A characterization of the Nash bargaining solution

Nir Dagan, Oscar Volij, and Eyal Winter

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We characterize the Nash bargaining solution replacing the axiom of Independence of Irrelevant Alternatives with three independent axioms: Independence of Non-Individually Rational Alternatives, Twisting and Disagreement Point Convexity. We give a non-cooperative bargaining interpretation to this last axiom.

Social Choice and Welfare 19:811-823 (2002)

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Since Nash (1950), a bargaining problem is usually defined as a pair <S,d> where S is a compact, convex subset of R2 containing both d and a point that strictly dominates d. Points in S are interpreted as feasible utility agreements and d represents the status-quo outcome. A bargaining solution is a rule that assigns a feasible agreement to each bargaining problem. Nash (1950) proposed four independent properties and showed that they are simultaneously satisfied only by the the Nash bargaining solution.

While three of Nash's axioms are quite uncontroversial, the fourth one (known as independence of irrelevant alternatives) raised some criticisms, which lead to two different lines of research. Some authors looked for characterizations of alternative solutions which do not use the controversial axiom (see for instance, Kalai and Smorodinsky (1975), and Perles and Maschler (1981)) while some other papers provided alternative characterizations of the Nash solution without appealing to the IIA axiom. Examples of this second line of research are Peters (1986b), Chun and Thomson (1990), Peters and van Damme (1991), Mariotti (1999) and Mariotti (2000), and Lensberg (1988). The first three papers replace IIA by several axioms together with some kind of continuity, the next two replace IIA and other axioms by one axiom, and lastly Lensberg (1988) replaces IIA with consistency, therefore a domain with a variable number of agents is needed.

In this paper we provide an alternative characterization of the Nash bargaining solution in which the axiom of independence of irrelevant alternatives is replaced by three different axioms. All three of them are already known in the literature but were never used together. One of them is independence of non-individually rational alternatives (INIR), which requires a solution to be insensitive to changes in the feasible set that involve only non-individually rational outcomes. This axiom neither implies nor is implied by IIA, but is weaker than IIA and Individual Rationality (IR) together (a solution is individually rational if it assigns each player a utility level that is not lower that its disagreement level.) The second axiom is twisting, which is a weak monotonicity requirement that is implied by IIA. The third axiom is disagreement point convexity (DVEX) which requires that the solution be insensitive to movements of the disagreement point towards the proposed compromise. This last axiom does not imply nor is implied by IIA. Further, the three axioms together do not imply IIA.

All the axioms used in this paper have a straightforward interpretation except, perhaps, for DVEX. This axiom, however, has an interpretation that is closely related to non-cooperative models of bargaining. Assume that the solution recommends f(S,d) when the bargaining problem is <S,d>. The players may pospone the resolution of the bargaining for t periods geting f(S,d) only after t periods of disagreement. From today's point of view, knowing that one has the alternative of reaching agreement t periods later is as if the new disagreement point is f(S,d) paid t periods later. DVEX requires that the solution be insensitive to this kind of manipulation.

Our result, though not its proof, is closely related to Peters and van Damme (1991). The main difference is that we replace their disagreement point continuity by twisting. In this way we get rid of a mainly technical axiom and replace it by a more intuitive and reasonable one.

The paper is organized as follows: In Section 2, we present the preliminary definitions and the axioms used in the characterization. Section 3 gives the main result. Section 4 shows that the axioms are independent. Finally, Section 5 discusses the related literature.

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  1. Frank A. Schmid, Equity Financing of the Entrepreneurial Firm, Federal Reserve Bank of St. Louis Review 83:15-28 (2001)
  2. Paola Manzini and Marco Mariotti, The Effect of Disagreement on Noncooperative Bargaining, Journal of Economic Theory 107:490-499 (2002)
  3. Hannu Vartiainen, Collective choice with endogenous reference outcome, Games and Economic Behavior 58:172-180 (2007)
  4. Abraham Diskin and Dan S. Felsenthal, Individual rationality and bargaining, Public Choice 133:25-29 (2007)
  5. Geoffroy de Clippel, An axiomatization of the Nash bargaining solution, Social Choice and Welfare 29:201-210 (2007)
  6. Xiaobing Gan, Yanhong Gu, George L Vairaktarakis, Xiaoqiang Cai, and Quanle Chen, A scheduling problem with one producer and the bargaining counterpart with two producers, in Bo Chen, Mike Paterson, and Guochuan Zhang (Eds.), Combinatorics, Algorithms, Probabilistic and Experimental Methodologies, pp. 305-316 Springer (2007)
  7. Xin Zeng, Xiaoning Jin, and Qiang Wang, Trading options in supply chain, in 2007 IEEE International Conference on Industrial Engineering and Engineering Management, pp. 1568-1573 (2007)
  8. Tanja Magoč and Vladik Kreinovich, A new simplified derivation of Nash bargaining solution, Applied Mathematical Sciences 3:1097-1101 (2009)