Integrating the Nash Program into Mechanism Theory Walter ... - Core

Integrating the Nash Program into Mechanism Theory

Walter Trockel∗ Institute of Mathematical Economics (IMW), Bielefeld University

Abstract The present paper provides a method by which the Nash Program may be embedded into mechanism theory. It is shown that any result stating the support of any solution of a cooperative game in coalitional form by a Nash equilibrium of some suitable game in strategic form can be used to derive the mechanism theoretic Nashimplementation of that solution.

∗

I am grateful for helpful comments from participants at seminars at UCLA, UC Berkeley, CalTech, Duke University, UC Irvine, USC, VirginiaTech, Washington University, Brown University, Harvard/MIT, Bilkent and Bogacizi Universities and, in particular, to Anke Gerber. I also profited very much by a referee’s and, in particular, an associate editor’s comments. The hospitality of the University of California, Los Angeles, and of the University of Arizona at Tucson as well as financial support through a sabbatical grant of the Volkswagen Foundation are gratefully acknowledged.

1

Introduction

The present paper connects two strands of game theory. One of them, that can be traced back to von Neumann and Morgenstern (1944) is concerned with the interrelation of cooperative and non-cooperative game theory and the solutions they offer. The other one, mechanism theory, arose from two seminal papers of Leonid Hurwicz (1969, 1973) but has his roots in the famous debate on centrally planned versus free market economies envolving economists like Lange, Lerner, Taylor and von Hayek. Mechanism theory is based on the separation of a non-cooperative game into the common rules and the individual preferences. This feature allows it to design game rules simultaneously for a whole class of potential player populations or, put in a different way, of different type profiles for a fixed set of players. Once socially desired outcomes can be associated with solutions of cooperative games, any insights into the interrelation between cooperative and non-cooperative games and their solutions are potentially valuable for a designer or planner. The first strand mentioned above has been called Nash Program (cf. Binmore 1987, 1997) because John Nash in his seminal papers (1951, 1953) had explicitly emphasized the importance to analyse the interrelation between cooperative and non-cooperative approaches and, in fact, had started himself by his analysis of the Nash bargaining solution. Bergin and Duggan (1999) call this research agenda “non-cooperative foundation”, although the quotations of Nash below indicate a more symmetric role of the parts of classical game theory. The main result of the present article, the proposition in section 4, is a structure theorem, that clarifies the exact relation between the two strands of game theory mentioned above, which I therefore call Embedding Principle. The perception of the relation between the Nash Program and mechanism theory as reflected in the literature is quite controversial (cf. Serrano (1997)). It is therefore important to begin the analysis as well as the verbal discussion only after the two original passages have been quoted, which build the historical basis for the Nash Program. In Nash (1951) one finds: “A less obvious type of application (of non-cooperative games) is to the study of cooperative games. By a cooperative game we mean a situation involving a set of players, pure strategies, and payoffs as usual; but with the assumption that the players can and will collaborate as they do in the von Neumann and Morgenstern theory. This means the players may communicate and form coalitions which will be enforced by an umpire. It is unnecessarily restrictive, however, to assume any transferability or even comparability of the pay-offs [which should be in utility units] to different players. Any desired transferability can be put into the game itself instead of assuming it possible in the extra-game 2

collaboration. The writer has developed a“dynamical” approach to the study of cooperative games based upon reduction to non-cooperative form. One proceeds by constructing a model of the preplay negotiation so that the steps of negotiation become moves in a larger non-cooperative game [which will have an infinity of pure strategies] describing the total situation. This larger game is then treated in terms of the theory of this paper [extended to infinite games] and if values are obtained they are taken as the values of the cooperative game. Thus the problem of analyzing a cooperative game becomes the problem of obtaining a suitable, and convincing, non-cooperative model for the negotiation. The writer has, by such a treatment, obtained values for all finite two-person cooperative games, and some special n-person games”. The work Nash is referring to in his last sentence of the above passage is Nash (1953) where he writes: “We give two independent derivations of our solution of the two-person cooperative game. In the first, the cooperative game is reduced to a non-cooperative game. To do this, one makes the players’ steps of negotiation in the cooperative game become moves in the noncooperative model. Of course, one cannot represent all possible bargaining devices as moves in the non-cooperative game. The negotiation process must be formalized and restricted, but in such a way that each participant is still able to utilize all the essential strengths of his position. The second approach is by the axiomatic method. One states as axioms several properties that it would seem natural for the solution to have and then one discovers that the axioms actually determine the solution uniquely. The two approaches to the problem, via the negotiation model or via the axioms, are complementary; each helps to justify and clarify the other.” The Nash Program tries to link two different ways of solving games. The first one is non-cooperative. No agreements on outcomes are enforceable. Hence players are totally dependent on their own strategic actions. They try to find out what is best given the other players are rational and do the same. In this context the Nash equilibrium describes a stable strategy profile where nobody would have an interest to unilaterally deviate. Nevertheless there is an implicit institutional context. The strategy sets define implicitly what choices are not allowed, namely those outside the strategy sets. The payoff functions reflect which strategies in the interplay with others’ strategies are better or worse. It is not said explicitly who grants payoffs and how the physical process of paying them out is organized. But there is some juridical context with some enforcement power taken for granted. There is no interpersonal comparison of payoffs involved in the determination of good strategies. Each player only compares his different strategies contingent on the other players’ different strategy choices. As applications in oligopoly show, institutional 3

restrictions of social or economic scenarios are mapped into strategy sets and payoff functions, thereby lending them an institutional interpretation. Yet, totally different scenarios may considerably be modelled by the same non-cooperative game, say in strategic form. This demonstrates clearly the purely payoff based evaluation of games. Payoffs usually are interpreted as reflecting monetary or utility payments. Associated physical states or allocations orrur only in applications and may be different in distinct applications of the same game. The second way to solve a game is the cooperative one via axioms as first advocated by Nash (1953). Again the legal framework is only implicit. Yet, now not only obedience to the rules is assumed to be enforceable but even contracts are. Mutual gains are in reach now as it becomes possible by signing a contract to commit oneself to certain behavior. In this context it is the specific payoff configuration which is of interest rather than the strategy profile that would generate it. In this framework it is reasonable, therefore, to neglect the strategic options and concentrate on the feasible payoff configurations or utility allocations on which the players possibly could agree by signing a contract. Again the formal model does not specify the process by which physical execution of a contract is performed. Again it is the payoff space rather than some underlying social scenario on which the interest rests except in applications of game theory. In contrast to the non-cooperative approach now players are interested in what other players receive. Although utilities or payoff units for different players are in general not considered comparable typically there are tradeoffs that count. The axioms that are fundamental in this context reflect ideas of fairness, equity, justness that do not play a role in the non-cooperative model. But a process of negotiation with the goal to find an agreement makes it necessary for each player to somehow judge the coplayers’ payoffs. If very different underlying models lead to the same cooperative game in coalitional form it is only the solution in terms of payoff vectors that is relevant. And this determines in any application what underlying social or physical state is distinguished. It becomes irrelevant in the axiomatic cooperative approach which are the institutional details. Important are only the feasible utility allocations. The Nash Program is purely game theoretic as it neither relies nor depends on any underlying social or economic model. Using the terminology of social choice theory the Nash Program ”operates in a purely welfaristic framework”. There is not any focus on decentralization in the context of the Nash Program simply because there is no social planner involved in the analysis. Therefore the question as to such an actor’s information about the players’ characteristics is meaningless. Also it is taken for granted in this context that there is common knowledge among the players about the data of the games, both the cooperative one and non-cooperative one. The Nash Program does not specify a formal model which would allow to derive information about one of the games and its solution resp. equilibria from the other one. Nevertheless any contribution to the Nash Program is considered to provide insights into some aspects of the cooperative game and its solution,

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that are not obvious from the underlying axioms, or of the non-cooperative game and its equilibria, which are not directly deducible from the strategic interaction. Any attempt to look at the Nash Program as a specific part of implementation theory must rely on an alteration of the framework discussed above. A designer or social planner has to be added to the model as well as a set of social states in which he is interested. The solution of the cooperative game is to be interpreted as the set of socially desirable states. The designer’s goal is to invent rules for games which would force any population of players to establish one of the desirable social states by playing an equilibrium of a game according to those rules. So while an additional agent, the designer, enters the scene when the analysis moves from standard game theory to mechanism theory, the players leave it. They are replaced by a class of potential player populations, for each of whom the designed mechanism is intended to be effective. The important question now arises, namely what exactly the designer knows about the player populations, the actions of the players and what he is able to enforce. The very idea of implementing a socially desired state via strategic interaction of the members of a society under certain rules implicitly restricts the power and the knowledge of the designer, who otherwise could act as a dictator and establish the desired state by fiat. Therefore the question of what a reasonable mechanism could be should be posed only for well specified information and enforcement contexts of the designer. The attempt to use results from the Nash Program for a planner’s design problem requires that the common knowledge of players about their mutual characteristics is taken for granted for any potential feasible population. This clearly excludes the framework of asymmetric information among players. Also one should only consider a scenario where the planner is unable to enforce the desired state by fiat and where he, because of the design of the mechanism for a whole class of player populations, is also unable to know the players’ preference profiles. It is assumed implicitly that the players’ participation in the game can be taken for granted. Given this fact, we may assume that an equilibrium will be established, at least when it is unique, because of the idea that it is selfenforcing. This whole aspect of an appropriate enforcement is quite delicate but is excluded from the formal modelling. For an inspiring discussion of this point we refer to Hurwicz (1994, pp. 11-12). As soon as refinements of the Nash equilibrium are considered there arises an additional problem, namely that of the agreement among all players in all potential populations on the same refinement. Apart from the different sets of acting agents a crucial difference between the purely game theoretic framework of the Nash Program and the framework of implementation theory is the explicit occurrence of an outcome space in the latter. Formally, the games, including specifications of the players’ preferences, are replaced by game forms containing all the information about the game except that information concerning players’ preferences 5

or payoff functions. A non-trivial factorization of players’ payoff functions into an outcome function, common to all players in all populations, and individual utility functions defined on the outcome space enables us to distinguish between observability of the result of players’ actions as opposed to their resulting payoffs. The question as to what extent the possibility of observation is crucial for enforcement depends on the degree of selfenforcement power of the considered equilibrium concept. For instance, a dominant strategy equilibrium could be considered a perfect substitute for observability whenever the players’ participation can be taken for granted. In this case the rationality that is assumed for all players guarantees the desired result. A similar claim could be made, even if with weaker justification, for a unique Nash equilibrium. The very fact that the presence of an outcome space is an additional ingredient in mechanism theory as compared to the Nash Program indicates that these two can hardly be considered ”equivalent”, as claimed, for instance, in Dagan and Serrano (1997, Abstract). Also the objectives of both theories are quite different. While the explicit judgement of the adequateness of a certain game form in view of the enforcement and informational restrictions of the planner has no meaning in the Nash Program a certain result in the Nash Program might provide the theorist with some valuable insight but could still be useless for a planner in a specific context. Also a formal result in the Nash Program could possibly lack any informational value for the game theorist and simultaneously, for a planner amount to the enforcement of the desired state by fiat. Let us take, for example, the following modification of Nash’s simple demand game: The two players choose real numbers, which, when suitably coordinated, result in a feasible bargaining outcome. If both choose their respective coordinate of the Nash solution they receive these as their payoffs. Otherwise both receive zero. This game obviously supports the Nash solution of the bargaining problem by the dominant strategy equilibrium of the game. However,it provides no valuable information about the specifics of the Nash solution, because any other solution can be supported in an analogous way. For the planner in the derived implementation theoretic context this is tantamount to his direct forcing any population of any bargaining problem to agree on the Nash solution. Yet, though the objectives of both theories are different, it appears natural to exploit results yielded in the Nash Program for the purpose of mechanism design, or as Serrano (1997) has put it, ”to adapt the mechanisms in the Nash Program as standard game forms of the theory of implementation”. In fact, in his paper Serrano provides such an adaptation. But for this purpose he uses the explicit introduction of a specific underlying production economy. In his framework Serrano then establishes several impossibility results for the implementability in Nash equilibrium of cooperative solution concepts, among them the Shapley value as well as the Kalai-Smorodinsky and the Nash bargaining solutions. The basis for these results is the lack of Maskin-monotonicity of these solutions interpreted as social choice rules. In a similar way Dagan and Serrano (1998) add a set of physical outcomes to the structure

6

of a cooperative game in order to adapt the Nash Program into the theory of implementation. From their analysis in a specific context they derive the general conclusion that ”major solution concepts in coalitional games (e.g. the Nash bargaining solution, the NTU-Shapley value) can be derived strategically only by considering the possibility of random outcomes: either chance moves, mixed strategies or pure strategy equilibrium refinements based on trembles must appear in the analysis”. A critical discussion and analysis of the possibility to Nash-implement the Nash bargaining solution is contained in Trockel (2000). In the present paper I propose a general procedure of embedding the Nash program into the theory of implementation. That procedure enables us in our framework to transform any support result from the Nash Program into an implementation result in mechanism theory. In particular, our method enables us to implement the Nash and the Kalai-Smorodinsky solutions in dominant strategy equilibria (cf. Trockel (2000) and Haake (2000), respectively).

2

Nash-Implementation

The integration of the Nash Program into implementation theory requires two main steps. First, we have to derive a social choice rule from the solution concept under consideration. Any solution in the welfaristic context is represented for any game in its domain by the payoffs to the players. In contrast, a choice rule maps preference profiles on the outcome space to outcomes. Consequently, the first step involves the choice of an appropriate outcome space. Secondly, we have to relate the non-cooperative game theoretic support of that solution to the implementation of the derived social choice rule. This step involves the factorization of certain functions, a formal method quite prominent in mechanism theory. The implementation of a social choice rule by equilibria of games relies on the factorization of payoff functions, i.e. their representation as compositions of utility functions with an outcome function. The famous Revelation Principle is based on the factorization of a social choice function into the composition of a map from the space of preference profiles into the strategy space with the outcome function of the given mechanism. The factorization we shall employ is somewhat more complex. We need simultaneous factorizations of two different functions sharing one common factor, namely the profile of utility functions on the outcome space. We begin by providing the basic notions from implementation theory and game theory. We shall deal with games and game forms for n > 1 players. So N = {1, ..., n} is the set of players 0 positions. Σi 6= ∅ denotes the strategy set for any player in position i ∈ N . The joint strategy space Σ1 × ... × Σn is denoted Σ. A player is characterized by his payoff function πi : Σ → R, where the subscript i indicates the player’s position i ∈ N . 7

The tuple Γ := (Σ1 , ..., Σn ; π1 , ..., πn ) is an n−person game in strategic form. Removing the population of players while keeping the rules of the game is achieved by replacing the payoff functions by an outcome function. A mechanism (or game form) is defined by a tuple (Σ1 , ..., Σn ; h), where h : Σ → A is an outcome function associating with any strategy profile in Σ a state in the outcome space A 6= ∅. A game (Σ1 , ..., Σn ; π1 , ..., πn ) or a game form (Σ1 , ..., Σn ; h) are also denoted (Σ; π) or (Σ; h), respectively. Now any population of n individuals with preferences on A, representable by utility functions ui : A→R, i = 1, ..., n, transforms the game form (Σ1 , ..., Σn ; h) into the game (Σ1 , ..., Σn ; π1 , ..., πn ) via the compositions πi := ui ◦ h, i = 1, ..., n. We abbreviate (u1 ◦ h, ..., un ◦ h) by u ◦ h. Depending on the population specified by the utility functions on the outcome space A one may distinguish certain states as desirable. This idea leads to a social choice rule, or in Hurwicz’s (1994) terminology, a desirability correspondence, F, which associates with any feasible profile u of utility functions on A a subset of A. The determination of which profiles of utility functions on A are considered feasible is crucial for the derivation of implementation results. The implementation of a social choice rule in equilibrium relates for any feasible u = (u1 , ..., un ) the set F(u) of desirable outcomes to the set N E(Σ; u ◦ h) of equilibria of the game (Σ; u ◦ h). Here N E is the set of Nash equilibria. We say that the mechanism (Σ; h) weakly N ash − implements the social choice rule F on the class U ⊆ (RN )A of (feasible) utility profiles, if for all u ∈ U : N E(Σ; u ◦ h) 6= ∅ and h(N E(Σ; u ◦ h)) ⊆ F(u). Weak N ash-implementation becomes full N ash-implementation if h(N E(Σ; u ◦ h)) = F(u) for all u ∈ U . A large part of the literature uses the concept of full implementation requiring equality rather than inclusion. It is full implementation for which Maskin (1999) gave a complete characterization via the properties of Maskin-monotonicity and no veto power. A careful discussion of the pros and cons of both notions of implementation can be found in Thomson (1996). Jackson (2001) stresses the fact that our (weak) implementation of a social choice rule implies full implementation of some subcorrespondence. We favorize the (weak) implementation, also used by Hurwicz (1994), for the following reasons: First, (weak) Nash-implementability does not require Maskin monotonicity. However, the second reason is in fact more important. As long as the social choice rule perfectly describes desirability there is no reason to discriminate between different socially desirable states. Each one is an equally good representative of social desirability. Even if each of the desirable states can be realized by some equilibrium of a game, at most one of these equilibria will be played and thus only one of the desirable states is realized. So performance of the social planners’ goal is independent of whether he designs a mechanism that weakly or fully Nash-implements the social choice rule. 8

In the ideal case of implementation by a unique Nash equilibrium the only case without a remaining coordination problem for the planner and the players, only weak Nashimplementation is possible. The only exception is a framework where the social choice rule is a social choice function. But then weak and full implementation coincide anyway. Next, we need the concept of a (cooperative) game in characteristic or coalitional form. Such a game is a correspondence V that associates with any non-empty subset S of N a subset V (S) ⊂ RS . These sets V (S), S ⊆ N , are interpreted as the sets of feasible payoffs of players playing in positions i ∈ S when acting or cooperating in coalition S. Denote by G C a set of n-person coalitional games. A solution on G C is a correspondence L associating with any coalitional game V ∈ G C a non-empty subset L(V ) of RN , such that for any x ∈ L(V ) there is a partition {T1x , ..., Tmx x } of N such that ∀ jx ∈ {1, ..., mx }: (xi )i∈Tjxx ∈ V (Tjxx ). This formula expresses the feasibility of the utility allocation x for the game V . We denote the set of feasible utility allocations for V by F(V ). For any solution L on G C we denote by SL the set of selections of L, i.e. of functions fL : G C −→ RN such that ∀V ∈ G C : fL (V ) ∈ L(V ). We do not distinguish between a singleton-valued solution and its unique selection. Notice, that then any selection of a solution is itself a solution. Let Λ denote the set of singleton-valued solutions l on G C .

3

Integrating the Nash Program into Mechanism Theory

The literature does not provide much insight into the exact relation between the Nash Program and mechanism theory. In Serrano (1997) one finds the statement: “The Nash Program and the abstract theory of implementation are often regarded as unrelated research agendas”. And Bergin and Duggan (1999) write: “Nevertheless, because the implementation-theoretic and traditional approaches both involve the construction of games or game forms whose equilibria have specific features, considerable confusion surrounds the relationship between them”. In fact there are instances in the literature where the term “implementation” is used in a framework of non-cooperative games where the mechanism theoretic aspect is not addressed at all. Sometimes games forms are simply confused with games. Starting from this situation Serrano (1997) attempted “to clarify the role of the mechanisms used in the Nash Program for cooperative games”. Notice, that even this statement contributes to the confusion by using “mechanisms”, a technical synonym for “game form” in the descrip9

tion of non-cooperative foundations that, prior to and without the intended clarification of the relation between the two agendas, cannot justifiably be termed that way. An extension of Serrano’s approach is contained in Dagan and Serrano (1998), where in contrast to traditional terminology games in characteristic function form are distinguished from games in coalitional form. The latter are induced by the former ones via adding outcome functions admitting it to define solutions as mappings to outcomes rather than to payoff vectors. These general outcomes extend Serrano’s model, where characteristic forms are supplemented by physical allocations resulting from some production economy. The most general model of non-cooperative foundation based on the explicit modelling of physical environments is due to Bergin and Duggan (1999). Also here cooperative solutions are alternatively defined as mappings resulting in outcomes rather than in payoff vectors. In contrast in the present paper the outcome space needed for an implementation is derived endogeneously from the data of the classes of games considered in the traditional non-cooperative foundation of an axiomatic cooperative solution. An application to the Nash bargaining solution is contained in Trockel (2000). Before we can possibly relate the Nash Program to mechanism theory it is important to clearly see the differences, the formal ones as well as those in intention and interpretation. Let me recall the latter ones first. The Nash Program relates two alternative ways to model how rational players with partially conflicting goals interact do determine their payoffs. One is a strategic model in which due to lack of commitment power every player non-cooperatively acts on its own. The other one is a coalitional form where for each coalition the feasible payoff vectors are described and axiomatically determined payoff vectors can be contracted on and enforced. In both approaches the acting persons are players. There is no explicit notion of a society or a social planner or designer. In mechanism theory, in contrast, the only acting person is a designer or social planner. He uses the body of game theory to design general rules forcing any potential population of players from a given pool to realize social states he declares “desirable” for that population by playing equilibria according to those rules. The players are objects of thought of the designer. So the designer, due to lack of information, enforcement power, and monitoring options, tries to decentralize social decisions uniformly for all feasible populations, trusting to the self-enforcing power of equilibrium. On the formal level, apart from the players versus designer difference, there are two other crucial differences. One lies on the cooperative side. Following social choice theory mechanism design is based on social choice rules, and therefore, is interested in social states rather than in payoff vectors. Clearly, considering payoff vectors as social states provides one specific degenerate example. The other difference is on the non-cooperative side. To give a non-cooperative procedure for determining social states one has to replace the payoff functions in a game in strategic form by an outcome function associating with any strategy profile some social state. 10

The link between this outcome based approach and the payoff based Nash Program is provided by any population of individuals via their utility functions on the outcome space. These utility functions composed with the outcome function create payoff functions, thereby completing the game form to a game and making the individuals of that population players of this game. Our next goal is it to illustrate the Nash Program, and the problem of Nash-implementation by some diagrams. They are modifications of diagrams used by Bergin and Duggan (1999). Furthermore we shall illustrate diagrammatically the difficulties to integrate both diagrams into one and even make this larger one commute. But beforehand we need some terminology and notation. Let G C and G N C be sets of cooperative NTU-games in coalitional (or characteristic) form and of non-cooperative games in strategic form, respectively. As before we denote the product of players’ strategy sets by Σ and assume that they are the same for all considered non-cooperative games. We comment on the restrictiveness of this assumption in Section 4. Let A be some non-empty set, interpreted as an outcome space and h : Σ −→ A an outcome function. U denotes some set of profiles u of utility functions defined on A, and L and L denote a cooperative solution for G C and a social choice rule on A, respectively. The use of the symbols “L” and “L” indicates that we finally want the social choice rule L to suitably represent the solution L. % (

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Figure 1 very clearly shows the differences between the Nash Program and implementation theory, despite their obvious structural similarities. A link of the two agendas would require the definition of L as a suitable representative of L. This in turn requires the specification of a suitable outcome space A. Last not least the relation between G C and U needs clarification. Figure 2 is a slight modification of Figure 3 in Bergin and Duggan (1999), that represents both situations, the one where one starts with profiles of utility functions and seeks the “right” induced game and the other one, where for a given coalitional form game one looks for profiles of utility functions generating this game. Bergin and Duggan term the two problems “effectivity” and “supportability”. Clearly, starting with a result from the Nash Program and trying to embed this into implementation theory requires to solve the supportability problem. In a situation where both sets G C and U may be identified the supportability and the effectivity problem are solved simultaneously. The whole problem of embedding the Nash Program into mechanism theory is represented by Figures 3 and 4. The inner boldface parts of these diagrams represent the implementation problem, while the outer parts represent the Nash Program. The diagrams have to be interpreted as follows: 576

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In mappings between sets in Figure 3 we use always the arrow “−→” and do not distinguish between (point-valued) functions and (set-valued) correspondences. In the corresponding diagram of Figure 4 where elements are mapped to elements or subsets we use “−→” again. Here the use of “∈” versus “⊆” indicates whether a point-valued mapping or a correspondence is considered. Notice also that in Figures 3 and 4 the question marks and the symbols in boxes like A, L, U and h indicate the explananda, i.e. the notions that have to be explained. The embedding of the Nash Program into mechanism theory is only possible if these explananda can be consistently defined such that the diagrams commute. 12

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∃i ∈ N : ui V (l) 6= ui V (l∗ ). Pareto-efficiency of l∗ excludes that for all i ∈ N : 0 0 ui V (l) ≥ ui V (l∗ ) with strict inequality for at least one i ∈ N . Hence, there must be 0 0 0 some j ∈ N with uj V (l∗ ) = uj V (l0 ) > uj V (l), which implies Maskin-monotonicity. 

17

6

Concluding Remarks

The present paper demonstrates a possible way of embedding the Nash Program into implementation theory. Thereby it provides the basis for some implementation results [cf. Trockel (1999, 2000, 2001), Haake (2000) and Naeve (1999)] which contrast several impossibility results in the literature. This paper does not claim, however, that looking at the Nash Program as a part of mechanism theory is particularly natural. As argued in Section 1, the goals of the Nash Program and of implementation theory are very different. Also, the informational context of the Nash Program relates only to decentralization aspects rather than to information eliciting aspects of implementation theory. This paper shows, however, that even under the heroic assumption of a purely welfaristic framework for game theoretic analysis we are not automatically condemned to deprive ourselves of potential applications of mechanism theory. It may also be questioned whether the outcome space and the mechanism employed for our above embedding lemma are very reasonable from a practical point of view. Such considerations, however, lead us immediately back to the question to what extent the presently established modelling of implementation via game forms is an adequate one, a question that led Hurwicz (1994) to suggest “genuine implementation”. Again, to answer that question we would have to deal with the problem of what “reasonable” mechanisms are. Our understanding of this problem is still in the beginning [cf. Jackson (1992), Jackson, Palfrey and Srivastava (1994), Jackson and Palfrey (2001)]. At least, our method is not dependent on integer or modulo games as it applies to every game that supports a certain solution.

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