent's decision making scope can lead to big changes in behavior

D.G Saari's work may be of interest if only as an awful warning- vide http://socioproctology.blogspot.co.uk/search?q=dark+matter