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A universal meta bargaining implementation of the
Nash solution
Walter Trockel
Institut fu¨r Mathematische Wirtschafts forschung, Universita¨t Bielefeld, Postfach 100131, 33501 Bielefeld, Germany (e-mail: wtrockel@wiwi.uni-bielefeld.de)
Received: 31 July 2000/Accepted: 19 March 2001
Abstract. This paper follows van Damme (1986) in presentinga meta bar-gaining approach that justifies the Nash barbar-gaining solution. But in contrast to van Damme’s procedure our meta bargaining game is universal in the sense that allbargaining solutions are allowed as strategic choices in the meta bar-gaining game. Also our result holds true for any numbernof players.
1 Introduction
The concept of a meta bargaining game had been introduced by van Damme (1986). It’s purpose is it to clarify what could be a reasonable solution of a cooperative two person bargaining game if players have di¤erent ideas about the ideal solution concept to be applied. The meta bargaining game is a non-cooperative game in strategic form generated by a given non-cooperative bargain-inggame. It is characterized by the fact that the players’ strategy sets are sets of bargaining solutions. The Nash equilibria of the meta bargaining game determine distinguished solutions of the cooperative bargaining game.
Other work on meta bargaining games induced by van Damme’s article are Anbarci and Yi (1992), Marco et al. (1995) and Naeve-Steinweg(1999). The support of the Nash solution of a two person bargaining game by all Nash equilibria of an induced meta bargaining game due to van Damme and its generalization due to Naeve-Steinweg may be seen as contributions to the Nash program which is concerned with non-cooperative foundations of co-operative solutions.
The idea to support (implement, realize, justify) axiomatic solutions of co-operative games by Nash equilibria of non-coco-operative games goes back to the work of John Nash (1951, 1953). Virtual (or asymptotic) support results for the Nash solution have been provided by Nash (1953) in his smoothed demand game and by Binmore et al. (1986). Rigorous versions of Nash’s somewhat vague treatment have been given later on by Binmore (1987), van Damme (1991) and by Osborne and Rubinstein (1990). Direct implementations are contained in Howard (1992) and Trockel (2000).
The number of equilibria is a delicate feature in implementation theory and mechanism design. For an extensive discussion see Trockel (2002b). Uniqueness of equilibria, although optimal from the point of view of coordinating strate-gic equilibrium play, is in conflict with full implementation of non-singleton-valued social choice rules. In our meta bargaining game despite multiple equilibria we have unique equilibrium payo¤s. It is the latter that matters in our context as the theoretical justification of a solution as an equilibrium out-come of the meta bargaining game is looked for rather than a game which is really played and where the multiplicity of equilibria would be a serious draw-back requiringsome coordination!
2 The meta bargaining game
The meta bargaining game we are going to define combines van Damme’s (1986) idea of usingsets of solutions as strategy sets with the game in Propo-sition 1 of Trockel (2000), which established support to the Nash solution by its unique Nash equilibrium. This combination of two di¤erent approaches with distinct interpretations and intentions results in a new interestingpropo-sition. This extends van Damme’s meta bargaining approach to cover all
solutions as strategies. At the same time it providesfullimplementation of the Nash solution, where Trockel (2000) only yields weak Nash implementation. First we formalize the general framework. Next we recall Trockel’s (2000) game. Then we construct our corresponding meta bargaining game.
For notational convenience we restrict ourselves to the case of two-person bargaining games. The same reasoning, however, can be used for the case of general n-person bargaining games as the method by which the meta bar-gaining game is deduced from the game in Trockel (2000), is independent of the specificn, and the proof there holds true for anynAN.
A two-person bargaining game is a non empty, compact convex and com-prehensive [that means: ðSR2
þÞXRþ2 HS] subsetS ofRþ2 which is
inter-preted as the set of payo¤ vectors which the players are able to obtain by cooperation. In case of non-cooperation the players receive their respective coordinate of the threat pointd AS. Like van Damme we restrict ourselves to the case of d ¼0AR2. Moreover we normalize Ssuch that projiðSÞ ¼Si¼
For any bargaining gameSthe setqS:¼ fxASj ðfxg þR2þÞXS¼ fxgg is the (strong) Pareto e‰cient boundary ofS.
This is exactly the framework used by van Damme (1986). Although our analysis can be performed for this framework with some additional notational and terminological e¤ort we restrict this framework for convenience in the fol-lowingway.
We assume that for any xAqS there exists a unique normal vector pðxÞAR2
þþsuch that the inner product pðxÞ x¼1. We call this vector pðxÞ
the e‰ciency price system associated with x. The set of all such bargaining
games is denoted by S. Now any mapping f :S!R2:S7!fðSÞAS is a bargaining solution.
Two bargaining solutions f;gare calledðS;iÞ-equivalent for givenSAS,
i¼1;2 if fiðSÞ ¼giðSÞ. TheðS;iÞ-equivalence class of a bargaining solution
f is denoted½fS
;i,i¼1;2.
We denote byFthe set of all bargaining solutions. Notice, that van Damme (1986), as well as Naeve-Steinweg(1997), denotes by Fa much smaller sub-set of the sub-set of all bargaining solutions, all elements of which satisfy certain axioms.
The Nash solution is denoted fN and is defined by ffNðSÞg
:¼
Now consider for any SAS the followingtwo-person game in strategic
form
The game GðSÞ is a modification of Nash’s (1953) simple demand game. It generates the same payo¤s as Nash’s game for consistent strategy choices
xAS but dinstinguishes payo¤s of the two players in a more subtle way in case of inconsistent strategy choices. For a thorough discussion and interpre-tation see Trockel (2000).
By Proposition 1 in Trockel (2000) for any SAS the game GðSÞ has a
unique Nash equilibrium, which coincides with fNðSÞ. Notice that fNðSÞ
coincides with the unique equilibrium strategy profile as well as with the re-sultingunique payo¤ vector.
Next we derive a specific meta bargaining game GGðSÞ from the game
GðSÞ. For this purpose we defineUG
UiGðf;g;SÞ ¼piððf1ðSÞ;g2ðSÞÞ;SÞ; i¼1;2:
GGðSÞis then defined asGGðSÞ:¼ ðF;F;U G
1 ð;;SÞ;U G
2 ð;;SÞÞ.
3 Equilibria of the meta bargaining game
As for anySASthe gameGðSÞhas as its unique Nash equilibrium the point fNðSÞ ¼ ðf1NðSÞ;f
N
2 ðSÞÞthe Nash equilibria of GGðSÞare given by the set ½fN
S;1 ½f
N
S;2. All these equilibria result in the same payo¤ vector fNðSÞ.
We collect this insight in the following
Proposition.For any SASthe pairðf;gÞAFF is a Nash equilibrium if and
only if f1ðSÞ ¼fN
1 ðSÞand g2ðSÞ ¼f2NðSÞ.The unique equilibrium payo¤ vector is fNðSÞ.
Although any Shas many equilibria it is onlyðfN ;f
NÞ, which is an
equilib-rium for eachSAS. The situation is exactly as in van Damme (1986) apart
from the fact that we allow forallbargaining solutions to be strategies.
4 Relation to mechanism theory
The support result provided by our Proposition can be seen as belonging to the realm of the Nash program. The question then is how it is related to imple-mentation in the sense of mechanism theory.
Mechanism theory is concerned with game forms (¼mechanisms) rather than with games. That means that support results have to be established simultaneously for a whole class of games all of which are induced from the same game form by di¤erent populations of players.
Formally one has to find a suitable factorization of the players’ payo¤ functions into an outcome function from the strategy space to an outcome space and the players’ utility functions on the outcome space. But to find a suitable outcome space is only one problem. A second one is it to represent the solution that is to be implemented by a suitably defined social choice rule.
The relation of the Nash program to mechanism theory has been thor-oughly discussed in Trockel (2002a) and (2002b). The first of these papers contains the ‘‘embeddingprinciple’’, a method which allows it to transform any support result of the Nash program into a proper implementation result in the sense of mechanism theory. See also Dagan and Serrano (1998), Serrano (1997) and Bergin and Duggan (1999).
Applyingthat embeddingprinciple to our present Proposition we get the followingmechanism theoretic implementation. The same method would apply to van Damme’s result thereby justifyinghis use of the term ‘‘mechanism’’.
A:¼ ðR2ÞSis theoutcome space. That means that any bargaining solution considered as a behavioral norm constitutes a social state.
Let
and
evS :A!R2:f 7!evSðfÞ ¼fðSÞ:
We define anoutcome function hby
h:FF !A:ðf;gÞ 7! ðproj1evð;fÞ;proj2evð;gÞÞ:
Next we derive from the gameGðSÞa specific meta bargaining gameGGðSÞ.
For this purpose we define UG
i ð;;SÞ, i¼1;2 by U G
i ðf;g;SÞ:¼
pS
i hðf;gÞðSÞ
For the purpose of Nash implementation we still have to represent the Nash bargaining solution by a suitable social choice rule. This has to be a correspondence defined on a class of profiles of preferences or utility functions associatingwith any such profile a subset of the outcome space A. This subset is interpreted as a set of socially desirable states given the prevailing prefer-ence profile.
By identifyinganySASwith the mapevSwhich associates with any social
state f AAthe utility vectorevSðfÞ ¼ ðf1ðSÞ;f2ðSÞÞof the two players in the bargaining gameSderived from the solution f we can define ourNash social
choice ruleby
½fNðÞ:S)A:S¼evS) ½fNS:¼ ½fNS;1 ½fNS;2
It is then an immediate consequence of the proposition in Trockel (2002a) that
½fN
This chain of equalities still holds true if the two first arguments of UiG are replaced, respectively, by arbitrary members of½fNS;1 and½fNS;2.
5 Concluding remark
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