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Let us with Hobbs suppose that life in a state of nature, without agreement on basic social values (“Murder is bad”), without benefit of our common heritage of knowledge and technology, without, in short, an implicit social contract would be "solitary, poor, nasty, brutish, and short.” Given an imaginary presocial period of unlimited discussion, what sort of rule would our predecessors choose to govern subsequent societies? Let us take the rather unpleasant state of nature as the disagreement point should no agreement be reached.

Various philosophers have argued that some particular rule would be chosen if given a choice among such meta-rules from a variously defined impartial positions. Sidgwick argues that utilitarianism would be chosen by an ideal sympathetic impartial observer, Harsanyi that we will choose it if we do not know which member of society we shall be. Rawls argues that a different and very egalitarian solution results from behind a carefully constructed “veil of ignorance.” I believe Nash has the best thought-out position on how to form a social contract, though if he proposed that his axiomatic bargaining theory be used to solve the “What would we do in the original position” problem I am not aware of it. He proposed as the general bargaining solution the unique solution that satisfied a plausible set of axioms, and later scholars have found that the same solution arises under many other sets of axioms. See:

W. Thompson, T. Lensberg (1989). "Axiomatic Theory of Bargaining with a Variable Number of Agents."

H. J. M. Peters (1992). "Axiomatic Bargaining Game Theory."

A. E. Roth (1979) . Axiomatic Models of Bargaining.

Consider three individuals, Normal, Depressed, and Passionate. We compare their utility functions from a God’s-eye point of view, an omniscient narrator. Depressed is a sort of black hole, absorbing goods and producing misery. Passionate has desires similar to Normal, but ten times as strong. These examples suggest, and Nash concluded, that the agreement point for bargaining should be invariant under the sort of transformations that produce Depressed and Passionate. Nash suggested choosing a bargaining solution to maximizing the product of the differences in utility between the disagreement point and the bargaining solution. A little thought will show that if a point s maximizes UN(s)*UD(s)UP(s) and we replace one of those utility functions with a + bU(s), for a and b constants, maximization of the revised product will choose the same point. Note further, the Nash bargaining solution requires that we know the differences in utility functions that produce observable differences in behavior (e.g. choices over lotteries) but not that we know anything about utility functions which is in principle unobservable (which is to say, we can not and need not distinguish between a utility function and any affine transform of that function).

So, here is my question: My intuition is that the Nash-bargaining social contract is more egalitarian than the utilitarian social contract. However. I am having difficulty proving it to my satisfaction, and I would very much like to see such a proof, if I am right, and a disproof or counterexample if I am wrong,

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    $\begingroup$ What is your notion of "more egalitarian?" $\endgroup$ Oct 23, 2023 at 1:44

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