Scale invariance versus translation variance in Nash bargaining problem

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. -----------------------------------------------------------------------------------------------------------------[1] Two-person Bargaining Problem John Nash’s article “The Bargaining Problem” relates to the classical economic problem of two parties freely bargaining with each other. In this situation, subsets of two original sets of items belonging to two individuals are possibly exchanged voluntarily if the sum of utilities is increased for both players. The two opponents or players are each in possession of items that have utilities to both, and who engage in direct bartering without the use of money. Each wishes to convince the other to give away as many and most valuable items as possible in exchange for the fewest and least valuable items. A proposal of exchange involves a specific offer as well as a specific demand, where ‘offer’ and ‘demand’ refer to the details of the proposal. For example: “I offer the set R and I demand in exchange the set K”. We shall use the following notation for the utility functions of X and Y: the-person-possessing-the itemITEMthe-person-for-whom-utility-is-considered. For example: xAy would mean the utility to Y of an item named A which X possesses originally. Each possible exchange, whether reasonable or not, can be drawn on the Ux vs. Uy plane, where Ux and Uy refer to the marginal utilities of X and Y respectively, that is, the sum of utilities after an exchange is made minus the original sum of utilities of the items they owned, or equivalently, the sum of utilities of new items gotten (gained) minus the sum of utilities of items given out (lost). It is very clear that only points in the first quadrant on the boundary/periphery (and not on the axes) are candidates for a solution, because any points inside (where there is room to go upward and/or to the right) are worse off for both players as compared with the boundary/periphery. For example, (3, 6) would not be considered at all if (5, 7) is also available with a different exchange, since the latter point offers more for X as well as more for Y. The existence of (3, 7) would also preclude the inferior point (3, 6) even though it’s only worse off for Y. Moreover, Nash introduces the possibility of X and Y tossing a variable bias coin, with probability p on (0, 1) to decide between any two points on the periphery, hence creating a continuous line there. Note that this line includes linear connections between all points, and not only between adjacent ones on the periphery, thus resulting in an overall curve that is not the same as simply connecting adjacent points on the periphery, but rather a curve that encloses more area. The only question that remains then is “where exactly on that boundary?” and Nash’s answer is that we look for an algorithm that maximizes Ux*Uy. Without any loss of generality, in this article, we would deemphasize or downplay the continuous line approach taken by Nash and instead focus only on those discrete points of actual deterministic exchanges. One severe consequence or limitation though of the discrete approach is that there are cases where maximum Ux*Uy is not unique, whereas the continuous approach always guarantees uniqueness; hence an essential part of Nash’s edifice was the joining of the points with this crooked yet continuous line. Examples of multiple maximum Ux*Uy with the discrete approach abound. [2] Scale Invariance Principle It is noted that Nash’s solution point is scale invariant. The algorithm of maximizing Ux* Uy would always point to the same transformed point (and the same items being exchanged – the same deal) regardless of scale. This is so because the maximum of a set of real numbers is scale-invariant in the sense that the same point (original/transformed) always serves as the maximum. IF Max{X1, X2, X3, ... , Xn} = Xi (for some unique i in the index to n ) THEN Max{Q*X1, Q* X2, Q*X3, ... , Q*Xn} = Q*Xi (same i as above!!) (where Q is any real positive number).