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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
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.
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).
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.
References
Anbarci N, Sun CJ (2011a) Weakest collective rationality and the Nash bargaining solution. Soc Choice Welf 37:425–429
Anbarci N, Sun CJ (2011b) Distributive justice and the Nash bargaining solution. Soc Choice Welf 37:453–470
Fleurbaey M, Salles M, Weymark JA (eds) (2008) Justice, political liberalism, and utilitarianism: themes from Harsanyi and Rawls. Cambridge University Press, Cambridge
Kalai E (1977) Proportional solutions to bargaining situations: interpersonal utility comparisons. Econometrica 45:1623–1630
Kalai E, Smorodinsky M (1975) Other solutions to Nash’s bargaining problem. Econometrica 43:513–518
Karagözoğlu E, Rachmilevitch S (2018) Implementing egalitarianism in a class of Nash demand games. Theory Decis 85:495–508
Lensberg T, Thomson W (1988) Characterizing the Nash bargaining solution without Pareto-optimality. Soc Choice Welf 5:247–259
Mariotti M (1999) Fair bargains: distributive justice and Nash bargaining theory. Rev Econ Stud 66:733–741
Nash JF (1950) The bargaining problem. Econometrica 18:155–162
Rachmilevitch S (2015) The Nash solution is more utilitarian than egalitarian. Theory Decis 79:463–478
Rawls J (1971) A theory of justice. Harvard University Press, Cambridge
Roth AE (1979) Axiomatic models of bargaining. Springer, Berlin
Sen A (1970) Collective choice and social welfare. Holden-Day, San Francisco
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 6:284–306
Yaari ME (1981) Rawls, Edgeworth, Shapley, Nash: theories of distributive justice re-examined. J Econ Theory 24:1–39
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Rachmilevitch, S. Egalitarianism, utilitarianism, and the Nash bargaining solution. Soc Choice Welf 52, 741–751 (2019). https://doi.org/10.1007/s00355-018-01170-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00355-018-01170-6