A Two-Step Proportional Rule for Division with Multiple References

Proportionality is a primary principle generally accepted when dividing a commodity between a set of agents characterized by their reference with respect to a certain characteristic. However, when multiple characteristics have to be taken into account, it is not clear how to define proportionality. We propose a two-step proportional rule for the class of division problems with multiple characteristics. The rule is based on the best expectations of the agents and incorporates the extensions of two crucial properties which are inherent to proportionality: the proportions obtained with respect to the different references cannot be improved simultaneously, and the result does not depend on the scale in which each of the characteristics is measured. We also prove that the two-step proportional rule can be understood as the result of a negotiation in which the Kalai–Smorodinsky solution is applied to a bargaining game between parties each one aiming to maximize the proportions that the set of agents attain with respect to the references of the corresponding characteristic.