The Bargaining Problem

The idea of exchange, the basis of economics, is nearly as old as man, and deal-making has been the stuff of legend since the Levantine kings and the pharoahs traded gold and chariots for weapons and slaves.  Despite the rise of the great impersonal capitalist marketplace, with its millions of buyers and sellers who never meet face-to-face, the one-on-one bargain-- involving wealthy individuals, powerful governments, labor unions, or giant corporations-- dominates the headlines.  But two centuries after the publication of Adam Smith's The Wealth of the Nations, there were still no principles of economics that could tell one how the parties to a potential bargain would interact, or how they would split the pie.

[John] Nash took a completely novel approach to the problem of predicting how two rational bargainers will interact.  Instead of defining a solution directly, he started by writing down a set of reasonable conditions that any plauusible solution would have to satisfy and then looked at where they took him.

This is called the axiomatic approach-- a method that had swept mathematics in the 1920's, was used by von Nuemann in his book on quantum theory  and his papers on set theory, and was in its heyday at Princeton in the late 1940's.  Nash's paper is one of the first to apply the axiomatic method to a problem of the social sciences.

Nash's theory assomes that both side's expectations about each other's behaviour are based on the intrinsic features of the bargainning situation itself.  The essence of a situation that results in a deal is "two individuals who have the opportunity to collaborate for mutual benefit in more than one way."  How they will split the gain, he reasoned, reflects how much the deal is worth to each individual.

He started by asking the question, Whhat reasonable conditons would any solution --any split-- have to satisfy?  He then posed four conditions and, using an ingenious mathematical argument, showed that, if his axioms held, a unique solution existed that maximized the product of the player's utilities.  In a sense, his contribution was not so much to "solve" the problem as to state it in a simple and precise way so as to show that unique solutions were possible.

from A Beautiful Mind by Sylvia Nasar
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