Implementations of the Nash solution based on its Walrasian characterization

Abstract

The present paper provides three different support results for the Nash bargaining solution of n-person bargaining games. First, for any bargaining game there is defined a non-cooperative game in strategic form, whose unique Nash equilibrium induces a payoff vector that coincides with the Nash solution of the bargaining game. Next this game is modified in such a way that the unique Nash equilibrium that supports the Nash solution is even in dominant strategies. After that an n-stage game in extensive form is presented whose unique subgame perfect equilibrium supports the Nash solution of the bargaining game. Finally, the support results are shown to induce implementation results in the sense of mechanism theory.

Two support results for the Kalai–Smorodinsky solution in small object division markets

“…In effect, there remains a "max-min problem" for agent 1 as this is also incorporated in the definition of the KS solution. This kind of tradeoff was also observed by Trockel (2000) in a welfaristic context.…”

“…One way is to follow a purely welfaristic approach, which means that one only considers the possible allocations of utility provided in the bargaining situation and takes this set (together with the status quo point) as the primitives of the cooperative model. Along this line, Trockel (1999) discusses support results for a class of bargaining solutions including the KS solution (cf. also Haake (2000)).…”

“…However, Trockel (2002) shows that for solutions concepts from cooperative game theory, a support result as discussed above can be "transformed" into an implementation result in the spirit of implementation theory (see also Serrano (1997), Dagan & Serrano (1998) and Bergin & Duggan (1999)). This is the content of the so-called embedding lemma.…”

On Maskin monotonicity of solution based social choice rules

“…Howard (1992) argues that, due to a lack of Maskin monotonicity, the Nash bargaining solution fails to be Nash implementable. That a suitably defined Nash bargaining social choice rule is in fact Nash implementable has been demonstrated by von Damme (1986), Naeve (1999), and Trockel (2000Trockel ( , 2002b.…”

“…As we demonstrate in Section 5, the solution based choice correspondence ϕ ν does satisfy Maskin monotonicity. In fact, weak and full Nash implementation of the Nash bargaining solution based on social choice rules have been established in Trockel (2000Trockel ( , 2002b.…”