Abstract
We analyze the implications of Nash’s (Econometrica 18:155–162, 1950) axioms in ordinal bargaining environments; there, the scale invariance axiom needs to be strenghtened to take into account all order-preserving transformations of the agents’ utilities. This axiom, called ordinal invariance, is a very demanding one. For two-agents, it is violated by every strongly individually rational bargaining rule. In general, no ordinally invariant bargaining rule satisfies the other three axioms of Nash. Parallel to Roth (J Econ Theory 16:247–251, 1977), we introduce a weaker independence of irrelevant alternatives (IIA) axiom that we argue is better suited for ordinally invariant bargaining rules. We show that the three-agent Shapley–Shubik bargaining rule uniquely satisfies ordinal invariance, Pareto optimality, symmetry, and this weaker IIA axiom. We also analyze the implications of other independence axioms.
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Notes
There is no reference on the origin of this rule in Shubik (1982). However, Thomson (forthcoming) attributes it to Shapley. Furthermore, Roth (1979, in pp. 72–73), mentions a three-agent ordinal bargaining rule proposed by Shapley and Shubik (1974) which, considering the scarcity of ordinal rules in the literature at the time, is most probably the same bargaining rule.
The class of ordinally normalized problems can be interpreted as the ordinal counterpart of the class of 0–1 normalized problems (that is, agents’ disagreement payoffs are normalized to \(0\) and aspiration payoffs to \(1\)) for cardinal bargaining rules. Both classes have the property that any physical bargaining problem has a utility image in this subclass (and therefore, any problem outside the class is equivalent to a member of this class).
The Shapley–Shubik and the Safra–Samet solutions to arbitrary bargaining problems are defined as the limit of a sequence constructed on the problem’s Pareto surface.
For example, Kıbrıs (2004b) utilizes the fact that on the class of ordinally normalized problems, the Shapley–Shubik rule coincides with the Egalitarian rule.
Note that the issue is not the existence of a normalized class but that of constructing one that is desirable in the aforementioned sense. In fact, Sprumont presents a highly asymmetric construction for more than three agents and notes that “it may be of little use to define attractive solutions”.
Without Pareto optimality, the two requirements can contradict. Consider for example \(d=(0,0)\), \(S=conv\{(0,0),(1,0),(0,1)\}\) and \(S^{\prime }=conv\{(0,0),(\frac{1}{2},0),(0,\frac{1}{2})\}\). The problems \((S,d)\) and \( (S^{\prime },d)\) are related to each other through both an affine transformation of the agents utilities and by a contraction of \(S\) to \( S^{\prime }\). For example if \(F(S,d)=(\frac{1}{4},\frac{1}{4})\), scale invariance requires \(F(S^{\prime },d)=(\frac{1}{8},\frac{1}{8})\) and IIA requires \(F(S^{\prime },d)=(\frac{1}{4},\frac{1}{4})\).
Mathematically, this new problem will be a non-transferable utility game where the feasible set of a two-agent coalition is the projection of the grand coalition’s feasible set on their utility subspace. Note that any three-party negotiation in which the outcome is determined through majority voting will be of this form.
Note that in this case, the first round of negotiations lead to two extreme divisions for these two agents. Therefore, this second round can be interpreted as an attempt of the agents to insure themselves against the outcome in which they receive nothing.
In this example, the feasible set contracts in a way that all left out profiles are the ones that assign Agent 1 a smaller payoff compared to the other two agents. Therefore, it is only intuitive that Agent 1 be better-off as result of such a contraction.
This is not very surprising. The aspiration points (together with the disagreement point) are of significance for cardinal bargaining (since any bargaining problem can be cardinally normalized via these points). However, they do not have a similar function in ordinal bargaining.
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I would like to thank Arzu Kıbrıs and Atila Abdulkadiroğlu for comments and suggestions.
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Kıbrıs, Ö. Nash bargaining in ordinal environments. Rev Econ Design 16, 269–282 (2012). https://doi.org/10.1007/s10058-012-0134-6
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DOI: https://doi.org/10.1007/s10058-012-0134-6