Abstract
I characterize the individually-rational utilitarian bargaining solution by combining several classical axioms with a novel axiom, monotone step-by-step negotiations (monotone SSN). One of the axioms involved in the characterization is conflict freeness, which imposes Pareto optimality on problems that include their ideal point; when conflict freeness is replaced by weak Pareto optimality, only one additional solution becomes admissible—the egalitarian solution. I also show that in Kalai’s (Econometrica 45:1623–1630, 1977) SSN-based characterization of the proportional solutions, SSN can be weakened to monotone SSN if feasible set continuity is assumed.
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Notes
All concepts will be defined formally in Section 2.
CF, which pertains only to a small class of problems, is the only efficiency axiom invoked in the characterization of the individually-rational utilitarian solution. In an earlier paper (Rachmilevitch 2015) I showed how CF can replace the Pareto axiom in a characterization of the asymmetric Nash solution.
See, e.g., the book by Fleurbaey et al. (2008).
Vector inequalities: uRv if and only if \(u_iRv_i\) for all i, for both \(R\in \{>,\ge \}\); \(u\gneqq v\) if both \(u\ge v\) and \(v\ne v\) hold.
\(\text {conv}X\) denotes the convex hull of X.
One has to make appropriate assumptions on the domain in order to guarantee that \(U_\alpha \) is single-valued; a sufficient condition for that is strict comprehensiveness, namely that the Pareto frontier has strict curvature.
The stronger version of this axiom is Pareto optimality (PO), that requires \(f(S,d)\in P(S)\).
To be precise, Kalai does not impose TINV explicitly but implicitly, because the disagreement point is normalized to the origin in his model.
Note that \(y_2=x_2\).
Additionally, it is easy to check that any asymmetric individually-rational solution satisfies all the axioms but SY, on every domain where this solution is well-defined.
The smallness of r is required for the left inequality, the right inequality holds for any \(r>0\), because \(b>1\).
Note that the permutation of \((S''',d')\), namely \((\pi S''',\pi d')\) where \(\pi (a,b)\equiv (b,a)\), is a “tall and narrow problem” in which the player who has less to gain from bargaining receives his ideal payoff. As was proved above, in this problem the solution point is the egalitarian one, but this is clearly not the case because the line that passes through \(d'=\gamma (c,b)\) and (c, b) has a slope different from 1—the slope is \(\frac{b}{c}\).
For brevity, I do not spell out the n-person counterparts of all definitions, axioms, etc., trusting that no confusion will arise.
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I am grateful to several anonymous referees for helpful comments.
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Rachmilevitch, S. Step-by-step negotiations and utilitarianism. Int J Game Theory 50, 433–445 (2021). https://doi.org/10.1007/s00182-021-00755-3
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DOI: https://doi.org/10.1007/s00182-021-00755-3