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Theory and Decision

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01.02.2014

Randomized dictatorship and the Kalai–Smorodinsky bargaining solution

verfasst von: Shiran Rachmilevitch

Erschienen in: Theory and Decision | Ausgabe 2/2014

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Abstract

“Randomized dictatorship,” one of the simplest ways to solve bargaining situations, works as follows: a fair coin toss determines the “dictator”—the player to be given his first-best payoff. The two major bargaining solutions, that of Nash (Econometrica 18:155–162, 1950) and that of Kalai and Smorodinsky (Econometrica, 43:513–518, 1975), Pareto-dominate this process (in the ex ante sense). However, whereas the existing literature offers axiomatizations of the Nash solution in which this ex ante domination plays a central role (Moulin, Le choix social utilitariste, Ecole Polytechnique Discussion Paper, 1983 ; de Clippel, Social Choice and Welfare, 29:201–210, 2007), it does not provide an analogous result for Kalai–Smorodinsky. This paper fills in this gap: a characterization of the latter is obtained by combining the aforementioned domination with three additional axioms: Pareto optimality, individual monotonicity, and a weakened version of the Perles–Maschler (International Journal of Game Theory, 10:163–193, 1981) super additivity axiom.

Vorheriger Artikel Theory and implementation of coalitional analysis in cooperative decision making

Nächster Artikel Mutual Fund Theorem for continuous time markets with random coefficients

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Vector inequalities: \(xRy\) if and only if \(x_i R y_i\) for both \(i\in \{1,2\},\,R\in \{>,\ge \}\); \(x\gneqq y\) if and only if \(x\ge y\) & \(x\ne y\).

Given two vectors \(x\) and \(y\), the segment connecting them is denoted \([x;y]\).

A natural strengthening of this axiom is Pareto Optimality (PO), which requires \(\mu (S,d)\in P(S)\) for all \((S,d)\in \mathcal{B }\).

i.e., the set of functions \(f\) of the form \(f(x)=\alpha x+\beta \), where \(\alpha >0\).

If \(f_i:\mathbb{R }\rightarrow \mathbb{R }\) for each \(i=1,2,\,x\in \mathbb{R }^2\), and \(A\subset \mathbb{R }^2\), then: \((f_1,f_2)\circ x\equiv (f_1(x_1),f_2(x_2))\) and \((f_1,f_2)\circ A\equiv \{(f_1,f_2)\circ a:a\in A\}\).

See the aforementioned references for descriptions of these solutions.

That \(KS\) satisfies MD is obvious; the fact that \(N\) satisfies it was proved by Sobel (1981).

A related result has been obtained by de Clippel (2007), who showed that \(N\) is characterized by MD and one other axiom—disagreement point convexity (see his paper for the details).

The justification for BFP is that the equal areas to the north-west and south-east of the focal point \(d+(a,b)\) can be viewed as representing equivalent concessions.

\(cS\equiv \{cs: s\in S\}\).

See Thomson (1994) for a detailed discussion of these interpretations.

\(S+T\equiv \{ s+t : s\in S, t\in T\}\).

Myerson (1981) considers essentially the same axiom, but in a slightly different model; in his model there are only feasible sets, and no disagreement points.

See the Online Appendix.

Anbarci, N. (1998). Simple characterizations of the Nash and Kalai/Smorodinsky solutions. Theory and Decision, 45, 255–261.CrossRef

Anbarci, N., & Bigelow, J. (1994). The area monotonic solution to the cooperative bargaining problem. Mathematical Social Sciences, 28, 133–142.CrossRef

de Clippel, G. (2007). An axiomatization of the Nash bargaining solution. Social Choice and Welfare, 29, 201–210.CrossRef

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

Moulin, H. (1983). Le choix social utilitariste. Ecole Polytechnique Discussion Paper.

Myerson, R. B. (1981). Utilitarianism, Egalitarianism, and the timing effect in social choice problems. Econometrica, 49, 883–897.CrossRef

Nash, J. F. (1950). The bargaining problem. Econometrica, 18, 155–162.CrossRef

Perles, M. A., & Maschler, M. (1981). The super-additive solution for the Nash bargaining game. International Journal of Game Theory, 10, 163–193.CrossRef

Sobel, J. (1981). Distortion of utilities and the bargaining problem. Econometrica, 49, 597–619.CrossRef

Thomson, W. (1994). Cooperative models of bargaining. In R. J. Aumann & S. Hart (Eds.), Handbook of game theory, vol 2 (pp. 1237–1284). Amsterdam: North-Holland.

Titel: Randomized dictatorship and the Kalai–Smorodinsky bargaining solution
verfasst von: Shiran Rachmilevitch
Publikationsdatum: 01.02.2014
Verlag: Springer US
Erschienen in: Theory and Decision / Ausgabe 2/2014
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-013-9367-2

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