A characterization of the lexicographic Kalai–Smorodinsky solution for n=3

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Abstract

We propose the axiom of individual monotonicity under individual rationality and combine it with the axioms of efficiency, symmetry, invariance under utility transformation, independence of irrelevant alternatives other than the ideal point, to characterize the lexicographic Kalai–Smorodinsky solution for 3-person bargaining problems.

Introduction

In the axiomatic theory of the n-person bargaining problem, Kalai and Smorodinsky [2]provided a characterization of the Kalai–Smorodinsky solution (KS solution) [3]for the 2-person case. Thomson [5]extended the result to the n-person problem under certain conditions and studied the KS solution for allowing variations on the number of agents as well [6]. Imai [1]studied the properties of the lexicographic Kalai–Smorodinsky solution (lKS solution). It is known that the lKS solution is exactly the KS solution for 2-person problems. Hence, the lKS solution can be viewed as an another extension of Kalai and Smorodinsky's result. For more comprehensive surveys, see Imai [1], Roth [4], and Thomson [7].

Imai considered the following five axioms: efficiency (E), symmetry (S), invariance under utility transformation (Inv), independence of irrelevant alternatives other than the ideal point (IIIA), and individual monotonicity (IM), and he shows that the lKS solution is the only solution satisfying these five axioms.

The individual monotonicity axiom adopted by Imai differs from Kalai and Smorodinsky's axiom in taking projection of the entire agreement set instead of individually rational parts of the same set. However, the entire agreement set might include points which do not satisfy individual rationality. Because it is impossible for all players to accept points not in the individually rational part of the entire agreement set and a solution for bargaining problems is thought as unanimous agreement, Imai raised the following question: Can we characterize lKS solution by the same axioms except that Axiom IM is modified to take projection of the individually rational part?

We will propose the axiom of individual monotonicity under individual rationality (IMd) which is a revised version of Axiom IM in accordance with Imaoi's suggestion and combine it with the other 4 axioms to characterize the lKS solution for 3-person problems. Our proof also holds for problems bounded below by the disagreement point. This question is also raised by Imai. Unfortunately, our technique can't be applied when n≥4. We will propose an example to illustrate this. In Section 2, we introduce the model, basic definitions, and some facts. In Section 3, our main result is proved.

Throughout this paper, we follow the notations and conventions used in Imai's paper as much as we can.

Section snippets

The model, basic definitions and some facts

An n-person bargaining problem is given by a player set N={1, 2,…,n}, n≥2, an agreement set, SRn, and a disagreement point, dS. Fixing N, B=(S, d) denotes a problem.

For x, yRn, xy means xi>yi for all iN, xy means xiyi for all iN, and x>y means xy but xy. We denote by x·y the inner product of x and y.

For S, a subset of Rn, P(S) denotes the Pareto frontier of S, i.e., P(S)={xS: (x+R+n)∩S={x}}. We consider the class of problems, B, consisting of all B=(S, d) such that S is convex,

The proof of the main result

Our characterization result of the lKS solution is summarized in the following theorem:
Theorem 3.1. The lKS solution is the unique solution on B3 satisfying Axioms E, S, Inv, IMd and IIIA.

The fact that the lKS solution satisfies these five axioms can be obtained by applying Imai's proof [1]. Before showing that if a bargaining solution f satisfies Axioms E, S, Inv, IMd and IIA it coincides with lKS for all BB3, we need make the following remarks and some observations concerning the

Some remarks

We can easily see that only the individually rational part of the agreement set is essential in our proofs throughout. It is natural to say that our result is also true for problems bounded below by the disagreement point.

It is easy to see that Imai's procedure takes 1 step only for problems with strictly comprehensive agreement sets. Hence, the KS solution and the lKS solution coincide for this class of problems. Hence, these axioms can also be used to characterize the KS solution for 3-person

References (7)

  • H. Imai

    Individual monotonicity and lexicographic maxmin solution

    Econometrica

    (1983)
  • E. Kalai et al.

    Other solutions to Nash's bargaining problem

    Econometrica

    (1975)
  • H. Raiffa, Arbitration Schemes for Generalized Two-Person Games, University of Michigan,...
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