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Social Choice and Welfare

Erschienen in:

06.01.2021 | Original Paper

No individual priorities and the Nash bargaining solution

verfasst von: Shiran Rachmilevitch

Erschienen in: Social Choice and Welfare | Ausgabe 4/2021

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Abstract

A bargaining solution satisfies no individual priorities (NIP) if the following holds: if x is the selected utility allocation and \(\pi x\) is also feasible, where \(\pi \) is some permutation, then \(x=\pi x\). I characterize the Nash bargaining solution on the basis of this axiom, non-triviality (the disagreement point is never selected), and scale covariance. An additional characterization is presented for the 2-person case, in which NIP is weakened and symmetry is added.

Vorheriger Artikel An axiomatic characterization of the Slater rule

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Further mathematical assumptions on S will be specified in Sect. 2.

Axioms will be formally defined in Sect. 2.

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

Though bargaining papers typically assume either that the solution is a function or a correspondence, there are papers that allow the solution to be, in principle, a correspondence, but then impose, as an explicit axiom, that it be single-valued in some situations. For example, Dagan et al. (2002) require that the solution be single-valued on symmetric sets.

I employ the notation “f(S)” instead of “f(S, d)” since the disagreement point is fixed at the origin throughout my analysis.

Moreover, in the 2-person case NIP is equivalent to the combination of weak NIP and symmetry (this is proved in Lemma 3 below).

Weak NIP is satisfied, for example, by the dictatorial solutions; the dictatorial solutions violate maximal symmetry.

Vector inequalities: uRv iff \(u_iRv_i\) for all i, for either \(R\in \{>,\ge \}\); \(u\gneqq v\) iff \(u\ge v\) and \(u\ne v\).

The smallest comprehensive set containing X is denoted \(\text {comp}X\).

Of course, if \(n=2\) then “for all \(j\notin \{1,n\}\)” is redundant.

This is not be true without comprehensiveness. For example, if \(S={\text {conv}}\{\mathbf{0 },s\}\) for some \(s>\mathbf{0} \), then \(\pi x'\) would generically be outside of \(S'\) (it is in \(S'\) iff the latter is a subset of \(\{(r,\ldots ,r):r\ge 0\}\)).

If \(f_i(S)\ne f_j(S)\) and f satisfies NIP, then \(\pi ^{ij}f(S)\notin S\), so W.NIP has no bite.

To see this, let \(D^1(S)=(r,0,\ldots ,0)\) and consider, w.l.o.g, \(\pi ^{12}\). If \((0,r,0,\ldots ,0)\in S\) then \(a_2(S)\ge r=a_1(S)\) and therefore \(a_1(S\cap \pi ^{12}S)=r\); therefore \(D^1(S\cap \pi ^{12}S)=(r,0,\ldots ,0)\).

This is illustrated, for example, by the dictatorial solution. This solution can be viewed as a correspondence on Mariotti’s domain, and it clearly violates maximal symmetry. As was proved in the previous footnote, it does, however, satisfy W.NIP.

Until the end of the present section, whenever I write “Corollary 1 is an improvement on X,” the meaning is that the improvement is in the 2-person version of my bargaining domain.

A detailed discussion of these axioms can be found in the respective papers.

Other contributions to this research program include Lensberg and Thomson (1988), and Rachmilevitch (2015).

IR is the requirement \(f(S,d)\ge d\). I have not spelled it out earlier (e.g., in Sect. 2) because it holds trivially on my domain.

This is an improvement on Roth’s result because Roth’s SIR implies both NT and IR, and the combination of SY and IIA implies NIP (in the 2-person version of my domain). To see the latter implication, let S be a feasible set and let f satisfy SY and IIA, and let \(f(S)=(a,b)\), where \(a\ne b\). Assume by contradiction that \((b,a)\in S\). Then IIA implies \((a,b)=f(S\cap \pi S)\)—in contradiction to SY.

In addition to works mentioned earlier, other works on this subject include Mariotti (2000b) and Xu (2012).

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

Dagan N, Volij O, Winter E (2002) A characterization of the Nash bargaining solution. Soc Choice Welf 19:811–823CrossRef

Kalai E (1977) Proportional solutions to bargaining situations: interpersonal utility comparisons. Econometrica 45:1623–1630CrossRef

Kalai E, Smorodinsky M (1975) Other solutions to Nash’s bargaining problem. Econometrica 43:513–518CrossRef

Lensberg T, Thomson W (1988) Characterizing the Nash bargaining solution without pareto-optimality. Soc Choice Welf 5:247–259CrossRef

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

Mariotti M (2000a) Maximal symmetry and the Nash solution. Soc Choice Welf 17:45–53CrossRef

Mariotti M (2000b) An ethical interpretation of the Nash choice rule. Theory Decis 49:151–157CrossRef

Mori O (2018) Two simple characterizations of the Nash bargaining solution. Theory Decis 85:225–232CrossRef

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

Rachmilevitch S (2015) Nash bargaining with (almost) no rationality. Math Soc Sci 76:107–109CrossRef

Roth AE (1977) Individual rationality and Nash’s solution to the bargaining problem. Math Oper Res 2:64–65CrossRef

Roth AE (1979) Axiomatic models of bargaining. Lecture notes in economics and mathematical systems, vol 170. Springer, BerlinCrossRef

Sen A (1970) Collective choice and social welfare. Holden-Day, San Francisco

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

Thomson W (1994) Cooperative models of bargaining. Handb Game Theory Econ Appl 2:1237–1284

Xu Y (2012) Symmetry-based compromise and the Nash solution to convex bargaining problems. Econ Lett 115:484–486CrossRef

Titel: No individual priorities and the Nash bargaining solution
verfasst von: Shiran Rachmilevitch
Publikationsdatum: 06.01.2021
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 4/2021
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-020-01302-x

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