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Social Choice and Welfare

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02-01-2019 | Original Paper

Egalitarianism, utilitarianism, and the Nash bargaining solution

Author: Shiran Rachmilevitch

Published in: Social Choice and Welfare | Issue 4/2019

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Abstract

A bargaining solution satisfies egalitarian–utilitarian monotonicity (EUM) if the following holds under feasible-set-expansion: a decrease in the value of the Rawlsian (resp. utilitarian) objective is accompanied by an increase in the value of the utilitarian (resp. Rawlsian) objective. A bargaining solution is welfarist if it maximizes a symmetric and strictly concave social welfare function. Every 2-person welfarist solution satisfies EUM, but for \(n\ge 3\) every n-person welfarist solution violates it. In the presence of other standard axioms, EUM characterizes the Nash solution in the 2-person case, but leads to impossibility in the n-person case.

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Further assumptions on the structure of a problem will be specified in Sect. 2.

The letter “R,” which denotes the minimum operator, stands for “Rawls”. In his Theory of Justice Rawls (1971) promoted the view that a just society should maximize the well-being of its least well-off member. In the present model, this principle translates to maximizing the utilities-minimum.

Whenever I write “n-person case” or “n-person bargaining”, I mean \(n\ge 3\).

That there are substantial differences between 2-person and n-person bargaining is known ever since the work of Shapley (1969). Recently, Karagözolu and Rachmilevitch (2018) showed, in the context of a model different from the one studied here, that it may matter whether the number of bargainers is greater than 4 or not.

Discussions of Shapley’s approach can be found in Yaari (1981) and Rachmilevitch (2015).

This notion of dominance is due to Suppes (1966) and Sen (1970).

Vector inequalities are as follows: uRv if and only if \(u_i R v_i\) for all i, for both \(R\in \{\ge , >\}\); \(u\gneqq v\) if and only if \(u\ge v\) and \(u\ne v\). Given a non-empty set \(X\subset {\mathbb {R}}_+^n\), the smallest comprehensive problem containing it is denoted \(\text {comp}(X)\).

Given a permutation \(\pi \) on \(\{1,\ldots ,n\}\), \(\pi S\equiv \{(s_{\pi (1)},\ldots ,s_{\pi (n)}):s\in S\}\). A problem S that satisfies \(S=\pi S\) for every permutation \(\pi \) is called symmetric.

The functions U and R do not adhere to this definition, as they are not strictly concave. They can be viewed, however, as limit cases: they correspond to the limits \(\rho \rightarrow 1\) and \(\rho \rightarrow -\infty \) of \([\sum _i(x_i)^\rho ]^{1/\rho }\).

PO and SY exclude non-welfarist solutions that satisfy EUM in some trivial way (e.g., D, \(D^i\)).

He derived the result for \(n=2\), but the generalization to \(n\ge 3\) is straightforward.

For example, this solution assigns \(\text {comp}\{(1,1)\}\) the point (1, 1), but assigns \(\text {comp}\{(1,1),(1+\epsilon ,0)\}\) the point \((1+\epsilon ,0)\), for every \(\epsilon >0\). Hence, it violates EUM (to check that it satisfies IIA is easy).

Theorem 3 has a similar flavor to a result of Roth (1979), who showed that when WPO is deleted from Nash’s (1950) axiom-list, a joint characterization of N and D obtains. Other papers that provide WPO-free axiomatizations of N include Lensberg and Thomson (1988) and Anbarci and Sun (2011a).

As I mentioned in the proof of the previous theorem, it is straightforward that D satisfies the axioms. That N satisfies EUM follows from Proposition 1, since N is welfarist; that is satisfies the other axioms follows from Nash (1950).

For more on this idea, see Mariotti (1999).

\(U=U^{\frac{1}{2}}\) and \(R=R^{\frac{1}{2}}\).

The only axiom which is omitted from the table is CF. The column associated with it is identical to the one of PO.

Anbarci N, Sun CJ (2011a) Weakest collective rationality and the Nash bargaining solution. Soc Choice Welf 37:425–429CrossRef

Anbarci N, Sun CJ (2011b) Distributive justice and the Nash bargaining solution. Soc Choice Welf 37:453–470CrossRef

Fleurbaey M, Salles M, Weymark JA (eds) (2008) Justice, political liberalism, and utilitarianism: themes from Harsanyi and Rawls. Cambridge University Press, Cambridge

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Karagözoğlu E, Rachmilevitch S (2018) Implementing egalitarianism in a class of Nash demand games. Theory Decis 85:495–508CrossRef

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Mariotti M (1999) Fair bargains: distributive justice and Nash bargaining theory. Rev Econ Stud 66:733–741CrossRef

Nash JF (1950) The bargaining problem. Econometrica 18:155–162CrossRef

Rachmilevitch S (2015) The Nash solution is more utilitarian than egalitarian. Theory Decis 79:463–478CrossRef

Rawls J (1971) A theory of justice. Harvard University Press, Cambridge

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Shapley LS (1969) Utility comparison and the theory of games. In: La Décision: Agrégation et Dynamique des Ordres de Préf’erence, Editions du CNRS, Paris, pp 251–263

Suppes P (1966) Some formal models of grading principles. Synthese 6:284–306CrossRef

Yaari ME (1981) Rawls, Edgeworth, Shapley, Nash: theories of distributive justice re-examined. J Econ Theory 24:1–39CrossRef

Title: Egalitarianism, utilitarianism, and the Nash bargaining solution
Author: Shiran Rachmilevitch
Publication date: 02-01-2019
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 4/2019
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-018-01170-6

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