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On the Interpretation of the Nash Bargaining Solution and Its Extension to Non-Expected Utility Preferences

Ariel Rubinstein
Zvi Safra
William Thomson

Abstract

The paper reexamines the foundations of the axiomatic Nash bargaining theory. More specifically it questions the interpretation of the Nash bargaining solution and extends it to a family of non-expected utility preferences. A bargaining problem is presented as $\langle X, D, \geqslant_1, \geqslant_2 \rangle$ where $X$ is a set of feasible agreements (described in physical terms), $D$ is the disagreement event and $\geqslant_1$ and $\geqslant_2$ are preferences defined on the space of lotteries in which the prizes are the elements in $X$ and $D$. The (ordinal)-Nash bargaining solution is defined as an agreement $y^\ast$ satisfying for all $p \in \lbrack 0, 1 \rbrack$ and for all $x \in X$: if $px >_1 y^\ast$ then $py^\ast \geqslant_2 x$ and if $px >_2 y^\ast$ then $py^\ast \geqslant_1 x$ where $px$ is the lottery which gives $x$ with probability $p$ and $D$ with probability $1 - p$. Revisions of the Pareto, Symmetry, and IIA Axioms characterize the (ordinal)-Nash bargaining solution. In the expected utility case this definition is equivalent to that of the Nash bargaining solution. However, this definition is to be preferred since it allows a statement of the Nash bargaining solution in everyday language and makes possible its natural extension to a wider set of preferences. It also reveals the logic behind some of the more interesting results of the Nash bargaining solution such as the comparative statics of risk aversion and the connection between the Nash bargaining solution and strategic models.