# Implementation in Nash Equilibrium (I): General Results

Chapter

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## Abstract

In this chapter we will study the implementation of social choice correspondences by means of Nash equilibria. As you will remember from the previous chapter, the motivation for studying this type of implementation is to obtain positive results by relaxing the concept of equilibrium: instead of requiring the equilibrium strategy for each agent to be a good response to any possible strategy of the other players, the Nash equilibrium requires it to be a good reply for those who constitute a good reply of the other players (see Definition 3, Chapter 2).

## Keywords

Utility Function Nash Equilibrium Social Choice Optimal Decision Social Choice Function
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## References

- The following articles offer a general view of implementation in Nash equilibrium: A. Postlewaite (1985), ‘Implementation via Nash Equilibria in Economic Environments’, Chap. 7 in L. Hurwicz, D. Schmeidler and H. Sonnenschein (eds),
*Social Goals and Social Organization*(Cambridge University Press)Google Scholar - J. Moore (1992), ‘Implementation, Contracts and Renegotiation in Environments with Complete Information’ in J.J. Laffont (ed.),
*Advances in Economic Theory*(vol. I), VI World Congress of the Econometric Society (Cambridge University Press). The latter paper should be consulted for matters concerned with two-person implementation, renegotiation and implementation of social choice*functions*(see also exercises 24 and 27).Google Scholar - The characterization of social choice correspondences which are implementable in Nash equilibria was first studied in E. Maskin (1977), ‘Nash Equilibrium and Welfare Optimality’, mimeo, MIT.Google Scholar
- The following papers present proofs that the no veto power and monotonicity imply that the social choice correspondence is implementable in Nash equilibria when there are more than two agents. S. Williams (1984), ‘Sufficient Conditions for Nash Implementation’, IMA mimeo, Minneapolis.Google Scholar
- R. Repullo (1987), ‘A Simple Proof of Maskin’s Theorem on Nash Implementation’,
*Social Choice and Welfare*, 4, pp. 39–41.CrossRefGoogle Scholar - T. Saijo (1988), ‘Strategy Space Reduction in Maskin’s Theorem: Sufficient Conditions for Nash Implementation’,
*Econometrica*, vol. 56, no. 3 (May), pp. 693–700.CrossRefGoogle Scholar - R.D. McKelvey (1989), ‘Game Forms for Nash Implementation of General Social Choice Correspondences’,
*Social Choice and Welfare*, 6, pp. 139–56.CrossRefGoogle Scholar - One problem which has drawn attention from the theorists is that of finding a condition which at the same time is necessary and sufficient for implementing social choice correspondences in Nash equilibria. See J. Moore and R. Repullo (1990), ‘Nash Implementation: A Full Characterization’,
*Econometrica*, vol. 58, no. 5 (September), pp. 1083–99CrossRefGoogle Scholar - V. Danilov (1992), ‘Implementation Via Nash Equilibrium’,
*Econometrica*, vol. 60, pp. 43–56CrossRefGoogle Scholar - T. Sjöström (1991), ‘On the Necessary and Sufficient Conditions for Nash Implementation’,
*Social Choice and Welfare*, vol. 8, pp. 333–40.CrossRefGoogle Scholar - The following papers can be consulted on the revelation principle in Nash equilibrium: R. Repullo (1986), ‘On the Revelation Principle under Complete and Incomplete Information’ in K. Binmore and P. Dasgupta (eds),
*Economic Organizations as Games*(Oxford: Basil Blackwell)Google Scholar - D. Mookherjee and Reichelstein, S. (1990), ‘The Revelation Approach to Nash Implementation’, Mimeo Graduate School of Business, Stanford UniversityGoogle Scholar
- D. Mookherjee and S. Reichelstein (1990), ‘Implementation via Augmented Revelation Mechanisms’,
*Review of Economic Studies*, July, pp. 453–75Google Scholar - H. Matsushima (1988) ‘A New Approach to the Implementation Problem’,
*Journal of Economic Theory*, 45, pp. 128–44.CrossRefGoogle Scholar - The theorems relating the Nash and Lindahl equilibria were published in: L. Hurwicz (1979), ‘On Allocations Attainable Through Nash Equilibria’ in J.J. Laffont (ed.),
*Aggregation and Revelation of Preferences*(Amsterdam: North-Holland), also published in the*Journal of Economic Theory*, vol. 21, pp. 40–65 (1979); W. Thomson ‘Comment’ in Laffont (ed.),*Aggregation and Revelation of Preferences*Google Scholar - D. Schmeidler (1982), ‘A Condition Guaranteeing that the Nash Allocation is Walrasian’,
*Journal of Economic Theory*, vol. 28, pp. 376–8.CrossRefGoogle Scholar - A similar result to that of proposition 6 can be found in L. Hurwicz, E. Maskin and A. Postlewaite (1984), ‘Feasible Implementation of Social Choice Correspondences by Nash Equilibria’, Mimeo, Department of Economics of Minnesota and Pennsylvania.Google Scholar
- A question not studied in this chapter is the characterization of the allocations generated by Nash equilibria of manipulating games. The original contribution is due to Hurwicz. The interested reader may consult the following paper and the references there: W. Thomson (1984), ‘The Manipulability of Resource Allocation Mechanism’,
*Review of Economic Studies*, vol. 51, pp. 447–60.CrossRefGoogle Scholar - The theory of implementation without commitment has been developed in: E. Maskin and J. Moore (1987), ‘Implementation with Renegotiation’, Mimeo, Harvard UniversityGoogle Scholar
- B. Chakravorty, L. Corchôn and S. Wilkie (1992), ‘Credible Implementation’, Working paper, Bellcore 1992Google Scholar
- S. Baliga, L. Corchön and T. Sjöström (1994), ‘The Theory of Implementation when the Planner is a Player’, Working Paper, Harvard University, 1994.Google Scholar
- On signaling games the reader may consult D. Fudenberg and J. Tirole (1991),
*Game Theory*(Boston: MIT Press), pp. 446–59.Google Scholar

## Copyright information

© Luis C. Corchón 1996