Abstract
We characterize all n-person multi-valued bargaining solutions, defined on the domain of all finite bargaining problems, and satisfying Weak Pareto Optimality (WPO), Covariance (COV), and Independence of Irrelevant Alternatives (IIA). We show that these solutions are obtained by iteratively maximizing nonsymmetric Nash products and determining the final set of points by so-called LDR decompositions. If, next, we assume the (set-theoretic) Axiom of Determinacy, then this class coincides with the class of iterated Nash bargaining solutions; but if we assume the Axiom of Choice then we are able to construct an additional large set of discontinuous and even nonmeasurable solutions. We show however that none of these nonmeasurable solutions can be defined in terms of set theoretic formulae. We next show that a number of existing results in the literature as well as some new results are implied by our approach. These include a characterization of all WPO, COV and IIA solutions—including single-valued ones—on the domain of all compact bargaining problems, and an extension of a theorem of Birkhoff characterizing translation invariant and homogeneous orderings.
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References
Billingsley P (1968) Convergence of probability measures. Wiley series in probability and mathematical statistics, John Wiley & Sons, New York
Birkhoff G (1948) Lattice theory. American Mathematical Society Publications, American Mathematical Society, New York
Conley J, Wilkie S (1996) An extension of the Nash bargaining solution to nonconvex problems. Games Econ Behav 13: 26–38
d’Aspremont C (1985) Axioms for social welfare orderings. In: Hurwicz L, Schmeidler D, Sonnenschein H (eds) Social goals and social organizations. North-Holland, Amsterdam
Debreu G (1964) Continuity properties of Paretian utility. Int Econ Rev 5: 285–293
Denicolò V, Mariotti M (2000) Nash bargaining theory, nonconvex problems and social welfare orderings. Theory Decis 48: 351–358
Fishburn PC (1972) Even-chance lotteries in social choice theory. Theory Decis 3: 18–40
Hamel G (1905) Eine Basis aller Zahlen und die unstetige Lösung der Funktionalgleichung f(x + y) = f(x) + f(y). Math Ann 60: 459–462
Harsanyi JC, Selten R (1972) A generalized Nash solution for two-person bargaining games with incomplete information. Manag Sci 18: 80–106
Herrero MJ (1989) The Nash program: non-convex bargaining problems. J Econ Theory 49: 266–277
Kahneman D, Tversky A (1979) Prospect theory: an analysis of decision under risk. Econometrica 47: 263–291
Kaneko M (1980) An extension of the Nash bargaining problem and the Nash social welfare function. Theory Decis 12: 135–148
Kaneko M, Nakamura K (1979) The Nash social welfare function. Econometrica 47: 423–435
Machina MJ (1982) “Expected utility” analysis without the independence axiom. Econometrica 50: 277–323
Marek VW, Mycielski J (2001) Foundations of mathematics in the twentieth century. Am Math Month 108: 449–468
Mariotti M (1998a) Nash bargaining theory when the number of alternatives can be finite. Soc Choice Welf 15: 413–421
Mariotti M (1998b) Extending Nash’s axioms to nonconvex problems. Games Econ Behav 22: 377–383
Maschler M, Owen G, Peleg B (1988) Paths leading to the Nash set. In: Roth AE (eds) The shapley value: essays in honor of Lloyd S. Shapley. Cambridge University Press, Cambridge UK , pp 321–330
Mycielski J, Swierczkowski S (1964) On the Lebesgue measurability and the Axiom of Determinateness. Fundam Math 54: 67–71
Nash JF (1950) The bargaining problem. Econometrica 18: 155–162
Naumova N, Yanovskaya E (2001) Nash social welfare orderings. Math Soc Sci 42: 203–231
Podnieks K (2007) http://www.ltn.lv/~podnieks/gt.html
Roth AE (1977) Individual rationality and Nash’s solution to the bargaining problem. Math Oper Res 2: 64–65
Shubik M (1982) Game theory in the social sciences: concepts and solutions. MIT Press, Cambridge
Starmer C (2000) Developments in non-expected utility theory: the hunt for a descriptive theory of choice under risk. J Econ Lit 38: 332–382
Xu Y, Yoshihara N (2006) Alternative characterizations of three bargaining solutions for nonconvex bargaining problems. Games Econ Behav 57: 86–92
Zame WR (2007) Can intergenerational equity be operationalized. Theor Econ 2: 187–202
Zhou L (1996) The Nash bargaining theory with non-convex problems. Econometrica 65: 681–685
Acknowledgements
We thank participants of several seminars for useful comments. In particular, we thank Luc Lauwers, Marco Mariotti, Natalia Naumova, Peter Wakker, Jan Mycielski, and William Zame for comments on earlier drafts of this paper. We are indebted to Wim Veldman and Arnoud van Rooij of the Mathematical Institute, Radboud University Nijmegen, The Netherlands, for pointing our attention to the Axiom of Determinacy. Also, the comments of two anonymous referees helped to improve the paper considerably.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Peters, H., Vermeulen, D. WPO, COV and IIA bargaining solutions for non-convex bargaining problems. Int J Game Theory 41, 851–884 (2012). https://doi.org/10.1007/s00182-010-0246-6
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DOI: https://doi.org/10.1007/s00182-010-0246-6