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The Nash bargaining theory with non-convex problems. (English) Zbl 0876.90102
In his seminal work on axiomatic bargaining theory, Nash (1950) proved that a solution for all convex bargaining problems always maximizes the product of individuals’ utilities if and only if it satisfies four axioms: symmetry (S), Pareto optimality (PO), invariance with respect to positive affine transformations (INV), and independence of irrelevant alternatives (IIA). One drawback of his work is, however, that it deals with convex bargaining problems only. The standard defense for this convexity restriction is that the feasible utility set can be convexified by randomization when all individuals are expected utility maximizers. But this argument is not compelling for at least two reasons. First, randomization may not always be possible or relevant in all bargaining situations. Second, if individuals involved in some bargaining situations are not all expected utility maximizers, then randomization cannot convexify the feasible utility set. The need to modify Nash’s bargaining theory in light of the second point becomes more apparent as more advances have been made in axiomatic bargaining theory with non-expected utility individuals. The goal of this paper is thus to remove the convexity restriction from Nash’s work.

##### MSC:
 91A12 Cooperative games
##### Keywords:
non-convex problems; axiomatic bargaining theory
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