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Unformatted text preview: Chapter 9 Bargaining Theory In this section we present the basics of bilateral bargaining theory. We begin by discussing the axiomatic model of Nash (1950). We then describe a simple version of the strategic model developed by Rubinstein (1982). We then generalize the simple strategic model, and analyze the relationship in this general context between the Nash and Rubinstein solutions. Finally, we examine bargaining in nonstationary environments, including cases where preferences are changing over time. 9.1 An Axiomatic Bargaining Model Consider two agents, labelled j = 1 ; 2 , trying to come to an agreement over alternatives in some arbitrary set A . Each agent j has a von Neumann - Morgenstern utility function u j dened over A [ f D g , where D represents the outcome if they fail to reach an agreement. From these, we can construct the set of all utility pairs that result from some agreement, S = f [ u 1 ( a ) ; u 2 ( a )] R 2 : a 2 Ag ; as well as the pair d = ( d 1 ; d 2 ) , where d j = u j ( D ) is referred to as the disagreement point or threat point . 1 Following Nash (1950), we take the pair ( S ; d ) to dene a bargaining problem, and assume that S is compact and convex, that d 2 S , and that for some s 2 S we have s j > d j . We are interested in a bargaining solution , by which we mean a function f that species a unique outcome f ( S ;d ) 2 S for every bargaining problem ( S ;d ) . Rather than specifying an explicit model of the bargaining procedure, the idea behind the axiomatic approach is to impose properties that one wants a bargaining solution to satisfy, and then look for solutions with these properties. Nash species four axioms, which we simply state without comment (see Osborne and Rubinstein , e.g., for a discussion). A1. Invariance to equivalent utility representations: If we transform a bargaining problem ( S ;d ) into ( S ; d ) by taking s j = j s j + j and d j = j d j + j , where j > , then f j ( S ; d ) = j f j ( S ;d ) + j . A2. Symmetry: If the bargaining problem is symmetric, in the sense that d 1 = d 2 and ( s 1 ;s 2 ) 2 S () ( s 2 ; s 1 ) 2 S , then f 1 ( S ; d ) = f 2 ( S ;d ) . A3. Independence of irrelevant alternatives: If ( S ; d ) and ( S ; d ) are bar- gaining problems with S S , and f ( S ;d ) 2 S , then f ( S ; d ) = f ( S ; d ) . A4. Pareto eciency: If ( S ; d ) is a bargaining problem with s;s 2 S and s j > s j , j = 1 ; 2 , then f ( S ; d ) 6 = s . Nash shows that there exists a unique bargaining solution satisfying these axioms, and it takes the simple form f ( S ;d ) = arg max ( s 1 d 1 )( s 2 d 2 ) ; (9.1) where the maximization is over s 2 S , and is subject to the constraints s j d j , j = 1 ; 2 . The maximand on the right hand side of (9.1) is called the Nash product , and the solution f is called the Nash bargaining solution ....
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- Spring '02