### Axiomatic Approaches to Coalitional Bargaining Sergiu Hart

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Introduction

The simplest bargaining situation is that of two persons who have to agree on the choice of an outcome from a given set of feasible outcomes; in case no agreement is reached, a specified disgreement outcome results. This two-person pure bargaining problem has been extensively analyzed, starting with Nash (1950).

When there are more than two participants, the n-person straightforward generalization considers either unanimous agreement or complete disgreement (see Roth (1979)). However, intermediate subsets of the players (i.e., more than one but not all) may also play an essential role in the bargaining. One is thus led to an n-person coalitional bargaining problem, where a feasible set of feasible outcomes is specified for each coalition (i.e., subset of the players). This type of problem is known as a game in coalitional form without side payments (or, with nontransferable utility). It frequently arises in the analysis of various economic and other models; for references, see Aumann (1967, 1985a).

Solutions to such problems have been proposed by Harsanyi (1959, 1963, 1977), Shapley (1969), Owen (1972), and others. All of these were constructed to coincide with the Nash solution in the two-person case. Unlike the Nash solution, however, they were not defined (and determined) by a set of axioms.

Recently, Aumann (1985b) has provided an axiomatization for the Shapley solution. Following this work, further axiomatizations were obtained: for the Harsanyi solution by Hart (1985), and for a new class of monotonic solutons by Kalai and Samet (1985). The purpose of this chapter is to review and compare these three approaches.

The discussion is organized as follows. The mathematical model is described in Section 14.2, and is followed by the definitions of the solutions in Section 14.3. The axioms that determine these solutions are presented in Section 14.4, and Section 14.5 includes some general remarks together with a comparison of the solutions in terms of the axioms of Section 14.4.

It should be emphasized that this chapter includes only minimal discussion of the various concepts; it is intended as a summary and a directory to the existing literature on the subject. In particular, the reader should consult the papers of Aumann (1985b), Hart (1985), and Kalai and Samet (1985) for extensive presentations and comments.

• Game-Theoretic Models of Bargaining, Alvin E. Roth (editor), Cambridge University Press (1994), Chapter 14, 305-319