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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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.
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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
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DOI: https://doi.org/10.1007/s00355-018-01170-6