Scale invariance versus translation variance in Nash bargaining problem

TL;DR

This paper revisits Nash's bargaining solution, emphasizing the importance of scale invariance over translation invariance in utility functions, and explores implications for bargaining scenarios and ownership situations.

Contribution

It introduces the principle of scale invariance for utility functions in the bargaining problem and discusses its implications, contrasting it with translation invariance.

Findings

Scale invariance applies to utility functions in bargaining.

Translation invariance leads to contradictory behavior.

Ownership scenarios can be explained by scale invariance.

Abstract

Nash's solution in his celebrated article on the bargaining problem calling for maximization of product of marginal utilities is revisited; a different line of argument supporting such a solution is suggested by straightforward or more direct reasoning, and a conjecture is raised which purports uniqueness of algorithm, namely his solution. Other alternative inferior algorithms are also suggested. It is argued in this article that the scale invariance principle for utility functions should and could be applied here, namely that utility rescaling u'=a*u is allowed, while translations, adding a constant to utility functions u'=u+b could not be applied here, since it is not invariant and leads to contradictory behavior. Finally, special situations of ownership and utilities, where trading is predicted not to take place at all because none is profitable are examined, and then shown to be consistent with the scale invariance principle.