# Implementation in Nash Equilibrium (II): Applications

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## Abstract

In the previous chapter we presented a general mechanism that implements in Nash equilibrium any social choice correspondence satisfying monotonicity and no veto power. A general criticism of the Nash equilibrium approach to the theory of implementation is that, in the case where a *correspondence* is implemented, this kind of equilibrium requires a good deal of coordination among agents in order to select those strategies corresponding to a particular equilibrium.^{1} This problem is well-known in game theory: when playing, say, *The Battle of the Sexes* agents may, by means of strategies that are part of different Nash equilibria of the game, achieve outcomes that are not Nash equilibria. In our case, if the mechanism is run by a real person, she can suggest what strategy agents have to use (truth-telling). This is of course *cheap talk*, but it makes slightly more palatable the assumption that agents know the particular Nash equilibrium that is going to be played.

## Keywords

Nash Equilibrium Market Game Strong Equilibrium Walrasian Equilibrium Feasible Allocation## Preview

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## References

- A criticism of integer games can be found in: M.O. Jackson (1992), ‘Implementation in Undominated Strategies: A Look at Bounded Mechanisms’,
*Review of Economic Studies*, vol. 59, pp. 757–75.CrossRefGoogle Scholar - On mixed strategies see A. Rubinstein (1991), ‘Comments on the Interpretation of Game Theory’,
*Econometrica*, 59, 4, pp. 909–24.CrossRefGoogle Scholar - Sections 5.2 and 5.3 rely heavily on L.C. Corchbn and S. Wilkie (1990), ‘Doubly Implementing the Ratio Correspondence by Means of a Natural Mechanism’, Mimeo, Bellcore and Universidad de AlicanteGoogle Scholar
- L.C. Corchbn and S. Wilkie (1989), ‘Implementation of the Walrasian Correspondence by Market Games’, Mimeo, Rochester, September, 1989. Revised November, 1994.Google Scholar
- The following papers present particular mechanisms for implementing the Lindahl correspondence with three or more agents: T. Groves and J. Ledyard (1977), ‘Optimal Allocation of Public Goods: A Solution to the “Free Rider” Problem’,
*Econometrica*, 45, pp. 783–809CrossRefGoogle Scholar - L. Hurwicz (1979a), ‘Outcome Functions Yielding Walrasian and Lindahl Allocations at Nash Equilibrium Points’,
*Review of Economic Studies*, vol. 46 no. 2, pp. 217–25CrossRefGoogle Scholar - M. Walker (1981), ‘A Simple Incentive Compatible Scheme for Attaining Lindahl Allocations’,
*Econometrica*, 49, pp. 65–73.CrossRefGoogle Scholar - The paper by Hurwicz also considered the implementation of the Walras correspondence. The case of two agents is studied in L. Hurwicz (1979b), ‘Balanced Outcome Functions Yielding Walrasian and Lindahl Allocations at Nash Equilibrium Points for Two or More Agents’ in J. Green and J.A. Scheinkman (eds),
*General Equilibrium Growth and Trade*(New York: Academic Press).Google Scholar - The issue of the so-called ‘completely feasible implementation’ of the Lindahl and Walras correspondences when the feasible set is unknown to the designer has been considered by L. Hurwicz, E. Maskin and A. Postlewaite (1995), ‘Feasible Nash Implementation of Social Choice Rules when the Designer Does not Know Endowments or Production Sets’, in J.O. Ledyard (ed.),
*The Economics of Informational Decentralization, Complexity, Efficiency and Stability*(Kluwer Academic Publishers).Google Scholar - Postlewaite and D. Wettstein (1989), ‘Continuous and Feasible Implementation’,
*Review of Economic Studies*, 56, pp. 603–11; and L. Hong, ‘Nash Implementation in Production Economies’,*Economic Theory*(forthcoming).Google Scholar - Implementation of the Walrasian correspondence by means of market games has been considered by: J.P. Benassy (1986), ‘On Competitive Market Mechanisms’,
*Econometrica*, 1986 no. 54, pp. 95–108CrossRefGoogle Scholar - P. Dubey (1982), ‘Price—Quantity Strategic Market Games’,
*Econometrica*, no. 50, pp. 111–26CrossRefGoogle Scholar - and L.-G. Svensson (1991), ‘Nash Implementation of Competitive Equilibria in a Model with Indivisible Goods’,
*Econometrica*no. 51, 3, pp. 869–77.Google Scholar - Schmeidler obtained double implementation with an abstract mechanism. See D. Schmeidler (1980), ‘Walrasian Analysis via Strategic Outcome Functions’,
*Econometrica*, 48, pp. 1585–93.CrossRefGoogle Scholar - In some cases it is possible to dispose of the inefficient Nash equilibrium by means of certain kind of trembles. See, for instance, M. Bagnoli and B. Lipman (1989), ‘Provision of Public Goods: Fully Implementing the Core through Private Contributions’,
*Review of Economic Studies*, 56, pp. 583–602.CrossRefGoogle Scholar - For implementation in Strong Equilibrium see H. Moulin and B. Peleg (1982), ‘Stability and Implementation of Effectivity Functions’,
*Journal of Mathematical Economics*, 10, pp. 115–145CrossRefGoogle Scholar - and B. Dutta and A. Sen (1991), ‘Implementation under Strong Equilibria: A Complete Characterization’,
*Journal of Mathematical Economics*, 20, pp. 49–68.CrossRefGoogle Scholar - The pioneer contributions to fair division are D. Foley (1967), ‘Resource allocation and the Public Sector’,
*Yale Economic Essays*, 7, pp. 45–98Google Scholar - S.C. Kolm (1972),
*Justice et Equité*(Paris: CNRS)Google Scholar - E. Pazner and D. Schmeidler (1978), ‘Egalitarian-equivalent Allocations: a New Concept of Economic Equity’,
*Quarterly Journal of Economics*, 92, pp. 671–87.CrossRefGoogle Scholar - On implementation of fair solutions see W. Thomson (1992), ‘Divide and Permute and the Implementation of Solutions to the Problem of fair Division’, Working Paper, University of Rochester, June (forthcoming in
*Games and Economic Behavior*)Google Scholar - W. Thomson (1993), ‘Monotonic Extensions’, Working Paper, University of Rochester, NovemberGoogle Scholar
- A. Sen (1987), ‘Approximate Implementation of Non-Dictatorial Social Choice Functions’, Mimeo, Princeton UniversityGoogle Scholar
- S.C. Suh (1993), ‘Doubly Implementing the Equitable and Efficient Solution’, Working Paper, University of Rochester, July. See also the references of Chapter 6.Google Scholar
- Implementation of matching rules is discussed in T. Kara, and T. Sönmez: ‘Nash Implementation of Matching Rules’
*Journal of Economic Theory*, (forthcoming) and T. Sönmez, ‘Implementation in Generalized Matching Problems’,*Journal of Mathematical Economics*(forthcoming).Google Scholar - Implementation with ‘nice’ mechanisms is the concern of the following papers B. Dutta, A. Sen and R. Vohra ‘Nash Implementation through Elementary Mechanisms in Economic Environments’.
*Economic Design*, 1995, pp. 173–204; T. Sjöström ‘Implementation by Demand Mechanisms’.*Economic Design*(forthcoming) and Saijo, T., Y. Tatamitani and T. Yomato: ‘Toward Natural Implementation’,*International Economic Review*(forthcoming).Google Scholar